Knihovna PřF MU 3145016110 JI U Um 0 .ľ-Ri.T/ J. L. p.u;a.. V DF.rlĚ, lúIIttSli * Invenľ'irnl 6falo............... utuKDlura................... tnvtnUrnl tis. KS ........-......Utk. 3145016110 PLASMA DIAGNOSTICS WITH MICROWAVES i WILEY SERIES IN PLASMA PHYSICS SANBORN C. BROWN, advisory editor research laboratory of electronics massachusetts institute of technology PLASMA DIAGNOSTICS WITH MICROWAVES heald and wharton • plasma diagnostics with microwaves mcdaniel ■ collision phenomena in ionized gases M. A. Heald I Jcpartment of Physics Swarthmore College Swart hmore. Pennsylvania ('. B. Wharton < leneral Atomic Division (ii'iicral Dynamics Corporation San Diego, California Universita J. E. PurkynS přffodovftdeická fakulta Knihovní slřadls HUv. Inv. 6. £y.Oá>Q . Depo v knih. ,. ;fá>'H--/2Jh. ZJ&s Ústav. Inv. 6.„7?1......~JL. Ugitatura m John Wiley & Sons Inc., New York - London ■ Sydney Copyright £) 1965 by John Wiley & Sons, Inc. All Rights Reserved. This book or any plrt thereof must not be reproduced in any form without the written permission of the publisher. Library of Congress Catalog Card Number: 64-23839 PRINTED IN THE UNITED STATES OK AMERICA In Jane and Gloria, who have played well the roles of both wile and midwife during this long labor. Preface vii Preface The subject of plasma diagnostics is concerned with making significant, nonperturbing measurements and expressing them in numbers. The term "diagnostics," of course, comes From the medical profession. The word was first borrowed by scientists engaged in testing nuclear explosions about fifteen years ago to describe measurements in which they deduced the progress of various physical processes from the observable external symptoms. The word crept into the jargon of the then-classified Sherwood Program, the AEC program of controlled nuclear fusion research. Now the term "plasma diagnostics" is applied to a wide variety of plasma measurements. The term implies that the diagnostic measurement does not itself change the state of the plasma, that two or more diagnostic measurements may thus be made simultaneously and, in most cases, that time-resolved data can be obtained from a single transient event. In the case of microwave diagnostics, the interpretation often is difficult and requires not only an understanding of the formal theory of electromagnetic interactions with plasmas, but also development of an intuitive skill in selecting meaningful simplifications. Many of the cases of wave propagation are much too complicated to permit exact formulation and solution; a feeling for how things scale from similar, more tractable cases is often essential. Consequently, we shall find it necessary to develop a fairly complete basic theory of many facets oTwave interactions on which to base our scaling and approximating to specific cases. Historically, the subject of microwave diagnostics is not new. The laboratory experiments of Balthazar van der Pol (1920) to demonstrate that charged particles have a large influence on electromagnetic wave propagation did much to settle an impnrlanl controversy of the day, vt whether or not an ionosphere was responsible for distant radio propagation. His calculations of plasma conductivity and refractive index yielded the same equations to be found in our Chapter 1 for the Lorentz plasma, including a density-dependent term that was later named the plasma frequency by Tonks and Langmuir (1929). Subsequently, workers in the field of ionospheric propagation, with such landmarks as Appleton (1932) and Mitra (1952), have perfected to a high degree the use of radio-wave probes for sounding the ionosphere. The improved microwave technology following World War II opened new expanses of the frequency spectrum. The pioneering theoretical work of Margenau (1946) and experimental work of Biondi and Brown (1949) and others in the M.l.T. group, in developing resonant cavity techniques, rekindled interest in plasma measurements with electromagnetic waves. Faraday-rotation measurements with waves beamed through controlled fusion plasmas, performed by R. F. Post and others in Berkeley in 1952, stimulated the development of microwave diagnostics as ,i standard measuring technique in Project Sherwood research (Wharton 61 al., 1955; Heald, 1956). Laboratory experimental techniques have come a long way since the days of van der Pol, who measured the shift in standing waves on a Lecher wire terminated by small capacitive discs immersed in the plasma. His microwave source was a Blondlot arc, running under kerosene, producing n few milliwatts of damped 200 Mc wave trains. Nevertheless, embellishments of those early day techniques are still used for diagnosing low-density plasmas, and several of the experiments described in Chapters 5 and 6 have recognizable similarities. Our aim in writing this book has been to bring together, on the one hand, a summary of the basic theory of the interaction of electromagnetic waves with plasmas and, on the other, a description of the practical experimental techniques that exploit this interaction. The book is written mainly in the context of the plasmas of controlled fusion research, which mv characteristically hot (implying a high degree of ionization and low Inlerparticle collision rates) and large (relative to the wavelength of an ■ lectromagnetic wave at the plasma frequency w„). However, most of Hii material is relevant also to the plasmas found in the fields of M.H.D. bower generation, space vehicle propulsion and communication, iono-Iphcric radio propagation, microwave devices, classical gas discharges, jin• I radio astronomy. We have limited the detailed discussions to "high-ih quency" techniques that use waves at frequencies of the order of the .lit lion plasma frequency. Also, we have given somewhat more attention in free-Space beam techniques than to those employing resonant cavities Mini waveguides. viii Preface The formal theory ofelectromagnetic interactions in plasma often makes use of mathematical techniques that are beyond the experimentalist's training. We make no apologies about writing the book from an experimentalist's viewpoint, so that many of the theoretical discussions are somewhat cavalier and inelegant. On the other hand, we have endeavored to go beyond the mere displaying of useful theoretical formulas. The theory presented is developed sufficiently to indicate the assumptions and ranges of validity of the more sophisticated theoretical presentations in the literature as well as to serve as a primer for these treatments. Thus for instance, we present a hydromagnetie treatment of wave propagation in Section 3,3 although, in general, more useful "practical" results are obtained by the corresponding kinetic treatment of Section 3.4. The hope, then, is that this volume may be of some value to those seeking an introduction to plasma physics beyond the narrow topic of microwave diagnostics, without offending the specialist seeking solutions to specific problems. The similarities and differences of electromagnetic and spacecharge wave propagation in bounded plasmas and in infinite, homogeneous plasmas are carefully delineated, including the dependence of wave properties on temperature and magnetic field strength. Because the effects of finite temperature on spacecharge wave propagation have been included, we are able to give an introduction to wave growth and electrostatic instabilities and to collisionless (Landau) damping. Some of the effects of instabilities and turbulence on wave scattering and electromagnetic radiation are also discussed. There is considerable experimental material included. We have attempted to present the broad picture of wave propagation and radiation experiments, still retaining enough detail to permit an experimentalist new in the field to proceed with a diagnostic experiment. The descriptions of the techniques, therefore, are fairly complete. Numerous illustrative examples have been chosen from the literature and from our own work. Some of the latter are new and are published here for the first time. Our subject is specialized; this volume was not visualized as a text in formal courses, although portions of it may well fit into certain courses. No formal problems have been included although problem material is present. In the plasma physics field, workers will come from a number of diverse backgrounds. An argument or notation that is familiar and elementary to a microwave tube engineer, for example, may not be familiar to a specialist in gaseous discharges. And the terminology of fluid mechanics or ionosphere research may be unfamiliar to workers in the former fields, and vice versa. The literature abounds with different Rotational conventions, often not explicitly slated, Consequently, we Preface ix have attempted to maximize this book's use for reference and individual study by defining many terms, by cross-referencing, and by including or giving references to elementary and applied material from diverse fields that might be passed over in a textbook. The first six chapters are devoted to the propagation of externally generated waves in a plasma, the first five being primarily theoretical. Chapter 1 presents the well-known Appleton-Hartree theory of wave propagation, including a brief discussion of the effect of heavy-ion motion. The next two chapters examine the role of collision processes, especially electron-ion Coulomb collisions, and summarize the modifications that occur when electron thermal speeds are comparable with the wave phase velocity. Chapters 4 and 5 consider the boundaries and spatial non-uniformity of real plasmas. Chapter 4 deals with the propagation of free-space beams, and includes a discussion of the choice of antenna systems with which to probe a plasma sample of given size. Chapter 5 meanwhile deals with a plasma confined in a resonant cavity or waveguide, or acting as a waveguiding structure itself, for either electromagnetic or spacecharge waves. Finally Chapter 6 gives an extensive discussion of the practical applications of these active microwave probing techniques. Chapters 7 and 8 present, respectively, the theory of microwave radiation generated by a plasma, by both thermal and nonthermal processes, and the practical application of passive radiometric techniques. Chapter 9 is devoted to extensive descriptions and photographs of much Of the hardware and special circuits useful for microwave diagnostics, but not readily available commercially. With some regret we have omitted a listing of commercial suppliers of generators, detectors, and components, although references to listings in the literature have been cited. While this information is of great use to an experimentalist entering the field, we feared that a list would soon be obsolete and subject to unintended favoritism. A brief survey of a number of other plasma diagnostic techniques besides microwaves is given in Chapter 10. The main emphasis has been placed on techniques that yield information similar to that obtainable with microwaves, that is, plasma electron properties, although other techniques were included for completeness. Correlative measurements are indicated where possible. A summary of wave propagation in general dielectric materials is given in Appendix A. This can serve as a starting point for those readers un-hi miliar with microwaves or needing a review. A brief summary of some of I lie properties of tensors and matrices is given in Appendix B, since these operations are used repeatedly throughout the text. An unusually large number of references is cited, many from the very recent research literature. To a large extent, this book is a review of research in progress rather than a text in a well established field. Therefore we have felt it necessary, especially in the theoretical chapters, to provide the means of following up our introductory discussions with more detailed research papers. It is inevitable that many of these references will be superseded by new work in the near future. A bibliography of important general references is given at the end of the book. Literature references in the text are identified by author and year. Bibliographic details are then given in listings following the general bibliography. The references for Chapters 1 to 8 and the appendices are compiled into a single alphabetical author list. However, since the material covered in Chapters 9 and 10 is somewhat foreign to that oT the other chapters, separate reference lists are given for these two chapters. The reference lists may be used as an author index. We have attempted to follow the notation and terminology of plasma waves admirably systematized by the M.T.T, group (Allis et al., 1963), except in a few cases where strong tradition decrees otherwise. Familiarity with our notation for general wave propagation may be obtained by a quick glance through Appendix A, We have tried to avoid obscure normalized parameters, so often found in theoretical journal papers, in an attempt to preserve some physical insight into the equations. Because we make much use of complex notation for familiar coefficients (such as conductivity, dielectric constant, and refractive index) we have used the special symbol y (as 5, H, and u) to indicate explicitly when a quantity is complex. Considerable use is made of vector and tensor notation; a brief review is given in Appendix B. Rationalized inks units have been used throughout. However, numerical graphs and formulas are often stated in the vernacular of centimeters, gigacyles, and kilogauss. In particular, energies normally are stated in electron volts, or in units of the Rydbcrg energy constant (^ = 13.6 eV) when this is relevant to the physics. Temperatures are expressed in energy units {kT)\ those of other preference may note that 1 electron volt fS 1.16 x 104 degrees Kelvin. In preparing this book we have been assisted greatly by our colleagues at the Plasma Physics Laboratory (Project Matterhorn) of Princeton University, at the Lawrence Radiation Laboratory of the University of California, and at the John Jay Hopkins Laboratory of General Atomic in San Diego, as well as by numerous other associates who have taken the time to point out some of our errors and shortcomings. Professor S. C. Brown, of M.T.T., has read the manuscript and contributed helpful suggestions. Swarthmore, Pennsylvania M. A. Heald San Diego, California C. B. WHARTON Contents 1 Electromagnetic Wave Propagation in a Cold Plasma, 1 1.1 Introduction, 1 1.2 Plasma oscillations and the plasma frequency, 2 1.3 Electromagnetic wave propagation (no magnetic field), 4 -J.3.1 Elementary case neglecting collisions, 4 . L3.2 Conductivity with collisional damping: Lorentz conductivity, 6 ^r3.3 Propagation in a Lorentz plasma (no magnetic field), 6 1.3,4 Low-loss plasmas (v << wy): the three frequency regions, 7 \A.~i-5 The critical electron density, 12 1.4 Wave propagation with magnetic field, 12 1.4.1 Wave propagation along the magnetic field; circularly polarized waves, 12 1.4.2 Faraday rotation of angle of polarization, 19 1.4.3 Arbitrary direction of propagation: coordinate systems, 20 1.4.4 Magnetic field at angle 6 with respect to propagation: Appleton's equation, 21 1.4.5 Wave polarization, 24 1.4.6 Propagation across the magnetic field, 25 1.4.7 The conductivity tensor, 29 1.4.8 Conductivity in rotating coordinates, 31 1.4.9 Summary of principal waves, 34 xi 1» xii Contents 1.4.10 Propagation at an oblique angle: the QL and QT approximations, 38 1.4.11 Index, velocity, and ray surfaces, 45 1.4.12 Refractive index contour maps, 49 1.5 Ion motion effects, 50 1.5.1 Conductivity with ion motions, 51 1.5.2 Principal waves including ion motions, 52 1.5.3 Oblique propagation with ion motions, 56 2 Collision Processes, 57 2.1 Introduction, 57 2.2 Elementary considerations of collision processes, 58 2.2.1 Collision cross sections and frequencies, 58 2.2.2 Velocity dependence of cross sections, 62 2.3 Effect of collisions on electron motion, 64 2.4 Analysis of particle interactions, 66 2.4.1 Boltzmann equation, 67 2.4.2 Elementary Boltzmann theory of plasma conductivity, 69 2.4.3 Effective collision frequency, 71 2.5 Coulomb interactions, 76 2.5.1 Debye shielding, 76 2.5.2 Coulomb collisions, 79 2.5.3 Effective coulomb collision frequencies, 82 2.5.4 The logarithmic term, 85 2.6 Nonlinear effects, 89 2.6.1 Criterion for linearity, 89 2.6.2 Breakdown, 91 2.6.3 Luxembourg effect, 91 2.6.4 Other nonlinearities, 92 2.6.5 Incoherent scattering, 93 3 Waves in Warm Plasma, 95 3.1 Introduction. 95 3.2 Magnetic permeability of a plasma, 96 3.3 Hydromagnetic calculation of plasma waves, 98 3.3.1 Moment equations, 98 3.3.2 Hydromagnetic dispersion relations, 100 3.4 Kinetic (Boltzmann) theory of waves, 104 3.4.1 Propagation along the field, I oh 3.4.2 Propagation across I he field, I I I Contents xiii 3.4.3 No magnetic field, 112 3.4.4 Plasma or electrostatic waves, 112 3.5 Landau damping and wave absorption, 113 3.6 Relativistic plasmas, 115 4 Wave Propagation Through Bounded Plasmas, 117 4.1 Introduction. I 17 , 4.2 Simple adiabatic analysis of a plasma slab, 120 4.2.1 Average electron density, 120 4.2.2 Adiabatic measurement of density profile, 123 4.2.3 Reflections from cutoffs and resonances, 125 4.3 The slab with sharp boundaries, 127 4.4 lnhomogeneous plasmas, 130 4.4.1 Isotropic inhomogeneous plasmas, 133 4.4.2 Anisotropic inhomogeneous plasmas, 136 4.5 The geometrical optics of a uniform cylindrical plasma column, 137 4.5.1 Transmission loss by refraction, 137 4.5.2 Other sources of loss, 141 4.6 The antenna problem, 141 4.6.1 Fresncl zones, 142 4.6.2 Collimalion, 146 4.6.3 Optimization of antennas, 148 4.6.4 Validity of the geometrical-optics, slab model, 150 5 Guided Wave Propagation, 155 5.0 Introduction, 155 5.1 Measurements on plasmas contained in resonant cavities, 155 5.1.1 Measurement of plasma admittance, 158 5.1.2 Measurement of plasma density and collision frequency, 159 5.1.3 Special cavity modes for high density plasmas, 162 5.1.4 Experimental techniques, 163 5.2 Waveguides containing plasmas, 163 5.3 The to-/3 diagram, 167 5.4 Nonuniform plasma in a waveguide, 168 5.5 Spaccchargc waves, 170 5.5.1 Spaccchargc waves in a cold, drifting plasma, 171 5.5.2 Spacecharge waves in plasma columns of finite radius, 174 xlv Contents Contents xv 5.5.3 Spacecharge waves in a plasma column in a magnetic field, 179 5.5.4 Surface spacecharge waves on a drifting plasma column, 182 5.6 Spacecharge waves in a warm plasma, 183 5.6.1 No magnetic field, 183 5.6.2 Spacecharge waves in a warm, magnetized plasma, 188 6 Microwave Propagation Experiments, 192 6.1 Transmission-attenuation and reflection experiments, 192 6.1.1 Microwave circuits for transmission and reflection measurement, 192 6.1.2 Transient plasmas, 194 6.2 Frequency diversity, 197 6.2.1 Frequency diplexers, 199 6.2.2 Polarization diplexers, 200 -6.3 Phase-shift measurements, 200 6.3.1 Microwave interferometer, 200 6.3.2 Rf modulation envelope, 204 6.3.3 "Fringe-shift" or zebra-stripe interferometer, 206 6.3.4 Polar plot display, 210 6.4 Density distribution: profile measurements, 212 6.5 Magnetic field effects, 219 6.5.1 Ordinary and extraordinary waves; density profiles, 220 6.5.2 Faraday rotation, 223 6.5.3 Whistler mode propagation, 225 6.5.4 Propagation at angle 8 to magnetic field, 227 6.5.5 Doppler-shifted gyrofrequency in drifting plasmas, 228 6.6 Propagation through fluctuating plasmas, 229 6.7 Microwave scattering experiments, 232 6.7.1 Incoherent scattering, 232 6.7.2 Scattering from plasma fluctuations, 234 6.7.3 Scattering from small plasma columns, 240 7 Microwave Radiation from Plasma, 242 7.1 Introduction, 242 7.2 Strict blackbody radiation, 242 7.3 Bremsstrahlung in a transparent medium, 245 7.3.1 Radiation hy a single electron, 246 7.3.2 The Gaunt factor, 248 7.3.3 Quantum considerations, 250 7.3.4 Electron shielding, 252 7.3.5 The Gaunt factor and \nA, 253 7.3.6 Summary of microwave bremsstrahlung, 254 7.3.7 Atom bremsstrahlung and total radiation, 256 7.4 Radiation transport and the gray body, 257 7.4.1 Energy flow in an inhomogeneous medium, 257 7.4.2 The Einstein coefficients, 259 7.4.3 The partially transparent plasma, 261 7.4.4 Correlation of emission and conductivity theories, 262 7.5 Radiation from a slab and Kirchhoff's law, 263 7.5.1 Effect of antenna gain, 264 7.5.2 Surface reflection, 266 7.5.3 Kirchhoff's law, 270 7.6 Cyclotron radiation, 272 7.6.1 Total radiation, 272 7.6.2 Radiation anisotropy (nonrelativistic), 273 7.6.3 Line shape (nonrelativistic), 274 7.6.4 Radiation by a single rclativistic electron, 275 7.6.5 Spectrum (relativistic), 276 7.6.6 Effect of collective electron motion, 278 7.7 Cerenkov radiation, 280 7.8 Coherent radiation, 285 Plasma Radiation Experiments, 287 8.1 Radiation from dense plasmas: blackbody radiation, 287 8.2 Radiation from a plasma in a magnetic field, 291 8.2.1 Magnetic mirror radiation experiments, 292 8.2.2 Absorption-radiation experiment in a pulsed mirror machine, 293 8.2.3 Absorption-radiation measurements in a waveguide or cavity, 297 8.3 Swept-frequency radiometers, 299 8.4 Radiation of nonthermal origin, 299 8.4.1 Instability-generated radiation, 300 8.4.2 Nonthermal cyclotron radiation, 302 Microwave Hardware and Techniques, 305 '» I Transmission lines, 305 9.1.1 Waveguide considerations, 305 xvi Contents Contents xvii 9.1.2 Open waveguide transmission lines, 310 9.1.3 Free-space radiation, 312 9.2 Special components, 313 9.2.1 Phase shifters, 314 9.2.2 Hybrid junctions, 315 9.2.3 Polarization and frequency diplexers, 316 9.2.4 Filters, 318 9.2.5 Circular polarizers, 319 9.2.6 Resonant cavities, 324 9.3 Antennas and radiators, 329 9.4 9.5 9.6 9.7 Horns, 329 Lenses, 331 Dielectric rod antennas, 334 Reflecting antennas, 336 Slot radiators, 338 Antenna pattern measurements, 341 Signal sources, 344 9.4.1 Klystrons, 344 Traveling wave oscillators and amplifiers, 344 Magnetrons, 345 Tunnel diodes. 345 Harmonic generators, 345 Signal detection, 346 9.5.1 Video crystal detectors, 346 Superheterodyne receivers, 349 Parametric mixer-amplifiers, 351 Miscellaneous receiver systems, 352 Microwave radiometers, 353 Measurement of receiver performance, 355 Vacuum system considerations, 356 9.6.1 Vacuum materials, 356 9.6.2 Transmission-line windows, 357 9.6.3 Movable probes, 360 9.6.4 Microwave absorbers, 361 Circuitry considerations, 362 9.7.1 Electronic circuits, 362 9.7.2 Circuit interference and stray pickup, 364 9.3.1 9.3.2 9.3.3 9.3.4 9.3.5 9.3.6 9.4.2 9.4.3 9.4.4 9.4.5 9.5.2 9.5.3 9.5.4 9.5.5 9.5.6 10 General Plasma Diagnostic Techniques, 366 10.1 Tabulation of some useful diagnostic techniques, 366 10.2 Optical and infrared probing, 368 10.3 Conductivity probes, 374 10.4 Langmuir probes, 378 10.4.1 Single Langmuir probe, no magnetic field, 10.4.2 Single probe with a magnetic field, 383 10.4.3 Single probe: miscellaneous effects, 384 10.4.4 Double floating probes, 384 10.4.5 Double probes, no magnetic field, 384 10.4.6 Double probes in a magnetic field, 385 10.4.7 Double probes: miscellaneous effects, 386 10.5 Plasma wave and resonant probes, 386 10.6 Magnetic probes, 387 10.7 Ballistic probes, 387 10.8 Optical spectroscopy, 388 10.8.1 Constituent identity and state, 388 10.8.2 Stark broadening, 388 10.8.3 Doppler broadening, 389 10.8.4 Doppler shift, 390 10.9 Bremsstrahlung and recombination continuum, 391 378 Appendix A Review of Electromagnetic Wave Propagation, 392 A.l Basic relations for a linear medium, 392 A. 1.1 Complex dielectric constant or conductivity, 394 A. 1.2 Complex propagation constants, 396 A.2 Microscopic relations, 400 A.2.1 The microscopic field in a dielectric, 401 A.2.2 The microscopic field in a plasma, 402 A. 3 Propagation in an anisotropic medium, 403 Appendix B Tensor and Matrix Algebra, 406 B. l Vectors, tensors, and matrices, 406 B.2 Addition, multiplication, and inversion of tensors, 408 Cicneral References, 411 References for Chapters 1-8, 412 References for Chapter 9, 430 References for Chapter 10, 435 Subject Index, 439 CHAPTER 1 / often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thought,, ad Danced to the state of Science, whatever the matter may be. Lord Kelvin (1824-1907) Electromagnetic wave propagation in a cold plasma I. I Introduction A real-life plasma is a very complicated phenomenon. Viewed as a incdium Tor the propagation of electromagnetic waves, a plasma in a magnetie field is refractive, lossy, dispersive, resonant, anisotropic, non-reciprocal, nonlinear, and inhomogeneous. We begin by accepting as many simplifications as possible and then in later chapters introduce itTmements to the elementary treatment. In this chapter we consider the theory of electromagnetic wave propagation in an infinite, uniform, Lorentz plasma. A Lorentz plasma denotes a \iiiiplified model in which it is assumed that the electrons interact with 11u 11 other only through collective spacecharge forces, and that the heavy piisilive ions and neutral molecules are at rest. In effect, the ions and tteutrals are regarded as a continuous stationary fluid through which the • KM runs move with viscous friction. In addition, we use a simplified ■ \ii.i lysis that infers the properties of the plasma medium from the motion "I individual representative particles, thereby making implicit assumptions About the nature of the inlerparticle collision processes. Both these implications ignore statistical correlations in the positions and velocities "I the plasma electrons, refinements which are considered in Chapters 2 111 • I \. In particular, as implied by the adjective "cold," we first neglect ellccls which depend explicitly on electron temperature.1 These refinements can usually be incorporated as small corrections to the effective 1 Novell hclcss, as pointed oul in Chapter 3, it cannot be assumed thai the electron llli'imal velocities are zero. Therefore the term "temperate" is perhaps more lippinpiialc than "cold." 1 2 Electromagnetic wave propagation in a cold plasma Chap. I 1.2 Plasma oscillations and the plasma frequency 3 plasma parameters within the format of the simplified theory. Thus the simple model gives results which are at least qualitatively correct for more complicated models. Theeffectsof plasma boundaries and inhomogeneity are considered in Chapters 4 and 5. 1.2 Plasma oscillations and the plasma frequency Before proceeding with the basic topic of electromagnetic wave propagation, it is helpful to consider briefly the phenomenon of electron plasma or spaeecharge oscillations. The physical nature of plasma oscillations can be seen from a simple argument originally given by Tonks and Langmuir (1929). We consider an initially uniform electron gas of density n. By some external means, let a one-dimensional perturbation occur such that electrons at position x are displaced in the x direction by a small increment £(x), as shown in Fig. 1.1. The local density of electrons then departs from the uniform density n by the increment m dx (1.2.1) Since the net charge density p was originally zero, the perturbed charge density is op — — e on — ne -t-> dx (1.2,2) Oscillating plane FIG. I.I Geometry of one-dimensional pert 11 rhu lion lending to plasma oscillation. where the electron charge is — e. The new net charge is related to the existing electric field by Gauss's law dE d$ dx dx which can be integrated immediately, giving (1.2.3) (1.2.4) within an arbitrary constant. The electric force on each electron is then F= -eE=~— Š, (1.2.5) simplicity, we neglect the viscous damping forces which arise from collisions between the electron and heavy particles. Newton's equation dI' motion is then mš + — Š = Fext, (1.2.6) where m is the electron mass, t — d^^j&t2 is the electron acceleration, and l\,.,t is the external force required to produce the perturbation. If this external force is suddenly removed, (1.2.6) shows that the electrons oscillate about their equilibrium positions with simple harmonic motion id the plasma frequency %\ */ lnezV (1.2.7) :'') moo h 100 III 1 1 1 . 1 1 1 ' 1 1 I 1 1 1 1 L_ - - K~ 1 1 1 1 1 1 ■1 1 1 1 A s -1 1 lOOju 1 mm 1 cm Š 10 cm IQlO 10ll 101S 1013 10M 10» 1016 10" Electron density ft [cm ~3 J JG. I.Z Cyclic plasmu frequency tu9/2w as a function of electron density. 4 Electromagnetic wave propagation in a cold plasma Chap. I 1.3 Electromagnetic wave propagation (no magnetic field) 5 Numerically, [cps] = 8979(«[cm-3])^ (1.2.8) Figure 1.2 shows this relation for the microwave region. This oscillatory behavior is known as a plasma oscillation. The generalization of this argument to three dimensions has been discussed by Dawson (1959). The arbitrariness of the spatial distribution of £(x) in the above argument implies that a plasma oscillation does not transfer energy; that is, a disturbance does not propagate beyond the region in which it is excited. This property ceases to be true if a significant electron temperature exists (Chapter 3), or if the plasma is bounded or contains gradients of electron density (Chapter 5). In the case of bounded plasmas, depolarizing effects displace the macroscopic resonance frequency from the plasma frequency (for example, to w„/V2 for the transverse dipole mode of a cylinder, tiip/vS for a sphere). The disturbance is carried along as a convected wave in a drifting plasma (see Section 5.5). 1.3 Electromagnetic wave propagation (no magnetic field) 1.3.1 Elementary case neglecting collisions. The interaction of an electromagnetic wave with an electron gas can be presented in the following simplified form, which continues to neglect collisional damping. Instead of prescribing a spatial displacement £(x), as in the previous section, we prescribe a net electric field E(f). This net electric field is the sum of an external field imposed by sources outside the plasma and the internal field associated, with the electron spacecharge by (1.2.4). Neglecting all other external forces, we have for the equation of motion, in place of (1.2.6), m 1= -eE. (1.3.1) Assuming an oscillatory electric field varying as exp jmt, the steady-state solution is mar The resulting current density is J = —nef = —i-E, which is of the Ohm's-law form J=aE, having a conductivity j — 3 (1.3.2) (1.3.3) (1.3.4) I or wave propagation in a linear medium, as shown in Appendix A, a complex conductivity may be replaced by a complex dielectric constant which, in this case, is e0u> e0ma)2 ti)3' (1.3.5) where wp is the plasma frequency defined by (1.2.7), and the symbol U is used to denote explicitly a complex quantity. This analysis may be applied to a plane electromagnetic wave traveling in the 2 direction and varying as exp(yW—yz), where y = a+jfi is the complex propagation coefficient, and a and /S are the attenuation and phase coefficients.2 The dispersion relation for such a wave is y=JK -• (1.3.6) For low frequencies towJ>, the wave is propagated without attenuation: « = 0 I In- phase and group velocities, in this case, are 1 2\ ¥■ (1.3.8) (1.3.9) (1.3.10) Which are respectively greater and less than the vacuum velocity of light c. The refractive index p, for high frequencies, J I ''' (1.3.11) mi I haii unity, in contrast to the index of ordinary dielectrics. Api'i'inM* a lor ii review of I he basic electromagnetic wave theory and a ili .i ii........if our noliilion. 6 Electromagnetic wave propagation in a cold plasma Chap. 1 1,3 Electromagnetic wave propagation (no magnetic field) 7 1.3.2 Conductivity with coiUsional damping: Lorentz conductivity. The influence of discrete positive ions and neutral molecules in a plasma can be represented to good approximation by including a viscous damping term, proportional to velocity, in the electron equation of motion. With this addition to (1.3.1), the equation of motion becomes ml=-eE-vmi. (1.3.12) The form of the damping term vmk anticipates the fact, discussed at length in Chapter 2, that direct physical meaning can be given to the parameter v as a collision frequency (strictly, collision frequency for momentum transfer).. In brief, we argue that on the average an electron loses its directed momentum m£ at each collision. Thus, if the electron averages v collisions per second, — vmi; represents the time rate of change of momentum and, hence, the statistical average force exerted on the electron by the massive ion-neutral component of the plasma. More careful examination of the averaging process (Section 2.3) shows that this form of damping term is strictly correct only when the collision frequency is independent of electron velocity, a rather special case. For the present, we need only regard v as a phenomcnological damping constant, having the dimensions of radian frequency. The steady-state solution of (1.3.12) for oscillatory fields, obtained by the substitution 8j8t —> jw, is eE mu>{oi —jv) The current density /= — net yields a complex conductivity 5 = CTr+.M: ne* ne* v—ju) m(v + ju>) (1.3.13) (1.3.14) which is known as the Lorentz conductivity Lorentz dielectric constant is m + or The equivalent complex 1 — 60W oj(oj — jv) (1.3.15) 1.3.3 Propagation in a Lorentz plasma (no magnetic field). The propagation of plane electromagnetic waves in a uniform medium (of relative permeability unity) may be expressed in terms of a complex dielectric constant k; (1.3.16) where y is the complex propagation coefficient occurring in the phase factor i\\p(/W — yz), and a and /3 are the attenuation and phase coefficients, n■'.pectively.3 It is useful also to define a complex refractive index f* = V--JX= -JY - = *7 (1.3.17) Where ft and x are the (real) refractive index and attenuation index, respectively. Thus ft M (1.3.18) (1.3.19) he sign of the square roots in (1,3.16) and (1.3.17) is taken such that 0 iiiul ft are positive. For a resume of the relevant basic theory of electro-bgnetic waves, see Appendix A. Using the Lorentz dielectric constant (1.3.15) in appendix formulas I \ 46) and (A.47), we obtain explicitly: (1.3.20) - ......'H-i('-^n[(^h(^my (1.3.21) I In- propagation coefficients a and fjj are obtained from (1.3.18) and il '< Evaluation of I he effective collision frequency v is discussed in Hclions 2.4.3 and 2.5.3. Low-loss plasmas (v« a>„): the three frequency regions. We consider |l il function of frequency the electrical properties of a plasma charac-ii ' il by the parameters plasma frequency wv (proportional to square [tnl of electron density n'-) and effective collision frequency v (in general, ii linulion of electron density and temperature; see Sections 2.4.3 and i assuming the Lorentz conductivity (1.3.14) and no magnetic field. 1 I* use of a highly ionized, high-temperature plasma for which v«Qim VVi' i'ii11 distinguish three frequency regions, ll"i i ili, when- I is Hie complex annular wave number used by many authors. I i \ppwidlx A. 8 Electromagnetic wave propagation in a cold plasma Chap. 1 1.3 Electromagnetic wave propagation (no magnetic field) 9 LOW frequencies cu < v. In this region the conductivity is largely real and, to a first approximation, is e2 , (1.3.22) per '' mv a familiar relation from the elementary kinetic theory of conductors. Expanding in the limit co«v, v2«p2, and using (A.52) and (A.53), we obtain: 3f3 ui2 \ x~ o\ W 2o>.2) X+8a>2 + 2a>p*J c a), (1.3.26) (1.3.27) (1.3.28) Thus, the penetration depth of this evanescent wave is practically constant in the interior of this frequency region, and is comparable to the wavelength of a free-space wave at the plasma frequency—a few millimeters in typical laboratory plasmas. high frequencies w>cop. Here, the plasma becomes a relatively low-loss dielectric. In the limit v2«cu2 — 2(a}2 — w,,2)2/^*, using (A.50) and (A.51), we obtain: v^w2{^-M 2^2 o J 2w2{w2 (1.3.29) (1.3.30) (1.3.31) Nole that the refractive index is quite insensitive to collisional damping unci that the attenuation is very small for the assumed conditions. Numerical values of a, fi, fi, and y are shown as functions of frequency in Fig, 1.3 for a typical laboratory plasma. Fig. 1.4 shows the attenuation length S as a function of frequency for two different collision frequencies 11ii temperatures). The magnitudes of the quantities plotted change ......>tlily and slowly at the boundaries of the three frequency regions with I In- exception of the attenuation parameters, a, v, and S, across the plasma resonance, ca = atp. Expansion of the complete expression (1.3.21) for v in llie neighborhood oftu = Wj, shows that the attenuation length S changes 104 to2 10 lo- ur 1 III III 1 1,1)1 III 1 1 11 1 1 1 1 - l^Ul-UU 1 1 ■-iUI ilüCU IL - "'I "> 1 11 M r — - ß[< n 1 "V . X 1 \ - A ^-Ol = (lip fß _ 1 1 1 1 1 1 1 i i 1 1 \a »v 1 1 .IG" 10E 10b 10' 10s 10a 10' 10' HI Transmission frequency oi/2ir [cps| W'j. 1,3 Propagation constants as a function of frequency for a typical laboratory i in hydrogen gas, Electron density I0111 cm"3; electron temperature 10 eV. i Im ihn I iVciiucncy domains are shown at (he top. Symbols are defined in the text. 10 Electromagnetic ware propagation in a cold plasma Chap, 1 10? 1 10' 10* i—r r J_L 10 10^ 10J Frequency w/iop FIG. T.4 Attenuation length S of electromagnetic wave in a highly ionized plasma, as a function of frequency, for two temperatures T?x \ 0'1\, such that f1/ibp = 3- 10™3 and i'2/™!> = 10"'1. In many laboratory devices the thickness of the plasma is of the order of 100 cjuip. °3 ■1 0 1 0c/up = nu/ap 1.3 Electromagnetic wave propagation (no magnetic field) 11 by the large factor (2cu„/i')''i in the small frequency change (2vco?)-'. Thus, li fractional change of a few per cent in frequency or, alternatively, in electron density ( tiel'raclive and altuiuiiilion iiulitrtrs as I'lini/lions ol'cleciron delisily, 12 Electromagnetic wave propagation in a cold plasma Chap. 1 are vlll = cajfi and vs — daijdfi, the value of the diagram in obtaining these quantities is seen immediately. Further use of o>-3 diagrams, normalized to various quantities, is made in connection with spacecharge waves in Chapter 5. s 1.3.5 The critical electron density. For rf measurements in a time-varying laboratory plasma, the physical situation is better described with the following emphasis. Consider a fixed test frequency «j, to which there corresponds a critical density >ic, defined by the plasma frequency relation urn (1.3.32) For densities below this critical value, the medium is a nearly transparent dielectric; above, the medium is opaque and highly reflecting. The indices p. and x are shown as functions of electron density in Fig. 1.6. 1,4 Wave propagation with magnetic field We now consider the uniform Lorentz plasma to be immersed in a static magnetic field B0. The situation is considerably more complicated because of the vector relations involved in the magnetic force on a moving charge ?vxB, The equation of motion, corresponding to (1.3.12), becomes the Langevin equation m\=— eE(0 - e\ x B0 - vmv (1.4.1) where v is the vector velocity of the electron replacing the one-dimensional Note that here, as before, we neglect the time-dependent (wave) magnetic field associated with the time-dependent electric field. Since the magnetic force compares to the electric force as vjc, this neglect is usually well justified, unless the plasma approaches relativistic temperatures in which case other corrections are necessary as well (Chapter 3). 1.4.1 Wave propagation along the magnetic field: circularly polarized waves. Before proceeding with a general solution, it is instructive to examine the special case of propagation along the field, which we take to be the z direction. Thus the wave electric field is expected to be in the x-y plane. The vector equation oT motion (1.4.1) represents three scalar equations. For an oscillatory electric field varying as exp jtot, as usual, these equations are explicitly (> + v>v+ (||k= -4 & (1.4.2a) \ ml m (1.4.2b) (1.4.2c) 1.4 Wave propagation with magnetic field 13 The third equation is not coupled to the other two. Physically, it would represent a possible electromagnetic wave propagating perpendicular to I he magnetic field or, alternatively, a spacecharge oscillation of the sort discussed in Section 1.2. ]n either case, the transverse motion is not affected and thus (1.4.2c) may be ignored. The first two equations are i oupled together by the yxB term in a way that obscures the dependence of VOil E, needed to evaluate the conductivity and propagation constants.4 The x-y symmetry of (1.4.2a) and (1.4.2b) suggests consideration of minting or circularly polarized fields. The notation used to express such vectors can be a matter of some subtlety. Consider a vector rotating with lime in a right-handed sense about the z axis. The y component is equal in magnitude to the component and lags it by 90°. In terms of the usual convention that the actual physical quantity is given by the real part of a complex quantity (phasor), we may write, for instance, Ex = Eq exp jait Ey — ~jEa exp jcut (1.4.3) (1.4.4) where Ea is a time-independent amplitude. Similarly, left- and right-hand rotating unit vectors would be written (1.4.5) where a*, a„ are the unit vectors in the „v and y directions. It is understood lluil any quantity with which these unit vectors are used contains the time liulur cxp(-f-jW). We can now express an arbitrary field (of unrestricted polarization) in ItCt'ins of circularly polarized, rather than cartesian, components. Such an in biliary field may be expanded formally in either system, with the identity3 E = a.xEx + siyEy + »gEz A +a Ml V2 + aW2 (1.4.6) Hv equating coefficients of the cartesian unit vectors, we find & Et + Er P ;Et- Er V2 V2 (1.4.7) In Sivliim 1.4.7, a general method is developed for representing the conductivity as i Icimor. Wi' chouse Ihe numerical factor of \ 2 in the circularly polarized terms largely for Mill"iiis of mutational convenience. In effect, wc thus regard Ex, E\, etc., as rms Ui|ililuilus, Other conventions will he found in the literature. 14 Electromagnetic wave propagation in a cold plasma Chap. 1 and, conversely,6 E,= A/2 V2 (1.4.8) Note that a wave traveling in the +z direction with right-hand circular polarization in time has a left-handed space dependence. In the steady state, the electron velocities resulting from circularly polarized electric fields will also be circularly polarized, r ■ \ V, (1-4.9) (1.4.10) For these circularly polarized velocities the v xB0 term may be evaluated as = ±j^^,±My)= ±jB0y; (1.4.11) that is, the term appears formally as being "parallel" to v so that (1.4.2a) and (1.4,2b) become independent of each other, (ju + y+j'Jo), v m J m ' (■ .eB0\ e - I ]us + v-j-\vr=--Er. V m J m (1.4.12a) (1.4.12b) Using — nes- J = crE, we then obtain two conductivities for the circularly polarized fields 1 I r in v+j(w — ±-X-ii_* + W(«±« (1.4.16) < ii cularly polarized electromagnetic waves, traveling in the ±z direction, 11 up circularly polarized electric fields, to which the conductivities or .electric constants just calculated apply. The geometry of these waves lllown in Fig. 1.7. Therefore, the refractive and attenuation indices of lie'.e cyclotron waves may be calculated from the dielectric constant using ,46) and (A.47): |T-Re(«,,K) |f I -V-> + Oib) 2\ «>[(o>±a>by + v:i] (1.4.17) + i(! tt>„3(<0 + cu„) u>[(o> ± to,)2 + v2]/ n \w[(co ± a>b)z + v2] 1 'Mil'. iiiinlily is not to be confused with the Lurmor frequency, which occurs in other I problems involving precession of electrons in a magnetic field, and is defined ii (thai is, one-half the cyclotron frequency). 16 Electromagnetic wave propagation in a cold plasma Chap. 1 Xi,t= -Im(«!./'-) a>„2(o> ± wb) = - 1 21 w[{cj±wby+v2\ (1.4.18) 1 ±toby+vi] + If Numerical illustrations of (1.4.17) and (1.4.18) are given by Figs. 1.8 and 1.9. Further examples are given in Sections 1.4.9 to 1.4.12. The evaluation of the effective collision frequency v is discussed in Sections 2.4.3 and 2.5.3. 5.0 4.0 3.0 2.0 1.0 1.0 2.0 3.0 ■1 1 1 1 I 1 1 1 1 i ii i i i i i i i i i \ \ \ XjBf/Ml' = 2.0 V ^•>>oj>^-^__^ —-—Z ^-5^^----__- >"* iii! • 1 \ o.2\ __—■—V" (,/« 1.5 FIG. 1.8 Refractive and attenuation indices as functions of magnetic field for circularly polarized waves propagated along the field, for various plasma densities. Right-hand polarization (sense of electron gyration), solid curves; left-hand, dashed. No collisions (i=0.0.l. If I lie collision frequency is sufficiently small, as in Fig. 1.8, simplified ex-ions may be obtained. For propagating frequencies («r > 0) in the limit "</\, 18 Electromagnetic wave propagation in a cold plasma Chap. 1 1.4 Wave propagation with magnetic field 19 approximations (A.50) and (A.51) yield L "•("> ± "VJ The refractive index (1.4.20) has a zero (cutoff condition) for txiv%\ofi = 1 + oijw, and a pole (resonance condition) at the cyclotron resonance (1.4.20) (1.4.21) (1.4.22) (1.4.23) for the right-hand wave, which is the sense of electron gyration. Fig. 1.10 sketches the zeros and poles in the uj*—u>t plane; the cut-off regions are indicated by cross-hatching. The refractive index of the left-hand circularly polarized wave (+ sign in propagation formulas), which rotates in an opposite sense to electron gyration, shows no cyclotron resonance. The refractive index is always less than one. At low fields and high densities the wave is cut off. The refractive index of the right-hand wave (— sign), as a function of magnetic field, is seen in Fig. 1.8 to start at p.< 1, decrease to zero (cut off), go through a resonance and, finally, return asymptotically to unity. At high Electron density (wp/cS)"=n/nc FIG. 1.10 Propagating regions of left and right circularly polarized waves along magnetic field (no collisions). Waves are cut off in shaded regions. fields (wbjoj > 1) a right-hand wave always sees ;i> 1, and near the resonance region, coJoj > 1, sees a very high refractive index. The wavelength is then very small and the wave velocity slow. This condition of right-hand polarization and to„ > m is often called the "whistler mode" of propagation (Helliwell and Morgan, 1959). Such waves tend to be ducted along the magnetic field lines, the propagation vector being confined to small angles in respect to the field lines (see Section 1.4.11). Most laboratory plasmas, however, have appreciable losses at the conditions appropriate for whistler propagation, and the wave attenuation is very large. Further mention of the whistler mode is made in Chapter 6. 1.4.2 Faraday rotation of angle of polarization. The significance of the independence of (1.4.12a) and (1.4.12b) for circularly polarized waves is that a general wave, with arbitrary state of polarization, propagates along I he magnetic field in a plasma in a manner which may be analyzed by (a) resolving the actual wave into two counterrotating circularly polarized waves, (b) following the independent propagation of these components, 1.0 2.0 3.0 Electron density (ojp /a))2 = n/nc i ii. I.I I Faraday rotation index An — ii., — nr, neglecting collisions, for uj,,/ = 0.5 ItiidlX = (A^dlk (1.4.24) where d is the distance traveled in the plasma. An example is given in Fig. 1.11. Note that this rotation is with respect to the magnetic field and independent of the sense of propagation along the field. Thus, the propagation is nonreciprocal (Goldstein, 1958). The attenuation of the two counterrotating components will also be different in general. When the difference is significant, one of the components in the emerging wave is smaller than the other, so that the wave is elliptically polarized. If one of the waves is completely cut off, as shown at densities greater than 0.5 and 2.5 in Fig. 1.11. the emerging wave will be purely circularly polarized, and no Faraday rotation effects will be measurable. 1.4.3 Arbitrary direction of propagation: coordinate systems. We are concerned with waves propagating in the direction specified by the vector FIG. 1.12 Alternative coordinate systems for propagation in an arbitrary direction to the magnetic field. propagation constant v.9 The vector nature of the equation of motion (1.4.1) forces us to define a coordinate system with care. Two such coordinate systems are common. We shall have occasion to use both. In one case, the z axis is aligned with the wave propagation direction y, and the static magnetic field B0 is taken in the y-z plane (Fig. 1.12a). In the other, the z axis is aligned with B0, and y is taken in the x-z plane (Fig. 1,12b). In both cases, 6 is, in magnitude, the angle between y and B(). Thus, 0 = 0 corresponds to propagation along the field, 0 = 90° to propagation across it. 1.4.4 Magnetic field at angle B with respect to propagation: Appletorts equation. We now seek a general solution for propagation at an arbitrary angle with respect to the magnetic field, using the coordinate system of Fig. 1.11a in which the propagation is in the z direction with phase factor expfjW —yz), and the static magnetic field lies in the y-z plane [Bn = IS;. (0, sinfl, cost?)]. Expansion of the force equation (1.4.1) yields: r ■ , ii eBn s eBa 0"+"K+™ cose vv—— 4i£ m --COSf? Vx+ {ja> + v)uy m —- sin 8 v, m ±E m 'J + (jw + v)v,—--E, m (1.4.25a) (1.4.25b) (1.4.25c) Identifying the cyclotron frequency (1.4.15) and replacing velocity by current density (J= — ney), we have (jto\v)Jx\ u>b cos(? Jv — ojb sinOJz — — Ex ■ oib COSd Jx + (joj + i>)Jv ne rn wj; Sinň Jx in written in matrix form /co + v ojb cos# — aib COS# j"-> + v ojh sinů 0 tie" -oib sinfi 0 Ex Jy = Jz. (1.4.26a) (1.4.26b) (1.4.26c) (1.4.27) This is seen to be an expression of Ohm's law, with a tensor reciprocal-.....ductivity (resistivity),11 ů-I-J = E. (1.4.28) " I lie d i ruction of y is the waoe-nnrmul direction, perpendicular to planes of constant I'll.i-.v. II Is not necessarily the direction of energy flow. Sec Section 1.4.11. " Ati olementary discussion of tensor algebra is given in Appendix 13. 22 Electromagnetic wave propagation in a cold plasma Chap. 1 1.4 Wave propagation with magnetic field 23 Tn principle we could invert the matrix tr_1 by standard methods to obtain the conductivity tensor a, find the equivalent dielectric constant tensor k = 1 — /o/W0, and then substitute in the anisotropic dispersion relation (A.74). This approach is followed in Section 1.4.7. Here, however, we use a different but equivalent approach, which is somewhat less complicated algebraically, ft is possible to obtain a formal expression for the conductivity tensor in terms of the propagation constant from Maxwell's equations. We then demand that the two expressions for the conductivity tensor be self-consistent. We assume plane waves having the phase factor exp(/W — yz), where y = a+jfi=j[ip,0Hv 0= —jw[j,0H;. yHv=Jx+jwe0Ex (1.4.29a) (1.4.29b) (1.4.29c) (1.4.30a) (1.4.30b) (1.4.30c) We note that the wave can have no longitudinal component of H, regardless of the direction of the static magnetic field; however, a longitudinal component of E is not excluded so that the Poynting vector need not be in the direction of y (see Section 1.4.11). Eliminating Hx, Hv, we have, in matrix notation, -,;2-i 0 0 0 0 ~E* w = A. which is an expression for Ohm's law in the form a-E = J. Now for (1.4.28) and (1.4.32) to be self-consistent E=o-1.J = d1a-E (1.4.31) (1.4.32) or wliere 1 is the unit or identity tensor. Equation (L.4.33) represents three simultaneous homogeneous equations; the determinant of the coefficients must vanish to yield a solution. Substituting from (1.4.27) and (1.4.31) and carrying out the matrix multiplication (see Appendix B), we have (J<*> + v)(p? — 1)4-jw/joi ct>i,(}lz — 1 )cos6 f"o sin(? tti^/i2 — l)sin0 0 — (J(j)+v)+jci}vzjo) 0. (1.4.34) We now have a straightforward but complicated equation to solve for p., the complex refractive index. It is helpful to define some normalized quantities to simplify the algebra: X = oip2joyi Y=tiibjoi T = 1 -jfjoj Y,.= Ycoa6 (1.4.35) M2 = p-\ YT=Y sin0 The determinantal equation (1.4.34) becomes T-I-A7AP -jYL -jYr/M* jYL Y+Jf/M2 0 -jYr 0 -(T-X)IM2 M6 ' X = 0. (1.436) II is most convenient to solve for XjAf2. Rewriting and factoring out W'(\' — X)jX, we have the quadratic rich is I=_fT_JlLl;f, 42 L 2(T-X)\ + [4(T-X)2 llic solution of which is X_ Y 4 -\Vi it 114.1 finally /la=l+M2=l L 2(T-Ar)J-L4(r- ■r (1.4.38) (1.4.39) 2(T- X)\ ~ L4(T- Xf I Ims Ihc result of solving the determinantal equation (1.4.34) is I . . v_ ((Uj,!i/n)a) sin2fj 4(1— CJLt^lu}'-' —JVJO})2 W2 (a-l-/«). _ f (co<,*/">4)s«n*0 4(i-VK->M <»2 An interesting property is (/f+)0R_)=I. (1.4.44) (1.4.45) I or the special case studied in Section 1.4.1 of propagation along the field, K ■ +j as required. The coefficient R specifies the wave polarization except for the longitudinal component From the third equation of (1.4.33) [using (1.4.34)], we may evaluate a longitudinal polarization coefficient q /:.. _ tob(ji2 — l)sinfl ' " Ex~ -(j«> + v)+jo>P2l«> j(wp2luj2)(w,Jio)s\n8 l-JT- (oJb2loj2)sm20 CO 2(1 — 0)p2/u)E — jvjw) 4(l-VK->/"')2 "2 (1.4.46) I lie wave polarizations vary in a complicated manner with the many I'.i1.1meters. Curves for R have been published in connection with Ionospheric investigations (Snyder and Helliwell, 1952; Consoli et al., I'K.I). I 4,6 Propagation across the magnetic field. The important special cases "I propagation along (0 = 0) and across (0 = 90°) the magnetic field are 26 Electromagnetic wave propagation in a cold plasma Chap. 1 1.4 Wave propagation with magnetic field 27 easily obtained from the general Appleton equation (1.4.40). The 0 = 0 case has already been discussed at length in Sections 1.4.1 and 1.4.2. For 0 = 90°, again two characteristic waves are obtained: ordinary extraordinary f-ord = 1- 7« 1 -jvjw (1.4.47) 2 =1- ex — 1 ' ai 1 — top2/cu2 —jvj(i> = 1 -J op2[(a,2 - m2)^2 - o>2 - a>„z) + v2w2] VE - w* - wb2 - i^2)2 + v2(2a>2 - V): v:>pz[w/ + o>2(o>2 - 2o>2 + a>,? + v2)] oj[(o\coz - w2 - w,3 - v2)2 + v\2u>2 (1.4.48) The labels ordinary and extraordinary are conventional in ionospheric terminology, but are in conflict with the terminology of crystal optics and some magnetohydrodynamic literature (Allis, Buchsbaum, and Bers, 1963). Examination of (1.4.40) and (1.4.44) shows that the ordinary wave is linearly polarized with E parallel to the magnetic field. The ordinary wave is so named because it has the same dispersion relation as if no magnetic field were present [compare (1.3.15) and (1.4,47)].11 The field-free case was discussed at some length in Sections 1.3.3 and 1.3.4. The extraordinary wave is polarized with E perpendicular to the magnetic field and is linearly polarized in the sense that it is excited by a linearly polarized wave outside the plasma. However, from (1.4.46), we find that there is a component of E in the direction of propagation; thus, actually, E is elliptically polarized in the plane perpendicular to the magnetic field and including the direction of propagation. To obtain explicitly the indices \x, and x, if p2=L— jM, then by (A.46) and (A.47) ±L + (L2 + M2),/2 (1.4.49) In the limit where collisionai damping may be neglected, (1.4.48) reduces to r(i-M,>y-wvn* [ i-VK-VK J (1.4.50) "This independence no longer holds when finite electron thermal velocities are considered; sec Chapter 3. Electron density (cop/ü>)" = n/nc l''IG. J.13 Propagating regions for extraordinary wave across (0 = 90°), neglecting collisions. The wave is cut off in the shaded rcg magnetic field ions. 0.5 1.0 1.5 Magnetic field w&/<2 2.0 fUi, 1.14 Uel'raelive ami allemiation indices as functions of magnetic field, at bilious plasma densities, for extraordinary wave propagating across the Held, ivulivtinu collisions. 28 Electromagnetic wave propagation in a cold plasma Chap. 1 Cutoff of the extraordinary wave occurs at the two conditions to„2l2 i 1 +r the index is real (waves propagate) if the magnetic field is sufficiently large. For a>„a/co2 < 1, there are cutoffs and resonances, depending on the value of wb. At oj2\oP-=Y the index remains at unity for all nonzero values or magnetic field. Figure J. 15 shows the index of refraction plotted For ioh > oj, note that the index remains 0 0.5 1.0 1.5 2.0 Electron density (wp/a\)" = n/nc FiG. 1.15 Refractive and attenuation indices as functions of electron density, at various magnetic field strengths, for extraordinary wave propagating across the field, neglecting collisions. 1.4 Wave propagation with magnetic field 29 real (waves propagate) even at densities greater than twice the normal cutoff density. A linearly polarized wave incident on a magnetized plasma and propagating across the field is converted, in general, into an elliptically polarized wave. The situation is analyzed in the same way as the Faraday rotation of Section 1.4.2 by resolving the incident wave into component characteristic waves which propagate with different velocities. The effect is analogous to the Cotton-Mouton effect in classical optics. 1.4.7 The conductivity tensor. We seek a general conductivity relation between current density J and electric field E, for a plasma in a magnetic field. Because of the vector nature of the magnetic force gvxB, this conductivity is anisotropic with respect to the magnetic field and is therefore a tensor. Although the conductivity tensor is a general relation, valid, within the assumptions of this chapter, for arbitrary electric fields of any origin,12 it offers in particular an alternative procedure to that of Section 1.4.4 for calculating the propagation constants of characteristic electromagnetic waves in a plasma. To derive the conductivity tensor, it is more convenient to use the second coordinate system of Fig. 1.12, in which the z axis is aligned with the magnetic field. The expansion of the general equation of motion (1.4.1) is thus identical to (1.4.25) with the simplification 0-*0. The reciprocal conductivity is obtained immediately from (1,4.27) as in ne2 JO) + V wb 0 — a>6 Jto + v 0 0 0 y'cu + v (1.4.54) where the coefficient is understood to multiply each term of the matrix. A matrix may be inverted formally by transposing the co-factors, and dividing by the determinant (see Appendix B). Carrying out this operation, we obtain 6 (a-1)'1 ne2jm (jo, + v)[(joj+vy+^] <>' + ")2 ~ti>6(j) {JCD + V)2 0 0 0 (Ja} + vy + a)z (1.4.55) Kclincments are considered in Chapter 3 which introduce spatial dispersion, tlicrchy i lie conductivity is itself a function of wave direction and propagation onitarits, 30 Electromagnetic wave propagation in a cold plasma Chap. 1 and thus the general tensor conductivity is where a = -1" j (m 0" 0 0 0 °ll - . «e2 -> m (to-»2 —cub2 ,2 °x = -J - »6 /H (tU-»2-W(,2 . tie I <*!! = — / The corresponding tensor dielectric constant is —JKX 0 " K=i-y 6 = 0 0 0 where _ (qJp2/oJ2)(l ->H 2/,.,2 k„ = 1 - (1.4.56) (1.4.57) (1.4.58) (1.4.59) (1.4.60) (1.4.61) (1.4.62) (1.4.63) Ki-/i2cos20 -./«■* £2sin0cos0 2 /i2sin0cos0 0 k„-/i2sin20 This equation is quadratic in fir =0. (1.4.64) (1.4.65) Having now obtained general expressions for the conductivity and dielectric constant, we may find wave propagation constants by using the dispersion relation for plane waves in an anisotropic medium (A.74). Without loss of generality we take the propagation direction to lie in the x-z plane with direction cosines (sin9, 0, cos0); then (A.74) becomes 1.4 Wave propagation with magnetic field 31 with coefficients A = kx sin20 + k|; cos20 B = - (/cx2 - Kx 2)sin2(9 - k ^.(l + cos20) C^k/kx2-**2). Alternatively the equation may be solved for tan20, yielding (1.4.66) tan20= — (1.4.67) This latter form is particularly convenient, as it permits one to extract easily the propagation formulas for the special cases of 0 = 0 and 90°. Also one finds cutoff and resonance conditions as a function of angle by setting [l2 — 0 or oo, respectively. Both (1.4.65) and (1.4.67) are alternative forms of Appleton's equation (1.4.40). Further discussion and numerical examples are given in Sections 1.4.9 to 1.4.12. 1.4.8 Conductivity in rotating coordinates. Some algebraic simplification of the conductivity and dielectric constant tensors is obtained by expressing them in the rotating coordinates introduced in Section 1.4.1 (Astrom, 1950; Turner, 1954). By (1.4.6) an arbitrary electric field is resolved into the independent component vectors E,=(ax-jay) £TiV2 E2 = a,£, (1.4.68) rather than the usual cartesian components (Ex, Ey, Ez). Continuing in the coordinate system with z axis aligned with the magnetic field, and using (1.4.12), we find for the equations of motion [(>+v)+>b]i.'i = -- Mi [O + v)-joj„]vr = - - Er (joi + v)t>., = or in matrix form with J= — ne\ . in cm + oib —jv 0 0 0 > — i 0 0" a' = 0 _0 0 0 °1 - 77 e2 1 m -jf)+OJ„ tie2 1 m (a>~jv)—cub .ne2 \ Comparison with (1.4.57) to (1.4.59) shows al + Vr (1.4.71) (1.4.72) (1.4.73) (1.4.74) 17x-- 2 -(/«)+«>6/w and the relations 1 ->/* (1.4.76) (1.4.77) (1.4.78) (1.4.79) «i = - K|=Kj. + Kx (1.4.80) In this notation, Applcton's equation in the form (1.4.67) becomes (1.4.81) /.^ Wace propagation with magnetic field 33 The conversion of the conductivity or dielectric constant tensors from fixed to rotating coordinates may be accomplished directly by a unitary matrix transformation (Turner, 1954). Consider a vector which is given in terms of the usual cartesian components. There exists a transformation matrix U which operates on the vector to express it in components in a second coordinate system witliout changing its physical meaning. For example, the equivalence stated in (1.4.6) is given in matrix notation by 1 V2 T -j 0 1 J 0 0 0 V2 'Ex- 'El- Ey -- Er .Ez_ M. which is of the form UvE = E'. The reverse transformation is T 1 1 V2 0 j -J o .0 0 V2. -Er ~EX- = Ev of the form U and U "1 are reciprocal matrices such that u-i-u = u.u1=i. (1.4.82) (1.4.83) (1.4.84) (1.4.85) (1.4.86) Consider the relation J = ct-E; then, with the primes denoting the rotating coordinate system, CT'.E' = J,= U.J=U.a-E=U.o.U1E' and we obtain the transformation relation for the conductivity tensor ct^U-o-U-1. (1.4.87) If in cartesian coordinates in general (1.4.88) <*yX azx °zv (7*. 34 Electromagnetic wave propagation in a cold plasma Chap. 1 then in the rotating coordinates using (1.4.82) to (1.4.85) (axx + avy) +j(ax„ - ovx) (axx - asv) —j(oxs + ovx) axz-jay;.~ 2 2 a 2 V2 2 V2 which for (1.4.56) simplifies to 2 V2 VI 0 0 0 <*i. + o* 0 0 0 (1.4.89) (1.4.90) in agreement with (1.4.71) to (1.4.75). By the same transformation, the dielectric constant is obtained in the form (1.4.76). 1.4.9 Summary of principal waves. Before further consideration of the general case of propagation in an arbitrary direction, it is helpful to collect and summarize results for propagation in the principal directions along (0 = 0) and across (0 = 90°) the magnetic field. Since there are two characteristic waves for each direction, we have four principal waves. From the "tan20" form of Appleton's equation, (1.4.81), we obtain the indices by setting the numerator or the denominator equal to zero. Thus, in terms of the dielectric constant elements (1.4.77) to (1.4.80): 0 = 0: 6 = 90° Kl (1.4.91) (1.4.92) (1.4.93) (1.4.94) For simplicity, we here neglect collisions {vjw 0). We introduce special frequencies: r*t s - k/2)+[K/2)2+vi54 l«a = + K/2) + [K/2)2 + «,„*]» cutoffs resonance ajuh = K2 + to2) 2\% (1.4.95) (1.4.96) (1.4.97) where u>„ and o}„ are, of course, the electron cyclotron and plasma frequencies of (1.4.15) and (1.2.7). The resonance at is known as the 1.4 Wave propagation with magnetic field 35 upper hybrid (cf. Section 1.5.2). In terms of these special frequencies the principal propagation formulas become: 6 = 0: (at + oi1)(co ± co2)] - (9 = 90° fiord - to(oj + co),) J [(to2-at12){w2-w2 [ w2{co2-<4h) (1.4.98) (1.4.99) (1.4.100) This formulation permits quick identification of the frequencies for cutoff (fi^-0) and resonance ao) for the respective waves. Figure 1.16 sketches qualitatively the behavior of the propagation indices as functions of frequency. The plasma medium may be thought of as a sort of filter with pass- and stopbands (P'ower, 1956). A useful point of view in laboratory plasma physics is to regard electron density, rather than wave frequency, as the independent variable. All quantities may then be normalized to the (fixed) wave frequency. In particular, the density may be normalized to the critical density nc, defined in Section 1.3.5 (nin^w/jcu2). Accordingly, the special frequencies of (1.4.95) to (1.4.97) correspond to the special densities: (tiilne= l+tofc/tu {n2jnc= l—mjoi resonance nhJnc=l—tub2lo>z The propagation indices of the principal waves become: cutoffs (1.4.101) (1.4.102) (1.4.103) 0=0: (I 90": - (»-"\; » r(/u-«x»2-«)ii4 I. nc(nh-n) J (1.4.104) (1.4.105) (1.4.106) (1.4.107) Figure 1.17 shows qualitatively the behavior of the propagation indices as functions of density. Note, in particular, that for strong magnetic fields such that tab>ta, «a and nh become negative, so that the respective cutoffs Ii 1 0 Wave frequency u FIG. 1.16 Qualitative variations of refractive index with frequency for principal waves, showing slopbands (shaded) and passbands. The locations of cutoffs and resonances are unchanged as p. 36 ::::::i:::::i':!:::;B «2 rlh Electron density n. FIG. 1.17 Qualitative variation of refractive index with electron density for principal waves. Note disappearance of slopbands for R and X waves when u>bj I. 37 38 Electromagnetic wave propagation in a cold plasma Chap. 1 and resonances at n = n2 and nh no longer occur in (1.4.105) and (1.4.107). For low fields where uib2->/CO)2 co2 *i(£2-«l)(£3-«r) (P-2 - K\\)(Klfi2 - KlKr) (1.4.108) (1.4,109) Cutoffs (/j, -f> 0) and resonances (/j. -> co) for the various waves are sharply defined only as i>/to —0. They may be found from either (1.4.108) or (1.4.109): cutoffs St—1 (1.4.110) resonances 1 ■ a>2 l-K2/a;2)C0S26l (1.4.111) The cutoffs are thus independent of propagation direction, but the resonances are not. The poles and zeros of the refractive index, bounding the pass- and stopbands, are shown in Fig. 1.18. The most notable new feature is the second family of resonances in the upper right portion of the figure. Appleton's equation in the form (1.4.108) is algebraically awkward because of the square root. It simplifies in either of the two limits in which the radical can be expanded binomially. These are known as the quasi-longitudinal (QL) and quasi-transverse (QT) approximations depending upon whether the term involving cosf? or sinfl, respectively, is dominant (Booker, 1935; Whitehead, 1952). Putting P = (<«,,/«>HsinM0/cos0) , — KfKf s\n26 2(1 — W^ju)2 — jvjui) k||k, 2 cos# (1.4.112) 1,4 Wave propagation with magnetic field 39 .c 3 £ i 0 1 2 Electron density fcj^/oj)2 a n/n<._ FIG. 1.18 Cutoffs and resonances for oblique propagation, neglecting collisions. L and J? refer to the QL approximation, O and X to the QT. As an example, the nonpropagating regions for 0 = 30° are shaded (positive slope for O, negative slope for X). and expanding binomially to first order, we have: QL(|p2|«l): l-7-±-cos0(I+p+ip2+...) oj o) QT(|p2|»l): ['■ord i 1- »1- 1 —jyjo) ± (t0(,/a>)C0Sfl 1-/,-.+? cosf3(p 1+...) ___^2\^_ 1 — jfju) + (1 — u)p2/tu2 — /V/cu)cotH0 (1.4.113) (1.4.114) l-J:-,KyilnI('^-'+.-) s 1 - oj 1 — u>p2l2 — jvjcti t . i/ K2/ 1, neglecting collisions (vju> ->0). rtHtttttrltHK Ordinary wave . _ Extraordinary 1.0 1.5 Electron density = n/nc FIG. 1.19 Refractive indices for characteristic waves at various angles 0; iub!w = 0.5, vjui — Q. The shaded regions indicate the domains of the ordinary and extraordinary waves. relation (1.4.44) show that the QL waves are very nearly left and right circularly polarized,13 and the QT waves nearly linearly polarized (wave E in the plane of B0 and y for the ordinary wave, E perpendicular to this plane for the extraordinary). Except for a small range of conditions such 13 The handedness of the circular polarization is defined with respect to the magnetic field direction, nol the wiive propngal ton direction. 1.0 1.5 2.0 Electron density (ojp/u)2 = n/nc FIG. 1.2(1 Same as I ig. 1.19, hut w,/ai=l,5. Curves for propagation angles other than those shown lie within the cross-hatched regions, and can be estimated by interpolation. An interesting anomaly appears at a>„2/a>2=L The 0C curves are seen to interchange. The curve for the left-hand wave drops down abruptly ami becomes the right-hand wave; the right-hand wave goes through a resonance to become the left-hand wave. This purely mathematical difficulty arises from the fact that at ojr,a/tu2 = l the QL condition |p2[«l 42 Electromagnetic wave propagation in a cold plasma Chap. 1 -} £ ^ V FIG. 1.21 Polar plots of refractive index (index or slowness surfaces), (a) (u6/tu = 0.5, (6) tii„/«)= 1.5. Ordinary wave, dashed curves; extraordinary, solid; no collisions. 1.4 Wave propagation with magnetic field 43 -_w2 = 2.0 -1.1 ' \\ \0.5 itl \ i 90° | 1 /If / I / ,// i / V! H ir i 1 1 90° 3 2 1 0 1 2 3 ■ ti ÍLN CO -a I g" S s s ,s ■** § 3 a a cj US E a IB I 3 -a Pi tfl I) fi £ I ] 13 o 1 . 5 (N £ a - Si £ m a c eg — 3 O [/] C id U y > 13 >■ 1 u a; w » t3 :| c 5a _ >4M . H ~ in Q a 13 c I 5:J ' u '-3 § XI d o S c p ■3 £ 1 ■■■■ c 3 6 1-1 O EC i D 2 M ~ a cannot be obtained without collisions. There is thus a critical angle defined by sm2dcrn \co&0čr 2v (1.4.116) such that when 8 > 8crit the curves for the ordinary and extraordinary QT waves are continuous across the boundary o>2\oýl= 1 (as in Figs. 1.19 and 1.20 for 0^0), but when 0<0crtt the curves for the left and right circular QL waves are continuous (in Figs. 1.19 and 1.20 only for 0 = 0). When 0=8„lh the two curves intersect at u>v2layl= 1, and the two characteristic waves couple to each other. Further discussion of propagation characteristics when collisions are significant is found in Ratcliffe (1959), Chapters 9-10, and Budden (1961), Chapter 6. 1.4.11 Index, velocity, and ray surfaces. The refractive index [l= Re(/i) may be displayed with polar plots as a function of 6, the angle between the propagation direction and the static magnetic field (Clemmow and Mullaly, 1955). Figure 1.21 shows representative cases. Tn classical crystal optics this type of figure is known as the refractive-index or slowness surface.14 Alternatively, one may make a similar polar plot for the reciprocal refractive index-—that is, the phase velocity véjc. Figure 1.22 shows in this format the same cases as Fig. 1.21. This type of figure is known as the phase-velocity or wave-normal surface. A useful catalog of wave propagation characteristics as functions of electron density, magnetic field, and direction of propagation can be made by dividing the density-magnetic field plane into regions in each of which the phase-velocity surfaces have a distinct topology (Allis, Buchsbaum, and Bers, 1963; Stix, 1962). This chart is shown in Fig. 1.23. Resonances at oblique angles occur in the three regions with the "8" or "co" surface. A third type of polar plot is the ray surface, which is the shape of the Huvgens wavelet and has application in all problems of the energy flow, refraction, and diffraction of monochromatic waves. In an anisotropic medium the Poynting vector ExII (ray direction) is not parallel to the wave-normal direction y whenever there is a component of E parallel to y. If one considers plane wave fronts of all orientations crossing an origin at / = 0, the envelope of these wave fronts at t — 1 is the ray surface (Hines, 1951). The ray direction is displaced from the wave-normal direction (y) toward the magnetic field direction (B„) by an angle a where 1 dii tuna. — —~< H do (1.4.117) 1,1 The term "surface" is used since the plot may be visualized in three dimensions as a ligure of revolution about [he ()-=() axis, symmetric about the 0 = W plane. 46 Electromagnetic wave propagation in a cold plasma Chap. 1 Velocity surface Wave front 1TG. 1.24 Relation between phase-velocity and ray surfaces. The wave front is perpendicular to the radius vector to Q and tangent at P. The magnitudes of the radius vectors are the phase velocity and the ray velocity vTau. The radius vector to the ray surface in any ray direction is the ray velocity idm® (1.4.118) The ray surface is related to the phase-velocity surface in that the tangent planes of the ray surface are everywhere perpendicular to radius-vectors of the velocity surface (Fig. 1.24). The ray direction corresponding to a given wave-normal direction (y) is also the normal to the tangent plane of the refractive index surface. In a dispersive, as well as anisotropic, medium the group velocity, which arises from the dependence /a(">), is parallel to the ray velocity, which arises from the dependence /a(0). For a loss-free but otherwise general medium the dispersion relation [such as (A.74)] is a functional relationship between the angular frequency = i du> 9(o den -;)/u))s = n/nc FIG. 1.26 Contour map of real part of refractive index for the extraordinary wave propagating across the magnetic field (0 = 90°; .»/«.= I0"3). The ordinary wave is represented by the (<«P/co)2 axis (that is, a>(,/co = 0). classical crystal optics for further background (Ramachandran and Ramaseshan, 1961). The problem in the plasma context is discussed at length by Budden (1961), Stix (1962), and Brandslattcr (1963). A famous example is the guiding or ducting of waves in the "whistler mode" along --Ordinary wave - Extraordinary wave 2.0 Electron density (wp/w)2 = n/nc FIG. 1.27 Refractive index contour map for 0 = 20" (v/w = 0). 0 = 0 and 90° are shown in Figs. 1.25 and 1.26. The ridge of high index in the vicinity of resonance and the valleys of depressed index approaching cutoff are apparent. Cross sections at constant u>„2 yield the real parts of the curves in Fig. 1.8 for 8 = 0, and those in Fig. 1.14 for 0 = 90°. Cross sections at constant a>b yield the real parts of the curves in Figs. 1.11 and 1.15, respectively. Contour maps for propagation at other angles are shown in Figs. 1.27 and 1.28. Merc the "ordinary" resonance is seen emerging in the upper 50 Electromagnetic wave propagation in a cold plasma Chap. 1 © II 3 0 right corner. It does not extend to low density for any magnetic field, so that the only possibility of coupling to it from outside a plasma is' by evanescent waves through thin cut-off regions or by mode conversion (see Section 4.2.3). 1.5 Ion motion effects Our discussion thus far has neglected motion of the heavy positive ions. In the high-frequency domain, which is our principal interest, the ion 1.5 Ion motion effects 51 current is very small relative to the electron current, on account of the greater inertia of the ions. However, at lower frequencies (for example, near the ion cyclotron frequency) the ion current can be dominant. Coincidcntally, it usually happens that those frequencies for which ion motion is significant imply wavelengths comparable to the size of laboratory plasmas and, hence, boundary conditions must be considered simultaneously (Stix, 1962). In contrast, our basic viewpoint for high-frequency waves is that of such small wavelengths that the laboratory plasma can be treated as an infinite medium. 1.5.1 Conductivity with ion motions. The contribution of ion currents may be evaluated by a straightforward extension of the methods of Sections 1.4.7 and 1.4.8 (Astrdm, 1950; Allis, Buchsbaum, and Bers, 1963). The equation of motion of the Ath particle species is dv mk j=ft(E+v,.xB0)-mkvkyk, (1.5.1) where mk is the mass, qk the charge, and vk an effective collision frequency or damping term for the Ath species; E is the wave electric field and B0 the static magnetic field (as usual, neglecting the wave magnetic field). The corresponding current density is (1.5.2) where nk is the density of the Ath species. As in Section 1.4.7, the relations (1.5.1) and (1.5.2) can be expressed as a tensor Ohm's law Jk = ak-E. The total current density is then J=2J*=(2**)-E> and the total conductivity16 k The corresponding total tensor dielectric constant is » 2** tue0 tue0 (1.5.3) (1.5.4) (1.5.5) (1.5.6) '"This argument is often phrased in terms of mobility instead of conductivity. See footnote .1 in Appendix A. 52 Electromagnetic wave propagation in a cold plasma Chap. 1 In the circularly polarized coordinate system and notation of Section 1.4.8, the elements of the diagonalizcd dielectric constant tensor, replacing (1.4.77) to (1.4.79), are -2 >-2 CÜIO) — ÜJ, T -J**) where the generalized plasma and cyclotron frequencies are 2 _n^k '-Vi m,. (1.5.7) (1.5.8) (1.5.9) (1.5.10) (1.5.11) 1.5.2 Principal waves including ion motions. For the important special case of a two-component electron-ion plasma with equal densities/js=«; = h and negligible collision rate ve — vi—0, k,= l-kt—\ — KT KT (1.5.12) (1.5.13) (1.5.14) where belo>){\ — o>bilw) n-i\ncE2 (I -u>bch>){\ + tUbiM _(l-tt>g,K)(l-a&K) The refractive indices for the principal waves are 0 = 0: 2_ (tu —t^Xm + tda) _n1 — n Pi (w-riUtJfio-Wl,!) n1 0 = 90°: y , (lu + tritXm —ti)a) _n2-n ^ ~ (tk!u>Zh) (1.5.17) (1.5.18) (1.5.19) (1.5.20) (1.5.21) (1.5.22) (1.5.23) (1.5.24) (1.5.25) (1.5.26) (1.5.27) There is a new resonance for the left-hand wave at the ion cyclotron frequency, as we would expect intuitively. Jn addition, there is the new lower hybrid resonance for the extraordinary wave at wlh, where mlh and the modified oivh are the two positive solutions of the equation (Auer, Hurwitz, and Miller, 1958) »* - IM2 + a>(,.a.l!J[(co?')s-r -0, ' (1.5.28) WIK "1 dip cüf,,. a ah dii Wave frequency u 1.5 Ion motion effects 55 (r3 - r + 1)^ FIG. 1.29 Qualitative variation of refractive index with frequency for principal waves, including ion motions. Compare Fig. 1.16. Electron density (Up/w)2 = n/nc' FIG. 1.30 Characteristic shape of phase-velocity surfaces for various regions oT density and field, including ion motions with r^^Jut,,,. Compare with Fig. 1.23. The labeling of O and X waves changes across the dashed line, which corresponds to the condition kikt=k±k\\, 56 Electromagnetic wave propagation in a cold plasma Chap. 1 approximate solutions of which are given in (1.5.18) and (1.5.19), neglecting terms of order ajfjcalf, and higher. At high densities. cu6i. 1.5.3 Oblique propagation with ion motions. For propagation at arbitrary angles, cutoffs and resonances may be found from the "tan2t?" Appleton equation (1.4.81). Cutoffs, independent of angle, occur when frh or *r vanish. Resonances occur when tan2f? = — k[jk1 = — 2k:j/(k, + «r). For the two-component, collision-free case the resonance condition is K?=__ co2 (1 -u.ilca>(,1/a>2)sin261 + (1 - wfjw2)(l - w^^COsH (l"«4/w2)(1 r4/o,a) (1.5.30) A catalog of propagation characteristics is given in Fig. 1.30 by showing the topology of phase-velocity polar diagrams in the electron density-magnetic field plane (Allis, Buchsbaum, and Bers, 1963). This chart is to be compared with Fig. 1.23, which neglects ion motion. Resonances at oblique angles occur in the five regions having "8" or "co" velocity surfaces. Note that the scale of Fig. 1.30 is greatly distorted by the assumption of a very small ratio tujm,.. In a practical case, mebl,»oi. CHAP T E R 2 Collision processes 2.1 Introduction The preceding chapter developed the propagation characteristics of electromagnetic waves in a uniform ionized medium from the equation of motion of a single, "typical" electron. This analysis would be rigorous if the ionized medium were to consist of nothing more than free electrons which (/) do not interact with the background of neutral atoms and charged ions, and (2) possess thermal speeds negligible with respect to the phase velocity of the wave. Both of these qualifications refer to processes that permit exchange of energy between the electron gas and the electromagnetic wave. Therefore, it was physically reasonable to anticipate these effects by including a simple viscous damping term in the equation of motion. Qualification (/), at least, is approximately accounted for by assuming that the background gas, of neutrals and ions, can be represented by a continuous, stationary, charged fluid through which the electrons move with a drag force proportional to the velocity. The present chapter justifies the identification of the damping term with a collision frequency and investigates its physical meaning in terms of the discreteness of the background particles. Our principal interest is in the difficult problem of electron-ion (coulomb) "collisions." Although mention of collision processes implies particle motion and hence a nonzero temperature, the role of electron temperature in collision processes is essentially different from that implied in qualification (2) above. It is appropriate to describe as "cold" those plasmas for which qualification (2) is well satisfied. The cases of "warm" and "hot" plasmas are considered in Chapter 3. We again assume that the positive ions are so massive that they do not move under the action of the wave field. 58 Collision processes Chap. 2 2.2 Elementary considerations of collision processes 2.2.1 Collision cross sections and frequencies. Consider the interaction of a test particle (an electron, let us say) with a group of field particles (atoms and molecules, or ions). Initially, we assume that the mean free path is sufficiently long and the interparticle force sufficiently short-range that a collision can be treated as a discrete two-body interaction. This assumption is notably troublesome in the case of the long-range coulomb force. For a specific central-force law between two particles there exists, in general, a relation between the impact parameter b, the relative velocity v, and the scattering angle of the test particle tf> (see Fig. 2.1). The analysis in the general case is most readily carried out in center-of-mass coordinates and then transformed to the laboratory system. We shall be principally interested in the case where the field particle has large mass and small velocity compared to the test particle. In this case, v and are simply the speed and scattering angle of the test particle in laboratory coordinates. The probability per unit path length for the test particle to be scattered -- . sab particle i FIG. 2.1 Geometry of a collision. The impact parameter b is the distance of closest approach if there were no interparticle force. The scattering angle is the deflection of the orbit asymptote. 2.2 Elementary considerations of collision processes 59 through an angle between and fi + d

) db = ns 2tt b(v, £} § <#, (2-2.1) where nf is the number of field particles per unit volume. Tt is customary to write this probability in terms of a differential cross section qn(v, 4>)> having dimensions of area per unit solid-angle, such that dP^, = nf <7fl(t-, ) 2t7 suvj> d (2.2,2) where 2n sini/i d = dQ is the element of solid-angle. The total cross section qt is obtained by integration over all angles: =2w J qQ(v,)sindy/3 + r sm2tf>] lost by test particle Fractional forward momentum lost by test particle (1+r)2 1 —cosi£{l — r2 sin2[/i)''ä +r sin2<4 1 +r (2.2.7) (2.2.8) where (f> is the angle through which the test particle is scattered and r — m/M. In the interesting case of m«M: Fractional energy lost 2m M (1 —cos^) Fractional forward momentum lost 1 —cos<4 (2.2.9) (2.2.10) To obtain the average energy and forward momentum lost, it is necessary to average over the scattering angle Thus : <2n(v, <£)0 —cosr/>}sin I <}n(pi ^)sin^ d (2.2.11) In problems of energy and momentum transfer, the important collision cross section is the cross section for momentum transfer qjv) = 2tt J qa(d, £)( 1 - cos<£)sinc£ yqt(v). (2.2.12) Relations equivalent to (2.2.4) to (2.2.6) exist for the momentum-transfer cross section. In particular, there is the collision frequency for momentum transfer, ^=nfqm(v)i\ (2.2.13) which is generally the physically important collision frequency in problems of wave propagation. Indeed, for convenience, we shall normally omit the subscript m hereafter for this quantity. The functional forms qQ(v, <£), or equivalently b(v, ), may be computed, in principle, from the dynamics of the specific interparticle force law. As an example, consider an electron (regarded as a mass-point) colliding with 2 A more general discussion of collision geometries, using cenler-of-mass coordinates and allowing Tor field particle velocity, is given by Allis (1956, Section lb). The neglect of target particle veloeily is reasonable except in unusual eases where the ion temperature exceeds [lie electron temperature by a factor of I lie order of (MjmY/ii. 2.2 Elementary considerations of collision processes 61 1'IG. 2.2 Geometry of the "electron-molecule" hard-sphere collision. a massive hard-sphere molecule of radius R. If the electron impinges with an impact parameter b, it is easily seen (Fig. 2.2) that it is deflected by an angle if> for which cos# = ^-l. (2.2.14) Thus, we obtain for the three cross sections: b cblc sin<£ 4 2 Jo 2 Jo (1— cos0)siii^ cl=2m/?t=1 (2.2.15) (2.2.16) (2.2.17) (2.2.18) Both total cross sections are equal to the geometrical cross section, independent of velocity, and the average energy lost per collision is 2m! M, a well-known result. When the problem is treated by quantum mechanics, the total cross section is found to be greater by a factor of two to four, depending on velocity, on account of diffraction of the electron wave (Mott and Massey, 1949, pp. 38^10). For short-range electron-molecule (or electron-atom) collisions, with forces falling off more rapidly than 1/r3, it has been shown by Mott (Massey and Burhop, 1952, p. 3) that the total cross section qt (2.2.3) is bounded on account of quantum effects. Furthermore, for most common atoms or molecules and electrons of low to moderate energy (~1 eV), scattering is approximately isotropic, and the total and momentum-transfer cross sections do not differ appreciably. 62 Collision processes Chap. 2 As a second example, in the case of coulomb scattering by a massive ion of charge Ze, the well-known Rutherford formula gives (Symon, 1960, pp. 135-38) Ze2 ^4 tan 4^ = mv^b from which one obtains; _/ Ze2 \2 1 qfl~\%7t^mv2) sin4^ sin^ (2.2.19) (2.2.20) (2.2.21) (2.2.22) In this case, both integrated cross sections diverge at the lower limit (large b), on account of the dominance of small-angle deflections, so that an appropriate cutoff on the impact parameter must be applied. The same result is found from the quantum-mechanical treatment. The momentum transfer cross section qm (2.2.12) deweights small-angle deflections by the factor (1—cos<£) and, accordingly, diverges only logarithmically. The long-range nature of the coulomb force leads to many complications to which we shall return in Section 2.5.2. While our discussion has emphasized elastic collisions, most experimental situations clearly also involve inelastic collisions such as excitation, ionization, and dissociation (Francis, 1960; Brown, 1959). These processes can also, of course, be described in terms of cross sections and collision frequencies. The over-all collision frequency is then the sum of the component collision frequencies for each relevant process, i>=ri1qlv + n2g2v+ . . . =v1 + v2 + (2.2.23) Thus, for instance, collisions with impurity particles having a large cross section q may contribute significantly to the over-all collision frequency even though they are present in small concentration n. 2.2.2 Velocity dependence of cross sections. In order to discuss the dependence of cross sections on velocity, it is useful to assume a simple inverse-power force law of the form (Kihara et al., 1960; Mott-Smith, 1960) (2.2.24) where j is usually an integer, and further to assume that the momentum-transfer cross section is determined approximately by the impact par- 2.2 Elementary considerations of collision processes 63 ameter for which the potential energy equals the kinetic energy of the incident particle. Thus K (s- i)lf = h?iv2 -1~ 2 and 2K 1 2,'(s- 1) CC V ,-i;(s-d (2.2.25) (2.2.26) [(s—i)mv2_ vm(r)rjLvis-5i! v, for which the validity of (2.3.1) is not obvious. To do this, we again consider an average electron, but distinguish between the ordered velocity component produced by the wave iield and the random component of thermal motion (Appleton and Chapman, 1932). The equation of motion for the ordered component, during the time interval between collisions, is m£= ~eE0 exp joit. (2,3.2) We assume the boundary condition that this ordered velocity is zero at the instant of collision, /x. Thus eE eE' m~t CX P*'/W)[ 1 ~ eXp^ ~]' (2.3,3) 2.3 Effect of collisions on electron motion 65 where t — 1 — tv. This equation is, of course, valid only for the interval lx < t < where t2 is the time of the next collision. In order to find the average ordered velocity, it is necessary lo know the statistical distribution of the time intervals t, in the past, at which the electrons made their last collision. This distribution is entirely equivalent to that of the time intervals t, in the future, when the electrons make their next collision. Consider a group of electrons, N0 in number, at an arbitrary instant of time. The number N of these that have not undergone a collision after an interval r is obtained from dN dl" giving N=ATa exp(-vr). (2.3.4) (2.3.5) Therefore, the fraction making the previous collision within the interval r to r+dr, in the past, is dN --v exp( — vr) dr, (2,3,6) the familiar Poisson distribution. Combining (2.3.3) and (2.3.6), we obtain for the statistical average of the ordered velocity component at time t. (a eEn f™ =j — exp(>0 J [1 -exp(-jW)> exp(-yr) di eE0 m(v -rjto) exp fat. (2.3.7) But if we had written the equation of motion (2,3,2) with a damping term g£, m'i= -g£-eE0 expjW, (2.3.8) we obtain i=- eE° exp jcut. Comparison of (2.3,9) with (2.3.7) leads to the identification g = mv. (2.3.9) (2,3.10) It should be noted that (2.3.7) assumes that v is either independent of velocity or that the random velocity is large compared to the ordered component. In the latter case, a common one, v is an appropriate average over the electron velocity distribution (Section 2.4.3). Comparison with (2.3.1) shows that the assumption that the ordered velocity is zero after a collision is equivalent to the identification of v as the momentum-transfer 66 Collision processes Cluip. 2 2.4 Analysis of particle interactions 67 collision frequency. It should be noted that collision frequency v compares with the radian wave frequency u>, not the cyclic wave frequency , is usually taken to be the reciprocal of the momentum-transfer collision frequency v; thus, we substitute in the Boltzmann equation (2.4.4) (2.4.6) A second major problem is the mathematical complexity of solving the Boltzmann equation. It is usually necessary to expand the distribution function in a series, and retain only low-order terms. Two such expansions are commonly used. The Chapman-Enskog technique (Chapman and Cowling, 1951) assumes only small departure from local thermodynamic equilibrium due to some perturbing agent of strength measured by a parameter a; thus where fir, v, 0=/c(r, v2) + aMr, v, t) + a%(r, v, 0 + /0(r,^) = H(r)(^)%Xp(- mir\ 'ikfV (2.4.7) (2.4.8) the Maxwell distribution. This expansion does not converge rapidly in problems where the perturbed distribution is anisotropic. Where there exists a preferred direction in space, it is useful to expand in spherical harmonics In velocity space (Allis, 1956): fix, v, 0=2.^r< v> Pis$ =/o(<-'Wi(r, Ocos0+/2(r, u\ t) 3 C°*~ 1 + (2.4.9) where f0 is assumed isotropic but not necessarily Maxwellian. The volume element of velocity space, often written in the alternate notations, ds\=dav = dv = dvx dv-j di\,, (2.4.10) becomes explicitly in this case d3v=vasin8dvd8d (2.4.11) where v, 0, are the spherical coordinates in velocity space. Most physical problems are symmetric with respect to the azimuthal angle ; the trivial integration then yields the simplification4 ds\ —> 2ttv2 sind dv dB = — 2-nv2 dv d(co&ff). (2.4.12) It is useful to evaluate some moments of the distribution function /for the spherical harmonic expansion. Recalling that spherical harmonics are orthogonal, we obtain the normalization condition J/(r, t) rf3r= j 1 ./'(r, v) 2ttv2 d(co&8) dv =4tt j ,M§ v2) v2 dv = n(r) (2.4.13a) where n(r) is the particle density. Frequently it is more convenient to separate the space and velocity dependence, replacing/(r, v) by /7(r)/(v) with the normalization |/(v) d*y=4ir J" f0(v2) v2 dv = \. (2.4.13b) The average value of the velocity component in the preferred direction, vx = vcos8, (2.4.J4) is 7j^= J j (v cos0)/(r, v)2tto3 ^(cosfl) dv \* f,{T,tf), P<" being the associated Legenclre function. 70 Collision processes Chap. 2 but not necessarily Maxwellian. Using the spherical harmonic expansion (2.4.9) to first order, we have /(v, o=/o(/0 + ?^#-)exp>r, (2.4.17) in which the time dependence oi'/i is shown explicitly and costf is written as vxjv. Evaluating the collision term in the form (2.4.6) and substituting (2.4.17) in (2.4.16), we may solve for eE0lm dfo /i=- (2.4.18) v(v) + Jca OV where the small nonlinear term in 8(*\/i/r)£t'* has been discarded. The fact that/i is complex signifies that the electron velocity perturbation is not in phase with the electric field, an entirely reasonable result. The resulting current density for electrons of charge — e is J= — nevx = — ne jvxf(y, t) d = -~ne exp(yW) jj\(v) »a 4-rrne2 3m Ee exp(/a>f) r1 dv (2.4.19) v)+joi dv using (2.4.15) and the alternative normalization (2.4.13b). Equation (2.4.19) is equivalent to the complex conductivity 3 m J0 v(v 4rr_)ie^ r™ d\_# 3 Hi Jo do[v(v)+ja>. ) + jw di 1 fa(v) dv dv (2.4.20) the latter form obtained by an integration by parts. The corresponding complex dielectric constant is »2 ~ 1 +^ % \ •— dr. (2.4.21) e0mto 3 J a>—jv(v) dv Two important special cases of (2.4.20) and (2.4.21) may be obtained, readily, as follows. Collision frequency independent of velocity: *=1- 2 I ne' = 1 c0mtu (jt)'—jv w(m—j\') (2.4.22) (2.4.23) 2.4 Analysis of particle interactions 71 independent o(f()(v), results which coincide with the Lorentz conductivity (1.3.14). This illustrates the well-known Maxwell condition under which the transport properties of a gas are independent of the distribution function. No such simplification exists for other dependencies on velocity. Maxwellian velocity distribution: Assuming (m \ ^- / mv2 \ one obtains for (2.4.20) (2.4.24) 3 V ne-m , * . m* exp(-«2) du, (2.4.25) where u is the velocity normalized to the most probable speed, u = vl(2kTjmyK (2.4.26) In the case of a static magnetic field, a similar analysis summarized by Allis (1956) yields for the elements of the diagonalized (rotating coordinate) conductivity tensor (1.4.71): 4n ne2 3 m 4tt ne2 3 m riM&j^ (2.4.27) Jo v(v) dv f ' I , 'iff] dr. (2.4.28) Jo v{v)+j(u> + ojb) dv The corresponding dielectric constant tensor is then given by (1.4.76). 2.4.3 Effective collision frequency. The previous section has demonstrated that only for the special case or collision frequency independent of velocity does the conductivity take on the simple Lorentz form 1 m v+jco (2.4.29) Since most real collision processes are not of this special case, the conductivity depends upon the integral (2.4.20) involving the functions of velocity v(i>) and /0(7~, r).5 Because of the relative simplicity of the Lorentz formula, and the vast literature employing it, it is convenient to define an effective collision frequency vcff(T, w) which may be used directly in the Lorentz formula. This approach is easily followed in the limits of '■ The electron temperature Tmay here be regarded more generally as any appropriate scaling parameter of the velocity distribution, specifying the distribution even in non-Muxwclliiin cases. 72 Collision processes Chap. 2 very low and very high frequencies, with the results summarized in Table 2.2. In both limiting cases the frequency dependence drops out. Table 2.2 Effective collision frequencies eor extreme wave frequencies Direct-current limit, Radio-frequency limit, ft)»V Loren tz conductivity, Eq. (2.4.29) Boltzmann conductivity, Eq. (2.4.20) Effective collision frequency né\ 1 m v nein 3m J v(v) dm dv t'3 dv 47T/ÍP2 3«; 477 C 1 dm „3 - 47T f Velf— — — V(U) -;- V dV a J v(p) dv 3 1 m In particular, we shall be interested in the effective collision frequency in the high frequency limit, a useful parameter which wc identify by the special notation <» = v3 dv (2.4.30) where f=fa is the unperturbed electron velocity distribution function. This is, in fact, a kind of average collision frequency, like (2.2.28), but obtained with a different weighting of the velocity distribution. For a Maxwellian distribution (2.4.24), the general form (2.4.30) reduces to = Assuming a simple power law [see (2.2.27)] «<(.') = CV, (2.4.32) we may write (2.4.31) in terms of gamma functions (Jahnke and Emde, 1945) as /5+A l2kT\»* \ 2 i (2.4.33) 2.4 Analysis of particle interactions 73 For the same assumptions, the direct average collision frequency (2.2.28) is > = \<{r)fit) Ur; Jr C Maxwell distribution r(—\ Thus ,s 3+/_ (2.4.34) (2.4.35) special cases of which have been quoted in the literature (Molmud, 1959; Phelps, 1960). For each /, there will exist some characteristic velocity for which the velocity-dependent collision frequency equals one or the other of the averages (2.4.33) and (2.4.34). Figure 2.3 shows these characteristic velocities v1 and v2 defined such that t.(i;1) = v(ü3) = V. (2.4.36) -3 -1 0 1 Collision-frequency exponent I 1'IG. 2.3 Characteristic velocities for which v(o) equals the high-frequency effective collision frequency O) or the simple average collision frequency v, as a function of the power-law exponent /. 74 Collision processes Chap. 2 3.0 2.0 1.0 0.5 / = -2 -1 0.3 I 0.01 100 FIG. 2.4 Correction factors g and li for electron-molecule collision frequencies varying as i>'; (a) exponent / negative, (6) exponent / positive. They are not far different from the usual velocities characteristic of a Maxwell distribution, namely (Sears, 1953): (3kT\ ßkTY'"- root-mean-square - \ m 1 average - most probable Ttm m (2.4.37) (2.4.38) (2.4.39) This mathematical analysis in terms of an assumed power law (2.4.32) is physically meaningful so long as the empirical collision frequency approximates the power law over the range of velocities for which Ihe integrands of (2.4.30) and (2.4.34) are large. For practical purposes, this range may 2.4 Analysis of particle interactions 75 |w±W(,|/ T , (2.4.40) 1 fr.r = — / \ ■ -,. / ■ V (2-4.41) where the three g's and three /t's are multiplicative correction factors of order unity. For a given velocity dependence v(v), they are in fact functions of a single argument: gi ltJ=g(h±wb|/±"»l/<>'>)» (2.4.42) 76 Collision processes Chap. 2 2.5 Coulomb interactions 77 where the notation implies that for ga and ha, tu6 is taken as zero. The advantage of the normalization in terms of <\v} is that in the limit of high frequencies g and h go to unity. In the two extremes r *(0)= r (W) r r2 m (2.4.43) (2.4.44) g(co) = /2(oo)=l. The integrations from which g and h are calculated in simple power-law cases have been carried out explicitly by Molmud (1959) and by Sen and Wyller (1960), using slightly different notation. An equivalent formalism has been developed by Gurevich (1956). The integrations are also reducible to functions discussed and tabulated by Dingle et al. (1957). The resulting g and h factors are given in Fig. 2.4 for several power-law exponents. The expression of the conductivity in the form of (2.4.40) and (2.4.41) has the power that the g and /( factors can be generalized to include the effects of electron-ion and electron-electron collisions, as discussed in Section 2.5.3. Extensive calculations of various cases for partially ionized gases have been made by Shkarofsky (1961). It is also possible to use the g- and //-factor formalism for non-Maxwellian velocity distributions. We now summarize the procedure for computing conductivity and hence propagation constants, as functions of +01 b)!(.")>, and \to — wb\j(y'). For given oi and aib, the conductivity tensor elements may then be computed from (2.4.40) and (2.4.41), replacing the simple Lorentz elements (1.4.72) to (1.4.74). The corresponding dielectric constant tensor elements follow trivially from (1.4.76) and (1.4.80). The propagation constants may then be found from the Appleton equation (1.4.65) or (1.4.81). 2.5 Coulomb interactions 2.5.1 Debye shielding. We wish to investigate the manner in which the long-range coulomb forces in a plasma act to maintain charge neutrality. a problem similar to the behavior of ions in an electrolyte (Debye and Huckel, 1923). Consider a uniform plasma with n electrons and njZ positive ions, of charge Z, per unit volume to which a single additional ion of charge Q is added at the origin. Unlike charges will be attracted, like repelled. We wish to find a self-consistent solution for the electric potential \ji{r) at radius r in the vicinity of this excess ion. According to Boltzmann statistics, the plasma ions and electrons will now be distributed with the respective densities (we assume Tr,= Ti) nt(r) =- Qxp{-Z&pjkT% «e(r)=;i cxp(e>pjkT). The resulting net charge density is (except at the origin) p(r) = — ne[exp(eiljjkT) — exp( —ZejijkT)], which in the high-temperature limit Ze -■ kT (2.5.1) (2.5.2) (2.5.3) The potential and charge density are related by Poisson's equation VV=-P,V (2.5.4) Writing the Laplacian operator in spherical coordinates and substituting the approximation (2.5.3), we have 1 £ I 2#\_(j +Z)«eV r2 dr \' dr) ~ eQkT ' the appropriate solution of which is O exp(-r/Ar,) 4irec where A0 = (1+Z)«e2 (2.5.5) (2.5.6) (2.5.7a) For many applications in plasma physics, involving dynamic processes at frequencies of the order of and + d is cos-^ 9 \4iTe0m} t'JstnJ-^ r ,,asin3-^ which becomes in the limit of small deflections dt'./, > S-nnZ I e2 Y d t \4rr€am) v3 and 4> + d, and the expectation value of the resulting total deflection is = t Snnzl-r-^-—\ \ In 4> (2.5.15) where min and max represent the range of small deflections considered. Since the limits enter only logarithmically, they may be rather grossly approximated. Let <£mc*=90°, which does not greatly violate the small-angle approximation of (2.5.14). For min we take the angle for which the impact parameter equals the debye length AD, since for b>\D the shielding effectively masks the deflecting ion (Cohen, Spitzer, and Routly, 1950)e; from (2.2.19) Ze2 2VV 2 l2irenkT\i) in which the kinetic energy \mv2 has been replaced by its mean thermal value jkT. The argument of the logarithmic term becomes <^_3\/W4^7Y/* 1 8 ITT) ^-54**4», (2.5.17) where A3p is a parameter defined by Spitzer (1962, §5.2) as the ratio of the debye length to the mean impact parameter for a 90" collision bso. Specifically A„ 2 iAire0kT\^ I 3 kT I kT \* A = ^1__L. (±^okTy* 1 = 3 kT I kT \ ' >»„ (*rfcVD* = 1.55-101 Z(«[cm-3])!'i (2.5.18) Under common laboratory conditions ln/lSp~10 (see Fig. 2.5). Note that the number of electrons in a "debye sphere" (2.5.9) is lZASp. The time required to achieve a cumulative deflection of 90° is. from (2.5.15), Tbo~32 zTra (2.5.19) * The debye length cutoff is preferable to setting the impact parameter equal to the mean interionic spacingn~ 'i,for which case the logarithmic term is ln|( \»', I'o', I,,''..|, and the number of times per second a deflection by 90° is achieved is 1 32 / e2 VZn\nAK J__32 / e2 \2 Z»ln. "90-~t90~ tt \4-tt*0m) i'3 32 \nASp z-2- '/9(1 + - (2.5.20) Thus, the deflection rate (collision frequency) due to cumulative small-angle deflections is, typically, of the order of thirty times larger than that due to individual large-angle deflections. A more rigorous analysis in terms of velocity-space diffusion theory, in which velocity changes rather than angular deflections are considered, has been carried through by Chandrasekhar (1943) and by Spitzer (1962, §§5.2-5.3). In particular, Spitzer defines a "90° deflection time," the reciprocal of which is tt2 (2.5.21) this is to be regarded as an improved version of (2.5.20). In this theory the argument of the logarithm emerges as the ratio between maximum and minimum impact parameter and the definition A..s = XD!hgo given by (2.5.18) follows logically. However, these particular limits must be modified under a variety of circumstances, discussed in Section 2.5.4. In the following section we use the general symbol A to denote the appropriate ratio. In summary, we can divide electron-ion coulomb collisions into three classes (Bernstein and Trehan, I960). (7) Impact parameter h< impact parameter for 90u deflection ba0. In this case of close encounters, the electron-ion interaction may validly be considered as a discrete two-particle collision and treated by the usual collision integrals of kinetic theory (Allis, 1956; Dcsloge and Matthysse, I960). (2) bgo A.,. The effect of the debye shielding is to eliminate statistical (uncorrected) particle encounters. Particle motions on this scale represent correlated processes such as the propagation of waves. 2.5.3 Effective coulomb collision frequencies. It remains to be found how to incorporate into the conductivity formalism (2.4.40) and (2.4.41) the effects of electron-ion and electron-electron collisions. From the Fokker-Planck equation, or directly from the Boltzmann equation, it can be shown that an electron-ion collision frequency7 2 \2ZnlnA (^[cm/sec])'* (2.5.22) plays exactly the same role in the calculation of conductivity as the more straightforward electron-molecule momentum-transfer collision frequency 1 vn, defined in (2.5.21), is the reciprocal of Spitzer's "90° deflection time." 1000 at loo S 10 i y i i y | i y ' i 1 y ^$X X <$y y^ y X^* ~^X X iy $X X X ^X X ^X y* > \^y i0X y i 1 iy i i i/ i y^ y ■$X~ i y i 101 1012 1013 Electron density n [cm-31 101 1015 FIG. 2.6 The effective eleclron-ion collision frequency (2.5.23), The numbers shown arc f>„i>/2wZ \nA. The appropriate value of In/1 must be round from (2.5.30) and Fiys. 2.8 and 2.9. The 2tt normalization facilitates comparison with Ihe wave eye lie-frequency u>/2tr in propagation formulas. See also Plgi 2.10. veJv) (2.2.13) (Ginsburg, 1944; Cohen et a!., 1950; Allis, 1956; and Shkarofsky, 1961). Thus, from (2.4.33) for /= —3 the effective electron-ion collision frequency at high frequencies is for a Maxwellian velocity distribution (see Fig. 2.6)e 40)"= / e2 \2 Zn ln/1 <«W< / e2 \2 Zr, \4ireJ mVl = 2.90.10-^^1^ sec-. pmm - ■ (Z5-23) In performing this integration, we have followed common practice by ignoring the velocity variation implicit in A.9 The proper evaluation of InA is discussed in the next section. Tn the common case of a partially ionized gas we have from (2.2.23) VlvtaiW^V^V) + vei(v), (2.5.24) <"(i,taJ> = <^řn> + , (2.5.25) the latter form following because of the linearity of the operation defined by (2.4.30). The critical degree of ionization above which coulomb collisions are dominant is given by = , which of course depends on the velocity dependence of the particular electron-molecule interaction and is usually a strong function of temperature. At room temperature the critical ionization for common gases in low-pressure discharges may be as small as ~ 10~7; at kT= 1 eV, —TO-4 (Lin et al., 1955; Anderson and Goldstein, 1955). Using (2.5.23) and (2.4.43) in the d-c limit of the conductivity (2.4.40), we obtain £(0) = 3tt/32 and __ne2 1 aic~^n~ g(QKvei> _4v/2 (47TE0)a (kryÁ 7TS/* m = (4W3)C/(0) where/(0) is the value of the electron velocity distribution function for u = 0. Thus, the high-frequency effective collision frequency is insensitive to departures from the Maxwellian distribution affecting only the high energy end. " If the velocity dependence of ASv, A{v) = \D(T)!bsa(v)b=0), Spitzer and Härm (1953) and Kelly (I960) have used a numerical factor yE to express the ratio of the conductivity including electron-electron collisions to that for a Lorentz gas 2.0 y£H, Z) = , Z) a(cu/<», CO) At extreme frequencies yt.(0, 1) = 0.582, yB(oo, Z)=l. (2.5.27) (2.5.28) The ratio remains of order unity at intermediate frequencies. As shown by Hwa (1958) and Kelly (1960), the major physical effect of electron-electron collisions on wave propagation is to broaden and damp the cyclotron resonance somewhat. Electron-electron collision effects may safely be neglected at high frequencies such that |a> + wb\ /<»» 1. 1.0 0.5 0.3 X Lorentz (Z= ~) Z= 1 Z= 1 0.01 100 FIG. 2.7 Effect of electron-electron collisions on the correction factors g anci h for a fully ionized gas with no magnetic field. The Lorentz gas calculation ignores electron-electron collisions; they are included in the curves for singly charged ions. 2.5.4 The logarithmic term. Most formulations of the electron-ion interaction process involve an integration over the impact parameter b that yields a result proportional to \n(bm!lxibmi^). The ratio A = bmiix!bmin and its logarithm are divergent unless we invoke physical arguments to cut off the range of integration (Theimer, 1963). The arguments sketched in Section 2.5.2 led to the identification of bmax with the debye length AD (because of shielding) and of bmin with the mean impact parameter for a 90° deflection 580 (because of the relative ineffectiveness of close encounters). The argument of the logarithmic term then becomes Spitzer's Asp = ■W^so, as given by (2.5.18). Aside from numerical refinements to As„ of order unity, there are two general effects which alter the form of A, one depending on electron temperature and one on wave frequency (De Witt, 1958). We discuss these in turn. The temperature effect pertains to the lower limit bmin. At electron temperatures above about 80 eV the deBroglie wavelength ftjtnv exceeds the 90" impact parameter b^-'/.c'-'jAm^iir''' and replaces it as the limit 86 Collision processes Chap. 2 bmin. That is, the electron is diffracted rather than classically deflected by the ion, and the interaction must be treated by wave mechanics (Marshak, 1940). The classical and quantum limits have been discussed extensively by Oster (1961b). Numerical calculations of the transition have been made by Greene (1959) and Oster (1963a). Meanwhile the frequency effect pertains to the upper limit bmax. Above the plasma frequency o>p, the distance traveled by the electron in one period VtJcu replaces the debye length XBxvtl!jCy/ hwv fid), v ftcu 4 kT Y kT fio) * e = 2.1 \ 8; y= 1.781 (Euler's constant); Rll = 13.6 cV (Rydberg energy constant); dsp = halbi>Q, see (2.5.18). common microwave propagation experiments is the high-frequency, low-temperature result (see Fig. 2.8) (Scheuer, I960) (fcr[eV])K = 6.2 10-1 (2.5.30) Z wj2tr[Gc] where y= 1.781 is Euler's constant and R„= 13.6 eV is the Rydberg energy constant. Figure 2.9 shows the corrections to be applied to (2.5.30) in the transition regions between the limiting cases of Table 2.3. The temperature dependence is taken from Greene's (1959) revised calculations. The frequency-electron-density dependence is taken from Dawson and Oberman (1962). The "bump" in the neighborhood of to = cop is due to the generation of plasma oscillations. These two corrections are independent in the Rayleigh-Jeans approximation (fiw«kT) (Oster, 1963a). The resulting collision frequency and attenuation coefficient, in the unshielded limit co»cop, are shown in Fig. 2.10. To summarize the effect of coulomb collisions on microwave propagation in a plasma without magnetic field, we recall that the formalism of (2.4.40) is used with the effective coulomb collision frequency (2.5.23). The g and h correction factors are given to good approximation by the /= -3 case of Figs. 2.4 and 2.5. The In/1 term appearing in (2.5.23) is evaluated using the A0 of (2.5.30) and Fig. 2.8 modified if necessary by 88 Collision processes Chap. 2 1000 j 10 100 1000 Wave frequency oi/ZttZ'1 [Gc] FIG. 2.8 The logarithm lnyl0 of the high-frequency, low-temperature cutoff ratio (2.5.30), for ions of charge Z. The temperature correction of Fig. 2.9 is shown here by the dashed curves; the density correction must still be obtained from Fig. 2.9. the two independent corrections given in Fig. 2.9. For most practical purposes one may use for A the asymptotic limits of Table 2.3 throughout the range of parameters, changing from one limit to the other at the points where the asymptotic forms are equal, namely, AT=y3ZX = (77 eV)Z2 = (890,000° K)Z2 (2.5.31) ,1 II n.6 "L '2e ;0.328: (2.5.32) Finally, we note that further modifications of A are required in strong magnetic fields for which the gyroradius is smaller than the debye length, that is, when wb>a>p (Sweet, 1959). The situation is complicated not only by excitation of longitudinal plasma waves but also by the fact that the cutoff conditions arc no longer isotropic in a magnetic field, so that -0.5 c -1.0 -1.5 2.6 Nonlinear effects <--- \ \ \& Shielding (n)~*\ \ <% \ — \ >*\\ Quantum (kT)--K \\ \ V 1 1 \ \ 1 1 \ 1 0.01 10 100 Electron temperature kT/Z? [eV] 0.1 1 Electron density n/nc = (up/üi)2 1000 10 FIG. 2.9 Quantum and shielding correction terms to be applied to the low-temperature, high-frequency In A] of (2.5.30). The value of each correction subtracts from ]nAB, which may be read from the solid lines of Fig. 2,8. The two corrections are independent. From Greene (1959, revised) and Dawson and Obcrman (1962), different forms of A enter the collision frequency terms of the various elements of the dielectric constant tensor (Silin, 1962; Eleonskii et al., 1962; and Oberman and Shure, 1963). However, for most practical purposes the field-free values may be used. 2.6 Nonlinear effects A number of nonlinear processes occur in plasmas, leading, for example, to harmonic generation and interaction between two waves (Ginzburg and Gurevich, 1960). The best known, the so-called Luxembourg effect, arises from a change in the effective collision frequency as a result of electron heating by the electromagnetic wave. Thus, it vanishes for the special case of collision frequency independent of velocity. Other types of nonlinearities arise directly from the electron dynamics and the non-transverse electric fields in a nonuniform plasma or a plasma in a magnetic field. Further discussion of nonlinearities can be found in Chapter 6. 2.6.1 Criterion for linearity. A rough criterion for the validity of the linearized theory, already noted in Section 2.4, is that the velocity increment due to the electric field be small compared to the random thermal velocities; that is, —r-a «— (2.6.1) 90 Collision processes Chap. 2 1000 \ \ \ \ \ \10"M \ \ ■ \ \ \ \ 2a/,,2 \ \ \10~32 \ \10" \ \ \ = 10_scps-cm3 \ \ \ \ \ \ \ 10 100 Wave frequency o)/27r [Gc] 1000 FIG. 2.10 Solid curves. Effective electron-ion collision frequency for a hydrogenic plasma (Z = l), as a function of frequency and temperature, assuming no shielding (uj» tu,,) but including the temperature correction of Fig. 2.9. The numbers given on the contours arc to be multiplied by the electron density n in cm"3 to obtain the cyclic (not radian) frequency v\1tt in cps. Compare Fig. 2,6. Dashed curves. The corresponding power attenuation coefficient 2a = 2x">jc in the high-frequency limit of (1.3.30). The numbers given arc to be multiplied by (»[cm"'])! to obtain 2a in decibels/cm. Attenuations derivable from this graph arc in general very small because of the restriction on electron density implied by the condition ci>p. For plasmas with ionic charge Z use the scaling relations: l«r>. ir)]z=z-s[a(«/^2. kT/z2)h,L This criterion permits the discard of the nonlinear term in the Boltzmann equation expansion (2.4.18), but is not sufficiently restrictive to prevent significant heating of the electron gas. In each collision with heavy particles an electron's coherent motion is randomized. The heating effect, although small, is thus cumulative. From a calorimcti'ie viewpoint 2.6 Nonlinear effects 91 the electrons are heated by the electric field and cooled by collisions with heavy particles. The power dissipated per unit volume in the electron gas by the electromagnetic wave is -^urE2, where ar is the real part of the conductivity and E the peak amplitude of the wave field. From (1.3.14) in the limit of high frequencies (o>2»vz; no magnetic field) the power dissipated per electron is then ve2E2j2m^\ v being the effective collision frequency for momentum transfer. The electrons lose energy to the molecules and positive ions at the rate (2mjA1)v(TkTe~f/]"1 (Spitzer, 1940). Or, if the disturbing wave is sinusoidally modulated such that the attenuation varies over the range a±Aa,a. wanted wave passing through a disturbed region of length d emerges with an amplitude-modulation index 4 Aa d. When the plasma is in a magnetic field, the real conductivity is resonant at the cyclotron frequency for certain modes. The evaluation used for u.r in (2.6.1) is no longer correct, and significant heating is caused by a nearly resonant disturbing wave at power levels well below the criterion of (2.6.2) and (2.6.3). Experimental techniques exploiting the Luxembourg effect have been used extensively to measure the velocity dependence of collision frequencies (Rao et al., 1961). It might also be noted that modifications of plasma properties by means other than a disturbing electromagnetic wave (for instance, electron-stream-driven longitudinal plasma oscillations) can modulate a wanted probing wave (Rosen, 1960; Baranger and Mozer, 1961). This aspect is discussed further in Section 2.6.5 and Sections 6.6 and 6.7. 2.6.4 Other nontinearities. The presence of an electromagnetic wave changes the electron velocity distribution function f{\) which, in turn, may alter the propagation characteristics. These matters may be investigated in detail by kinetic theory methods (Epstein, I960, 1962; Chen, 1962; and Sodha and Palumbo, 1963). We have already discussed the change of collision frequency due to electron heating, the Luxembourg effect. However, in addition, the a-V„/(v) term of the Boltzmann equation (2.4.4) is inherently nonlinear in time, a feature discarded in the expansion (2,4.18). Consequently, if the criterion (2.6.1) is violated, a single wave propagating in such a medium generates harmonics (Rosen, 1961), 2.6 Nonlinear effects 93 We have assumed throughout that the oscillating electric field, and consequently the electron velocity distribution function, are uniform in space. This assumption is invalid when a-c spacecharge is associated with the wave, as is generally the case for the nontransverse-electric waves in plasmas in magnetic fields and for inhomogeneous plasmas (Sections 1.4.5 and 4.4). The assumption also obviously ignores the wave nature of the electric field, a matter of importance even in the linear theory, and discussed at length in Chapter 3. When JE and/(v) are space dependent, for one or more reasons, additional terms in the Boltzmann equation are nonlinear (Ginzburg, 1959; Wetzel, 1961; Whitmer and Barrett, 1961, 1962; and Baird and Coleman, 1961). The density modulation, for example, may cause phase modulation of a probing wave, analogous to the Luxembourg amplitude modulation. Finally, we note that the vxB force arising from the wave (a-c) magnetic field is inherently nonlinear, but may usually be neglected for nonrelativistic plasmas. 2.6.5 Incoherent scattering. In conclusion, we discuss briefly the process of incoherent scattering, which arises from thermal fluctuations rather than nonlinearities in the usual sense. Under the influence of an electromagnetic wave, a free electron oscillates and reradiates, a process known as Thomson scattering, the classical analog of the Compton effect (Heitler, 1954, §§5, 22, and 33). This reradiation from an assembly of initially stationary electrons is coherent and produces the change in phase velocity described macroscopically by the refractive index (Ratcliffe, 1959, Chapter 3). At a finite temperature, however, thermal fluctuations in the electron density give rise also to an incoherent scattering of the incident wave. The total power radiated into unit solid angle per unit volume of the scattering medium is j=W"Ivtf® (2.6.5) where r0 = e2/4™0mc2 = 2.8 ■ 10"1S meter is the classical electron radius, n the average density of scattering electrons, / the intensity (watts/m2) of the incident wave, and & the angle between wave polarization and direction of observation (Fejcr, I960). There is no frequency or temperature dependence of the total power so long as the incident wavelength is large compared to the debye length. The corresponding amplitude attenuation coefficient of the primary wave, a^^r02n, is negligible for most laboratory plasmas (Sampson, 1959). The frequency spectrum of the scattered radiation is temperature dependent, and may be obtained from detailed calculations, which are especially complicated by a magnetic field (Renau etal., 1961; Farley etal., 1961) . Harmonics of the incident frequency are generated (Vachaspati, 1962) . Experimental measurements of backscatter from the ionosphere 94 Collision processes Chap. 2 have been used to determine electron density and temperature as a function of altitude (Bowles, 1961; Bowles et al., 1962). Modifications in the scattered intensity occur when the plasma is in a nonequilibrium state. For instance, the electron and ion temperatures may differ (Salpeter, 1963) or electron-stream-driven plasma oscillations may be present (Drummond, 1962). A laboratory experiment has been performed by Kino and Allen (1961). The presence of nonthermal fluctuations in the plasma also influences the low-frequency conductivity (Yoshikawa, 1962). A more complete discussion is given in Section 6.7. C HAPT E R 3 Waves in warm plasma 3.1 Introduction The preceding chapters have assumed a cold plasma in which electron thermal motion could be neglected.1 Specifically, the wave nature of the electromagnetic field has been ignored; that is, the phase velocity and wavelength of the wave have been assumed infinite. We shall use the term "warm" to designate the case in which the temperature is considered explicitly but for which nonrelativistic mechanics is still appropriate. The terra "hot" will be reserved for the relativistic case. Several new phenomena, described below, appear when the plasma is assumed warm. (I) Spatial variations (gradients) in density and temperature over the wavelength drive particle currents, which are in addition to those driven directly by the electric field. This effect, in addition to modifying the propagation of electromagnetic waves, provides a new class of waves variously known as plasma, electrostatic, spacecharge, or electroacoustical waves. Tn suitable limits these waves are longitudinal, analogous to sound waves in un-ionized gases. In this class are the modes commonly referred to as "plasma oscillations" (Bohm and Gross, 1949). In the presence of a magnetic field or density gradients, spacecharge waves may couple to electromagnetic waves, as discussed in Chapter 5. 1 Even in a cold plasma, however, the electron velocity must be greater than the velocity increments produced by the electromagnetic field (Section 2.3) if the presence of the wave is not to distort the distribution function significantly, fn recognition of this lower bound on velocity, the term "temperate plasma" has been used (Allis, Buchsbaum, and Bers, 1963). In a sense, however, this is more an upper bound on Ihc cleclric field than a lower bound on thermal velocity. 95 96 Waves in warm plasma Chap. 3 (2) A group of electrons having thermal speeds close to the wave phase velocity can exchange energy with the wave by the processes of Landau damping and Cerenkov radiation, processes which on a macroscopic scale take place in the linear accelerator and traveling-wave tube. For electromagnetic waves, a slow phase velocity (high index of refraction) is found only when a static magnetic field is present. The clectroacoustic waves may be slow even without the field. (J) The presence of a static magnetic lield introduces a new scale of length, the gyroradius, which may be comparable to the wavelength. Under this condition the relation between current and electric field (that is, the conductivity) is in general no longer a function of a point in the plasma, but depends upon the spatial variation of the field and the past history of the particles reaching that point (Drummond, 1958; Drummond et al., 1961). This chapter summarizes the results of theoretical analyses which take these effects into account. 3.2 Magnetic permeability of a plasma To calculate propagation constants in a cold plasma, we have considered all particle motions explicitly. Thus the plasma represents a continuous medium having the dielectric constant and magnetic permeability of free space and the complex tensor conductivity a defined by J = ct-E. Alternatively, it is possible, for a particular frequency w, to regard the plasma as a dielectric medium having zero explicit conductivity, the permeability of free space, and the complex dielectric constant (3.2.1) With the introduction of finite electron temperature in the presence of a static magnetic field, we recognize that the gyrating particles possess magnetic moments and exhibit diamagnetism (Astrom, 1958; Neufeld, 1963). An intuitive approach to including the effects of finite temperature would be to compute an appropriate magnetic permeability, which could then be used in addition to the cold-plasma dielectric constant in the dispersion relation obtained from Maxwell's equations. The magnetic moment of an electron orbiting in a magnetic field is ■IA _ (eaib\ \2rr) 0'52)=- B (3.2.2) where / and A arc the current and area generating the magnetic moment, 3.2 Magnetic permeability of a plasma 97 rb = 2-nvju>h is the gyroradius, and % is the velocity component perpendicular to the magnetic field.2 The magnetization vector (magnetic moment per unit volume) for a thermal distribution of electrons is thus By definition .... nkl" _ M=> —-p B. / nkT\B (3.2.3) (3.2.4) where the coefficient in parentheses is formally the reciprocal relative permeability. Clearly, no simple proportionality exists between H and B, so that permeability is not in general a valid concept. However, in the special case where the static magnetic field B0 ls large compared to the wave magnetic field, and the two field components are spatially orthogonal, as in propagation along the static field, the relative permeability «m is essentially constant: /, nkT\~' Bo*) The parameter -MS* =(i + ^)-1- 2fj.0nkT kT ,.2 B02 (3.2.5) (3.2.6) represents the ratio of material to magnetic field pressure, and occurs frequently in the theory of plasma confinement (Glasstone and Lovberg, I960, p. 52).3 Note that /? is the square of the ratio of the gyroradius to the wavelength of a free-space wave at the plasma frequency. For plane, transverse waves in a medium with zero conductivity, Maxwell's equations give the dispersion equation V? = kkm, (3.2.7) ■ In passing, we note that the magnetic moment of a gyrating particle is an adiabatic Invariant or the motion, a feature exploited in magnetic mirror devices (Glasstone and Lovberg, I960, p. 337; Lenard, 1959). This is essentially a consequence of the conservation of angular momentum. :l This parameter, conventionally represented by the ambiguous symbol p, is not to be confused with the rclativistic velocity ratio vjc, nor with the phase propagation coefficient. 98 Waves in warm plasma Chap. 3 3.3 Hydromagnetic calculation of plasma waves 99 where ^ is the index of refraction and k the dielectric constant. Hence we infer in this special case, using (1.4.20), F2= 1 1 + \o>J mc2 (3.2.8) i ± K/w). This naive argument cannot be expected to have the validity of a treatment which considers explicitly the effect of the wave on the electron velocity distribution—that is, considers particle motion and thus conductivity rather than dielectric and diamagnetic properties. In particular, the permeability argument ignores resonance effects when w ~ wh. The correct treatment is outlined in Section 3.4, from which it may be seen that the temperature correction in (3.2.8) is of the right order of magnitude. 3.3 Hydromagnetic calculation of plasma waves The kinetic (Boltzmann) theory for waves of finite wavelength in a warm plasma is mathematically difficult. To provide insight we first attack the problem using the hydromagnetic approximation, mentioned in Section 2.4. This treatment necessarily excludes the phenomenon of Landau damping; therefore, we must require that the value of the electron velocity distribution function be small at the wave phase velocity. In addition, we shall assume only high-frequency oscillations so that the ion motion may be neglected. We neglect collisions and all nonlinear terms. 3.3.1 Moment equations. Given the Boltzmann equation (2.4.4) (3.3.1) we multiply through by any function ^(v) and integrate over all velocity space. The integrals for the left-hand side are (Spitzer, 1962, appendix) where J A(y) a, ) i^d3v = Čt dtv ex' c 8x 8vx — n (nA) c dv. «(r,ř)= /(r v, t)d3y (3.3.2) (3.3.3) (3.3.4) (3.3.5) and the bar denotes an average over the velocity distribution as, for instance, JA(v)Xr, v, 0 rf3v A(r, 0 = : »(r, t) (3.3.6) If first A(y) is taken to be unity, we obtain the equation of continuity |-ivr.J = 0 (3.3.7) for electrons of charge — e and the macroscopic current density J=-nev. (3.3.8) The integral (3.3.4) is zero for electromagnetic forces; the integral over the collision term is also zero, since collisions cannot alter the local density. Second, we take A(y) to be the momentum my, and assume that the acceleration arises only from electromagnetic fields E and B, to obtain the equation of momentum transport* --^ + V.*l»-|-«eE-JxB = 0, e dt (3.3.9) in which * is the pressure tensor nmyy arising in the integral (3.3.3), and the collision term has been neglected. Maxwell's equations provide further relations between E, B and n, J. The remaining task, characteristic of the hydromagnetic formulation, is to evaluate the pressure tensor One approach is to approximate it by a scalar pressure For high frequencies, pressure changes will occur adiabatically so that we would expect p = (na1-'kT)n\ (3.3.10) and therefore V.«P^V/J = yAT?n, (3.3.11) where y is the ratio of specific heats and n0 and Tthe equilibrium electron density and temperature. For strictly longitudinal plane waves the compression is essentially one-dimensional, implying y= 3 (Spitzer, 1962, §3.2). This approximation leads to the correct dispersion relation for the space-charge waves in the low magnetic field limit but does not indicate any temperature perturbation of the electromagnetic waves (Allis, Buchsbaum, and Bers, 1963). A more accurate treatment of the problem is to form an additional moment equation, which may then be rather crudely approximated. Integrating the Boltzmann equation multiplied by the dyadic /hvv, we obtain the equation of motion of the pressure tensor. Neglecting 4 In the more general treatment, the corresponding equation for ions is also obtained. The sum of these two equations yields the macroscopic equation of motion; the difference, the generalized Ohm's law relation (Spitzer, 1962, §2.2). The treatment given here is appropriate for high frequencies when ion motion can be neglected, and Ini weak Melds when a linearized equation is adequate. 100 Waves in warm plasma Chap. 3 all nonlinear terms and those involving the magnetic field, one obtains (Bernstein and Trehan, 1960) I 8t e the divergence of which is 8 kT [V-J1 + YJ + (VJ)<] = o, 8 kT (V.*)-— [2VV-J + V.VJ]=0. (3.3.12) (3.3.13) 8t ' e The approximations made in this equation are essentially the assumptions of low temperature and low magnetic field.6 3.3.2 Hydromagnetic dispersion relations. We now assume a plane wave traveling in the direction of the complex vector propagation constant y, and expand the variables in the form J = Ji exp(./u)/ — 7'r) E=Ei expO'cu/ - y • r) B = B0. (3.3.14) The products of first-order quantities, and of all higher-order quantities, will be neglected. Thus the magnetic component of the wave field may be ignored. Equations (3.3.9) and (3.3.13) yield J — Ji -J — (2yy • Ji + y-tJi) + «o*E, + B0 x 31 = 0. e eui Rearranging, we have [. ok x k.T ,. 1 + / ^— +-g (2yy - + 7 • y) oi moj2 (3.3.15) (3.3.16) eB (3.3.17) where cup2 =- tO(, = — e0m m The coefficient of Jj is a tensor resistivity, similar too"1 in (1.4.28). It may be inverted to obtain the conductivity and dielectric constant tensors. A medium for which the dielectric constant is a function of the propagation constant is said to exhibit spatial dispersion (Neufeld, 1961). Without loss of generality, we take the propagation constant to lie in the x-z plane and the magnetic field along the z axis; then f=i^(f,o,9 «6 = (0,0, (Ob) (3.3.18) 5 We have neglected the magnetostatic field terms, which are of first order, simply because of algebraic complexity. Therefore the solution will be correct only to first order in temperature for infinitesimal field and, alternatively, for any field at zero temperature. Terms involving the product of temperature and magnetic field will not in general be correct. This restriction does not apply to the kinetic theory results quoted in Section 3.4. 3.3 Hydromagnetic calculation of plasma waves 101 where ji. is the complex index of refraction and £, £ are the direction cosines of y. The coefficient of J± in (3.3.16) becomes in matrix form l-8(l+2f) JY -2m -JY -8 0 o 1-8(1+2H. (3.3.19) (3.3.20) where Y=u>bja> 8=/xa/cT/mc3. It is to be noted that the parameter S is essentially the square of the ratio of electron thermal velocity to wave phase velocity, which we require to be small (see footnote 7). The reciprocal or (3.3.19) is the conductivity (1-8)2(1-3S)- y2[1-S(l+2D] 1-S(2 + 2H /y[l-S(l+2£2)] + S3(l+2£2) -jK[l-S(l+2e2)] 1-4S + 3S2 2^8(1-8) j2Y&h 2^8(1-8) -j2 ms 1-8(2 + 2^) + S2(1+2P)- T2J (3.3.21) where the coefficient is understood to multiply each clement of the matrix. The corresponding dielectric constant is (3.3.22) These expressions reduce to (1.4.56) to (1.4.60) in the limit 8 —> 0. The propagation constant is determined by substitution of (3.3.21) in the electromagnetic wave equation V x V x E + 1 82E c2 dt2 + ,^(ct-E) = 0, (3.3.23) which for a wave of the form exp(j

0. To first order in temperature and magnetic field (see footnote 5) K/tu)a ^2.r=l- 1 - {kT!mc2)[\ -km2] ± k/co) r, km2 in {*]s+3SV -({[l-(cu>)2]2-KH2}-4[l-KH2]S + 3S2) = 0. (3.3.31) For small 8 this equation is of the form (A+a)iJL2-(B + b) = 0, (3.3.32) " Since 8 contains //2, this bracket is in fact a cubic in (P, one of whose roots violates the assumption s« 1 and is extraneous. 104 Waves in warm plasma Chap. 3 where a«A and b«B, and has the formal solution , fi/, a b \ (3.3.33) The perturbed electromagnetic wave thus has the index i-K/a>)a-KMa where we have ignored magnetic field in the temperature term in accord with footnote 5. Since the index of refraction for the spacccharge wave is relatively large (although bounded by the requirement & = fi.2(kTjinc'2)« I), we obtain an approximation to the dispersion relation by equating the coefficient of ^ to zero. Since (a>„/a/)2« 1, this coefficient, in the form given in (3.3.29), may be rewritten within our approximation (M»8(i4) (1-3)(1-3S)[1-KH2]' whereupon the dispersion relation becomes (see Fig. 3.1) (l-3S>[l- KM2] - KM2 sl-38-KM2 - KM2=o; l-(e -KM2 or 3kTjmc2 3kT (3.3.35) Mathematical subtleties are involved in obtaining this dispersion relation in the limit where both temperature and magnetic field are taken to be small quantities (Gross, 1951 ; Bernstein, 1958). We note that this space-charge wave occurs only for frequencies near K2 + a>2yA x(s)] ds ,= -H*-yf (1 - cos h = p,\kTjmc2) 7«ww'a. £=sin0 £ = cos0 106 Waves in warm plasma Chap. 3 As in Section 1.4.8 by using the unitary matrix transformation where U = a' = 1 -j 0 " 1 j 0 0 0 v% rotating coordinate syste 11 o ./ -J o 0 0 V2 (3.4.7) (3.4.8) tensor is symmetrical and is diagonal for the special case of propagation along the field. The result is (Bernstein, 1958): m rr = -±]e0. The propagation constant is obtained from the dispersion relation (3.3.24) and may be written in the form A*p.i-B*fi2 + C* = 0, (3.4.10) where /i is the complex refractive index and C*=\k\ (3.4.11) with |k| denoting the determinant of the dielectric constant tensor. It must be remembered that the refractive index is contained implicitly in the 3.4 Kinetic (Boltzmann) theory of waves 107 dielectric constant elements, so that (3.4.10) is in general transcendental. Expanded to first order in the quantity $=ji.\kTlmc2), the elements of the dielectric constant tensor are, in fixed coordinates (Sitenko and Stepanov, 1957) :7 lO-JV O) (to —jt>)2 — to2 x|l+£2S [(a)_>)2_ajfi2]2 co-jv to (to —jv)2 -co2 ^[(co-pf + 3^] + 1 2 1 f. . ... 3fr'2 *y nyx co to —Jv J (co-»2 (w-»2- W ('u-»3-"'iJ2 f .>2[3(^->)2 + ^2] , \ mai,,2qi„[3(fc> -»2 - o>„2] (3.4.12) The integrals of (3.4.6) are in general complex, even when collisions are neglected (v^-0). However, the first-order expansions (3.4.12) are, neglecting collisions, pure real or imaginary in such a manner that propagation is either unattenuated or evanescent. Noncollisional damping has again been excluded. We now catalog indices of refraction for the principal electromagnetic waves. 1 The expansion parameter 8=^%kTlmc% which appears in both hydromagnetic Bnd kinetic theories of warm plasmas, using nonrelativistic mechanics, is directly proportional to the square of the ratio of thermal velocity to that of light. If a Srst-order expansion of a nonrelativistic theory is to be valid, we must require /»a»l. This is of course precisely the condition under which the warm plasma results nre significantly different from the cold case. 108 Waves in warm plasma Chap. 3 3.4 Kinetic (Boltzmann) theory of waves 109 3.4.1 Propagation along the field (£=0, £ = 1) (Platzman and Buchsbaum, 1962, 1963; Willett, 1962). fit. r = (*xx ±JKxy)t - 0 = Wll. rt)t - 0 =i-^rexp[^(i±7-^-HA = 1-7^ J" (1 -iS^ + iSV- . . .) exp[-/(l + K-7-^] A /l to2 2 jfeT 3m4 4/A:r\2 \ x r k« -» ± ^ «k2+k« -»± *>b]4 ;i WW + • • ■ j [f \IU + U »m (3.4.13) l «[(«> -a) ± /1 [(«j -a) ± <"„]3 /Me3/ The upper signs refer to the left-handed (ion gyration sense) circularly polarized wave; the lower to the right-handed (electron sense) wave.0 Within the limitations of this expansion and the nonrelativistic mechanics, the temperature correction is significant for only the right-handed wave near cyclotron resonance. The effect, shown in Fig. 3.2, is to exaggerate the resonant increase in /x for a> co + tob-\- ^tovjc (3.4.14) 8 Examination of the higher order terms in the expansion shows that a first-order approximation with the correction term V [(<"-j") ± ">:,f in the numerator is more accurate for low densities away from resonance. The form as shown, with the correction term in the denominator, is more appropriate under conditions when I and the magnitude of the correction is significant, The same 07 0.8 0.9 1.0 Normalized magnetic field o>b/ta FIG. 3.2 Refractive index for the right-hand wave propagating along the magnetic field, in the vicinity of cyclotron resonance. The dashed curves show the departures from the collisionless Lorentz theory (solid curve) produced by (a) short-range collisions, and (6) high temperature. The behavior of the high-lemperature curve close to /, +/iUWjC J_«i Jo =j jo *exPl-/(l ± Y)s] j_ d», exp[-^-; —j =y jo ds exp[-/( 1 ± Y)s--j;-] j_K exp|_--^-j =; P i/.vexp[-y(l+ y>-i8jra] Ja where vUí--(.kTI">) and ť-WJc* — s- 110 Waves in warm plasma Chap. 3 where/*(j>a) is the distribution of electron velocities in the axial direction (not necessarily Maxwellian). For small velocities (low temperatures), the integrand can be expanded to obtain tl»±ťOj J co + oif, +jííjjvjc J = -£-[l+0, co + (Oj, (co + co,,)2 +——;-■ (3.4.15) where v2 is the mean-square axial velocity. For an isotropic Maxwellian distribution v,2 = kTjm and the expression is identical to (3.4.13) and when o»b -s- 0 to (3.4.22). Incidentally, the formulation (3.4.14) emphasizes the fact that the high-temperature phenomena arise physically because of the doppler shift between the laboratory and the electron reference frames. It is also of interest to compare the relative size of perturbations of the refractive index arising from ordinary collisions and from the finite temperature correction. As an example, we obtain for propagation along the field, to first order: o((D ±t»b) (<" ± <*>bf l <^±"i(j-«'p2/£t> J W r=0, Tj^O: Qjp2, coco,,2 s kT co(co ± u>b) (co + cOf,)3 mc2 (3.4.16) We note that the corrections are in opposite directions. In the most important case, near cyclotron resonance in the right-hand mode (Fig. 3.2), approximate cancellation occurs when 3 /v\2_ a kT 4 \co/ ~" mc2 (3.4.17) Assuming electron-ion collisions, using (2.5.23), and approximating p2 by co„2/co(co„-co)»l, we find the finite temperature effect dominates when the inequality co (A-7'[eV])4. -> io- (3.4.18) co,, —co «[cm 3] is satisfied. Changes in the wave attenuation are discussed in Section 3.5. 3.4 Kinetic (Boltzmann) theory of waves 111 3.4.2 Propagation across the field (f= 1, £=0). Etf |1 B0 (Dnestrovskii and Kostomarov, 1961): -/Sjf f * exp[ - /(1 ->/<»>-(8/ Y*)(l - cos Ys)] ds -/^-exP(-S/y2) ^ I»(S/n exp[-j(i-nY-jvlo>)s]ds -^exp(-S/r2) s - My^ co2 „=^„(1-^)-"^ -^exp(-S/r2) wn, v 2(i->h[.(8/ni l ->/co ^ n4\ (1->h2-«272J JjL- U __°l_ a2 — [(co->)2-cob2][(co 1 + ; _ JjfLW -PY-Aa,-2)'1' \mc2) kT) (3.4.19) co(co-»J/ F" 1 (co-»[(co-»2-co62] mc}f In(x) = I_n(x) is the modified Bessel function of the first kind, and use has been made of the identity 00 exp(xcosj')= 2 ln(x) cxp{-jny). (3.4.20) Er, ±B0: p2_{KXXK1IV "i~ Kxv\ \ Kxx /t=0 ti-kht-kh2 =o\ l-icop/co^-fcojco)2 }/ co/[(co2-co!,2)2-r-co,J2(7to2-4a,p2)-8co„^J kT\ (co2-4cobH)(co2-coJ,2-wb2)2 /W2J The simplification of ignoring collisions in the latter form is made because of algebraic complexity. The ordinary (parallel polarization) wave is no longer independent of magnetic field but, rather, shows resonances at all integral harmonics of the cyclotron frequency. Likewise, the extraordinary wave shows both the expected upper hybrid resonance at co2 = o>b2 + w2 and additional resonances at harmonics of wb. The results obtained here may be compared with those obtained from the hydromagnelic calculation of Section 3.3.2. Further physical insight into the origin of the high-temperature modifications may be found in the discussion of Druinmond (1958). 112 Waves in warm plasma Chap. 3 3.4.3 No magnetic field (cu,, — 0): F2 = (*)*>,,= Q =i ť/í v r u(íí)-» L i + kT {a>~jvf *l me2 1 (o 2 jtn ■[l___^—1/fl+ (3.4.22) Since for this case the refractive index never exceeds unity, the correction is numerically significant only for temperatures requiring relativistic mechanics, as discussed in Section 3.6. 3.4.4 Plasma or electrostatic waves. For propagation along the field the dispersion relation is Kz2~ 1 +<7sJj<' -I). (3.4.23) From (3.4.6): v r •j(w-» [ J to2 . 15*. 1 — L ("-.ao2 J S2 + cu(u) — /jj) Therefore to lowest order in temperature mc2 (oi —jv)2 3ÍČŤ K)25^exP[-(t,J/^)2S]{lo[(^,)3Sl + 2 j %MfgPl (3.4.27) which exhibits resonances at all harmonics of the gyro frequency. Various limiting cases are considered by Bernstein (1958). Equation (3.4.27) does not converge properly in the limit of small magnetic field. An expansion of the first form of (3.4.26) for small &>.,, as well as small §, and neglecting collisions, leads to the dispersion relation p?=^ [i - KH2 - KH2n (3.4.28) which agrees with (3.3.28), (3.3.35), and (3.4.25), and Fig. 3.1. A fuller discussion of spacecharge waves in warm plasmas, including the treatment of wave vectors at arbitrary angles to the magnetic field, is given in Sections 5.5 and 5.6. 3.5 Landau damping and wave absorption Dissipative absorption of a wave appears mathematically as an imaginary term in the dielectric constant k and, hence, in the square of the complex refractive index, where fi and x Ste the (real) refractive and attenuation indices, respectively. From (A.46) and (A.47) l. Discarding higher order real terms in the expansion (3.4.15), we obtain for propagation in the left and right circularly polarized electromagnetic waves along the magnetic field ■(«±«Jc| tu(fi) + Ul,,j O) jj. (3.5.4) where the one-dimensional distribution function is in the Maxwellian case /fe)=(^r)™p(-mv,*mn (3.5.5) Thus, for low temperatures, the attenuation constant is, from (3.5.3) (Sagdeyev and Shafranov, 1958; Cullen, 1960), where řu(aj + 0),,)J (3.5.6) This case has been studied in detail by Scarf (1962), Willett (1962), and Platzman and Buchsbaum (1962, 1963). This result is an example of noncollisional damping known gcnerically as phase mixing or line scale mixing (Gershman, I960). In the context of plasma oscillations, the phenomenon is usually called Landau damping. It is closely related to the inverse process of Ccrenkov radiation. Physically, it arises because of synchronism between particle and wave velocities. Electrons in one region of the wave move, as a result of thermal motion, into adjoining regions where the phase of the wave is different. On the average, particles riding with the wave extract more energy from the wave (linear accelerator effect) than they give to it (traveling-wave tube effect); this energy is then gradually shared with nonsynchronous particles by collisions. Thus, thermal energy is increased at the expense of wave energy (Dawson, 1961). In the presence of a magnetic held, the effect may also be considered as arising from the doppler shift of the wave frequency as seen by the moving electron, and is frequently called cyclotron damping. This dissipative process was inherently excluded in the hydromagnetic calculation of Section 3.3.2 and discarded in the first-order expansions of (3.4.12). The application of theory to experiment is complicated by the fact that the strong interaction between wave and synchronous electrons distorts the distribution function near the wave velocity, whereas collisions 3.6 Relativistic plasmas 115 tend to restore Maxwellian distribution. Hence, the magnitude of noncollisional damping depends inherently upon the collision rate (Platzman and Buchsbaum, 1961). In cases where a highly non-Maxwellian velocity distribution is somehow maintained, such that there are more fast particles than slow in the neighborhood of the wave velocity (implying a "double-humped" velocity distribution), then wave amplification or growth can occur, as calculated in Section 5.6. Examples are the traveling-wave tube and various plasma instability processes (Drummond and Pines, 1961). Most theoretical treatments of noncollisional damping have been concerned with the longitudinal plasma waves, rather than the transverse electromagnetic waves (Landau, 1946; Bohm and Gross, 1949; and KJldal, 1961). Furthermore, the more common problem has been to assume an initial disturbance and follow its decay in time; the frequency o>, rather than the propagation constant y, is assumed complex. The problem of determining a complex y inherently requires consideration of driven waves. To contrast with (3.5.6), we may compute the ordinary collisional attenuation, in the special cases of propagation along the magnetic field (left and right circular polarizations) or across the field (ordinary mode, putting cu(, = 0). From (1.4.21), (3.5.7) If electron-ion collisions are dominant, as in a highly ionized gas, then v oc a>v2j(kT)%, according to (2.5.23). For given to, uir, and ojb, there will exist some critical temperature below which collisional dissipation dominates and above which phase mixing dominates. Because of the exponential term in (3.5.6), this critical temperature is in the kilovolt region for electromagnetic waves in most laboratory plasmas, except near the gyroresonance. It is worthwhile to note, in passing, that the real and imaginary components of the propagation constant are linked by general considerations of causality, the mathematical formulation of which is known as the Kramers-Kronig dispersion relations (Kittel, 1958; Leontovich, 1961; and Pradhan, 1962). 3.6 Relativistic plasmas Since the expansion parameter S = ^2(AT//hc2) which appears in the kinetic treatment of "warm" plasmas, using nonrelativistic mechanics, is of order [x2(rjc)2, it may be necessary to employ relativistic mechanics if one is to retain terms of order (kTjmc2) for ur*. 1 (Silin, 1960-1962; Graben, 1963). Also, synchrotron radiation processes imply that the interaction between a wave and plasma at multiples of the gyrofrcquency should be 116 Waves in warm plasma Chap. 3 enhanced for a relativistic plasma (Beard, 1959). Imre (1962) has considered the problem of electromagnetic wave propagation in relativistic plasmas in detail. As an example, he obtains for propagation along the field, to first order in kTjmc2, 1 - w(u) + CO,,) kT 2(cu ± aib) mc 3 i + kT (w ± cob)'J mcs (3.6.1) which is to be compared with (3.4.13). Johnston (1962) has developed weakly relativistic expansions, obtaining, for example, for electromagnetic waves in a plasma with no magnetic field \ 2mc2J 1+- kT which is to be compared with (3.4.22). ■ C HAP T E R 4 Wave propagation through bounded plasmas 4.1 Introduction The preceding chapters have been concerned with the propagation of electromagnetic waves in an infinite plasma. We now consider the elfcct of plasma boundaries on the propagation. To study high-density discharges of arbitrary size and geometry, one is generally forced to use microwave beams directed through the plasma by means of suitable antenna systems. The alternative situation in which the plasma is located within a cavity or waveguide, or is itself a waveguide, is considered in Chapter 5. The "free-space" beam technique is favorable where the dimensions of the plasma are larger than the wavelength of an electromagnetic wave at the plasma frequency. Both classes of measurements are essentially limited to frequencies a>> u>p, the plasma frequency (except for special techniques exploiting a static magnetic field or a detailed independent knowledge of the density profile). Thus, the beam technique is most readily analyzed when 2 Íwp2' (4.1.1) where D is the dimension of the plasma. The first of these two independent conditions permits convenient simplifications in the analysis by avoiding the plasma resonance^ the second is essentially a diffraction condition which permits reducing the problem of propagation of a finite beam of electromagnetic waves through a finite plasma to a one-dimensional, plane-wave problem, as a first approximation. The propagation constant of a microwave beam in a plasma has been 117 US Wave propagation through bounded plasmas Chap. 4 4.1 Introduction 119 shown, in Chapter 1, to depend upon the magnetic field, electron density, and collision frequency, and indirectly upon the temperature. The following basic arrangements, sketched schematically in Fig. 4.1, are useful in the case of high-temperature, highly ionized plasmas (that is, v«cop). (7) Simple transmission or reflection. For electron densities nnc it is opaque and totally reflecting, where nc = (eamje2)ojz is the critical density.1 The transition between 1 In the presence of a magnetic field, the effective critical density may be altered. However, the situation is qualitatively unchanged. / Source 1 \ 1(a). Transmission i> Detector Detector Source Source \(b). Reflection Pha'seshifter Attenuator 2. Phase shift FIG. 4.1 Elementary microwave observation schemes. these conditions is sharp. Thus, in principle, this elementary technique indicates whether the plasma density is above or below the critical value. Measurement at a given frequency is capable of determining only one value of density. The sharpness of the transition implied by the sudden change in the attenuation coefficient is not realized in practice because of the following factors. (a) For densities below but approaching critical, the dielectric constant discontinuity at a sharp boundary produces an increasingly strong surface reflection'(and corresponding reduction in transmission). (b) If the plasma is only a few wavelengths thick, interference effects occur between the surface reflections. (c) Inhomogeneous density distributions are not averaged in a simple manner. (d) Refraction and scattering by the plasma occur because of inadequacies in the one-dimensional, plane-wave approximation. If the plasma density is far above critical, an impinging signal is strongly reflected at the boundary. Therefore, motions of the effective boundary produce doppler shifts in the frequency of the reflected signal. (2) Phase shift, (microwave bridge or interferometer), ff the signal from an auxiliary transmission path, with adjustable amplitude and phase elements, is balanced against the primary transmission signal to give a null in the absence of plasma, the output signal of the waveguide (hybrid) junction is a measure of the attenuation and phase shift in the primary path due to the plasma. In the fully transparent region of electron density, where n«ns, a detected signal represents only phase shift which, in turn, is essentially a function of electron density only. Since the shift in phase can be calibrated, one has a continuous measurement of density between the upper limit of serious amplitude effects in the transmission path, and the lower limit of detector sensitivity. This technique is ideally suited to the observation of density as a function of time. The propagation of the microwave beam through the bounded plasma is most readily analyzed in two limiting cases: first, the gradual boundary, with density varying slowly over a wavelength, to which an adiabatic analysis may be applied; and, second, the sharp boundary which can be attacked as a boundary-value problem. A formally similar situation occurs in quantum mechanics, in which the first case is known as the WKB approximation (Bohm, 1951). The usual geometrical optics limit partakes of both the above limffs. It neglects reflections at the "sharp" boundaries which separate regions of different propagation characteristics and) thus, can be sell-consistent only for plasmas large compared to a wavelength. The models of plasma geometry that are most useful for 154 120 Wave propagation through hounded plasmas Chap. 4 4.2 Simple adiahatie analysis of a plasma slab 121 analytical purposes are the plane slab and the cylinder. We recognize, however, that most experimental plasmas will Tail to conform exactly to these limiting cases and simple geometries. 4.2 Simple adiabatic analysis of a plasma slab While most experimental situations approximate cylindrical symmetry, it is often possible to treat the plasma as a slab illuminated by plane waves, and thereby reduce the problem to one dimension. We can further simplify the situation by assuming that plasma properties vary slowly near thc boundaries so that reflection and interference effects are negligible, the adiabat ic a pprox i m at ion. 4.2.1 Average electron density. For a high-temperature, highly ionized plasma, for which v2«a>„2, dissipative attenuation is small. For simplicity, we here neglect magnetic field effects; they may be included readily by substituting the appropriate propagation formulas from Chapter I. The phase constants for vacuum and plasma are, respectively, ft 2w -HP: 2Y^2jt A (4.2.1) The phase advancement introduced by the plasma in a transmission path is then, in the adiabatic approximation, zf^=~ j(ß„-ß0)dx (4.2.2) where the integration is carried out along the direct path from transmitting to receiving antenna. To first order in njnc, (4.2.2) becomes Ad>-> *r~ n(x) dx "«n0 A«c J e2 f = ~- ax. 2eumco> J (4.2.3) Thus, for n«nc, the phase shift is linearly proportional to Hie electron density averaged along the propagation path. For a path length L we can write the average density Jo n^ ^X_2e0mc m A -r -3i i io a »/2»r[cpsl ztyfrad] «cm J] = 118.4—■— r r ' --• L [cm] (4.2.4) Assuming this first order approximation, we may evaluate the dynamic range of average densities which can be measured. Because of dissipative and nonlinear effects, eQm Amin "min — " L (4.2.6) where Aj>min is the minimum detectable phase shift, which depends upon the detector noise level and system stability. Hence I (4.2.7) The range of measurement scales with m, while the maximum density scales with to2. The situation is shown numerically in Fig. 4.2. Similarly, in the adiabatic approximation, the total attenuation, expressed as a power transmission coefficient 7\ in decibels, is given by 7'[dB]= -8.686 ^adx, which using (1.3.30) becomes in the limit v2«tuj,2«w2 4.3t'2 t[dB]-- e0mcu>- JK.v)«( x) dx, (4.2.8) (4.2.9) where v is the effective collision frequency, which generally depends on both density and temperature. The coefficient has the value 4.3ez/e0mc = 0.462 cm2/sec. In this common limit v3(*) dx+ . . Therefore (4.2.12) A^-lA^l £l ]>(*) dx+±% £s JV(*) dx+ ... (4.2.13) 4.2 Simple adiabatic analysis of a plasma slab 125 and we obtain for the first two averages of the distribution £ J«r/.v = 4{[^2-iWl-2^2)]+TL^ J«-rfr+...} (4.2.14) S J"' JxJT H-^«"^6 ^ J* **- ■ ■ ■}• (4.2.15) The usefulness of this approach is limited by the accuracy of the differential measurement Al—2A2. When this quantity can be successfully measured, (4.2.14) provides a refined evaluation of the average density and (4.2.15) an estimate of the mean-square density. Procedures for obtaining profile information have been developed by Motley and Heald (1959) and by Wharton and Slager (I960). Wharton and Slager use only the magnctic-field-independent parallel-polarization case. Their data-reduction procedure is to calibrate the peak electron density by means of the cutoff of a "low-frequency" wave, and obtain information from the simultaneously observed phase shift of a "high-frequency" wave. Motley and Heald, using different polarizations, calibrate the average density with the high-frequency wave, infer profile from the low-frequency wave. Because of the greater phase-shift non-linearity of the perpendicularly polarized wave near cyclotron resonance, the multiple polarization technique, when applicable, is somewhat more sensitive. The Wharton and Slager technique provides profile information only at the instants of time for which cutoff occurs; the Motley and Heald technique is limited to situations where the cyclotron frequency is comparable to the plasma frequency and is accurately known. Both methods benefit from additional phase-shift data channels at other frequencies and/or polarizations, at the expense of instrumentation and data-reduction complexity. Neither method is able to distinguish a hollow discharge from a peaked one. Experimental applications of these principles are discussed in Sections 6.4 and 6.5. 4.2.3 Reflections from cutoffs and resonances. Cutoffs, at which the index of refraction fx —0, and resonances, at which /i -*■ co, occur for certain combinations of frequency, density, and magnetic field. When a wave propagating in an inhomogeneous plasma impinges upon regions having these special characteristics, reflection and absorption must be considered even in the adiabatic approximation. Near the cutoff, the wavelength grows large, while near the resonance the wavelength becomes small. In both cases, the group velocity goes to zero. The analysis of this situation is formally identical to that resulting in the so-called turning-point connection formulas of the quantum-mechanical WKB approximation (SchilT, 1955). It can be shown that in the case of a cutoff the wave is 126 Wave propagation through hounded plasmas Chap. 4 4.3 The slab with sharp boundaries 127 reflected from the anomalous region with little dissipation (Denisov, 1958; Stix, 1960). The external behavior is thus very similar to that of a sharply bounded, high-density (n > ne) plasma. In the case of a resonance, however, the wave is largely absorbed. This distinction is of considerable significance for both reflection-type microwave probing measurements Vacuum Er" /Ht x = 0 Plasma / (a) Vacuum Ei Hi Er x = 0 Vacuum JE, (b) FIG. 4.4 Reflection and transmission at sharp boundaries, (a) Vacuum-plasma interface. (/») Plasma slah. and thermal radiation measurements, as well as for the nondiagnostic question of plasma heating by electromagnetic radiation. It will be noted from the graphs of Chapter 1 that, in general, for a given magnetic field, cutoff occurs at a lower density than the resonance. Thus, characteristically, waves entering the plasma from outside are reflected before reaching the resonance. The resonance may, in some cases, be made accessible by allowing the wave to enter the plasma in a region of high magnetic field (generally such that the cyclotron frequency o.)b>a>) which then decreases spatially within the plasma, so that the resonance is approached from the high-field side. In Section 6.5.4 an experiment using this technique is described. A situation of this sort has been exploited in the "magnetic beach" geometry for the dissipation of ion-cyclotron waves (Stix, 1958). If the regions of cutoff and resonance are close together within the plasma, relative to a wavelength, it may be possible for a sort of "tunnel effect" to occur in which the resonance region extracts energy from the evanescent wave passing through the cutoff. Tunneling or "bridging" may also take place by mode conversion processes (Ratclilfc, 1959, Chapter 17). Stix (1960, 1962) has shown that at a resonance high-temperature and ion-mass effects may reduce absorption, increase reflection, and excite other plasma modes. 4.3 The slab with sharp boundaries We again consider the interaction of a plane wave with a slab plasma. However, in contrast with the adiabatic case of Section 4.2, we now assume a homogeneous plasma with sharp boundaries, that is, the transition between vacuum and uniform plasma occurs over a distance much less than a wavelength. There exists a well-defined reflection coefficient at each interface, and reflection and transmission coefficients are determined by boundary conditions on the wave fields at the interfaces. Consider first the single interface of Fig. 4.4a. Waves traveling to the right are represented by the phase factor exp(j 1, the sense of Er is reversed and VSWR = «li = ll = Vo/v, We note, in passing, a convenient procedure for calculating the maximum transmission loss due to reflection. From standard transmission-line theory the maximum VSWR from two discontinuities is the product of the respective VSWR's (and the minimum, the quotient). Thus, the maximum transmission loss due to reflection from a slab can be obtained from standard charts assuming a single discontinuity with VSWR = ^ This procedure applies only if there is no dissipatlVO loss between discontinuities, The situation of practical interest is that of the slab of Fig. 4.46. By setting up boundary conditions similar to (4.3.2) at the two interfaces (or, alternatively, summing the infinite series of internally reflected waves), one finds (Stratton, 1941) the amplitude reflection and transmission coefficients f_Er p[l-exp(-2ýrf)] " £, l-p2exp(~2y is the phase angle of j5=|p| exp(/Y<) and 54 • / 2y (4.3.9) (4.3.10) cosi/i- (4.3.11) ii is to be^noted that the coefficients (4.3.8) to (4.3.10) are oscillatory functions of slab thickness d (or of frequency <*>) as a result of interference Of internally reflected waves. Likewise, the phase of the transmitted wave, which may be calculated from (4.3.7), is perturbed by interference. As a simplification, we may assume that the reflected waves are incoherent, thereby suppressing interference effects, and obtain* 4 Interference is suppressed by considering only power relations. The fraction r of (he incident wave is reflected at the first surface of the slab, the fraction a(l-r) [where u = exp(-2a = 0.003). The dashed curves assume incoherent Internal reflections, from (4.3.12) to (4.3.14). 132 Wave propagation through bounded plasmas Chap. 4 +30 Electron density fw^/aij2 = njn€ FIG. 4.6 Phase error of the transmitted wave, relative to the geometrical optics phase, for the same case as Fig. 4.5. in the presence of a static magnetic field, is much more complicated since propagation in an inhomogencous medium must be considered explicitly in terms of Maxwell's equations. In general, a-c spacecharge exists in regions of electron density gradients (Buchsbaum and Brown, 1957). For cold plasmas and for wavelengths long compared to interparticle distances and gyration radii, the local electromagnetic properties of the plasma medium may usually be represented by a space-dependent, complex dielectric constant ft(r), which is a tensor quantity on account of the anisotropy introduced by a static magnetic field (Drummond, Gerwin, and Springer, 1961). Hence, for fields varying as expyW, Maxwell's equations become VxE=->/l0H (4.4.1) VxH=>«0ft-E (4.4.2) V.(k-E) = 0 (4.4.3) Y-H = 0. (4.4.4) Taking the curl of (4.4.1) and using (4.4.2), we obtain the wave equation for E in the form 2 VxVxE=-jK-E. (4.4.5) The vector identity Vx VxE= Y(V-E)-V2E allows us to rewrite the equation for E as V2E + — k-E=V(V-E), (4.4.6) 4.4 Inhomogeneous plasmas 133 where in general the term on the right-hand side cross-couples the three components of E. Similarly, multiplying (4.4.2) by ft"1, then taking its curl and using (4.4.1), we obtain the corresponding wave equation for H in the form Vx[k"l.(VxH)]—2 H = 0. (4.4.7) Thus, either anisotropy or inhomogeneity causes the wave equations for E and H to be different and to contain terms which cross-couple the scalar field components. In the nonhomogeneous case, the difference arises physically from the fact that the wave impedance (that is, the ratio of E to II) changes even when the Poynting vector (the product of E and H) is approximately constant. From (4.4.6) and (4.4.7) one can deduce the nature of initially plane waves for various assumed forms of dielectric constant, directions of inhomogeneity, and directions and polarizations of the waves (Bachynski, 1960). The results are summarized in Table 4.1. For instance, even in the absence of a magnetic field, a wave propagating perpendicular to the density gradient is no longer transverse electromagnetic (TEM). 4.4.1 Isotropic inhomogeneous plasmas. In the special case with no magnetostatic field and consequently an isotropic, scalar dielectric constant k, (4.4.6) becomes V2E + ^r /cE-hV (V*)-E -0. (4.4.8) If, furthermore, we assume that the wave is initially plane and transverse and a changes only in the direction of propagation, then (Vk)-E = 0 and the wave equation reduces to d2E co2 ^ ,r n _ + _,(*) E = 0. (4.4.9) Indeed, the adiabatic approximation of Section 4.2 is simply a first-order solution of (4.4.9). For the same special case, (4.4.7) reduces to d2H 1 du dll (4.4.10) dxz c2 *^ " k(x) dx dx' the magnitudes of E and H being related by (4.4.1) as tu/a0 ax If an effective propagation constant y(x) is defined such that (Ostcrberg, (4.4.11) 1958) , 1 dE (4.4.12) 134 Wave propagation through hounded plasmas Chap. 4 then (4.4.9) requires that y satisfy the Riccati differentia] equation5 Hf + y2 + ^-(-v) = 0. (4.4.13) sIn a homogeneous medium (4.4.13) gives the familiar result y= ±jnVitajc. The condition for the validity of the adiabatic approximation is then seen to be or dy I «■> 1 dx <»'■ ~ 2c k'A dx c2 dx\ 1 dx 2« „ 4jt k dx c X where X is the local wavelength in the medium. That is, the relative change in k over a wavelength must be small compared to 4jt. The same condition is obtained from (4.4.17). Table 4.1 Effect of inhomogeneity and anisotropy on propagation of plane electromagnetic waves (Bachynski, 1960) Type of medium Uniform isotropic k~t =e Uniform anisotropic 1 e± eM 0\ k_1 = j -ex ex 0 j \ o o ej Inhomogeneous isotropic K-1 = e(r) Inhomogeneous anisotropic e±(r) ex(r) 0 X -ex(r) ex(r) 0 0 e,Xt)j Direction of inhomogeneity Wave type* O X L, R None TEM TEM TEM No ne TEM TM TEM Along initial y TEM TEM TEM Along initial E TM TM TM Along initial H TE TE TE Along initial y TEM TM TEM Along initial E NT TM NT Along initial H TE NT NT * 0 = ordinary, X= extraordinary (propagation across field); L, R = left/right-hand (propagation along field); TEM = transverse electromagnetic; TE = transvcrsc electric; TM = transverse magnetic; NT = nontransversc; the propagation coefficient y is in direction of wave normal. 4.4 Inhomogeneous plasmas 135 If (4.4.13) can be solved for y(x), the wave propagation is given by E(x) =£(0) exp J - J** y(x) dx] ■ (4.4.14) In general, (4.4.13) yields two solutions for y, corresponding physically to waves traveling in both directions. In fact, where k is pure real, one solution is the complex conjugate of the other. Reflection and transmission coefficients are obtained by matching boundary conditions in a manner analogous to the uniform slab problem of Section 4.3. For the simple model of a linear variation in electron density, for instance, (4.4.9) x = 0 0.3 0.2 \P\ 0.1 x = L 0.5 1 1 1 K (x > L) = 0.25 - 0.64 _ - 1.44 - - 4.0 —" i I 0.5 L/X l/x 1.0 1.0 1 1 1 1 - \ \k (x > L) = 0.25 V \ - - \ 4.0 X. ___ - "~\^\0.64 ---St- 1.44\Sv i i i i FIG. 4.7 Amplitude reflection and transmission coefficients for a linear-ramp variation of electron density », as a function of ramp length L for real dielectric constants of the form k = I -«/«,.; A is free-space wavelength. (Reproduced from Albini and Jahn, 1961, by courtesy of the Journal of Applied Physics.) 136 Wave propagation through bounded plasmas Chap. 4 4.5 The geometrical optics of a uniform cylindrical plasma column 137 may be solved directly in terms of Airy functions and computations made for linear ramp or trapesoidal profiles (Albini and Jahn, 1961; Wort, 1962). Figure 4.7 illustrates the dependence of reflection coefficient on ramp length and dielectric constant. Numerical calculations for other simple profiles have been made by Taylor (1961), Klein et al. (1961), and Hain and Tutter (1962). Interference effects, arising between reflections from the two sides of an inhomogeneous slab appear to be much more pronounced in the amplitude and phase of the reflected wave than for the transmitted wave. A somewhat similar problem has been considered in connection with tapered waveguides (Johnson, 1959). 4.4.2 Anisotropic inhomogeneous plasmas. In more general cases it is usually easier to deal with the magnetic vector, since it is always solenoidal. Once H is found from (4.4.7), E may be obtained from (4.4.2). Consider as a somewhat more general special case an inverse dielectric tensor in the form 0 0 0 0 (4.4.15) which is appropriate to a cold plasma in a magnetic field directed in the z direction. Further assume that the elements of ft-1 are functions of x only and that propagation is in the x direction with H-polarization alternatively in the y or z direction (ordinary or extraordinary waves, respectively). Expansion of (4.4.7) indicates that the magnetic field remains transverse for both cases, whereas (4.4.6) indicates that the electric field is transverse only for the ordinary wave. Assumption of a space-dependent, effective propagation constant analogous to (4.4.12) y(x)=~ 1 zVL H tlx' (4.4.16) leads to the differential equation for y(x) analogous to (4.4.13) (4.4.17) where «c_1 —«n_1(x) or kj."1^) for the ordinary and extraordinary wave, respectively. Numerical calculations for this anisotropic case have been made by Hain and Tutter (1962). The problem of an inhomogeneous cylindrical plasma is again more complex, since the wave equation must be dealt with in cylindrical coordinates. With a plane wave incident upon a cylindrical plasma, it is possible in principle to calculate the phase and amplitude of the scattered wave as a function of scattering angle (King and Wu, 1959). Since these quantities are readily measurable as a function of angle, the inverse problem of deducing the profile from scattering data provides an interesting technique for measuring plasma profiles (Shmoys, 1961; Kerker and Matijevic, 1961). 4.5 The geometrical optics of a uniform cylindrical plasma column A very approximate but useful model of common laboratory plasmas assumes a homogeneous cylindrical plasma several free-space wavelengths (of the probing microwave) in diameter, and yet neglects reflections at the boundary—the geometrical optics limit. The basic parameters of this geometry are defined in Fig. 4.8. The problem is assumed two-dimensional, the elements being of infinite extent normal to the paper. If the plasma is distant by at least a wavelength from the antenna, induction effects can be neglected and the situation treated as a radiation problem. If AjX»l D{k» l, geometrical optics is a valid approximation, and we can talk in terms of rays which, except for refraction, travel in straight lines. 4.5.1 Transmission loss by refraction. We now consider the effect of refraction (Heald, 1959a; Wort, 1963). Since the index of refraction of the plasma (no magnetic field, or parallel polarization) is p=(l-nfncy*sin(ei + 20a-01) 2Ä-Dcos(0, + 202-ö1)' (4.5.2) In many cases of practical interest it is reasonable to make small angle approximations. We obtain from (4.5.1) and thus HHH2(H+1K (4.5.3) Setting m = (l//x)-l, from (4.5.2) (2R-D)^2m-+(2m+\)9i = A~D (2m+l)£+2(m+1)0,1. (4.5.4) 4.5 The geometrical optics of a uniform cylindrical plasma column 139 .Unrefracted ray -Refracted ray FIG. 4.10 Effect of refraction in geometrical optics approximation. Solving for PjD, we have P A-12{2m+\)R+D}Bi D 4mR + D (4.5.5) For /x) 1 2(L + R)-D Eliminating 9{ in (4.5.5) and rearranging, we have finally P A[(L+R)-DI2] D 4mR(L + R) + D(L + 2R) (4.5.7) (4.5.8) (4.5.9) We recall that P\2 'is the largest entrance ordinate of rays that pass into the receiving aperture. Therefore, when PjD« 1 the cylinder is equivalent to 140 Wave propagation through hounded plasmas Chap. 4 a slab of thickness D. With the above evaluation of 0; the small angle approximations will be self-consistent if from (4.5.6) or using (4.5.9) P 2(L + JK)-D D 2(2/;/ + 1 )(Z. I R)-r!) A« 4mR{L + R) + D(L + 2R) (2m+l)(L + R) + D>2 (4.5.10) (4.5.11) Neglecting dissipation in the plasma, we obtain a reduction in amplitude at the receiving aperture because of the loss of highly refracted rays. This (power) transmission ratio is D(L + 2R) Pfo=i) 4mR(L+R) + D(L + 22?) (4.5.12) 15 10 - ---Refraction ■-Dissipation ----Reflection (max.) 5 0.5 Electron density (nfnj 1.0 FIG. 4.11 Loss of transmitted amplitude from refraction 4R{L + RtfD{L 4-2/0 = 3.6, dissipation vDjw'A-QA, and reflection. 4.6 The antenna problem 141 where we recall that 1 ■ m=--1 -.4-+. f (l-njn^- In, Figure 4.11 shows this transmission loss as a function of electron density for the particular case of 4R(L+R) D(L+2R) -3.6. 4.5.2 Other sources of loss. For comparison, we compute the dissipative loss in the plasma due to collisions. From (1.3.30), for nX and therefore A>2n'-X»X, and the intensity distribution near P is essentially that of geometrical optics; that is, uniform intensity falling sharply to zero in the geometrical shadow of the aperture. If a lens of focal length R is inserted at the aperture, a Fraunhofer diffraction pattern is obtained in the plane containing P, as in the familiar problem of the astronomical telescope. If, on the other hand, n is much less than unity, a Fraunhofer diffraction pattern is obtained at P even without a lens. This is the familiar far-field case of conventional microwave antenna theory. To the extent that n^A2j4XR«l, we have R»A'J-I4X. This is equivalent to the well-known rule for the far (Fraunhofer) field of an antenna, which is usually written7 R>A2j\ (4.6.4) and signifies that the maximum phase differential between "rays" is less than A/8, or that the aperture is less than one-fourth of the first Fresnel half-period zone (Montgomery, 1947). The total angular width of the central maximum of the Fraunhofer diffraction pattern is 2\jA. Therefore, the spatial width of the central maximum falling on a plane in the far field is (2AIA)R>2A, Thus, if .4» A the intensity distribution in the vicinity of P is quite smooth over distances of the order of a wavelength, as in the high n case but in contrast to the 1 A2jX without the use of collimating lenses as normally required in the optical region. That is, the "far field" of a radiation aperture, or an obstacle, is a much closer distance, in wavelengths, than for similar apertures in the optical case. Therefore, in many situations, far-field theory can be used to describe the microwave field. Meanwhile, the use of lenses becomes less powerful since the focal length F of the lens must be F2AS/X, corresponding to A/16 or one-eighth zone. Amplitude errors due to interference are then about 2% as opposed to 5% for the criterion given above. if the focusing effect of the lens is to influence the diffraction pattern appreciably. The so-called /'number of the lens is then f=FjA 1 for practical lenses. The far-field region can be effectively extended somewhat closer to the antenna aperture by using a lens to partially overcome the diffraction spreading (Sherman, 1962). The angular half-width of the central maximum of the Fraunhofer diffraction pattern is XjA. Tn geometrical optics a ray leaving the edge of an aperture of width A at this angle appears to originate at a point located a distance A'2j2X on the source side of the aperture, and therefore the insertion of a lens of focal length F=A2j2X will render this extreme ray parallel to the axis. The/number of such a lens is f=A\2X, (4.6.8) agreeing closely with the upper limit of (4.6.6). Table 4.2 summarizes the characteristics of the radiation field for various regimes of the parameters. The best collimation (~ A) is obtained Table 4.2 Field patterns and collimation of antennas Number of zones in aperture Small aperture A(\~i Large aperture AjX»\ n» 10 1 A) in Ihc AjX~\, n< I case (antenna far field) and the AjX»\, n»10 case wilh lens (geometrical optics). The latter, however, is a strongly 148 Wave propagation through bounded plasmas Chap. 4 converging wave passing through a focus. It appears best to design the experiment so as to avoid the low-order region (1D, appreciable energy passes around the plasma, reducing sensitivity and severely complicating interpretation. With D>A>\, in order to avoid induction field effects and Fresnel-zone interference effects, we must have the plasma located in the far (Fraunhofer) field of the antennas, R>A2jX. It is a well-known rule of antenna engineering that for a pair of antennas to be located in the far-field region, by the usual A2 (A criterion, the minimum insertion loss is of the order of !6dB (Montgomery, 1947). Since only about two per cent of the radiated power is received, the probability of interference from spurious reflected signals is high. We are, therefore, interested in pushing as close to the near field (Fresnel zone number «~ 1) as possible without encountering severe amplitude and phase disturbances from interference. This leads to the alternative of small (nondirective) antennas relatively close to the plasma or large (directive) antennas farther back. 4.6 The antenna problem 149 w= (t)s The concentration of rf energy produced in the field of a horn antenna depends upon two factors: the width of the wavepacket launched, and the angle of spread of the wave. Empirical plots of intensity contours in the field of millimeter horn antennas indicate that one half of the energy is confined within a beam width +(^)T (4-6-9) where a and b are correction factors depending on geometry and aperture illumination and departing only slightly from unity. For a given A and R, this is minimized when A~(^XR\'A (4.6.10) giving 2b A W^iabARfK (4.6.11) This condition corresponds to an aperture of about one half a Fresnel half-period zone at R. The insertion loss between two such antennas spaced 2R apart is about 8dB, depending upon the other dimension of the antenna aperture. Sometimes mechanical constraints of the apparatus will prescribe R, in which case A is determined by (4.6.10). If both A and R are at the experimenter's disposal, it is necessary to consider the role of the diameter D of the plasma column. The relative beam size W\D varies as R'^jD, whereas the relative spreading of the field over the plasma I) ldW\ varies as DjR. Since we wish W«D«R, we arbitrarily take D = {WminR^MabAR*yvK (4.6.12) Recapitulating, given D and A and assuming a = b = 1, we choose V2 (4.6.13) This heuristic argument is founded on the vague assumption that there is some virtue in minimizing the beam width at the plasma by choice of A and then compromising in the choice of R such that D R The effect is to prescribe a situation in which the plasma is located slightly 150 Wave propagation through bounded plasmas Chap. 4 inside the conventional far-field boundary. We have seen, however, that under these conditions diffraction anomalies should not be very severe. Note that when Z>/A»l, R«D2j\ and, therefore, the plasma approximates an infinite slab as far as diffraction is concerned. By reciprocity and symmetry arguments, we conclude that transmitting and receiving antennas should be identical. The preceding discussion has been based on the assumption of simple horn antennas without lenses. It has also assumed a "long" horn (L>A2jX) and L>2R, which may be impractical. Thus, two uses for lenses emerge: (7) to permit a less-than-long horn, in accord with conventional practice; and (2) to focus the beam or at least over.-collimate to compensate partially for diffraction. If a horn is "long," its far-field (R>A2jX) pattern cannot be appreciably narrowed by addition of a lens. However, in the previous section we have discussed the use of a lens to focus the energy at a distance R-l)JD/A (neglecting internal interference effects), where ji< 1 is the refractive index of the cylinder (air) relative to the paraffin, and A is the wavelength in the paraffin; and (b) the unperturbed wave passing around the cylinder of amplitude (l-D/ff)^ phase 0. W is the effective beam width at the plasma. and from (4.5.9) r L+2R ' A[(L+ R)-D!2] D 4^--lj R(L + R) + D(L + 2R) The resultant wave is then expD'2»r{p-l)Z>/AJ (4.6.14) (4.6.15) P_ W b-W expfjO)=C exp(/ A) (4.6.16) where C and A are the amplitude and phase shift of the resultant wave. We have c={w+ ([-f) +2[wT{] If «*[2wG*-i)o/A] zl^^tan" sinl>n>-])Z>/A] cos[2w(/t- l)Z>/A]+(l-£) (4.6.17) (4.6.18) & -180 2 3 Diameter D/S (in paraffin) FIG. 4.18 Results of simple geometrical optics theory for the conditions of Fig. 4.16, exhibiting "loss" of 360°. 2.25 L 2.0 .1 1.8 1.6 1.4 Dielectric constant k 1_ 0 0.1 0.5 0.2 0.3 0.4 Simulated electron density (nfaj FIG. 4.19 Phase shift a* a function of dielectric constant of cylinder in paraffin-analog experiment, simulating plasma of varying density. Cylinder diameter 4.6 wavelengths. 154 Wave propagation through bounded plasmas Chap. 4 The theoretical C and Aj> are plotted in Fig. 4.18 for parameters corresponding to the experimental conditions of Fig. 4.16. In order to obtain these relations for Candzl<£, several small-angle approximations have been made, diffraction was completely neglected, and interference effects inside the cylinder were disregarded. In spite of the crudity of the geometrical optics analysis, the numerical agreement is reasonably good. This "lost-wavelength" effect could cause misleading results in a plasma experiment in the uncommon situation in which the plasma is created with a small diameter (<2A) which subsequently grows larger.8 More commonly, the plasma is created with a relatively large diameter (>3A) and then grows denser (due to increased ionization) or smaller (due to some form of magnetic compression). In these cases, the transition from the vacuum (no plasma) phase-shift condition, as the plasma develops, appears to be unambiguous. By inserting rods of various known dielectric constants in a fixed diameter hole in the paraffin environment, the data of Fig. 4.19 was obtained, simulating varying plasma densities (Rosen, 1959). On the basis of this study, we conclude that the slab analysis is satisfactory for a cylindrical plasma diameter of at least three wavelengths, provided the antenna system is chosen judiciously. n The expanding-diameler situation could occur during a plasma decompression event or an expanding cylindrical shock. CHAP T E R 5 Guided wave propagation 5.0 Introduction The effects of finite plasma dimensions on wave propagation were discussed in Chapter 4. The boundaries were found to cause reflections and refraction of transmitted waves and, in some cases, to affect the radiation patterns of antennas. In most cases, the boundaries led to problems, rather than being beneficial to the propagation experiments. In the present chapter, we discuss another class of bounded plasmas; in this case, boundaries arc essential to the wave propagation. Resonant cavities and waveguides have metallic walls that carry currents and, thus, set up propagation modes. The electromagnetic fields penetrate the enclosed plasma, whose conductivity, in turn, affects the mode cut-off frequency. Measurements of wave phase shift or resonant frequency and loaded Q then can be related to the plasma properties. Plasmas having vacuum or dielectric boundaries can support space-charge-wave modes and, thus, can act as waveguides. Certain space-charge-wave modes propagate along the plasma surface (surface waves), while others are carried within the plasma (body waves). When a magnetic field is"present, the waves tend to be a combination of both types. Electromagnetic waves and spacecharge waves may propagate simultaneously along the same bounded plasma. Under certain conditions, the different wave types may couple to one another but, in general, the coupling coefficients are rather small. 5.1 Measurements on,plasmas contained in resonant cavities The resonance properties of a cavity containing a lossy dielectric can be staled in terms of Q 155 156 Guided wave propagation Chap. 5 5.1 Measurements on plasmas contained in resonant cavities 157 Q = o0 (energy stored) 2n (energy stored) average power loss energy lost per cycle (5.1.1) The stored energy is calculated from a volume integral of the fields contained in the cavity; the power is dissipated in both the wall losses and the dielectric losses. When the lossy dielectric filling the cavity is a dilute plasma, of complex conductivity a, the change in Q and the shift in resonance frequency due to introducing the plasma are given by a perturbation equation (Slater, 1946) (or <2o) 1 j «(r)E2(r) dV (5.1.2) dV where the 0 subscript represents unperturbed values and the 1 subscript the perturbed conditions, that is, with plasma present. The intrinsic or unloaded Q, Qv, is not measurable, however, since we must couple to the cavity through some kind of impedance. The equivalent coupling circuit, representing effects of the coupling orifice or loop, the wall losses, dielectric losses, and coupling line or P & p ft t FIG. 5.1 (a) Equivalent circuits for an empty resonant cavity, and (/>) one containing a plasma. The subscript 1 denotes plasma parameters. Other symbols are defined in Ihe text. FIG. 5.2 Standing-wave ratios, expressed in decibel format, in the feed line for (a) an empty and, (b) a plasma-filled resonant cavity in the vicinity of resonance, showing detuning. Symbols are defined in the text. Subscript 1 denotes plasma conditions. waveguide losses, is sketched in Fig. 5.1. The reflection coefficients are shown in Fig. 5.2. Ye is the matched characteristic transmission line admittance at /VTRamoand Whinnery, 1953). The network Xs represents the coupling reactance, and Gs the coupling losses. Xs is generally negligible and, in the remainder of the analysis, will be omitted (Slater, 158 Guided wave propagation Chap. 5 1946). The cavity across 7T" is represented by parallel reactances L and C and conductance G. The cavity is coupled to a transmission line by an admittance ratio A2:\. In the reduced equivalent circuit, the quantities are normalized to Y,: g^GM l=LYJA2 g = A2GjY0 c=CAzIYe. (5.1.3) The conductance ga across PP' at resonance, when the susceptance of l-c is zero (reactance is infinite), is obtained from go l-(#o/gs) (5.1.4) The dimcnsionless coupling coefficient J3C represents the ratio of energy stored in the cavity to that coupled into a matched transmission line. A system for which jic, and of the loaded Q, QL, for the two conditions (with and without plasma) then permit the calculation of the plasma admittanceyi—gi+jbx. This is best done with an impedance chart, such as the Smith chart (Southworth, 1959), since explicit calculations tend to be laborious. 5.1 Measurements on plasmas contained in resonant cavities 159 5.1.2 Measurement of plasma density and collision frequency. For low density plasmas (wz«w2) having low collision rates (v«a>) and no external magnetic field, (5.1.2) may be written in two parts: one dealing with frequency shift, and the other with change in Q (Buchsbaum and Brown, 1957), I 1 dV o.0~2 l+(»Ha " (5.1.8) (5.1.9) dV The spatial distributions of both the electron density and the cavity electric fields must be known. The field configurations of cavity modes, in general, are known. If the plasma density is uniform across the cavity, the evaluation of the integrals in (5.1.8) and (5.1.9) is straightforward (Buchsbaum and Brown, 1957). If the plasma consists of a small diameter to11 Electron density |cm"3' PIQ, 5,3 frequency rfiil't of TM0ao cylindrical cavity with a plasma post; r/o = l/10, ./;, 5555 Megacycles. (Courtesy S. C. Brown, M.l.T. Research Laboratory Tor Electronics.) 160 Guided wave propagation Chap. 5 column, much smaller than the cavity diameter, it may be looked upon as a lumped admittance, shunting the cavity. As an example, consider a cylindrical cavity excited in the TM[>2[J (linear accelerator) mode, containing a plasma post of diameter 1 /1 Oth that of the cavity. The curve for frequency shift vs. electron density is shown in Fig. 5.3 (Brown, 1958). Typically, Awjia of 10"4 is easy to measure by observing an oscilloscope trace. The minimum detectable density then is about one per cent of that detectable with a transmission interferometer having a path length of 10 wavelengths and a resolution A. When the electron density is nonuniform inside the plasma, as is nearly always the case, the frequency shift may be expressed in terms of an average density h and a geometry factor a, Aw -= — an. (5.1.10) This operation is valid only for low densities, that is, when the plasma can be treated as a perturbation in the cavity. The separation of (5.1.8) into the two parts represented by (5.1.10) is especially useful when the plasma density is to be studied as a function of time. The factor a, constant in time, then is calculated for various geometrical conditions, and the results tabulated or plotted (Oskam, 1957; Buchsbaum and Brown, 1957). A cavity mode that is particularly useful as well as easy to treat mathematically is the TM0io mode in a cylindrical cavity. The electric field is entirely axial (£3) and the magnetic field azimuthal (/7e). The frequency shift is given by Brown (1958) ■=a- I l+vjc (5.1.11) This equation applies also when there is an axial magnetic field, since E, || Bz. Other modes are useful as well. When an axial magnetic field is applied, the plasma becomes anisotropic. The cavity electric fields can be represented as right-hand and left-hand circularly polarized fields, much as described in Section 1.4. The cavity frequency response then splits into two peaks, instead of the one shown in Fig. 5.2. Some cavity modes are degenerate, with two (or more: see Fig. 9.19) modes occurring at the same frequency. The presence of the magnetized plasma in the cavity removes the degeneracy and the different modes appear at different frequencies. As an example consider the TM1U and TEon modes. For o>,,/ai«l the frequency shifts are given by Brown (1958); 5.1 Measurements on plasmas contained in resonant cavities 161 iAw\ _aa>£\ 1—a)6/tu W + "'2 y (1-«»Z+("M (Aoj\ a tu„a 1 + <°i)l(0 W — 2 cd2 (1+«.,,M2 + 0'W Aa> a a>2 (TMni) oj0 2 to2 1 -(o)6/c«.)s (TE0U) (5.1.12a) (5.1.12b) (5.1.12c) A study of the frequency dependence of the three modes yields information about the density spatial distribution. For higher densities and collision rates, the frequency shift tor the 11011 mode must be written in more complete form Aa> a a). ^2wJ [V2 + (u> + oJb)z][v2 + {u- +v there is no frequency shift A /due to plasma density variations. The cavity Q also is approximately minimum for this condition. This effect can be used to measure the collision frequency v (Hirshfield and Brown, 1958). FIG. 5.4 Frequency*hift of a TEojj mode cavity for high ^^^J and without magnet," field applied. (Courtesy S. C. Brown, M.l.T. Rcsea.ch Laboratory for Electronics*) 162 Guided wave propagation Chap. 5 5.1.3 Special cavity modes for high density plasmas. The TE011 mode is useful for high densities {wvjw> 1) when no static magnetic field is applied and the electron density gradient is radial (and, thus, perpendicular to the electric field). The circumferential electric fields induce no spacecharge fluctuations in the plasma (Persson, 1957). When a magnetic field is applied, the anisotropy permits radial currents and fields and the frequency-shift relations become complicated, as shown in Fig. 5.4. At very high densities and low collision rates, where the skin depth is much smaller than the cavity diameter, a high mode TE01M cavity may be used, with M as large as perhaps 8 or 10. The smaller the "active" volume of cavity, that Frequency meter detector (?) Indicator Oscilloscope Reflection signal input FIG. 5.5 M ici-owave cavity excitation and measurement system for studies of plasma contained in a resonant cavity. The ionizing signal couples to a different cavity mode, at a different frequency, from the measuring signal. Either the reflected signal (as shown) or VSWR may be observed. 5.2 Waveguides containing plasmas 163 is, that part of the cavity fields perturbed by the plasma, the smaller will be AcojAn and the less sensitive will be the measurement. It is important to note that these techniques for high-density plasmas, with the skin depth much less than the plasma diameter, require an independent knowledge of the plasma density (and temperature) profile if anything more than the plasma surface properties are desired. 5.1.4 Experimental techniques for cavity measurements. Often a cavity can be excited in two modes at once, at widely separated frequencies, such that there is no coupling between the two inputs. It is then possible to use one mode to break down a gas and heat the plasma with a high power source, and to use the other mode for diagnostics. A sketch of a typical system is shown in Fig. 5.5. The excitation source may be steady, or it may be pulsed, with the diagnostics done in the afterglow period. Ordinarily, the frequency of the signal source is swept back and forth over the resonant frequency of the cavity. The reflection coefficient or the standing-wave ratio is observed on an oscilloscope, and the shift in resonance frequency and cavity de-ging are recorded for various plasma conditions, ff the plasma is transient, the timing of the sweep may be delayed to sample various parts of the plasma transient event (Biondi, 1951; Sexton et al, 1959). A second technique is to fix the frequency of the signal source and wait until the cavity resonance frequency sweeps through, as the plasma density decays (Biondi and Brown, 1949). 5.2 Waveguides containing plasmas Waveguides also have normal modes and their cut-off frequencies will be shifted, much as the resonance frequencies of cavities are shifted, by the presence of a plasma inside. To find the effects of a plasma on the propagation characteristics, we employ some of the methods of cavity techniques and some of free-space propagation. When waves propagate along dielectric or conducting walls, they are no longer strictly TEM waves. The currents flowing in the walls lead to components of electromagnetic field along the direction of propagation, even when no magnetic field is present. As a clean example, consider a TE wave propagating in a plasma-filled rectangular waveguide with no external magnetic field present, as sketched in Fig. 5.6«. The wave equation can be separated into two parts, one describing the variations across the waveguide and one the variations along it. The wave equation becomes V"H = Ve2H-|-y2H= -wznom=-- (5.2.1) 164 Guided wave propagation Chap. 5 Plasma Plasma (a) (b) FIG. 5.6 Plasma-filled waveguides; (a) rectangular, (b) cylindrical. where Z = e0Z is the equivalent complex permittivity of the system and y is the propagation coefficient or complex wave number, which depends on the waveguide mode. In rectangular coordinates, the transverse term (V,3) is Ox oy2 (5.2.2) For TE waves, no Es component exists, and it is convenient to solve (5.2.1) for the Hz component /^ = //0(cos^xj(cosy.v) exp(>/-yz) (5.2.3) 5.2 Waveguides containing plasmas 165 where m is the mode number across x (or a) and n is the mode number across y (or b). From (5.2.1), (5.2.2), and (5.2.3), where /3c = iriode cutoff wave number Ac = mode cutoff free-space wavelength cDc = mode cutoff frequency. Here we neglect any losses in the waveguide walls. Solving (5.2.4) for the propagation coefficient, we have c2 "r ' J 2 Ki- (5.2.5) The value of <<, calculated in Section 1.3.2 for a Lorcntz plasma, is — i " r — I o 1 2 l + i>z/a>2 v/to (5.2.6) «>2 1 +v3/w3 To obtain the attenuation and phase constants, we take the square root of (5.2.5) as in Section 1.3.3 j9={+ 2 (as above) + 1[(as above)2-1- (as above)2],/«}'/» (5.2.8) where iae—c$e is the mode cutoff frequency in absence of plasma. These equations look similar to (1.3.20) and (1.3.21), except for the presence of the mode cutoff wave number ft,.. This factor tends to increase the plasma CUtoff frequency (rt'ow ><«,,) or, at fixed frequency, to cause the transmission to cut off at a lower density than for the free-space case. 166 Guided wave propagation Chap. 5 5.3 The cü-ß diagram 167 0.4 0.3 8 0.2 0.1 toe/üj = 0.9/ 0 oJc/o) = 0.0 (free space) I_L 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Normalized electron density (u,,/c«j)2 = n/«c FIG. 5.7 Wave attenuation for a plasma-filled waveguide as a function of plasma density, for various mode cutoff wave numbers. Attenuation in dB is dB = 8.69«; (u)a 0.5 ii 0.4 «» c. From (5.2.8), ojcp is seen to be (5.3.3) Figures 5.9 and 5.10 show w-fi diagrams normalized in two ways for the plasma slab and for the rectangular waveguide, where the collision frequency v is very small, the density is uniform, and no magnetic field is present. The slope of the radius vector to each point on the diagram is seen to be proportional to ot//3, which is the wave phase velocity %. The slope of the tangent at each point is proportional to dwjdfi, which is the wave group velocity vg. 5.4 Nonuniform plasma in a waveguide When we considered that the plasma was uniform across the waveguide diameter, we implied that it had no effect on the magnitude of the microwave fields, that is, that E and H were unperturbed. That, or course, is not true for dense plasmas, when io„ approaches w. Let us take the case for a rectangular waveguide again, in which the plasma has a diffusion-limited density distribution (Brown, 1959) ii(x, >0=n0(cos ^ xj (cos | >') (5.4.1) where nQ is the axial density. The corresponding dielectric constant is «(x, y) = 1 -1(cos I *)(cos I y). (5.4.2) The wave equation must now be written ■>2H. 82H, 0-:F- cy- y2+-^ p, for a uniform plasma"in waveguide; u),: = cold mode cutoff frequency, uiC!,=(wcz + u>payA = liol mode cutoff frequency. the simplified equation is not separable and cannot be solved explicitly for y as before, it resembles Mathieu's equation dHT dx T+(/' + ^i'^v)// = 0. (5.4.4) 170 Guided wave propagation Chap. 5 5.0 r FIG. 5.10 Dispersion (w-/3) diagram for eleclromagnetic waves, with propagation phase constant normalized to the mode cutoff wave number, for a plasma filling a waveguide; «jc=£„r = cold mode cutoff frequency. Various solutions to (5.4.4) exist, and series and numerical approximations are also possible (Madelung, 1943). 5.5 Spacecharge waves In Chapter 1, it was pointed out that spacecharge fluctuations in a cold, stationary, infinite, uniform plasma were nonpropagating. Any disturb- 5.5 Spacecharge waves 171 ance or departure from equilibrium is confined to the location of its origin, within the order of a debye length. There are conditions, however, under which these fluctuations can propagate and transfer wave energy away from the source. (/) The plasma electrons have a drift velocity, vQ. (2) The plasma is finite and possesses normal modes. (3) The electron temperature is finite. Let us now consider the characteristics of some of these propagating plasma waves under various plasma conditions. 5.5.1 Spacecharge waves in a cold, drifting plasma. Let the electrons have a drift velocity vn. If plasma oscillations (at wXojp) were present, they would be convected along at the drift velocity, and would constitute a convective wave. If no oscillations are present, but rather an externally applied electric field varying at frequency ai is present, we can also excite plasma waves arising from the resulting velocity fluctuations. First from a naive point of view, we recall that the spacecharge oscillation in a cold stationary plasma, discussed in Section 1.2, occurs at the plasma frequency top independent of its spatial variation. Thus we may choose a standing wave of arbitrary wavelength A, which in turn may be regarded as the superposition of two traveling waves with phase velocities Pfm ± Aoip/2jt. Now if we imagine that this oscillating plasma is moving past our observation point at a steady velocity t;0, the two traveling waves appear to us to have the same wavelength A, a doppler-shifted frequency u>, and the respective phase velocities v4, = v0±X+ojp v0. "slow spacecharge wave" (5.5.1b) Now on ajnore analytical level, we seek the dispersion relations for these waves under the action of an applied electric field. We rewrite the equation of motion of a Lorentz plasma, (1.4.1), in Eulerian rather than Lagrangian coordinates (that is, we watch the plasma stream flow past our fixed coordinate system, rather than use a coordinate system in which the plasma is at rest); the force equation now is (Marcuvitz, 1958) < v e —+v-Vy + v(v-v0)=--(E + vxB). i'I m (5.5.2) 172 Guided wave propagation Chap. 5 5.5 Spacecharge waves 173 Let us now assume the following. (J) The drift velocity v0a is constant, in the +z direction. (2) The induced a-c velocity variations are much smaller than the drift velocity. (J) The driving electric field is of the form ElL. exp(/W-yz), that is, a longitudinal wave propagating in the z direction. (4) No magnetic lield effect is present (vxB ~> 0). (5) We neglect collisions (v—>0). The electron velocity is of the form '-',(2, 0 m+viz e*PO' - yz) (5.5.3) where the subscript 0 represents the steady-state component and the subscript 1 the complex perturbation component. The second term of (5.5.2) may be evaluated to first order as oy (5.5.4) where y.j is the perturbation part of (5.5.3), in the z direction. Substituting in (5.5.2) and canceling the phase factor expfjW — yz), we obtain (5.5.5) In contrast to the case with electromagnetic (transverse) waves, the a-c current density Jls depends on the oscillating spacecharge as well as the oscillating velocity component; thus J., ~J0:,+Jiz exp(yW — yz) = -/lo^oz-efwo^+fli^?) exp(jW-rz), (5.5.6) neglecting higher than first-order terms. To evaluate the perturbation component nl of the electron density, « = + exp(y'cuf-yz), (5.5.7) we invoke the equation of continuity (3.3.7) _ , 8J„ dn V-Js- ™ = e — oz dt which becomes Substitution of (5.5.9) in (5.5.6) yields or n0e I -jyvcjw (5.5.8) (5.5.9) (5.5.10) (5.5.11) which in turn permits us to use (5.5.5) to find the (complex) conductivity In problems dealing with waves, it is useful to translate the conductivity into the equivalent dielectric constant 1 ?z 0) exp(-7j8z) exp(;W) (5.5.22) where n is the order of the angular mode number. Writing (5.5.21b) in cylindrical coordinates gives for the radial function

, which simplifies the solution for the constant C. The tangential electric fields (Elz and Eie) must also be zero along the conducting wall at r = b, but continuous across the boundary at r — a. The potentials inside and outside the plasma column are then found to be MM a oo is seen to be quite small, indicating that the presence of metal walls farther than about one column 0.8 0.7 0.6 5" 0.4 c a) 53 £ 0.3 0.2 0.1 11.11 III] Kb = 1 (vacuum) Transition case of thin- — walled dielectric (x = 4) with air space between - S, it and the metal wall — jjbja - 2 " __ ff-bja = ^^^-^— k{, = 4 (glass) /r*~bja = 2 i i i i III 4 5 6 Propagation constant {la 10 FIG. 5.14 The uj-,8 diagram for propagation of surface waves on a plasma surrounded by a vacuum interspace (t,,-en) between the column and the wall and for a material of dielectric constant « = 4 filling the space (sec Fig. 5.12 for geometry). 178 Guided wave propagation Chap. 5 diameter from the plasma has negligible effect. The propagation extends Irom zero frequency up to a cutoff frequency of (5.5.30) where «„ is the dielectric constant of the medium surrounding the plas as ma. 0.5 6/a = 2 Kb = 4 0.3 0.2 0.1 b/o = 2 4 6 8 Propagation constant ffa 10 12 FTG. 5.15 Phase characteristics for a surface wave mode of one angular variation (invelpiece, 1958). 5.5 Spacecharge waves 179 In Fig. 5.15 the co-/3 diagrams of a higher mode having one angular variation (the corkscrew mode with n=l) are shown. This mode does not extend to zero frequency, but has the same upper cutoff frequency as the fundamental mode, in fact, it is found that all of the surface-wave modes have the same cutoff frequency and that it is independent of the column or metal wall diameter, in contrast to the electromagnetic modes. This cutoff frequency, given in (5.5.30), is often referred to as the dipóle or column resonant frequency. The wave velocity, of course, depends on both a and b; for small a, the value of jS is large, and the velocity low, especially as to —> coco. An interesting analogy between the electromagnetic fields and space-charge-wave fields can be drawn. We pointed out that the magnetic component or the surface wave fields can be neglected. But, for a wave to propagate, the energy stored by the electric field must be interchanged with something and, in this case, it is the mass motion of the charges. For this reason, spacecharge waves are sometimes called electromechanical waves. The charges move back and forth in such a way that the electric field lines terminate on them, which in part explains the small penetration of the field outside the column and the difficulty of experimentally coupling to these waves without perturbing them. 5.5.3 Spacecharge waves in a plasma column in a magnetic field. The presence of a magnetic field changes the propagation characteristics considerably. The medium is now anisotropic, and the dielectric permittivity e becomes a tensor, e->£ = e0K. The permittivity derived for electromagnetic waves in Chapter 1 is perfectly general and applies also to spacecharge waves (Pines, 1960); (1.4.60) where the components for a collision-free plasma are (Section 1.4.7) Kl ->* 0 " K = 0 0 0 k\\ _ cop2/co2 , 1 - co, 2/co3 (co^/co^K/co) l-co„2/co2 To find the wave solutions for the magnetic-field case, then, we must rewrite (5.5.18) to include the anisotropic permittivity V-D^V-e-E^eoV-K-E^O (5.5.31) 180 Guided wave propagation Chap. 5 from which (5.5.20) becomes V-e-(- V01) = eoV-k-(-V«?1) = 0. (5.5.32) The form of (5.5.32) in cylindrical coordinates is analogous to (5.5.23) but with the permittivities included 1 d I g%\ „2 , Kl (5.5.33) and has solutions similar to (5.5.27) and (5.5.28), except that 32 is now replaced by jS2(«tl/kx). The electric field in the plasma is given by the gradient of the potential m In the z direction, this is ^Äff^j^ forr<« (5.5.34) where £(0) is the axial field at r=0 and I„ is the modified Bessel function 1 he radial electric field is E 801-J 8El* lr 8r | Br (5.5.35) The azimuthal field is similarly obtained E - 80i="^z 15 dB ßr 8r =^0)^Ia ßr (5.5.36) The last three equations are seen to be analogous to (5.5.25). They point out that all electric field quantities are derivable from the axial field. Tnvelpiece (1958) finds a propagation equation similar to (5.5.29), but with the permittivity components included, Kb ßa*b In(ßa)Kn(ßb) ~ K(ßb)Kn(ßa) '■M2T] (5.5.37) Substituting in the permittivity components of (1.4.60) leads to a complicated expression for the cutoff frequency which involves both wt and 5.5 Spacecharge waves 181 FIG. 5.16 Phase characteristics of spacecharge waves (ui-ß diagram) for a plasma column bounded by a dielectric and metal, in an axial magnetic field of various magnitudes. wp. The oj-fi diagrams in Fig. 5.16 show the trends for various relationships of wb to co,,. When cob is very large, most of the wave energy is carried by charge accumulation within the plasma column, with little surface rippling. These waves are called body waves, and have a cutoff frequency approaching oiv, as do one-dimensional spacecharge waves. Conversely, for (o„ —> 0, the wave energy is carried by surface rippling alone, with little spacecharge bunching within the plasma. The cutoff frequency Tor these surface waves then approaches «up/(l + K|,)''2. The surface waves cannot propagate if the dielectric space (vacuum or material) is absent between the plasma and the metal wall since, then, no charge accumulation can occur. The usual electromagnetic waves can propagate, of course, and a mixing of electromagnetic and spacecharge waves is usually present. For values of u>„lp between 0 and co, and at various values of fia, the wave mechanisms and wave types undergo smooth transitions. At large pa the electric field is almost pure Es and a mixed electromechanical and TM-type wave ensues. For smaller j3a, the wave goes from TM to mixed TE to pure TE (Bevc and Everhart, 1961). The upper branch shown in Fig, 5.16 is a backward wave whose characteristics are not influenced by the geometry and whose cutoff frequency is Ez exp(yW-yz), we may expand the velocity function as f(t, v, 0 ^/0(v) + /;(v) expfjW~yz), (5.6.3) where fc is the equilibrium distribution function, and (5.6.2) becomes to first order (5.6.4) The divergence equation relates the electric field and the charge density, V.EoE=-<0y£B=~ne J/i dsv, (5.6.5) where the uniform positive-ion background has been assumed to cancel the equilibrium electron density. Substituting/! from (5.6.4) into (5.6.5) and canceling the arbitrary amplitude $k, we have ne2 r m J «ca fidfu/iiUz) di:_y dv„ di.K jw - yus The integration over the transverse velocity components vx and j| may be carried out formally, to yield for the dispersion relation 1 (dfQs/di)s) dvg oj+jyvs (5.6.6) where f0s(vs) is the one-dimensional equilibrium distribution. 1 The adiabatic assumption allows the gas temperature to fluctuate due to density fluctuations. The neglect of this effect (the isothermal approximation) leads to a Tactor of 1/3 discrepancy in the thermal term in the (mat dispersion relation (Bernstein and Trehan, 1%0). 5.6 Spacecharge waves in a warm plasma 185 For generality we may assume that both the frequency w and the propagation coefficient y are complex, (5.6.7a) & — cu„. The product of the group and phase velocities for these thermal spacecharge waves is independent of frequency, and equals three times the one-dimensional mean-square thermal velocity, 3kT m = 3co„2AD2 (5.6.15) 188 Guided wave propagation Chap. 5 1CT E •ä 10-2 10" 10" ti i i: i ph. ,1 1 1 1 ips li 1 iiiih 1 1 ii 1 1 1 ii1. ii 1 1 1 11 ill - ^\ 'MINI "'S II 1 III \^ 1 1 1 1 1 il i Tthjii _l>kii' 1 rrHlll 10B 109 10'° 10" 101S Electron density |cm_3| i 01 101 FIG. 5.20 Debye length in a plasma vs. electron density, as a function of electron temperature (see also Fig. 2.5). 5.6.2 Spacccharge waves in a warm, magnetized plasma. To derive the wave dispersion relation in this case, the Boltzmann equation (5,6.1) again is used but the vx B term must be included in the electric field cf+v.Y^tE + vxB].^^ (5.6.16) The Vlasov form of (5.6.16) (that is, omitting the collisional term) has been solved (Berstein, 1958) in the small signal case by perturbation analysis, using Laplace transforms. The waves can no longer be treated as one-dimensional because the medium is anisotropic. Again, as in the discussion in Chapter 1 of electromagnetic waves at arbitrary angles to the magnetic field, we assign vector properties to the propagation coefficient y( ~/P)- The frequency will be complex to allow for wave growth or damping.2 After the Laplace and Fourier transforms have been applied to the distribution function and Maxwell's equations substituted in, the longitudinal wave part of (5.6.16) becomes iä2ß.E + E-Q-|S = ß-a (5.6.17) where w = a>+jaa '2 This linear treatment of course describes only the initial few e-foldings of any growing waves present, and not the final, saturation, equilibrium conditions of wave propagation. The latter would have to be arrived at with a nonlinear treatment (Drummond and Pines, 1961; Sturrock, 1961). 5.6 Spacccharge waves in a warm plasma 189 Q = a triple integral equation similar to (5.6.7) but dyadic and involving 0, the angle between p and B, and including the electron gyrofrequency oj„. a = a vector quantity involving E, B, and |), and the triple integral of the velocity distribution function over velocity space, evaluated at f = 0. The transverse wave part of (5.6.16) has already been discussed in Section 3.4, at which point Landau damping was introduced. Solutions of (5.6.17) can be found that yield either wave growth or damping, depending upon the slope of the tail of the distribution function (for example, a double-humped distribution leads to growth). The analogous situation for transverse waves may not occur, that is, the growth modes seem to be absent. The coupling between the longitudinal and transverse waves is, in general, small (Sturrock, 1961; Bevc and Everhart, 1961). The general solution or (5.6.17) is exceedingly complex. Some simplified cases yield convenient results, however, and we shall discuss these. For a small magnetic field and a Maxwellian distribution at low temperature, the dispersion relation is found by asymptotic expansion of (5.6.17). For the conditions w2»oj„2, a J, p2vfh, the expansion yields, to first order, a dispersion relation similar to (5.6.14) but with an extra term involving the gyrofrequency = o>/ + ^ß? + aJ2sin20 (5.6.18a) or 32=- j2 — tup2 — ctt,,2 sinaö (5.6.18 b) Equation (5.6.18) is plotted in Fig. 5.21 for various values of u>bjtap at an angle of 30". Since the product (p2) sin20 is really the parameter, the curves also apply to other angles such that the products are 0, 1, and 4. The imaginary part of <» gives the damping, analogous to (5.6.12). With dfojdv written explicitly Tor a Maxwellian distribution, the damping is (I) jife5 eXP( 2ß2AD2) [ 1 + sin3ö 24Č8AB2 cos20 [l + 2p2XD2-p2jXDz] where M* a,, cose /__1_\ \l) p3XD* eXpl 2ji2XD2j (5.6.20) (5.6.21) p2P2«p2xD2« 1 pe = (kTjm)'/2wh = electron gyroradius. The growth or damping rate in space is given by (5.6.13) as before. The small gyroradius causes the plasma to behave as if the electrons moved only along B, and to lowest order (5.6.20) is the same as (5.6.14). Another interesting feature of the finite pe case is that for 0 -¥■ ir/2, that is, propagation across B, the Landau damping approaches zero, and within the passbands there is no wave attenuation. There are stop bands, however. For particular values of cob and pe there are gaps in the propagation spectrum as J32XD2 is varied. For /32AD2«1 the spectrum for longitudinal waves propagating across the B field is given by Bernstein (1958) as 1-/3% a„ avs-i e2j32 sin2tf) x l0(^sin2fl)+- g W/V732sin20) 1 1 (5.6.23) Poles are seen to exist for o> — uib and its harmonics. A plot of w\cob vs. fi2XD2 is shown in Fig. 5.22. The poles are clearly evident bounding the passbands and stop bands. 3 M 2 1 1 1 1 I 1 -2 -1 0 FIG. 5.22 Typical «>-0 diagram for spacecharge waves propagating across the magnetic held (0 = 90°). CHAPTER 6 Microwave propagation experiments 6.1 Transmission-attenuation and reflection experiments One of the simplest microwave diagnostic experiments is a transmission-attenuation measurement (Wharton et al., 1955) in an isotropic plasma. Two radiators are arranged so that the path between them passes through the region to be studied, as sketched in Fig. 6.1a. When the plasma density in the path reaches a value high enough that Horn \^ antenna M *- 1 Signal detector (receiver) r ' m Video output signal Reflection signal detector Pad attenuator Video output, reflection signal Calibrated i i Impedance attenuator i i Directional transformer coupler (ret lee to meter) Frequency meter Signal source Video output, transmission signal FIG. 6.1 Block diagrams of basie microwave transmission experiments in a plasma. Diagram b shows the addition of a directional coupler to permit observing reflections from the plasma. coefficients. Two additional components added to Fig. 6.1a allow us to measure the over-all reflection from the plasma: a directional coupler and an additional detector. A typical composite circuit is sketched in Fig. 6.\b.1 The complete circuit includes a source of microwave power 1 The microwave components or "hardware" are described in some detail in Chapter Al this point, wp treat components as blocks having certain properties, in hi li Id in it up the circuits. Readers wishing to familiarize themselves with the waveguide delails are referred to Chapter 'J and to the references listed there. 194 Microwave propagation experiments Chap. 6 (usually, a klystron or backward-wave oscillator), a frequency meter, a calibrated attenuation standard, a directional coupler (which couples only the reflected signal into the "reflection signal detector"),an impedance-matching transformer by which the fixed reflections are canceled, a vacuum window and transmitting antenna, a receiving antenna and vacuum window, a level-setting pad attenuator, and a microwave detector. 6.1,2 Transient plasmas. Often the plasma to be studied is of transient duration. Shock waves and controlled-fusion containment experiments (Post, 1958) are examples. The electron density rises rapidly and then -cop = w2 (110 Gc) .Electron density n FIG. 6.2 Transient plasma event. The microwave transmission and reflection signals at a frequency of 90 Gc are shown, in relation to assumed density and collision-rate temporal variations (a), (b) Microwave transmission signal, detected by a silicon diode. Transmission path /. through plasma is 6 inches, (c) Microwave reflection signal, detected by a silicon diode. The term 100% refers to reflection from a copper cylinder placed at the location of the plasma, 6.1 Transmission-attenuation and reflection experiments 195 FIG, 6.3 Reflection and transmission signals for microwave propagation through a transient plasma. The frequency is 90 Gc. The amplitude of the reflection spikes corresponds to about 25% reflection, "lime scale is 20 microseconds per division. decays more slowly. As an illustrative example, let us consider the use of (he circuit of Fig. 6.16 to diagnose a transient plasma having no magnetic field applied.3 A typical event is sketched in Fig. 6.2a. The dashed line represents the density at which o)„ = a>. We estimate, from other measurements, a cosine spatial distribution of density, 23 wavelengths across at 90 Gc. A peak central density of 1.5- 10M cm-3 corresponds to a cut-oil' frequency of 110 Gc. This plasma column has density gradients steep enough to give small external and internal multiple reflections. The electron temperature in the event chosen reaches only about 2 eV, so that i'jo) is around 0.02; the internal reflections, thus, are partially damped by collision losses. The received signals, as detected by a square-law defector (silicon crystal diode) are sketched in Fig. 6.2b. The small Fluctuations, due to multiple reflections, are evident. The reflected signal will depend upon the steepness of the gradient, since (he signal must penetrate a lossy layer of plasma to reach the reflecting, inioil' plane and then pass through the layer again after being reflected, lite magnitude of the fluctuations, then, gives a qualitative estimate of the steepness of the gradient. A typical reflection signal is sketched in Fig. 6,2c, where 100%, corresponds to the reflection from a sheet of copper at I he following data were taken by one of us (CUW) at the University of California, I iiwiencc Uadialion/Laboratory, on a high-density pulsed plasma experiment dli Mir and Reagan, 1960), in the Controlled Fusion Program. We thank the University of California for releasing this unpublished data. 196 Microwave propagation experiments Chap. 6 6.2 Frequency diversity 197 the location of the plasma. The over-all reflection coefficient is seen to be less than 25% for these data. This is typical of many plasma experiments, including some controlled-fusion experiments; the plasma looks either transparent or surprisingly "black." Oscilloscope traces of reflection and transmission are shown in Fig. 6.3. The spikes of reflection correspond to about 25% and, presumably, arise from steep gradients due to turbulence. The rise rate of density is very fast compared to the decay rate. 6.2 Frequency diversity The simple reflection-transmission experiment, just discussed, gives information only when the electron density is near cutoff, that is, ojptzo). Additional temporal information can be gained by using more than one •rnimmmmmmffimmigm FIG. 6.4b Photograph of discharge chamber. Side arms containing the microwave horns and vacuum windows are shown. The two 70 Gc (4-mm band) horns are mounted in place. One of the 90 Gc (3-mm band) horns is shown removed. The black rcflcclionjjjss coating inside the chamber can be seen in the vicinity of the horns. The chamber internal diameter is 3 inches. (Photograph courtesy of the University of California Lawrence Radiation Laboratory, Livermore, Calif.) 198 Microwave propagation experiments Chap. 6 6.2 Frequency diversity 199 1.5 1.0 0.5 1.0 0.5 --------0)p = 0} at 110 Gc Plasma density as a function of time i\\ i ■ 1 i i 1 1 1 i i ill i i ui at 70 Gc -.->—i Time (arbitrary units) | (a) jjl i i -— /90Gc f /70Gc Microwave -V 11 J 1 transmission IV J signals FIG. 6.5 Transient plasma event. Microwave transmission and interferometer responses, using square-law (silicon diode) video detectors, at 70 and 90 Gc arc shown in relation to the assumed electron density variation in time. frequency, so that cutoff is reached at different times. Let us take the transient event again, but employ two identical transmission paths, one using 90-Gc and the other 70-Gc equipment, simultaneously. The arrangement of radiators used is shown in Fig. 6.4a.3 The horn radiators were \ inch in diameter, separated 3£ inches. The |-inch dimension, calculated from (4.6.4), avoids Fresnel interference problems but gives poor coupling between the horns. Radiators of 1-inch diameter were 3 See footnote 2. FIG. 6.6 Transmission amplitude signals for microwave propagation through a transient plasma. Frequencies of 90 Gc (top trace) and 70 Gc (bottom trace) were used. tried, giving much closer coupling, but the Fresnel interferences were excessive. Since the walls of the chamber are only a few wavelengths away from the transmission path, stray scattering of the diverging portions of the wave leads to spurious interferences. It was found necessary in this experiment to coat the walls of the chamber with a nonreflecting material4 to eliminate these interferences. A photograph of the chamber, showing one of the vacuum windows and the absorbing coating, appears in Fig. 6.46. The idealized density variation is sketched in Fig. 6.5a, with the cutoff conditions shown by dashed lines. The resulting transmission signals, for square-law detectors, are sketched in Fig. 6.5/j. Figure 6.6 shows an oscilloscope trace of measurements on a transient plasma event, recorded at 70 and 90 Gc. Small fluctuations (< 5%) are seen on the trace, evidence of internal reflections. The over-all reflection signal is fairly small. 6.2.1 Frequency diplexers. A feature that is immediately evident is that unless a plasma in the configuration of Fig. 6.4 is symmetrical about the axis (it often is symmetrical) the two transmission paths do not traverse the same thickness of plasma. To ensure the same plasma sample L for both frequencies, the two paths may be incorporated into one pair of antennas by frequency diplexers.5 The two signals are combined into one 1 See Seel ion 9.6.4 for descriptions of such materials. • Sec Section 9.2.3. 200 Microwave propagation experiments Chap. 6 6.3 Phase-shift measurements 201 waveguide, transmitted through the single path and, then, separated to two detectors. 6.2.2 Polarization Diplexers. Polarization diplexers, such as fin-line couplers,6 can also accomplish the function of frequency diplexing, not by using frequency-selective filters but by orienting the two wave polarizations normal to. each other. In an isotropic medium, free of nonlinear effects, the waves will not generally couple to each other, and will travel through the same effective path length. In an anisotropic plasma, the two waves of different polarizations will couple to different propagation modes. These effects are discussed in Section 6.5. 6,3 Phase-shift measurements Without much additional complication, the circuit of Fig. 6.1 can be expanded to become a phase-measuring interferometer, analogous to the Mach-Zehnder interferometer used in optics. A null path is necessary to provide a phase and amplitude reference with which to compare the transmitted signal (or the reflected signal). 6.3.1 Microwave interferometer. The interferometer or phase-bridge circuit (Wharton and Gardner, 1959), sketched in Fig. 6.7, is particularly useful in measurements on transient plasmas. The reference path and transmission path are kept the same electrical length to avoid differential phase changes if the klystron frequency drifts. The circuit shows three detectors. Only the one labeled "phase detector" gives phase-shift information. The "transmission-amplitude detector" plays the role of the detector in Fig. 6.1. In typical operation with transient plasmas the reference path is adjusted to null with the signal path in the absence of a plasma; the phase shift and amplitude then are observed as the plasma fills and then leaves the test path. The signal levels required depend upon the kind of detectors and video amplifiers used but, ordinarily, are in the microwatt-to-miUiwatt range.7 An interferometer that is initially nulled will produce a maximum signal when the angle of the aggregate transmission coefficient has shifted 180°, and will return to a null at 360°, repeating indefinitely as the angle rotates through successive values of n and 2ir. A square-law detector will yield a sinusoidal variation as the plasma density changes. 0 See Section 9.2.3 for this and other polarization diplexers. 7 Commonly, the detectors are silicon diodes. Up to about 100 microwatts, a silicon diode has a fairly faithful square-law response, that is, the output voltage is proportional to input power. The input-output characteristic then begins to straighten out, until at a few milliwatts the characteristic is nearly linear. Most microwave crystals are damaged by power levels in excess of 10 milliwatts. NULL PATH 0-360° Phase shifter Pad attenuator Crystal detector (phase) Video p ream p. 0-5 Mc Interferometer output signal Crystal detector (reflection) r- 1 6-dB 1 Coupler 1 T <* Interference /> [^comparator Receiving horn antenna Vacuum window in wave guide Direct transmission output signal 'IG. 6.7 Microwave interferometer and reflectomcter for plasma diagnostics. 202 Microwave propagation experiments Chap. 6 6.3 Phase-shift measurements 203 FIG. 6.8a Microwave interferometer responses to a transient plasma. Frequencies of 70 Gc (fop) and 90 Gc (bottom) were used. Sweep time was 100 (isec/cm. Small calibration pulses at 750 jascc indicate that the bridges were slightly off null at the time the traces were made. Oscilloscope traces of the interferometer responses to the same event as recorded in Fig. 6.6 ate shown in Fig. 6.8. Cutoff for the two frequencies is reached at different times as the density falls from maximum to zero. A different number of variations or fringes8 are noted for the two frequencies, due to the different number of effective wavelengths in the path. The attenuation of the signal is evidenced by the amplitude envelope of the fringes. In cases where the collision rate is high, so that signal cutoff is reached at a density somewhat below nc (see Section 1.4), the observed number of fringes is less than expected for a given path length and density distribution. Such a situation is shown in Fig. 6.8/). The top set of records of this figure was made from a discharge between plane electrodes in helium at a pressure of 80 microns. The responses at the bottom of the figure were made by adding 200 microns (partial pressure) of argon, to increase the collision rate from a value of ~0.001 to ~0.01cu at the time of u>p = oj. The damping is evident. The presence of the plasma under certain conditions increases the coupling between the horns above the vacuum level (not to be interpreted as amplification!), and introduces multiple interferences from stray scattering around the plasma. These effects are especially noticeable when the horns arc poorly aligned with each other, when the plasma column is only a few wavelengths across, or when there is an impedance mismatch. 0 Analogous to the fringes observed in an optical interference pattern on a Mach-Zchndcr or Fabry-Perot interferometer. ■ m • ...... J 11 Ü m ■ ■ 1 I H Ü ■i IN mm i I ■ M it I FIG. 6.8b Microwave interferometer responses to a transient plasma, showing the cll'cct of low and high collision frequencies. Transmission frequencies of 70 Gc {top) and 90 Gc (botiQfn) were used. The top pair of records was made in a helium discharge at a pressure of 80 microns. The bottom pair was made after 200 microns "I' argon had been added. 204 Microwave propagation experiments Chap. 6 6.3 Phase-shift measurements 205 19 1 H 11 11 Si 1 pw Im If? SI Kg H ■ it ■ m 391 FIG. 6.8c Microwave interferometer response to a transient plasma showing the effect of horn misalignment. The top record was made with the 70-Gc horns misaligned by 20" in a discharge chamber 8 centimeters in diameter. The 90-Gc horns (bottom record) were properly aligned. The effect of misalignment is shown in Fig. 6.8c to cause a distortion of the shape of the fringe envelope. Photographs of a 7^-band (25 Gc) interferometer and a 3-mm band (90 Gc) interferometer are shown in Figs. 6.9 and 6.10. Components arc arranged on panels for rack mounting. Coiled lengths of waveguide or flattened copper tubing9 are mounted behind the panel to compensate the length of the null path (or the plasma path, whichever is longer) to avoid differential phase changes if the klystron frequency drifts. 6.3.2 */ Modulation envelope. The interferometer circuit of Fig. 6-7 may be used with an rf modulated carrier. The klystron is amplitude modulated, at say 500 kc to 30 Mc, and the amplifiers following the various detectors are tuned to the modulation frequency. The system then resembles a superheterodyne receiver, with the i.f. frequency being that used to modulate the klystron; the name pseudosuperhet is sometimes applied to this system. The i.f. modulation envelope may be demodulated but, for frequencies below about 2 Mc, is better viewed directly on the oscilloscope. The transmission amplitude signal will appear as shown in 0 See Section 9.1. " Iff TTRFEROMETER FIG. 6.9. X-band (22 to 27 Gc) panel-mounted microwave interferometer for fringe-shift (zebra-stripe) presentation. (Photograph courtesy of the University of California Lawrence Radiation Laboratory, Livermore, Calif.) Fig. 6.11 (Buser and Buser, 1962). The interference fringes will appear as shown in Fig. 6.12. The Q of the tuned amplifier must not be too high if rapid fluctuations are to be followed. The amplifier band width must be at least 2jrm where rm is the time from crest to crest of an interferometer fringe. A useful preamplifier circuit is shown in Fig. 9.50. The no-plasma signal level is easy to see with the modulation present, even without d-c amplifiers; this makes a system that is easy to keep in adjustment. Tf the two arms of the interferometer bridge are approximately the same length, the incidental microwave frequency excursion of the klystron caused by the modulation will not cause problems with differential phase shift. The modulation envelope will be symmetrical about the base line if the amplifiers are linear. Stray pickup, which sometimes can shock-excite tuned circuits, wilf be asymmetrical and will cause the base line to bow; pickup is thus easy to distinguish from the real signal. 206 Microwave propagation experiments Chap. 6 6.3 Phase-shift measurements 207 FIG. 6.10 A 3-mm band (89 to 95 Gc) panel-mounted microwave interferometer. (Photograph courtesy of the University of California Lawrence Radiation Laboratory, Livermore, Calif.) 6.3.3 Fringe-shift or zebra-stripe interferometer. A more sophisticated interferometer is the "fringe-shift" interferometer (Wharton and Gardner, 1959; Hcald, 1959c). In this type of data presentation, the phase shift is plotted directly on the oscilloscope, and the effects of amplitude variations FIG. 6.11 An rf modulation envelope presentation of attenuation due to transmission through a plasma. The notches are time markers. (Photograph courtesy of R. Buser, Ft. Monmouth, N.J.) FIG. 6.12 The rf modulation envelope presentation of interferometer fringes for the same plasma event as that in Fig. 6.11. (Photograph courtesy of R. Buser, Ft. Monmouth, N.J.) Microwave source (swepl-freo,. klystron, etc) Repellet signal Ferrite isolator 10-dB Coupler JE*, Pad attenuator AAM Sawtooth generator (<1 (jsec) A l^A Pal! yVvV attenuator 00-360° Phase shifter Crystal video Video I detector proa m p. ]*«(Ž§j)_|_ (0-5 Mc) Video amplifier and dipper Coupler Video fringes "Wir „.. ... Clipped fringes •Tilllr Pad attenuator Z-axis input 10-Meter path Voltage isolator ^Voltage isolator (in waveguide) 10-Meter path Y-axis input Synchronization trigger from experiment to 0 oscilloscope FIG. 6.13 Fringe-sfiift or zebra-stripe microwave interferometer for plasma diagnostics. 208 Microwave propagation experiments Chap. 6 6.3 Phase-shift measurements 209 arc discriminated against. The circuit of Fig. 6.13 shows the reference or null path to be very much shorter than the plasma path, so that when the frequency of the klystron is swept back and forth, by varying the repeller voltage, the bridge will generate several maxima and minima, the number depending upon the difference in length of the two paths and the frequency excursion. The change in phase in a waveguide of length L, due to a small frequency excursion Aft is Ab). The radial extent of 4.1 cm (radius) was determined by pushing a movable probe into the chamber until it just began collecting current; data were normalized to that dimension. Results at five frequencies, in all, were recorded. Cutoff occurred at 17.6 Gc, indicating a peak density n0 of 3.8-1012 cm-3. The data presentation, showing the phase shifts for the four propagation frequencies, 18, 22.6, 24.6, and 33 Gc, appears in Fig. 6.22. Results are calculated from the phase-shift relation (4.2.2), derived in Section 4.2.2, A 0 = ' = —,A = -JN=).-'. cdo a These values are tabulated in Table 6.1 and plotted in Fig. 6.20. The plot of a trapezoidal distribution with b = dj6 fits within the experimental points, and probably represents a good approximation to the true distribution. A rough check with Langmuir probes showed the profiles to be flat-topped, giving further correlative evidence with the microwave data. Profile information from plasmas in a magnetic field may be obtained by methods discussed in Section 6.5.1, FIG. 6.21 Photograph of the Little Pig diagnostics correlation experiment. At (lie left is a 24 Gc (AT-band.) radiometer; right is a crystal video receiver for the interferometer. The vacuum flange on the left end carries one of the cathodes and an ion gauge. The front flange has a waveguide vacuum window, a voltage isolation section, and a Langmuir probe. The top flange contains a movable (Wilson) vacuum seal to permit the radiation-receiving antenna to be moved in and out of the plasma chamber. (Photograph courtesy of the University of California Lawrence Radiation Laboratory, Livermorc, Calif.) 216 Microwave propagation experiments Chap. 6 6.4 Density distribution: profile measurements 217 FIG. 6.22 Zebra-stripe data display of a transient plasma at four frequencies simultaneously. The frequencies are: 18 Gc, tower left; 22.6 Ge, upper left; 24.6 Gc, lower right; and 33 Gc, upper fight. At top right is the anode current trace, one ampere maximum. At lop left is the plasma radiation display, as received with a 24-Gc superheterodyne receiver. Time scale is 1 millisecond per major division. Phase-shift data at 1 = 3 msec is given in Table 6.1 and plotted in Fig, 6.20. 218 Microwave propagation experiments Chap. 6 Table 6.1 Tabulation of data from a plasma sample at five frequencies* 17.6 18.0 22.6 24.6 33 \ld 0.200 0.206 0.1 64 0.151 0.112 n0[nc 1.00 0.96 0.61 0.51 0.28 N 4.2 3.6 1.8 1.5 1.0 0.11 0.26 0.70 0.77 0.89 * Repealabilily of measuring N is ±i Iringe. The values of N arc taken from Tig. 6.22 at 1 = 3 milliseconds. Plasma diameter rf=S.l cm. FIG. 6.23 A seven-beam, 70-Gc focused probing system. Seven transmitting horns (/)), fed by dividing the power from a single klystron, are focused on the plasma by a lens (F). The seven receiving antennas (B) and split reference paths (C) feed signals to the waveguide junctions (D), which in turn supply the video detectors and preamplifiers (£). The on-axis vertical resolution of each beam is ~ 1 cm. (Photograph courtesy of R. 1. Primich, General Motors Defense Research laboratory, Santa Barbara, Calif.) 6.5 Magnetic field effects 219 Geometrical scanning can be used to obtain profiles if the microwave beam can be focused to a diameter considerably smaller than that of the plasma. Either Abel integration or trapezoidal approximations of the transverse variations in depth of plasma can be used. A multiradiator system, developed by Primich and Hayami (1963), provides focusing in the transverse plane by feeding a common 10-inch diameter, long focal length lens with seven horns. The resolution obtained on axis ( — 10 dB beamwidths) of each beam was 0.5 inch at 35 Gc and 0.25 inch at 70 Gc. A photograph of the apparatus is shown in Fig. 6.23. The multiple horns and the interferometer couplers appear at the right. The resolution is improved by having E-field polarizations of adjacent beams at 90° to each other. 6.5 Magnetic field effects When a magnetic field is present in the plasma, there are additional quantities to measure. When we propagate a wave across the field lines (# = 90°), we are able to couple to either the ordinary or the extraordinary wave. By using a square horn and waveguide and separating the two waveguide waves (E-lields at right angles with each other) with a polarization diplexer, we can transmit both waves simultaneously and make observations on a dual beam scope. When we propagate along the field lines (6 = 0°), we can measure the Faraday rotation (Section 1.4.2) either by rotating the receiving horn or, again, by using the fin-line coupler to compare the relative magnitudes of the two (.v and y) components. When the phase shift and Faraday rotation are combined, the bridge output shows amplitude fluctuations superimposed on the phase fringes. A 180° phase reversal also occurs with each half rotation. Losses in the plasma affect the two wave types differently, leading to ellipticity. An accurate measurement of Faraday rotation is thus difficult, since both the relative and absolute amplitudes of the .v and y components are changing. When one of the circular polariza-tions is attenuated to cutoff, the Faraday rotation ceases, and both the x and y components behave alike, since there is only one wave remaining. If circularly polarized antennas are used to study these waves, the two counterrotating waves can be studied independently. If the walls of the discharge chamber are close to the plasma, so that currents can flow on them, some ellipticity will result and true right-hand and left-hand waves do not exist. Also, when density and magnetic-lield gradients are present, the modes are no longer clean. We observed In Fig. 1.19 that at o> = a>v, when 0 is not quite 0° and the collision rate is very low (Section l?4.10), the left-hand wave jumps down to become the light-hand one (which is then cut off) and the right-hand wave jumps up 220 Microwave propagation experiments Chap. 6 out of cutoff to become the left-hand one. We must, therefore, use caution, as pointed out in Section 1.4.4, when labeling waves as "left-hand" and "right-hand." 6.5.1 Ordinary and extraordinary waves: density profiles. The ability to propagate two wave types at the same frequency over the same path allows us to obtain considerable additional data. For particular experimental conditions, the appropriate frequency (or frequencies) is obtained by reference to Figs. 1.2 and 1.13. The values of refractive index vs. plasma density and magnetic field for the ordinary and extraordinary waves are given in Figs. 1.6 and 1.15, respectively. These values, of course, apply only at one particular location within the plasma; if density and/or magnetic field gradients are present (as they invariably are in real experiments), the aggregate transmission properties must be obtained by the methods discussed in Sections 4.2, 4.3, and 4.4. In the large-plasma case, where the variations of density are gradual in the space of a wavelength in plasma, the adiabatic analysis applies to the total integrated transmission coefficient of the extraordinary wave as well as to the ordinary wave (Section 6.4). If the collision rate is low enough to be ignored, the phase shift for the extraordinary wave in a plasma path of extent d, may be approximated by A-. if* l4x)] dx where, from (1.4.50) r4- 1 —to/jc (6.5.1) (6.5.2) If, in addition, we are content to operate in the range in which w„jw < 1 and co„/ai< 1 — o>bjm. we may write the phase shift in the form of (4.2.2) in which the cutoff nc is no longer specified by w„2/ci>2 -> 1, but rather, on account of (1.4.52), by w^jw2^ \±o>bjw. The same data analysis as used for the ordinary waves in Section 6.4 then may be applied and profile information obtained to a fair approximation. A limitation is that the shapes of the index curves for ordinary and extraordinary waves are not exactly the same, the maximum discrepancy leading to errors in calculated density as large as 25% for waves at 90" to B, but no error for waves at 0" (along B) (see Section 1.4.9). The advantage of using the two polarizations is that two points on the -1 -2 A/27r (parallel) (/= 70 Gc) -3 FIG. 6.25 Phase-shift data plotted on theoretical curves for transmission through a plasma of trapezoidal density profile across a magnetic field. The values arc for phase shift of the extraordinary wave (perpendicular polarization) plotted against phase shift of the ordinary wave (parallel). Experimental points arc from a Stellarator discharge at Princeton University; w^o; = 1.25. (Courtesy of R. Motley and M. Heald, Princeton University, Princeton, N.J.) obtained from the B-] Stellarator (Coor et ah, 1958) at Princeton. The frequencies chosen were 35 Gc for the extraordinary wave (K J_ B), and 70 Gc for the ordinary (E || B), to optimize the sensitivity and measurable density range. The magnetic field intensity, and thus u>b, were known very accurately. The fringe-shift (zebra stripe) interferometer prcsenta- tions are shown in Fig. 6.24. The phase reversal as the refractive index goes through unity is apparent in the 35 Gc trace at 1200 ^sec. No phase reversal is seen in the 70 Gc trace, since the index for the ordinary wave is always less than unity. Plots of the calculated phase shift for the two waves at various values of hid, together with experimental points, are shown in Fig, 6.25. The points are seen to follow the plots of bjd—\ (uniform distribution) very closely. This is a reasonable conclusion for the Stellarator, since the plasma is defined by orifice plates at the diameter d to which these data were normalized. 6.5.2 Faraday rotation. When the ends of a magnetized plasma column are available so that antennas can be mounted with their radiation patterns looking along field lines as shown in Fig. 6.26, the circularly polarized waves can be studied, Eleetrodeless discharges (rf or pulsed) (Lisitano and Tutter, 1961) and plasma compression or confinement experiments (Consoli et al., 1961) usually lend themselves to such arrangements. Occasionally, discharge experiments are of a configuration that permits small radiators to be inserted directly in the discharge electrodes (Mahaffey, 1963) without upsetting the plasma uniformity. If no access from the ends can be arranged, it is sometimes possible to insert curved dielectric rod antennas from the sides, as shown in Fig. 6.27. In low-collision plasmas, having small electron orbit sizes, the rods cast a "shadow" along the field lines, but in dense plasmas, or plasmas whose ions or electrons have gyroradii larger than the probe diameter, the rods seem to give little perturbation. Rods made of boron nitride or glass-bonded mica have low loss and low vapor pressure, are highly directive, are refractory, and are easily machined. An example of the use of such rods with horn radiators is shown in Fig, 6.27. The propagation and radiation characteristics of dielectric rods are given in Section 9.3. To confine the radiation pattern to the desired volume, the curvature of the -d-c Magnet coils- Transmitter horn (^-oriented) X/2 Flat y quartz window X o o o o d D O f^^P^//; Plasma ^^^^^ o o V 0 o Fin-line ■""-^coupler y-Component detector :t-Component detector lrf Tank coil FIG. 6.26 Faraday rotation ex nc rime lit in rF-cxcitcd plasma in steady magnetic fluid. 224 Microwave propagation experiments Chap. 6 6.5 Magnetic field effects 225 //^Dielectric rod Wave-launching ■~ ' horn Dielectric rod— Vacuum r\7 chamber !-V 1 Vacuum window in waveguide F[G. 6.27 Curved dielectric rods used to launch waves along the magnetic Jield (or at an angle to it) in a plasma. bend must be gradual, and the diameter must be held large enough that most of the fields are within the rod until the taper at the end begins. Some photographs of curved dielectric radiators are shown in Fig. 9.37. Because the waves transmitted through the plasma seldom emerge with purely circular polarization, it is difficult to make quantitative measurements of Faraday rotation. Nevertheless, when differential attenuation or the two counterrotating waves is not excessive, a measurement of the total rotation gives information about the plasma density in a path along the experiment axis. If the magnetic field is known as a function of position, such as in low £ plasmas11 confined by external fields, the spatial distribution along the axis may be estimated by techniques similar to those described in Sections 4.2, 6.4, and 6.5.1. The total rotation is obtained by integrating (1.4.24). V={ J" ft/• 1 f (6.5.6) The Faraday rotation then becomes n(z) 1 nc \ + B(z)lB l «, 1- B(z)IB. (6.5.7) where nc is the density to give cutoff without a magnetic field and BR is the magnetic field to give gyroresonance. 11 ß, here, refers not to the phase constant, but is the ratio of the plasma kinetic pressure to magnetic pressure, ß = nkTj(B'2j2nB), a notation commonly used in con trolled-fusion research. 1.0 / Balmer series limit Phase shift Faraday rotation 1.5 2.0 2.5 3.0 Anode voltage of RF power oscillator, kv 3.5 FIG. 6.28 Electron density in rf discharge, measured by three methods, as functions of rf level. (Compiled from data presented in Lisitano and Tutter (1961) and von Gierke et al. (1961),) Equation (6.5.7) is difficult to integrate, but if B is constant or may be approximated by a stairstep function or a trapezoid, the same concept of an effective cutoff density as explained in Section 6.5.1 may be used for ü)„/ü)< 1 and a»6/«xL In addition, reference to Fig. 1.11 shows that over a considerable range of densities the linear approximation of (1,4.20) may be used. Using this approach in the data analysis of the experiment sketched in Fig. 6.26, Tutter (in von Gierke et al., 1961) obtained values of peak density and spatial distribution in reasonable agreement with optical and probe data (Schlüter, 1961) as well as with microwave phase-shift measurements across the plasma column (Lisitano and Tutter, 1961). Some compiled results are shown in Fig. 6.28. 6.5.3 Whistler mode propagation. When the transmission frequency is below but very near the cyclotron frequency and the collision rate is low, the plasma refractive index may be very high, especially if the density is high. The wave velocity and wavelength are then small, and the phase term ß is large (and also highly dispersive). The investigations (Storey, 1953) of very low frequency atmospherics (Helliwell et al., 1956) of descending pitch from -15,000 cps to 1000 cps led to the conclusion that waves generated by* lightning discharges were able to penetrate the ionosphere and travel along the earth's magnetic field lines, being ducted back 226 Microwave propagation experiments Chap. 6 6.5 Magnetic field effects 227 to earth in the opposite hemisphere. Because the higher-frequency waves travel faster (and, thus, arrive sooner), the descending pitch, of duration about 2 seconds, led to the name whistlers. The collision frequency is very low in the path of earth whistlers, and some of the attenuation is thought to be due to Landau damping (Scarf, 1962). In low-temperature laboratory plasmas, however, the collision rate is relatively high, and damping of the whistler mode is severe (Heald, 1960; Dellis and Weaver, 1962 and 1964), even at microwave frequencies. Slow-wave propagation at frequencies close to the electron gyrofrequency can occur in high-tcmperature, low-density plasmas, such as in magnetic-mirror compression experiments (Wharton, 1959). The difference between this low-density case and the true whistler case is that both left-hand and right-hand waves propagate (Ichtchenko, 1962), giving Faraday rotation, unless circularly polarized antennas in the desired sense are used. In the whistler mode the density is high enough that the left-hand circularly polarized wave is cut off (Fig. 1.25). An experiment at $ band (3000 Mc) (Gallet et al., 1960) in ZETA, a large toroidal high-density pinch experiment at Harwell, England (Thonemann et al., 1958), demonstrated that waves appear to be confined to channels or ducts, only a millimeter or so in diameter, if the launching probe is small. The geometry is sketched in Fig. 6.29. The data obtained from this experiment indicated a maximum refractive index of about 12, Transmitter probe line Receiver probe line FIG. 6.29 Whistler mode experiment in ZETA (Gilllel et al., I960). before the damping became excessive. Potentially, then, this mode of propagation could be used to trace out magnetic field lines in high-/3 plasmas and to indicate the presence of Alfvén waves and hydromagnetic instabilities by observing discontinuities in, and motions of, magnetic field lines in plasmas. A serious shortcoming of attempts to use whistler-mode propagation in laboratory plasmas is the difficulty of distinguishing between this transverse electromagnetic wave and longitudinal or mixed spacecharge modes; they all have some similarities in dispersion characteristics and may all be excited by a dipole antenna in or near a plasma. Further discussion of spacecharge waves in magnetized plasmas can be found in Sections 5.5 and 5.6. 6.5.4 Propagation at angle 8 to the magnetic field. Ionospheric observations of wave polarizations and dispersion as a function of reflection angle demonstrate that there is a measurable dependence on B. In laboratory plasmas, however, it is difficult to see the effect. The difficulty is due mainly to the presence of wall reflections and stray scattering masking the signals. Figures 1.19 and 1.20 show resonances occurring at densities and magnetic fields that are clearly functions of angle. Unfortunately, as pointed out previously, these resonating regions occur in the interior of the plasma, and are surrounded by lower-density, cutoff regions. In the ionosphere, where stray scattering is no problem and where mode conversion can generate propagating "bridges" across the cutoff region, the effects are clearly seen. A wave, by refraction in certain spatially varying magnetic fields, may be able to traverse a resonance region. For example, the propagation through a short magnetic mirror where both density and field gradients existed, was found, in unpublished work by Hill, Martin, and Wharton in 1957 at the Lawrence Radiation Laboratory, to have different resonance frequencies for 8 = 0°, 8x30°, and 8 = 90". The geometry is sketched in Fig. 6.30, where the dashed lines indicate the wave paths. The most interesting path is that at 30°. The wave enters the plasma at low density, but at 1 (see Figs. 1.25 to 1.28). Both the density and magnetic licld increase, at first, but then the field falls to the value at the center of the chamber. The angle B has been changing slightly along the path due to held curvature. At the center, 8 is the smallest although the density is (usually) high, so that resonance will be achieved here first as the density rises and last as the density falls. The paths at 9 = 0° and 90° yielded resonances at (di,/tu=l in their respective regions; the 8—30° path gave a resonance at a lower field strength. The foregoing experiment was far from a clean, quantitative one. 228 Microwave propagation experiments Chap. 6 Pulsed field coils 0° Microwave horn Magnetic field lines 90° Horn FIG. 6.30 Propagation at angle to the magnetic lield lines in a mirror-compression experiment. Nevertheless, the principles involved may be applied to certain classes of experiments where the parameters are under more precise control of the experimenter, in which case the results will be meaningful. Coupling to the ordinary resonance of Figs. 1.27 and 1.28 can occur only through wave coupling or by "tunneling" of evanescent waves through cutoff regions. The resonance in either case would be quite small and easily masked by other spurious fluctuations. If it could be detected, however, the measurement of its position in frequency would extend the measurable plasma density a factor of 4 or 5 at angles near 60°. 6.5.5 Doppler-shifted gyrofrequency in drifting plasmas. When the electrons in a plasma are drifting along magnetic field lines, the electron gyrofrequency is doppler-shifted in the laboratory frame of reference. A circularly polarized wave, directed along the field lines, will then have a different transmission coefficient in one direction than in the other, especially noticeable if the frequency is near the electron gyrofrequency. The coupling rods shown in Fig. 6.27 allow transmission from left to right or from right to left. If the plasma being studied is drifting in either direction, the measurement of transmission coefficient will be non-reciprocal. The amount of nonreciprocity will depend upon the derivative of the curve of refractive index vs. frequency,12 the maximum effect occurring at frequencies slightly below gyroresonance. The effect, which is somewhat analogous to the Fizeau effect, is not an easy one to measure in practice unless the magnetic field and frequency are 12 For example, see Fig. 1,9, obtained from (1.4.17). 6.6 Propagation through fluctuating plasmas 229 well known and constant during the measurement. The change in wavelength in the moving medium is AX= ± Xpv0jc (6.5.8) where A is wavelength, and \i is the refractive index, and v0 the drift velocity. The phase velocity in the moving frame is then — - + — — —• /x2 dX p, ju. dX c (6.5.9) u \sr aA (i /a ua c The phase shift observed in path length L from the left or right sense is thus 2nLfi_1 radians. (6.5.10) As an example, consider that the plasma electrons are drifting with 10 eV of directed velocity (r0 = 2 • 10s cm/sec). Near gyroresonance, A diijdA may be between 10 and 100, and p may be between 2 and 10, respectively. The differential phase shift for a given path length LjX may, thus, be between 0.07 (2ttL/A) and 23 (2ttL/A) for the example chosen. 6.6 Propagation through fluctuating plasmas A plasma whose electron density is fluctuating periodically may phase-modulate and amplitude-modulate an electromagnetic wave propagating through it. These effects are in addition to the Luxembourg effect (due to temperature fluctuations), discussed in Section 2.6. For example, an examination of the frequency spectrum of a signal transmitted through an rf-excited plasma invariably reveals side bands at frequencies displaced by the rf-excitation frequency and its harmonics (von Gierke et al., 1961). Figure 6.31 sketches such a spectrum, where fm is the rf driving frequency. The harmonics, if present, presumably arise from nonlinearities (Dreicer, 1961). /() - 2/m /o - fin ft ft +fm ft + 2/„ FIG. 6.31 Frequency spectrum of wave propagated through a plasma fluctuating at frequency fn> 230 Microwave propagation experiments Chap. 6 6.6 Propagation through fluctuating plasmas 231 Suppose that the electron density is fluctuating sinusoidally with a small amplitude a, at frequency o>mj2ir n(t)=n0(l+a cosojmt)=n0+n1(t), where ««1. (6.6.1) Ignoring damping, for the moment, the refractive index is then «i(0 = ^o2-^i2(0 (6.6.2) where the subscript 0 represents steady-state quantities and the 1 represents first-order perturbation quantities. If the frequency of fluctuation cura is slow, the wavelength of the disturbance in the plasma will be large, and the electromagnetic wave as a whole will be equally affected.1:5 There is not the limitation on modulation frequency here as we found in the Luxembourg effect, since here we do not have a relaxation time to consider. For I/j-!| smaller than about 0.2^n, the time-varying index may be approximated by expansion K0= W-i^\')YA -Mi - W(0/>02] ~ r*o(l - \a cosa>wf), (6.6.3) The perturbed electric field is then obtained from a perturbation solution of the wave equation, E(t) = E0 exp -J exp|y'cuu/ + a cos«j„/ (6.6.4) If a (or a,) is not small, these linearizations are not valid and the calculation must begin with substitution of n1 into the Maxwell equations, either in terms of a fluctuating conductivity or a fluctuating dielectric constant. The Poynting flux calculated from solutions of Maxwell's equations then will give the frequency spectrum and the component magnitudes. To find the frequency spectrum of (6.6.4), we must find its Fourier transform, E{o>). The transformation is of the form exp(/'.v cos0)= ^ Jnh(x) cos(/i0). (6.6.5) 13 If «, is large, so that the wavelength of the plasma disturbance is comparable to the electromagnetic wavelength and to the extent of the region being probed by the microwave beam, the present analysis will not be valid. The reader is referred to Sections 6.7.1 and 6.7.2 for the latter case. Expanding (6.6.4) according to (6.6.5) and taking the real part (since we are interested in the transmitted power) yields E(w) = E0 exp —j 7ttL.ii COSco0i -°){Jo(líf%) - Ji^p aj [sm(wQ + ajm)t + sin(«j0 - o»m)i] + Jzi^p aj [cos(w0 + 2wm)t - COs(a>a - 2wm)r] + h f—a} [sin(cu0 + 3cumV + sin(wc - 3wm)t] (6.6.6) Equation (6.6.6) is seen to have a frequency spectrum such as that sketched in Fig. 6.31, with the amplitudes of the side bands given by the Bessel coefficients. An examination of (6.6.6) shows that the fundamental frequency component vanishes if the argument of J0 is such that Jo-^0; that is, TrL/ioC/A equals 2.40; 5.52; 8.65 + mr. Likewise, the first side band vanishes for the Jt zeros, etc. In fact, however, since we have used a perturbation analysis, tacitly assuming that E remains essentially unchanged, the Bessel arguments must remain much smaller than unity. For larger arguments, the spectral appearance will be qualitatively similar but the relative magnitudes of the side bands will not be correctly given by (6.6.6). In warm plasmas a collisionless Luxembourg effect (see Section 2.6.3) may occur, when a wave suffering Landau damping propagates through a fluctuating plasma. If the fluctuations are in density, as in (6.6.1), we see from (3.5.6) that changes in tap influence the attenuation exponentially, leading to possibly large amplitude modulation for small density changes. If it is the slope of the velocity distribution, 8f0jdv, that is fluctuating, due to some nonthermal effect for example, the attenuation again is affected exponentially, as shown by (5.6.13). The amplitude modulation introduced by fluctuations in the collisionless damping produces sidebands similar to those shown in Fig. 6.31, although the spectrum may be asymmetrical, due to heavier damping at shorter wavelengths. The upper limit in modulation frequency, imposed by the relaxation time in collisional Luxembourg modulation, is not present in the collisionless case, since I here is no analogous relaxation phenomenon. A wave propagating through a fluctuating plasma then can suffer both amplitude and phase modulation simultaneously. 232 Microwave propagation experiments Chap. 6 Collisionless damping occurs for spacccharge waves as well, so that all of the above effects apply qualitatively to spacechargc wave propagation. One might expect, for example, to see an electron spacechargc wave modulated by a low frequency ion wave. If the fluctuations are at random frequencies (that is, noise fluctuations), the side bands will tend to run together into a continuous noise spectrum. The phase sense of the transmitted wave then becomes ambiguous. The ambiguity becomes worse for Hxlp-o increasing, which occurs as the density approaches cutoff. This effect often is called phase-sense scrambling, but should be distinguished from phase mixing, associated with Landau damping (Section 3.5). 6.7 Microwave scattering experiments The foregoing section has discussed a subject that could also be handled by treating the transmission as coherent forward scattering. Although scattering cross sections for free electrons are in general very small, in the case of a scattering angle 0=0 the phases of all the scattered wavelets are the same, and the properties of the scattered wave are identical to those discussed up to this point in terms of propagation through plasmas (Ratcliffe, 1959). For angles other than 8=0, however, we must specify the scattering cross sections and sum up all the wavelet components to find the scattered intensity in a given solid angle dQ. The total intensities are generally small, and stray reflections from walls and obstacles, as well as plasma radiation, may contribute as much (or more) power to the detector as the plasma scattering, unless great care is taken. Typically, the background must be down by at least 90 dB and often 120 dB to be able to detect the scattered signal with a good, broad-band superheterodyne receiver. If coherent detection is used,14 the background level can be considerably higher before the scattered signal becomes undetectable. With coherent detection, of course, the response time is necessarily long, which may be a disadvantage. 6.7.1 Incoherent scattering. particles, namely, electrons, Thomson cross section is The scattering from is called Thomson individual charged scattering. The total 9i = ^ro2 = o.6610-2S m2 (6.7.1) 14 The scatterer is modulated at a low frequency, for instance, 1000 cps, and a phase-sensitive detector compares the received signal with a sample of the modulating signal. Further elaboration is given in Section 9.5. 6.7 Microwave scattering experiments 233 FIG. 6.32 Coordinate system for microwave scattering. where r0 = e2/477£0m<-2 = 2.8-10 15 meter, the classical electron radius. The differential cross section is defined as dq energy scattered/unit solid angle-unit time incident energy flux/unit area-unit time = r02 sin2(9 (6.7.2) where & is the angle between the electron acceleration a and the direction of observation r, as shown in Fig. 6.32. In the absence of an external magnetic field (isotropic medium) the electron acceleration will be along E; in an anisotropic medium, this may not be the case. The angular distribution is given by (Jackson, 1962): sin2©=l-sin20cos2(-¥'). (6.7.3a) If the incident radiation is randomly polarized, the distribution is obtained by averaging over 9 sin20 = i(l+cos26»). (6.7.3b) The rate of incoherent reradiation (scattering) per unit solid angle, dQ, assuming that the electron displacement is small compared to wavelength, for s electrons and dQ solid angle is (Fejer, 1960) Psl = sl0 dqldQ = n VIQ dqjdQ = i«K/0r02sin2@ (6-7.4) where V is the scattering volume Ia is the incident intensity (watts/m2), /; is the election density in V. If the incident radiation is randomly polarized, the scattered power received is one half thai given in (6.7.4). 234 Microwave propagation experiments Chap. 6 As an example, consider the following case. n= 1012 electrons/cm3 f=lcm3 I0 = 25 watts/cm2 dQ— 1 steradian. The scattered power is 2-threshold at /=90 Gc. appears in Section 2.6.5. 10 12 watts, which is just above the detectable Further discussion of incoherent scattering 6.7.2 Scattering from plasma fluctuations of any wavelength. Two kinds of fluctuation scattering can be observed, both due to collective electron interactions. In both cases, the scattering cross sections are considerably enhanced over the Thomson cross section by the degree of coherency. For complete coherency, the scattered-signal intensity would be proportional to n2 instead of n. as shown in (6.7.4). The first type that we consider is incoherent backscattering; it has a cross section that is roughly proportional to the potential energy associated with plasma waves of any wavelength; the scattered frequency is doppler-broadened in proportion to the ion velocities (Rosenbluth and Rostoker, 1962). The second type (Drummond and Pines, 1961) also has a scattering cross section proportional to the potential energy of plasma waves, but is sensitive to angle and has scattered-frequency components in side bands spaced at multiples of the frequency of the plasma oscillations giving the scattering, similar to the case discussed in Section 6.6. In a given experiment, both kinds of scattering may occur at once, the scattered side bands being broadened by the doppler shift. The plasma fluctuations may be driven by an external frequency source, in which case the plasma column may oscillate in the dipolc resonant mode as well (sec Section 5.5). Or the oscillations may be due to an electrostatic instability, in which case the amplitude will be a function of position. Nonlinear effects can lead to the generation of harmonics of both the fundamental frequency and of the side bands. The scattered intensity is a function of angle and of frequency. The probing frequency to be scattered should be much higher than the frequency of plasma oscillations, perhaps an order of magnitude or so. For instability waves arising from electrons streaming through a plasma, the density fluctuations due to the plasma spacecharge waves are traveling essentially with the electron drift velocity, v0. The wave numbers (phase constants) of the incident wave, spacecharge wave, and scattered wave will 6.7 Microwave scattering experiments 235 'ft FIG. 6.33 Wave-number vectors for scattering from fluctuations in a plasma column. be related vectorially by the Bragg relationship, as sketched in Fig. 6.33 pHp^P, (6-7.5) where px = incident wave number, p2 = a>2\0jv0z — fluctuation wave number, p0 = final (scattered) wave number. For a wave vector incident at angle 6 to the plasma wave, the scattering angle is given by ft, m0—0l^=p\~$i cost? (6.7.6a) or, since ^IH^i! (6.7.6b) costft costf + sini/j sin(?=^ — cosfl. In a plasma that is only weakly unstable—that is, the e-folding growth coefficient is only slightly larger than the damping coefficient — the plasma wave amplitude will be essentially constant over several wavelengths of the electromagnetic scattering wave. The scattered waves will contain side bands, much as in the case described in Section 6.6, the frequencies being ojx and oj1±wm. If the plasma wave amplitude is large or the scattering is large, higher harmonics of wm will also be present. Plasma instability waves, obeying the Bohm dispersion relation, are well understood; the dispersion is expressed by ">m2 = <°22 = ">p2+/V 3/cre/«(e (6.7.7) The frequency of oscillation of such waves is near to,,, and lies between o^p and \fio>p, depending on the density distribution, temperature, and magnetic field. If the instability is due to counterstreaming (lows, such i t H a re * ■ n 5' i S ^ 3 0 o 1 § § I- rt — P- O » r» n c. FT I — c o d o cs g ct - ST p O Co p 2 SB •a o- P y: ft g „tJ § Sic » I& " » <» I 3 E a ET § <°' it c. 3 " ft 3 « 3 • *n P ö es! oq o — w ft E f> 2 £ 2 cr da O £3 O ca o in Q. sj 1+ cl ■ ft 1 - Si P "CO II g o "Cq S 3 1-1 H J1 sa sa " P. Q. II p' B" r~ GO 3 3 5 g >- d T To II -l II k> s oe *k 3 II Mc -•2d 33 3 to < O o o CD < he r 3 ™ P t3 las co 3 ~ ■JO p f- N — p pi < <« en -g — B' p ft ft - 3 > co CA p 3 c_ — ■ o a oq 3 -3 ft S- o — ft — 5' tm "2 0 t3 ft — ^+ ft" CA 3" ft P < ft OQ O Transmitter; 25 W. CW klystron Power-attenuater Plasma First AFC (£3- @Frequency meter Low-pass filter Second (AFC) local oscillator 1 Second AFC mixer f = 30 Mc 30 Mc AFC amplifier Rejection filter response Receiver band pass response fa h Scattering spectrum A.M. . 30 Mc IF amplifier Detector and video df= 10 Mc amplifier f=/o±30 Mc 30 Mc AFC discriminator AFC control on-off FIG. 6.34 Transmitter-receiver system used to study the scattering from an oscillating plasma. The receiver is tuned of the side bands and locked there by automatic frequency control (AFC), the difference frequency being determined second local oscillator. to one by the 238 Microwave propagation experiments Chap. 6 FIG. 6.35 A 35-Gc microwave scattering system for incoherent scattering from plasmas. The microwave source is a 25-watt c.w. klystron. The receiver is a sensitive superheterodyne, capable of either broad-band or coherent detection. The block diagram is shown in Fig. 6.34. (Photograph courtesy of General Atomic, San Diego, Calif.) The angle of scattering for the example assumed is obtained from (6.7.6), assuming |/33| = |&|. Let 0=45°. Equation (6.7.6) reduces to cos(<^-(7)=^-cosf? (6.7.6c) ■£-0 = 86° ■H131c 6.7 Microwave scattering experiments 239 A block diagram of a typical scattering experiment is shown in Fig. 6.34. A photograph of the 8-mm wavelength equipment used to observe scattering from plasma turbulence is shown in Fig. 6.35. 0.20 10 _L 0.6 0.8 1-0 Plasma density (u>p/y m 1.2 FIG. 6.36 Two-dimensional differential scattering cross section, per unit length and unit radius, as a function of (a) scattering angle and (h) plasma density for 90" scattering; 2na/X ='f. (From data compiled from Plat/man and Ozaki, I960, and Smythe, 1950). 240 Microwave propagation experiments Chap. 6 6.7.3 Scattering from small plasma columns. When the scattering plasma has a diameter comparable to a free-space wavelength, it may have a scattering cross section that is highly frequency-sensitive. Under certain circumstances, a resonance condition between plasma waves and external waves may occur, leading to a manyfoid enhancement of the scattering cross section (Herlofson, 1951). The resonance condition occurs for a plasma cylinder only when the incident electric vector is 'Will! Discharge / tube (a) 0.2 0,4 0.6 0.8 1.0 1.2 1.4 1.6 (b) FIG. 6.37 Microwave scattering from a resonant plasma column, (a) The small discharge tube mounted across an S-band waveguide. {/>) Relative power transmitted through the waveguide vs. normalized plasma current (Dattner, 1957). 6.7 Microwave scattering experiments 241 perpendicular to the axis of the cylinder z, in which case the electric field couples to the dipole resonance mode and the scattered magnetic vector H lies entirely in the z plane (Smythe, 1950; Boley, 1958). The scattering cross section as a function of angle between the incident wave vector and the plasma cylinder axis may also exhibit maxima and minima. Calculations for the differential cross section of a magnetized plasma cylinder (Platzman and Ozaki, 1960) for jS0a=l as a function of angle are shown in Fig. 6.36a. The cross section as a function of plasma density for "perpendicular resonance" a> = (cup2+ jkT)- 242 7.2 Strict blackbody radiation 243 where /; = /i/2tt, h = Planck's constant. This energy density represents an isotropic flux of electromagnetic energy flowing at velocity c. Thus, the radiation intensity (watts/m2) into the solid angle dQ is cVjjffilAtr) aw, and, 2BJu, J)=- 1 (7.2.2) An 47r3c2exp(//cU/A-T)-l the so-called Planck function, is the intensity per unit solid angle per unit radian frequency interval. In the Raylcigh-Jeans limit (/;to<, r)r- 4ttV (7.2.3) The prescript 2 denotes that two polarizations are included. The intensity in one of the two available polarizations is x»a=VBw (7.2.4) Meanwhile, the directive properties of a transmitting antenna (in the far field) are specified in terms of a gain function (7.2.5) where dxP is the power radiated into the solid angle dQ in the direction (8, <£) from the antenna and W is the total power radiated. The angles 9 and are those of a conventional spherical coordinate system with origin at some convenient point within the antenna and, as usual, <#2=sin0 dO d. By definition (JL) j G(8, *)dQ = j ~ 1. (7.2.6) By reciprocity, the angular response of a receiving antenna can be specified in terms of the same gain function. The effective area of a receiving antenna for wavelength A is given by (Schelkunoff and Friis, 1952, pp. 43-44 and Chapter 6) (7.2.7) S(8, lBl0S{B,)dQda> (7.2.8) 244 Microwave radiation from plasma Chap. 7 a well-known result, valid whenever a receiver is matched to a blackbody noise source (Nyquist, 1928; Knol, 1951).1 If the absolute power is measured with a receiver of known band width, the temperature of the emitter is readily obtained, according to (7.2.8). However, a number of conditions must be satisfied if this simple analysis is to be applicable. First, the emitted radiation must be in radiation equilibrium with the emitting medium; that is, the body must be truly "black" at the frequency band in question. To ensure this, the following conditions must be met. (/) The depth of the medium must be large with respect to the absorption length a'1. A partially transparent medium is often referred to as a "gray body," although in some usage this term also carries the implication that the optical thickness is independent of frequency. (2) The radiation must be able to escape freely from the medium and dissipate itself in the detector. If the nature of the medium is such that significant reflection occurs at the boundary, we may speak of a "silvery body." These considerations are implicit in the absorptivity or emissivity coefficient of Kirchhoff's law in radiation theory. In the case of a plasma for which the collision frequency is much less than the plasma frequency (v2«wp2), these conditions are often hard to meet. This question is considered further in Section 7.5. Second, there are conditions on the receiving antenna. With the notation /\ = distance from surface of blackbody to antenna A= linear dimension of the antenna aperture D = transverse dimension of the blackbody we have three additional requirements. (3) The antenna must be beyond the induction field of the blackbody2 /v>A. (7.2.9) (4) The blackbody must be in the far (Fraunhofer) field of the antenna R>A*jX. (7.2.10) 1 It is important to note the distinction between the specific intensity (per unit solid angle) of blackbody radiation (7.2.3), which is proportional to w2, and the power received by an antenna (7.2.S), which is independent of frequency because of the Aa term in (7.2.7). The fact that the gain function G is, in general, frequency dependent is of no consequence because of the normalization (7.2.6). 2 Under other conditions, we may wish to sample fields of modes that are trapped inside the plasma and have no radiation field (Dawson and Oberman, 1959). 7.3 Bremsstrahlung in a transparent medium 245 (5) The antenna must "see" only the blackbody (2XIA)R) exp( /W) d) - a(t) exp(-joiř) dt ff J - n f" {a(t)fdt = Tr f" \am{o>)\2dco (7.3.3) (7.3.4) (7.3.5) The total energy emitted in the encounter in the frequency band &a to ca + dit) is thus W„{<-<>) duj - e2p. TTjflj.fai)]2 da>. (7.3.6) For an electron in the coulomb field of an ion of charge Z, the acceleration is in general F(t) Ze2 1 "(0=: rn 4-iT0 lim KrlU) = (f V'" cxpi-x) 00. 97 ft \ LXj 248 Microwave radiation from plasma Chap. 7 Meanwhile, the number of electrons per unit volume having speed v is n,,f(v)4rTV2 dp; the total power emitted per unit volume at low frequencies is 2Pa(T) d 3V3 /(i?, 7") 4w dfe c/o) MaxwcllJEin distribution (2ir\«l e2 V n^Z2 [ V3 frm„x I 3 / V4#e^ ^ bmln. (7.3.12) 7.3.2 The Gaunt factor. The calculation of free-electron bremsstrahlung was originally undertaken by Kramers (1923), using a classical analysis similar to the above. However, he was principally concerned with higher frequencies, the dominant contribution to which comes from close encounters (large electron deflections). In a limit of moderate frequencies and low initial electron velocities (parabolic orbits), one obtains results identical to (7.3.11) and (7.3.12), except for the absence of the quantity in square brackets. This is a "white" spectrum, independent of frequency. It is possible, then, for arbitrary frequency and velocity (temperature), to write the results of any calculation in the forms: Power radiated per electron of velocity v: 2PJ, D *>J§ (ffyt$ JgfL > T) do, (7.3. .4) The refractive index ^ represents the effect of coherent motions of neighboring electrons (Westfold, 1950). Although introduced by Kramers (1923), the correction factors S and § are known as Gaunt factors (Gaunt, 1930). Kramers obtained, on the basis of calculations using the exact (7.3.7) in place of approximations such as (7.3.8) (Landau and Lifshitz, 1962, §70), ->^In(^U^ln(^-) (7.3.16) •' We here distinguish between electron and ion densities, instead of making the usual simplification n = ne = Zii:, in order to permit calculation of partial contributions to the total radiation when several species of ions of various charge numbers Z are present. 7.3 Bremsstrahlung in a transparent medium 249 where H(,i) is the first Hankel function (of imaginary order and argument), bB0=Ze2l47T€0mv2 is the impact parameter for a 90° deflection, and y = expC= 1.781 is Euler's constant. The integration over velocity for a Maxwellian distribution yields in the important low-frequency case (Oster, 1961b; Scheuer, 1960) m, y)—,-xV; (2r /'•-1 V3 (7.3.17) 1000 Wave frequency osfeirZ* |Gcl FIG. 7.1 The high-frequency (unshielded), velocity-averaged Gaunt factor §, for ions of charge Z. T>he solid curves include the temperature-dependent quantum correction of lig. 7.2. The dashed lines are the asymptotic forms (7.3.17) and (7,3.23). The shielding correction must be obtained from Fig. 7.2. 250 Microwave radiation from plasma Chap. 7 where vik — (/c 77 m)1/2, 590 =Ze2j4-n-e03kT, and the parameter A0, already defined in (2.5.30), is v8 2\V. 477e0(AT)% 4 £7 = 6.2-104 (yc7"[eV])% 4 A7_' / AT \- Z cu/2tt[Gc] (7.3.18) Numerical values of 80 are given in Fig. 7.1. It is to be noted that (7.3.15) to (7.3.17) follow directly from the exact classical dynamics without the necessity for imposing physical cutoffs on the impact parameter. A necessary criterion for this low-frequency limiting case is essentially 40» l (7.3.19) In laboratory plasmas, this inequality is usually well satisfied. The limitations of the analysis leading to (7.3.15) to (7.3.17) are: (/) classical mechanics, neglecting quantum and relativistic effects; (2) consideration of binary encounters only, neglecting shielding effects of other particles except as represented in the refractive index /a; (3) radiated energy small, such that the orbit is not perturbed; and (4) low frequencies, sufficient to make the argument of the logarithm very large. Results of more accurate calculations or other limiting cases are conventionally expressed in the form (7.3.13) and (7.3.14) with suitably modified Gaunt factors (Brussard and van de Hulst, 1962). The Q's are rather remarkably close to unity over a wide range of physical conditions, the most significant departure being at low frequencies. 7.3.3 Quantum considerations. The application of quantum mechanics immediately dictates, on energy grounds, that only those frequencies will be emitted for which na> <-\mv2~kT (v — initial electron velocity), except for the contribution of electrons captured into a bound (negative energy) atomic state. Thus, we can make the qualitative statement that the spectrum (7.3.14) is substantially independent of frequency, except for logarithmic dependence in S, up to m~kTjli, whereupon it falls to zero in a manner the details of which depend upon specific atomic processes (Kramers, 1923). This consideration implies that the velocity-averaged formula (7.3.14) may be improved by performing the integration from a minimum velocity c,„in, such that \mv2nin — />io, rather than from zero. 7.3 Bremsstrahtung in a transparent medium 251 Since the velocity integration was of the form [neglecting here the logarithmic velocity dependence of the Gaunt factor 8(i>)] 2A exp( — Ar2) v dv=l, Jo the modified integral is 2A exp(-Ar2) vdr = 2X cxp(- Ar2) v dv-2X exp(-Xv2)vdv Jvnin 49 S™ = l-[l-exp(-Aa.//c7}] = exp(-/2w/A-7j. (7.3.20) That is, the general expression (7.3.14) is to be multiplied by exp( —/ku/A'71) [note also (7.4.28)]. It is customary to write this factor explicitly in (7.3.14), rather than including it within the Gaunt factor S(oi, T). We anticipate, then, that the remaining S is rather insensitive to frequency, even when quantum-mechanical effects are considered, so long as {to>«kT. Quantitatively, this limit is ^[Gc]«2.4-105/t7TeV]. (7.3.21) This inequality is usually well satisfied for microwave frequencies in laboratory plasmas. Parenthetically, for the case of high frequencies lio)>kT, the electron orbits are highly perturbed and the contribution of transitions to bound states is important, so that a classical analysis is inadequate. One can obtain the bremsstrahlung cross sections from the exact (nonrelativistic) theory of Sommerfeld or the more tractable Born-Elwert approximation (Sommerfeld, 1951; Elwert, 1939). Brussard and van de Hulst (1962) argue that the effect of free-bound transitions can be approximated by omitting the exponential factor while keeping the 3 calculated for free-free transitions alone. Extensive numerical computations have been made for the high-frequency case (Greene, 1959; Karzas and Latter, 1961). The relativistic case is generally not of interest in laboratory plasmas (Heitler, 1954, §25; Koch and Motz, 1959). Meanwhile, when one treats the dynamics of the collision process itself by quantum mechanics, in analogy to Section 7.3.1, one obtains in the low-frequency limit (Gaunt, 1930) 4 $(a>, V) «1»« it ib» n l not J (7.3.22) 252 Microwave radiation from plasma Chap. 7 This result is most readily obtained by a calculation using the Born approximation, which is valid in the limit of high temperatures and low frequencies (Sauter, 1933). Comparison with (7.3.16) indicates that the impact parameter b90 has been replaced by A/y where K=ftjmv is the reduced de Broglic wavelength. Velocity averaging of (7.3.22) gives 5(a>, t) /if.) « IcT TT V3 /4m (7.3.23) The quantum mechanical result can be expected to apply when K>ba0 (a relation depending only on temperature and atomic number), since then the deflection caused by diffraction of the electron wave is greater than that of the classical collision process (Marshak, 1940). The crossover from the classical to quantum form can be estimated from the condition for equality between (7.3.17) and (7.3.23); we obtain kTxy3Z2Ry = (17 eV)Z2 = (890,000 °K)Z2, (7.3.24) where #,,= 13.6eV is the Rydbcrg energy constant. Note that the criterion may be stated in terms of the relation between kT (~electron energy) and the ionization potential for the (fully stripped) ion. Note further that the criterion (7.3.24) for the applicability of classical vs. quantum mechanics is of a different form from the Raylcigh-Jeans criterion (7.3.21). Thus, a purely classical analysis is justified only for temperatures such that /to«/rr«y3Z2/<1/. (7.3.25) 7.3.4 Electron shielding. A second important correction to the Kramers theory concerns the proximity of other particles when the plasma is not infinitely dilute. At frequencies of the order of o>P and below, the electron cloud is able to adjust itself so as to shield the scattering ion (Chang, 1962a). The radiating electron no longer "sees" ions farther away than the debye length AD. Qualitatively, we may say that the radiation contributions from individual scattering ions arc no longer independent and uncorrclated. The result of this effect appears only in the argument of the logarithm of (7.3.17), which is then multiplied by a factor of approximately wjo>p at frequencies less than ui„ (Burkhardt, Elwert, and Unsold, 1948; DeWitt, 1958; Ostcr, 1964). Consideration of shielding requires that we amend the use of the term "low frequency'' as used in the important limiting case given by (7.3.17). To avoid shielding but satisfy u>«.vtJljbno, \«a>!to„«A!iv = \DlbBD. (7.3.26) As noted in Section 2.5.2, Spitzer's ratio ASp is approximately the number 7.3 Bremsstrahlung in a transparent medium 253 of electrons in a sphere of radius equal to the debye length, and is normally quite large (~104) in common laboratory situations. 7.3.5 The Gaunt factor and \nA. The simple bremsstrahlung theory given in Section 7.3.1 led to results, (7.3.11) and (7.3.12), containing a term of the form \n(bmaxlbmin). A term of this same form arose in the theory of the electrical conductivity of an electron-ion plasma discussed in Section 2.5, where the generic notation InA was used. These two theories have rather different points of view. Bremsstrahlung is concerned with incoherent radiation by a (thermal) electron, accelerated in the field of an ion. The conductivity theory is concerned with the loss of directed (wave-induced) momentum by the electron, in being deflected by the ion. However, the two effects are very closely related (Theimer, 1963). The quantitative connection, invoking the principle of detailed balance, is made in Section 7.4.4. Comparing (7.3.11) and (7.3.12) with (7.3.13) and (7.3.14), we may make the identification (7.3.27) where the Gaunt factors and the ln/1 term may be thought of as equivalent -0.2 I £.-0.4 3 KB -0.6 -0.8 1 0.01 10 100 Electron temperature k'f/Z2 [eV] 0.1 1 Electron density n/nc = (cop/co)2 1000 10 FIG. 7.2 Corrections to the unshielded, classical Gaunt factor \S0 of (7.3.17) to lake account of quantum and shielding effects. The two corrections are independent and arc to be applied simultaneously. This figure represents the same data as Fig. 2.9. 254 Microwave radiation from plasma Chap. 7 7.3 Bremsstrahlung in a transparent medium 255 correction factors (slowly varying functions of plasma parameters) arising in the bremsstrahlung and conductivity theories, respectively. In Section 2.5.4, we discussed at length the form of the impact parameter ratio A = bmaxjbmin appropriate for various regimes of electron density and temperature. The same arguments and calculations apply here. For convenience, Fig. 7.2 reproduces Fig. 2.9 (Greene, 1959; Dawson and Oberman, 1962), labeling the scales as corrections to the high-frequency, low-temperature Gaunt factor (7.3.17). The two corrections are independent of each other in the Rayleigh-Jeans limit (/icu«AT) (Ostcr, 1963a). The analytic form of the Gaunt factor in various limiting cases of electron density and temperature may be obtained from Table 2.3, using (7.3.27). 7.3.6 Summary of microwave bremsstrahlung. To summarize the theoretical results for bremsstrahlung emission, we iirst recognize that three independent variables are involved: frequency, electron density (plasma frequency), and electron temperature. These may most conveniently be packaged in the following normalized parameters: hm\kT, which measures the importance of bound atomic states and the validity of the Rayleigh-Jeans approximation; co„/cu, which measures the importance of shielding; and kTjRv, which determines the applicability of classical vs. quantum mechanics.13 For microwave frequencies, where hmax = kTjh to obtain where S(D~1 is the Gaunt factor averaged over both electron velocity and frequency (Greene, 1959). The plasma is assumed transparent throughout, ignoring self-absorption and nonpropagation at low frequencies. For interesting temperatures, frequencies well above the microwave region contribute most of the radiated power. For moderate temperatures where the high-Z impurities are only partially ionized (Z-fold, for instance) they contribute approximately as Z2 to the total bremsstrahlung but only as X2 to the microwave bremsstrahlung. It follows, then, that the total power loss is sensitive to relatively small concentrations of high-Z impurities in a low-Z plasma. Furthermore, the presence of bound electrons permits enhanced radiation at the frequencies of the positive-ion's discrete spectrum, which may greatly increase the total radiation power loss but does not greatly affect the microwave radiation (Post, 1961). Likewise, the presence of a magnetic field introduces cyclotron radiation or "magnetic bremsstrahlung," which can be the dominant radiative loss mechanism for a hot plasma. This point is discussed further in Section 7.6. 7.4 Radiation transport and the gray body 257 7.4 Radiation transport and the gray body To handle situations between the two limits of blackbody radiation and transparent-medium bremsstrahlung, we must consider the effects of stimulated absorption and emission. Furthermore, we do not wish to be restricted to only the limiting cases of isotropic radiation and infinite plane waves, or to homogeneous media. We shall, however, consider in detail only those situations which can be satisfactorily described by ray concepts rather than detailed diffraction analysis. 7.4.1 Energy flow in an inhomogeneous medium. Let rj$- ) doj dQ (7.4.1) be the radiation power per unit perpendicular area flowing into the element of solid angle dQ in the (f?, ) direction, in the angular frequency band 1. However, relation (7.4.3) is no longer valid for a plane wave. " Many texts use the notation }..> = pJ4tt for Hie emission power per unit volume per unit solid angle. See Section 7.6. 258 Microwave radiation from plasma Chap. 7 it I FIG. 7.4 Refraction of a pencil at an interface. amplitude attenuation coefficient is a(s), the change in intensity along a ray path from all effects (except reflection at a discontinuity) is dfy_8Ia 2Pa Using (7.4.4), we obtain finally It is convenient to introduce a new distance coordinate, the optical depth {"observer t(s0)=\ 2 Wi). Emission of a photon of energy /ico= W2 — Wx is associated with a transition from level 2 to level 1, where ň = hj2n = 1.05-10~34 joule-sec is the reduced Planck's constant. The probability per unit time per unit solid angle that a system in level 2 will undergo a spontaneous transition to level 1 is denoted by A21. If the number of systems per unit volume in level 2 is n2, the total radiated power per unit volume is 4Trn2Azlfico; (7.4.10) this is the emission considered in Section 7.3.9 However, the presence of radiation of frequency a> will cause induced or stimulated emission and absorption. We assume that the probabilities of these induced processes are proportional to the intensity of the ambient radiation at the proper frequency. The probabilities (per unit time-volume-solid angle) for induced emission and absorption may then be written B2ila and B12Ia, respectively. The A and B coefficients represent the detailed interaction processes of the physical system with the radiation field and have the same value whether or not there is thermal, or even kinetic, equilibrium. However, for the special case of matter and radiation in thermodynamic equilibrium, we can make the following assumptions. (/) The relative population of the two states is given by the Maxwell-Boltzmann distribution n1 G1cxp(—W1jkT) G1 where G\ and C2 are the statistical weights or degeneracies of the respective energy levels, relevant only when dealing with bound atomic states. " In the ease of the continuous energy levels of interest in free-free transitions, we consider frequencies in the hand <■> to o> + dta. The coefficients A and li then contain implicitly the frequency interval <7.4.12) are substituted into (7.4.13), we obtain A R fli)> which may be described by a negative temperature in (7.4.11), This is the basic principle of the mascr and laser. 7.4 Radiation transport and the gray body 261 7,4.3 The partially transparent plasma. We wish first to relate phenom-cnological Einstein A and B coefficients to the specific processes for a highly ionized gas. In Section 7.3, we calculated the emission power density from bremsstrahlung; (7.3.13) and (7.4.10) may be identified as the same quantity, «2 -/'... i/w":4vr/;t-;«,T..,. (7,4.17) The summation over all available initial (upper) states corresponds to integrating over the Maxwellian electron velocity distribution; we obtain 2Pwdm=AwhoynAzl (7.4.18) where A.2^^\ A21f(\) d3v. Meanwhile, under conditions where the spontaneous emission may be neglected (for example, an externally generated signal propagated through a relatively transparent plasma), the observable amplitude attenuation constant a, defined by (see footnote 7) dlf, ds = - 2ala, (7.4.19) is related to the Einstein B coefficients by11 2a.la dm = &mh j [Bi2 exp(^) - diy = ham I"1 A 2 2D ^21' ** to (7.4.20) Canceling out la and using (7.4.18), we find that the ergiebigkeit parameter appearing in (7.4.8) and (7.4.9), has the value12 (7.4.21) the Planck function (7.2.2), Thus, under all conditions in which the matter is in local kinetic equilibrium (not necessarily in equilibrium with the radiation), (7.4.9) can be written as (M ={fil exp(-T)+f sBa[T(r)] exp(-r) rfr. (7.4.22) 11 The velocity integrations in (7,4.18) and (7.4.20) are both written in terms of the upper-stale dcnsil.y*ha, so that they arc fully comparable. ,B Closer examination reveals that litis general relation is only valid for relatively low-loss media, such lliul )c4«lkT) term in (7.4.20), from (7.4.11), is not valid, and the value of the ergiebigkeit parameter must be calculated explicitly. For instance, it may be evaluated for a Druyvesteyn distribution (Davies and Cowcher, 1955). As an application of (7.4.22), consider a sample of uniform temperature T±, but unspecified geometry. Tf the sample is illuminated from the rear with blackbody radiation of temperature T2, the observable radiation emerging from the sample will be independent of the optical thickness of the plasma when T1=T2. This feature has been exploited as a diagnostic technique under conditions where the density profile of a plasma is unknown and not easily measurable (Bekefi and Brown, 1961b). It is equivalent to the classical line-reversal technique of optical pyrometry. 7.4.4 Correlation of emission and conductivity theories. We may now compare the value of A31 computed by alternative theoretical models for the specific case of a highly ionized gas (coulomb collisions only) (Martyn, 1948; Westfold, 1950; Yamada, 1962; and Theimer, 1963). From the Kramers free-free bremsstrahlung theory of Section 7.3. using (7.3.14) and (7.4.18), we obtain — \3tt/ \47T£0c/ nZ/jcf mA{kTy*li, mv, T), for this case, must usually be obtained by numerical calculations (Greene, 1959; Karzas and Latter, 1961; and Brussard and van de Hulst, 1962). 7.5 Radiation from a slab and Kirchhoff's law As an introduction to the thermal radiation transport problem, we restrict ourselves to the case of a dilute plasma having the propagation properties of free space; that is, we assume cn2»wr,2>u2. FIG. 7.5 Geometry of slab radiation. 264 Microwave radiation from plasma Chap. 7 7.5.1 Effect of antenna gain. Consider an infinite homogeneous slab of plasma of thickness d and uniform kinetic temperature (Allen and Hindmarsh, 1955). We wish to calculate the intensity of thermal radiation received by an external antenna. The radiation emitted in a unit volume per unit solid angle by spontaneous transitions is 2pa(o), 7) d, (7.5.2) where 1B0y = ^Bo is the polarized Planck function. (7.2.2). The antenna distance R drops out, although we must assume that the plasma lies in. the 7.5 Radiation from a slab and Kirchhoff's law 265 far field of the antenna (R>Sma>_.l\). To proceed further, analytically, we must make some assumption as to the antenna pattern, the effective area S(d, 4) being proportional to the gain function G(8, jkT) (7.5.3) where the emissivity e is a function of the optical thickness parameter 2ad, and the antenna pattern G{8, = 42ad, G) kT (7.5.4) which may be compared with (7,2.8). For most practical purposes, the dependence on antenna properties is small. The thickness parameter 2ad 13 The cosine case gives the conventional emissivity (or the total intensity (watts/m2) radiated into the hemispheric solid angle on one side of the slab. 0.1 0.3 Slab thickness 2ad FIG. 7.6 Emissivity of dilute slab for the antenna types of Tabic 7.1; « = amplitude absorption coefficient, d = slab thickness; w' » wpa > v*, Table 7.1 Emissivity e of a dilute plasma slab for representative antenna types Antenna Hemispheric isotropic G0 = 3 dB Cosine pattern C0 = 6d.B High gain Ga> 10 dB Gain, G0, ) I *4 4 cose 0 V 2 8(0)/sin0 (Dirac delta ftinciton) t(x — lad) 1 — exp(-Jt) — x Ei(—x) l-(l-x)exp(~x) + x2 Ei(-x) I - exp( - x) 266 Microwave radiation from plasma Chap. 7 can be easily measured in an auxiliary transmission experiment. Since a cc 1/to2 for low frequencies, according to (7.4.24), there will exist a critical frequency a>0 for which 2 a(co = tu0, p = l) d= 1; that is, <3Znd (2tt)^ = 1.76-10 mv*c(kTf* V3 „1S SZ ři[cm~a] d[cm] (AT[eV])K 0 tupí/ 4s* 2irc (7.5.5) where 4^ = AD/AB0 is Spitzer's coulomb cutoff parameter (2.5.18). Below oi,) the slab radiation approaches blackbody and above, transparent-medium Cillie radiation. However, it must be recalled that the above analysis, neglecting refractive effects, is valid only for frequencies well above the plasma frequency. The criterion that this critical frequency be above the plasma frequency is OU \ 0. For a low-loss (v2«uiB2), moderately thick (d»cjlv) plasma, we have lad U) < CO, a) > CJ~ (2coJ3(//c»l) where the refractive index is p.= [1 -(<*JoffT* and the critical frequency oj0(d, u>p, T) is defined by (7.5,5). The thermal radiation spectrum of a plasma slab may be presented in two formats, shown in Fig. 7.7. The 10 This approximation is valid for a plasma which is neatly optically thick, d> 1/2«, or when a wide enough frequency band Ato is accepted to average out the resonances, d>2ircj)t Aai. It may also be reasonable when the change in propagation characteristics at the boundary, is "smeared out" or "bloomed" over an appreciable fraction of a wavelength, instead of being perfectly sharp, but then r is considerably reduced. Otherwise, our result will give only an average result which suppresses resonances (Uekeli and llniwn, 1961a). 1.0 0.5 0 1 y- Blackbody radiation V - N\0.5 \l.O :\l-5 Atop/too = 2.0 =:s=Si^-/^^----__^ Cillie radiation 1 1.0 2.5 3.0 1.5 2.0 Frequency (j/cjo m FIG. 7.7 Emission spectrum of plasma slab of low loss (V2 « wp2) and of thickness d » c/ Lit (7.5.14) In both cases, the emissivity e measures the fraction of the corresponding blackbody radiation level emitted. By way of contrast, it may be assumed that surface reflection is negligible (that is, the gradual boundary of Section 4.2). Then the emissivity is the high-gain case of Table 7.1 e= 1 — exp( — 2ad). This condition is also shown in Fig. 7.7. (7.5.15) 7.5.3 Kirchhoffs law. The experimentally measurable quantities for a slab are the (power) transmission and reflection coefficients, T and R, for an externally generated test wave. With the same assumptions made above as to incoherency of the internally reflected waves (see footnote 15), we obtain (Section 4.3) T (I — r)2 exp( — lad) 1-r2 exp(-4W) j r[l +(1 -2r) exp(-4«rf)] 1 — exp(-4aa') The corresponding absorption coefficient or absorptivity A is A=\-T-R= (l-/Q[l-exp(-2c^)] 1 — /• exp( — lad) (7.5.16) (7.5.17) (7.5.18) This identification of the absorptivity (of a test wave directed from observer to the plasma sample) with the emissivity (for thermal radiation from the sample to observer), which we have here demonstrated in a highly idealized case, is in fact a general principle, known as Kirchhoff's radiation law (Planck, 1914). We can argue, thermodynamically, that the fraction of a test wave directed at the plasma that is absorbed in the sample is equal to the emissivity, independent of details of propagation characteristics and plasma geometry, so long as the propagation characteristics are fully reciprocal. The formal statement may be made by identifying the emissivity with the quantity j Re(J. lv'-) d V J Re(E0xH0*)rf5 (7.5.19) where V and S are the volume and projected area of the sample, J and E are the a-c current and electric field existing inside the sample, and E0 and H0 are the test wave fields in the absence of the sample (Levin, 1957; Bekefi, Hirshfield, and Brown, 1959; and Bekefi and Brown, 1961a). Note that J may be related to E by the well-established conductivity theory. This formulation holds generally for homogeneous isotropic samples; caution must be used in extending it further. An important feature is that it is valid for boundary-value problems, so that we no longer need demand that ray optics be applicable. Thus, experimentally, we can establish a power balance whereby we measure R and T directly, in an auxiliary calibration experiment, and infer e = A.= 1 -R-T therefrom. While this technique automatically accounts for internal reflections and diffraction effects within the plasma, in practical situations other than infinite plane slabs it is often difficult to measure dependably all nonabsorbed power, because of scattering, refraction, and diffraction. In many cases where the interface reflection coefficient /■ and the optical thickness lad are simultaneously large, as for instance for oxwv, it is possible to approximate the emissivity by (Bekefi, Hirshfield, and Brown, 1959; Hirshfield and Brown, 1961) €Sl-r«l. (7.5.20) However, we note that under these conditions a small experimental uncertainty in r makes a much larger uncertainty in e. Also, since the received radiation comes only from the surface layer of the plasma sample, it may not be at all representative of the physical state of the interior. Wort (1964) has given a simple model for computing the emissivity of a turbulent plasma. In the presence of a magnetic field, which we have thus far ignored, the anisotropic propagation characteristics and opportunity for mode coupling and nonreciprocity require caution in the use of Kirchhoff's law (Martyn, 1948; Rytov, 1953; Bunkin, 1957; and Hirshfield and Brown, 1961). In terms of microscopic processes, in addition to ordinary bremsstrahlung from electron-ion encounters, we also have cyclotron radiation, which we discuss in Section 7.6. In the case of non-Maxwellian electron velocity distributions, the"*KiivhholT radial ion law must be suitably reinterpreted (Bekefi, Hirshfield, and limwn, 1961a; fields, Bekefi, and Brown, 1963). 272 Microwave radiation from plasma Chap. 7 7.6 Cyclotron radiation 273 7.6 Cyclotron radiation When a plasma is immersed in a steady magnetic field, the individual charged particles execute orbits which are in general helical. The particles are accelerated and radiate electromagnetic energy at the cyclotron frequency. Relativistic effects cause radiation at harmonics of the cyclotron frequency; this extension is often called synchrotron radiation (Schwinger, 1949; di Francia, 1959). The total cyclotron radiation is also known as magnetic bremsstrahlung (Trubnikov and Kudryavtsev, 1958). We note that ordinary bremsstrahlung occurs only during collisions, whereas cyclotron radiation occurs during the time interval between collisions. 7.6.1 Total radiation. The relativistic equivalent to (7.3.1) for the power radiated by an accelerated charge is (Panofsky and Phillips, 1962) where fS = vjc and the index of refraction of the surrounding medium is assumed to be unity. The acceleration of an electron resulting from the Lorentz (magnetic) force is e|vxB| evxB a = - where eB is the cyclotron frequency defined for the electron rest mass m0. total rate of radiation for one electron is then e^oi^v^ (7.6.2) (7.6.3) The " ~67r£oc3(i-/J2) (7'6'4) If the electron velocity distribution is isotropic and Maxwellian, the total power radiated per unit volume is 2 ,„ * neW (kT\ I, , 5 IkT\ 1 The prescript 2 signifies that the receiving antenna is assumed unpolarized. The total power from ordinary and magnetic bremsstrahlung may be compared; from (7.3.29) and the leading term of (7.6.5), where Ry, the Rydberg energy constant, equals 13.6 eV and H~l is the averaged Gaunt factor. In many practical situations the magnetic liclcl (ccuj„) and the particle pressure (~nkTocoj,,2kT) are correlated by confine- ment considerations such that oip2jo>b2 ub cost) The full width at half maximum is Otu = -— COStf ciih I mcJ (7.6.13) Doppler broadening dominates that due to electron-ion collisions when (7.6.14) (fcr[eV])a B[kG] cosfl^ Zn[cm-3] ~ " Doppler broadening, when obtained from the macroscopic absorption coefficient via Kirchhoff's law, is the result of the noncollisional (Landau) absorption process of Section 3.5 (Hirshfield and Brown, 1961; Rukhadze and Silin, 1962). Combined doppler and collisional broadening leads to a Voight line shape (Aller, 1953; Posener, 1959). (4) When the magnetic field is not uniform, the line is inhomogeneity-broadened. (J) Dense, high-temperature plasmas (high j3, in the language of controlled fusion research) can produce cyclotron radiation at frequencies somewhat below the expected frequency because of diamagnetism. The field depression, which may be quite large, generally is not uniform in space and time, so that this diamagnetic shifting and broadening is large and variable. The effect can be used to measure nkT and, since n is measurable independently, leads to a determination of T. 7,6.4 Radiation by a single relativistic electron. If the electron velocity is relativistic, the instantaneous radiation intensity is peaked in the direction of the electron's motion (Panofsky and Phillips, 1962). Qualitatively, this "headlight effect" concentrates the radiation in the direction perpendicular to the field and produces harmonics of the fundamental gyration frequency. Quantitatively, if the electron has no motion along the field, one obtains for the power radiated into solid angle dQ by a single electron of velocity ft=vie (Schott, 1912; Schwinger, 1949; and Landau and Lifshitz, 1962, §74) $ = sß sin 9. (7.6.15) where s= I, 2, 3,... is the harmonic number of the radiation (fundamental is s = 1) and Js and J'/ are the ordinary Bessel function and its derivative. If ihe electron has velocity components both parallel and perpendicular to the field, ft = r>ii/c and 3± = vxic, with ^^ftf+ftj2, the generalization of (7.6.15) to include the resulting doppler shift is (Trubnikov, 1958) i 2 Pflirfß-wv(i-ftcosi?)* rSsm ) iAi>\diJ •1- 1 -ft COSf?' (7.6.16) 276 Microwave radiation front plasma Chap. 7 7.6 Cyclotron radiation 277 Iii the special case of observation across the field (0 = tt/2), the term in Js' gives the intensity in the extraordinary polarization (Erf j_ B), which is dominant, while the term in Js gives the intensity in the ordinary polarization (Er/ |j B). Along the field (0 = 0), only the first harmonic remains, and the radiation is circularly polarized. The intensity ratio of successive harmonics may be obtained from (7.6.16) by expansion of the Bessel functions for the limit f«l, valid in the barely relativistic case, p\«I. We obtain IP, 1 / s Vs,. 1 / s \2s ft2 sin20 *Pt'~, 4 C—l) e 4(5-1) (l-ftcose)2' (7AI7) which, by hypothesis, is very small; the intensity decreases monotonically with increasing order. In particular, the ratio of second harmonic to fundamental, observed perpendicular to the Held, is (7.6.18) For high harmonics (s»l) in this weakly relativistic limit the envelope of the harmonic intensities may be seen from (7.6.17) to fall off exponentially, at 20 log, 0(2/e£) decibels per harmonic (e = 2.718). In the highly relativistic case, on the other hand, the intensity increases somewhat with harmonic number up to a broad maximum near — /32)~''2 (Landau and Lifshitz, 1962, §74). The radiation of a single relativistic electron consists of the series of harmonics where ,v= I, 2, 3,... is the harmonic number. 7.6.5 Spectrum (relativistic). For a group of electrons, with some assumed velocity distribution, the spectrum is a series of shifted, broadened lines. The shift is produced by the relativistic mass-increase (time-dilation) factor (I — fPfb and, for a Maxwellian distribution, amounts to J">SM/t = (7.6.20) The broadening arises from the same effects mentioned in thenonrelativistic case, plus the dispersion of the velocity distribution in the relativistic shift. The shape factor for the latter effect alone is approximately where y— 1 —(wjsoj„)>0; the full width at half maximum is 8**=] •KS)*"* (7.6.22) This is normally much smaller than the doppler width (7.6.13) except very close to perpendicular observation. To obtain the complete spectrum in the relativistic case it is necessary to integrate (7.6.16) over the electron velocity distribution /(P). This integration, which is very complicated in the general case, may be stated formally as 7cm(w, T) du> dQ = n ') d3p dw dQ (7.6.23) where jm is the power radiated per unit volume, solid angle, and frequency interval in the 5th harmonic, p = v/c, and 6(.v), the Dirac delta-function, is simply a formal way of stating the resonance condition between o>, p, and 0.ie The total radiation spectrum is then obtained by summing over harmonics jjf», T) d 1/ju.; that is, when the particle velocity exceeds the wave velocity. The radiation is emitted in the directions which form the elements of a cone of half-angle 0C. The radiation from a single particle is received as a short pulse (delta-function) with, consequently, a broad-band frequency spectrum. When the medium is dispersive, so that p — p(oj), the condition for constructive interference involves the phase velocity and (7.7.1) remains valid, with the characteristic angle 9C depending on frequency as well as particle speed. The pulse radiated by a single particle is of finite duration. The frequency spectrum is dependent on the particular variation of p(a>). When the medium is further complicated by anisotropy, so that p, depends also on the direction of propagation, the radiation pattern is no longer conical, except in the special case of particle motion along the optic axis. In the absence of a magnetic field, the refractive index of a plasma for electromagnetic waves is less than unity; Cerenkov radiation cannot occur. When a held is present, the index is greater than unity for certain frequency bands. We shall consider only the special case in which the electron motion is parallel to the magnetic held. Thus, the Cerenkov angle 0C coincides with the angle 0 of propagation (with respect to the field) and (7.7.1) becomes cos0 = - 1 (7.7.2) For low plasma temperatures, the index is given by the Appleton formula (1.4.40), so that the Cerenkov condition is (neglecting collisions) 1 /32 cos'^ I-- 1- ^ sin20 fAaVsin2ff i/-v; sin2 fly 1A } +■ cos (7.7.3) This relation may be solved for cos20, yielding (Kolomenski, 1956; McKensie, 1963) cos20= 2č2{(ío2 - cV)a|33 - ^WK/3a + (1 -^K2]} *{2(<„2 - w*f)*09 - wftWP" + (. I - 09Ka] ± ^wb\4(^ - = kTeCl^~A where S is the radiating surface area, and A is the wavelength. The power received by a high-gain antenna is given by (7.5.4), or in terms of (8.1.1) P« du ■-kT, — [l-exp(-2tWVj. 2ir (8.1.3) For incomplete opacity, wall reflections become troublesome and must be eliminated (sec Section 9,6,4 for absorbers). For complicated geometries 287 288 Plasma radiation experiments Chap. 8 8.1 Radiation from dense plasmas: blackbody radiation 289 an integration over the plasma volume must be performed, taking account of wall reflections, ir present (Wort, 1964). A quantitative determination of the radiation temperature of a dense plasma can be made with the radiometer circuit of Fig. 9.44, together with the transmission circuit of Fig. 6.7 (or that of Fig. 6.13). A microwave FIG. 8.1 A 3-mm (90-Gc) radiometer, including the klystron local oscillator the power supply and the 30-Mc i.f. circuit. The i.f. circuit can be operated with direct detection video output or with coherent detection and d-c output. (Photograph courtesy of the University of California Lawrence Radiation Laboratory, Livermore, Calif.) FIG. 8.2 A coaxial line, UHF swept radiometer, for use with an octave bandwidth, swept-frequency local oscillator. A band-pass filter in the signal input line is required, since otherwise the coaxial line passes extraneous noise and signals. (Photograph courtesy of General Atomic, San Diego, Calif.) system is pictured in Fig. 6.21 having only one klystron source; part of the signal provides local oscillator drive for the radiometer and part of the signal is used for the interferometer. No cross coupling between the two circuits exists, since the radiometer receiver is tuned to a frequency displaced by the i.f. frequency from the interferometer. Also the horns are cross-polarized, giving 20 dB of geometrical decoupling. A 3-mm radiometer (90 Gc) is shown in Fig. 8.1, including a 30-Mc i.f. amplifier and second detector that permits either direct video output or coherent detection with a long averaging time (see Section 9.5.5). A coaxial-line, swept-frequency radiometer, operating in the uhf bands with a swept-frequency local oscillator, is shown in Fig. 8.2. A band-pass filter is required in the signal input line to reject spurious signals, since a coaxial line does not have a lower cut-off frequency like a waveguide. Further discussion of radiometers and the associated hardware can be found in Section 9.5. Figure 8.3 demonstrates the relationship between opacity and radiation intensity. The lower trace is the transmission attenuation signal (the same event as recorded in Figs. 6.3 and 6.6) at 90 Gc. The upper trace shows the detected noise output from the radiometer of Fig. 8.1. Enhanced noise is seen to occur early in time, just as the hot, central core of the plasma is reaching cutoff, and later, just as it is coming out of cutoff. The noise level is low in intermediate times, since the cut-off, radiating shell is at a large radius, where the plasma is relatively cold. Also, when w„»w, the plasmtf is no longer "black" but "shiny" (see Sections 7.2 and 7.5). 290 Plasma radiation experiments Chap. 8 8.2 Radiation from a plasma in a magnetic field 291 FIG. 8.3 Plasma radiation intensity correlated with opacity. In the top trace the 90-Gc radiometer shows radiation signals during the times that the attenuation measurement in the bottom trace indicates WpKiu. The radiometer of Fig. 8.1 was used. Similar evidence is presented in Fig. 8.4, where the cutoff is indicated by the vanishing of the interference fringes (see also Wharton, 1961, p. 326). Three events are recorded, of consecutively higher-peak densities from top to bottom, all showing the increase in noise near cutoff. The data were made with the apparatus shown in Fig. 6.21, at 24 Gc. The peak amplitudes correspond to blackbody temperatures of 2 to 5 eV. A signal compression is inherent in superhet radiometers, since the output signal is proportional to the input voltage and thus to the square root of the noise temperature TN. The vertical scope deflection then is Dcc(rA-)-'s. The peak noise signal of Fig. 8.3 thus corresponds to a blackbody (electron) temperature of ~ 10 eV, which agreed within a factor of 2 to the value obtained spectroscopically.1 In general, noise temperature (blackbody radiation) measurements made in dense, high-collision-rate plasmas yield electron temperatures in good agreement with those obtained by Langmuir probes (Knol, 1951; Easlcy and Mumford, 1951) or spectroscopically (Harding et al., 1958). Even in high-temperature plasmas, where the collision rates are too low to provide the thermalizing mechanism for electrons, nevertheless the electrons often are found to have a Maxwellian distribution (Gabor et al., 1 Unpublished data of C. B. Wharton, .1. E. Katz, and D. Reagan, University of California, Lawrence Radiation Laboratory, Livermorc, Calif., 1961. FIG. 8.4 Plasma radiation intensity correlated with opacity. The radiometer response is maximum just before the plasma goes to cutoff, as indicated by the disappearance of the interferometer fringes. The apparatus of Fig. 6.21 was used. 1955), and in many cases the electron temperatures inferred from microwave radiation intensities compare favorably with temperatures measured by other methods (Dellis, 1958; also see Section 8.2.1). Occasionally, however, some obviously nonthermal radiation is observed, emanating from plasmas. Evidence for this type of radiation was reported in Section 7.8, and further discussion is given in Section 8.4. 8.2 Radiation from a plasma in a magnetic field An electron spiraling about a magnetic field line will radiate, due to its acceleration, as dcTn oust rated in Section 7.6, where radiation intensities and frequency spectra were calculated. In transparent plasmas, the 292 Plasma radiation experiments Chap. 8 8.2 Radiation from a plasma in a magnetic field 293 maximum radiation intensity was found to be in the plane of the orbits. In dense plasmas having radial density gradients (as real plasmas do), however, the cyclotron radiation is inhibited from escaping radially across the magnetic field, since it eventually reaches a cut-off region as the density falls toward zero at the edge (see Section 4.2.3). But, if the cut-off region is not extensive, some of the radiation generated in the hot interior can "tunnel" out and escape the plasma. Experimental evidence of this tunneling has been reported (Wharton, 1959; Motley et al., 1961). 8.2.1 Magnetic mirror radiation experiments. Cyclotron radiation emitted along the field lines is often observed. This can be explained in terms of Kirchhoff's law. The trapped magnetic bremsstrahlung is strongly absorbed at the QL resonance frequency (see Section 1.4.10) so that the plasma reaches an anisotropic radiative equilibrium (Beard, 1961). The radiation is able to diffuse along the magnetic lines or in directions in which the field increases in the appropriate manner to avoid the cut-off condition as the density decreases. The radiation escaping directly along magnetic field lines should be right-hand circularly polarized, since that is the wave that exhibits the large absorption. Measurements made on magnetic mirror machines at Livermore, Calif. (Wharton, 1958, 1959, and 1961), however, failed to show this polarization effect, presumably because of polarization coupling or wall reflections. Microwave receiving antenna - FIG. 8.5 Geometry for observing radiation from the end of a magnetic mirror plasma experiment. L is the radiation absorption length of the volume V. The region to the left of Fis cut off, that to the right is transparent, Cis the central region, M the mirror regions. S is the mirror separation. The radiation geometry for mirror or cusp experiments is sketched in Fig. 8.5. The radiation absorption length L denotes the distance in which radiation is effectively absorbed and reradiated in random phases. It is important that L be smaller than or comparable to the dimensions of the plasma for at least two reasons: (/) to establish quasi-blackbody conditions, and (2) to quench any coherent radiation by phase scrambling during the absorption-reradiation process. The phase scrambling2 is enhanced by a magnetic field gradient, such as found in a magnetic mirror, with a resulting action much like that of a magnetic beach (Stix, 1962). Absorption lengths typical of many controlled fusion experiments are from 0.5 to 5 cm. If the plasma extends for some distance into the magnetic mirror regions, M, it is this part of the plasma and not that in region Cthat will be sampled at a frequency giving cyclotron resonance in M. Region C can be probed by a second receiver, tuned to the lower frequency corresponding to the cyclotron frequency of that location. Waves can propagate in regions having higher magnetic fields, allowing the radiation from region C to pass unimpeded out through M. 8.2.2 Absorption-radiation experiment in a pulsed minor machine. Many magnetic mirror experiments have pulsed fields to provide magnetic compression and heating of the plasma (Post, 1958). Fixed-frequency radiation receivers thus can be in tune with the cyclotron frequency twice each pulse, once during the rise and once during decay. Even though the intrinsic cyclotron resonance may be sharp (Drummond, 1958; Hayakawa et al., 1958), the received signals tend to be broadened by the magnetic field fluctuations and gradients. Typical variations of the fields in the Tabletop TI experiment at Livermore (Post et al., 1960) with time are sketched in Fig. 8.6. The upper curve pertains to the field strength in the mirror region M, the lower curve to the field strength in the central region C. The low-/S plasma, of course, is confined in the region within the mirrors, that is, between the curves of Fig. 8.6. The Tabletop II experiment had a chamber 15 cm in diameter and 5.5 meters long, with a mirror separation S of 50 cm (see Fig. 8.5). The peak pulsed field was programmable up to 50,000 gauss. The plasma was provided by injection from titanium hydride sources into the evacuated chamber (Coensgen et al., 1958), the peak injected density before magnetic compression rising to about I01Z electrons/cm3. To determine the plasma opacity, an absorption experiment also was performed. The equipment is shown in Fig. 8.7. The receiver is a a To be distinguished from the phase mixing of Landau damping, which may also contribute to the establishment of radiative equilibrium, but not radiation, except through the inverse process, Ccrcnkov radiation (sec Sections 6.6 and 7.7). 294 Plasma radiation experiments Chap. 8 c 3 O O o " a s o V? [J _ 13 .2 1 o % 1 o rt c y h "P -s '5 .Eli' C tí = U C E I .g p c [ssneiJoipjJ OjiauSs?^ nipil/.Killlľ 331011 lllCl}t10 superhet, having a 10 Mc band width and 0.1 micro-microwatt threshold sensitivity, with detected noise viewed directly on an oscilloscope. Amplitude and geometry calibration were made by moving a standard 15.2 dB thermal noise source (with a small horn radiator attached) about inside the experiment chamber before it was evacuated (see Section 9.5.6). The transmitter klystron is modulated by random noise lo produce a microwave spectrum some 200 Mc wide. The effects of high-order Load isolator 8.2 Radiation from u plasma in a magnetic field 295 Horn-. I Z ~ '. Lx-Horrii-- nui [i-v ^ 24 Gt : :0 i Klystron oscillator Random noise generator Power supply Plasma Compressioi ~^TT7TrT7?77-/////7777T?-rTTTT, Experiment Energy storage bank ~10e joules n /Mom[ * >dh- 24 Gc Klystron local oscillator Power supply Trigger Load isolator 0-100 d B Pad Balanced mixer 30 Mc if amplifier (10 Mc B.W.) 5 Mc Video Oscilloscope FIG. 8.7 Microwave equipment used for a 24-Gc microwave gyro resonance absorption-radiation experiment on a pulsed magnetic mirror machine, (See Figs. 8.6 and 8.8 for experimental results.) waveguide modes and reflections in the vacuum chamber are thus minimized. The frequency stability of the receiver and transmitter are good enough that no AFC system is required. The transmitter power level is a few milliwatts; the receiver output is set to a steady arbitrary signal level by adjusting the 0 to 100 dB attenuator in the input. Two such systems were used simultaneously, one in the A-band (23 to 28 Gc) and one in the 8-mm band (32 to 40 Gc). All plasma radiation (of order !0"u watts maximum) is masked by the transmission signal (of the order of 10-r! watts). A A"-band reflection experiment at the transmitter end was performed, showing plasma reflection coefficients less than 1% under all conditions, when the plasma was both opaque and transparent. Absorption lengths are of the order of a centimeter or two (many plasma wavelengths, at resonance). When the radiation was to be viewed, the transmitters were turned off and the input attenuators turned to zero. The plasma experiment was operated again, exactly as before, but now the receivers viewed the plasma radiation. Typical receiver output responses are shown in Fig, 8.8. flic limes of resonance arc also shown on the iiekl plots of Fig. 8.6, 296 FIG. 8.8 Plasma microwave radiation data. Responses or radiometers to the cyclotron radiation from a magnetic mirror machine, («) at/= 35 Gc, (b) at/= .12 Gc, and (c) at /=24 Gc. Trace (BC), the transmission signal strength is seen to be twice as high as for the vacuum case, indicating enhanced coupling when the cyclotron frequency is above resonance, as would be expected for ducted cyclotron wave propagation (low-density case of whistler-mode propagation). It is highly unlikely that the enhancement is due to any kind of wave growth or amplification. The amplitudes of the first radiation peaks (at i~400 ^tsec), in many of the events, corresponded to blackbody temperatures of 15 keV (170 million degrees Kelvin). A direct energy analysis of escaping electrons (Ellis and Parker, 1958), assuming adiabatic trapping, yielded an average plasma electron energy of 17 keV at that time. The velocity distribution was not distinguishable from Maxwellian. The ions did not have a Maxwellian distribution, but their average energy was estimated at between 500 and 1000 eV. The average energy of escaping X-rays was within the range 10 to 100 keV, as estimated from absorber measurements. 8.2.3 Absorption-radiation measurements in a waveguide or cavity. The interactions between plane waves and plasma slabs can be simulated under certain conditions by enclosing the plasma in waveguides or resonant cavities (Buchsbatim et al., 1960). The boundary conditions can be satisfied, but the plasma ordinarily does not look "optically thick," so that local equilibrium is not established, except in the cases of high collision rates or strong resonances. Also, the sharp boundaries may permit charge separation, with attendant coupling between spacechargc and electromagnetic waves, or may allow wave tunneling through otherwise cut-off regions. For two cases, the radiation from bounded plasmas can be studied by applying corrections to the free-space relationships (Flirshfield and Brown, 1961): (/) weakly absorbing, tenous plasmas, utilizing a perturbation analysis, and (2) highly absorbing plasmas, such that the absorption length is much smaller than the plasma dimensions. In case / the plasma critical frequencies are shifted by the mode cutoffs (see Section 5.2) and flic total wave absorption is decreased by the increase in guide wavelength. In case 2, when Ihc absorption is large, the effects of boundaries 298 Plasma radiation experiments Chap. 8 8.4 Radiation of nonthermal origin 299 are small, and the resonance occurs at the Tree-space frequency given by (1.4.111) cos;ie) l (8.2.1) where 0 is the angle in respect to the magnetic field. It is usually not convenient in waveguide experiments to observe radiation at angles very far from 0 = 0" (along B) or 0 = 90° (across B). Solenoidal magnetic fields can be made very uniform, so that bz)K, are also well defined. The presence of the reflecting walls may increase the effective emissivity by multiple internal reflections, very much as in an optical hohlraum or integrating sphere (Wort, 1964). The total radiation transmission coefficient Trad analogous to (4.3.13), is obtained by summing over all of the S single-interface reflection coefficients r, for a system of dimension d Traa^O-r? J (rf-'txp(-2Sad) s = l , m exp(-2W) y 1 \-r2Q\p(-Aa.d) The total radiated power, compared to the blackbody power, is (I-/•)[!-exp(-2c«/)] P = P* 1 —r exp(—2«ď) (8.2.2) (8.2.3) If ad is small, the reflection coefficient is important; if ad is large, the reflection enters only in determining the effective emissivity of the sampling orifice. In the waveguide experiments of Hirschfield and Brown (1961), a time-integrating S-band Dicke radiometer, similar to the one shown in Fig. 9.44 and having a sensitivity of 10"17 watts, permitted measurements on tenuous plasmas to be made. Cyclotron resonance line profiles were made under various conditions of gas pressure. If the broadening of the line were to be explained as due to collisions, of constant v, then the line half-width would be hu>% = v, (8.2.4) If the broadening were pure doppler broadening, the half-width would be (2kTs mc." hi2j • (8.2.5) A combination would lead to a Voigt profile. The results obtained in this experiment showed much greater broadening in helium than would be expected by either of these effects, with also slight shifts toward higher frequencies than ojb for the known magnetic field. The conclusion was that either the magnetic field was not as uniform as they thought (0.5%), the electron temperature exceeded 4 eV, or some additional mechanism was contributing. The radiation intensities for blackbody radiation at densities from 1010 to 1012 cm"3 and for cyclotron radiation for densities between 10s1 and 1010 gave a noise temperature of 2 eV. At low densities (109 down to 107 electrons/cm3), the noise temperatures for cyclotron radiation rose to 7 or 8 eV. The data were all obtained at a fixed frequency by sweeping the magnetic field and the electron density. 8.3 Swept-frequency radiometers In some experiments the plasma conditions change radically as either the magnetic field or electron density arc varied. The swept-frequency radiometer pictured in Fig. 8.2 avoids some of the difficulties by providing a voltage-tunable octave band. Both side bands are received, since it would be difficult to sweep a tunable rejection filter in exact synchronism with the local oscillator. The balanced mixer must be carefully matched over the band to achieve optimum performance. A coaxial-line ferrite isolator and a band-pass filter in the input minimize the spurious responses. If single-sideband reception is required then the band-pass filter must be tunable and tracked with the local oscillator. If an integrating detection radiometer is used, the frequency sweep rate must be slow enough that the response can follow. Typical response times of Dicke radiometers are from 0.1 to 10 seconds. A swept intermediate frequency also can be used to scan over a spectrum. This permits a fixed-frequency local oscillator, with a high-Q cavity to stabilize it and remove much of its noise contribution to the mixer circuit. Single-ended mixers can then be used, instead of the more critical and expensive balanced mixers. For narrow frequency scanning, such as looking at resonance line profiles, the i.f. amplifier can be a conventional low-noise i.f. strip, followed by either a second mixer and swept local oscillator or a sweeping filter (Long and Butterworth, 1963). For wide frequency scanning, a low-noise traveling-wave tube or distributed amplifier, in conjunction with a swept-frequency tuned amplifier, can be used (Cohn et at., 1963). These techniques are particularly useful at millimeter wavelengths, where mixers are critical, especially if harmonic mixing is required (see Section 9.5). 8.4 Radiation of nonthermal origin It is often difiieiiTt to distinguish nonthermal from thermal radiation. True, when ihc radiation is observed in intense bursts or has an abnormally 300 Plasma radiation experiments Chap. 8 high harmonic content, the generation is clearly not of purely thermal origin. But the radiation emanating from a plasma having a non-Maxwellian velocity distribution or containing plasma waves or electrostatic instabilities may not appear abnormal. It is important to understand the basic radiation processes, then, to be able to attach much importance to various features of the emission. 8.4.1 Instability-generated radiation. Among early observations of nonthermal radiation were bursts in the VHF bands of fairly short-duration emanations from the sun, presumably associated with plasma oscillations in solar prominences (Kuiper, 1953). Similar large bursts at 8 mm wavelength have been observed in Stel-larators (Heald, 1956), presumably due to instabilities driven by runaway electrons. The intense "generation" of waves by the Stellarators was of sufficiently high power to endanger crystal detectors in microwave interferometers used for diagnostics. An electron beam-plasma interaction experiment was set up at Liver-more in 1959 to simulate the runaway electron interaction, but with controlled electrons from a gun. The gun was pulsed with 5 to 20 /xsec pulses 30 times per second, firing into the steady-state P-4 plasma (Hall and Gardner, 1961) as shown in Fig. 8.9. Radiation pulses, of microwatt intensity (that is, about 10e times the blackbody level) were detected during the electron pulses, in a narrow frequency band centered about the plasma frequency, at 34 to 36 Gc,3 depending on the electron density at the location of the beam. The radiation intensity varied smoothly with variation of the pulse current, the detectable threshhold occurring when the peak current was about 5 to 10 milliampcres. Two orientations were used for the pickup horns, as shown in Fig, 8.9, both using square cross-section horns and fin-line couplers to resolve the two polarizations. With the horn looking along the column (actually looking up at the plasma with a 20° angle from the axis), the radiation intensity was found to be about 5 times as large from up stream (radiation coming from the gun end) as from down stream. With the horn looking at 90° to the plasma, the radiation polarized along the direction of the beam was 6 times as intense as that polarized across the beam. The intensities measured at three ports along the column were essentially the same, although the frequencies varied by about 10%. The signals from two ports detected simultaneously with two wide-band 8-mm radiometers were uncorrected, when fed into a coincidence circuit (<2% coincidence over 10 seconds). a Here, the word "radiation" is used in the strictest sense, that is, far-field, electromagnetic reception, as distinguished from "noise," which may be picked up by a probe or antenna immersed in a plasma, and subject to near-field induction or spacecharge field fluctuations. 8.4 Radiation of nonthermal origin 301 Waveguide to Rotation drive to rotating langmuir probe Insert, showing details of mounting waveguide horns in the vacuum ports Break indicates 10 feet of chamber omitted Reverse field coil for the burial chamber ^&rna_c^rrjn___. Microwave!, horn I Wilson LT~ S6a' TT Wavegmde inside round pipe through u Wilson seal FIG. 8.9 Electron beam-plasma interaction experiment. The electron gun injects 20 fisec, 1-amp pulses of 25 keV electrons into the steady P-4 plasma column. The P-4 plasma was 99% ionized, with 1013 electrons/cm3 at 10 to 20 eV temperature. (From unpublished data of C. B. Wharton and A. L. Gardner, University of California, Lawrence Radiation Laboratory, Liver mo re, Calif.) When a Langmuir probe was inserted in the plasma in the space in front of the horn, the radiation power increased a hundredfold, and the frequency emitted became a function of the radial position of the probe.4 Presumably, the enhancement was due to conversion of some of the space-charge wave energy into currents on the probe which, in turn, radiated electromagnetic waves as a dipole. Since the beam-induced plasma oscillations are very nearly at the plasma frequency (Stepanov and Kitscnko, 1961; also see Section 5.5), a measurement of the radiation frequency gives a measurement of the local electron density. Correlative measurements of saturated ion current of the Langmuir probe (see '' Actually, the probe was swept through the plasma with a 15% duty cycle to prevent its burning up (Oardriír el al., 1961), so that the radial positioning was actually done hy tlmlhg I lie firing of the electron beam pulse with a variable delay. 302 Plasma radiation experiments Chap. 8 8.4 Radiation of nonthermal origin 303 Section 10.12) and of microwave phase shift at 64 Gc gave values of electron density and density profile, all agreeing remarkably well. Radiation was also viewed with a swept-frcquency S-band radiometer (Katz, 1959) to look for cyclotron radiation. Signals at a noise temperature of about 5 to 10 eV were received at a frequency of 2.6 Gc, which is the gyrofrequency in a magnetic field of 930 gauss (P-4 had Bx950 gauss). The signal was enhanced about twofold during about 25% of the electron beam pulses. We concluded that the coupling of the instability to cyclotron radiation was not large for those conditions. 8.4.2 Nonthermal cyclotron radiation. The cyclotron radiation spectrum, discussed in Sections 7.6.4 and 7.6.5, contains harmonics, whose relative intensities are strong functions of the plasma electron velocities. A typical harmonic spectrum for high energy electrons is shown in Fig. 7.8. For thermalized electron temperatures typical of most plasma experiments, including controlled fusion, harmonic numbers of 3 or 4 are about the theoretical limit. In several plasma experiments, nevertheless, harmonics as high as the 24th have been detected, with relative intensities that have little to do with conventional cyclotron radiation theories. In the experiment of Landauer (1961, 1962), up to the 24th harmonic of N=2, Ar=2 -> A' = 3, etc.) as -> 1. C HAP T ER 9 Microwave hardware and techniques 9.1 Transmission lines and fittings The ranges of electron densities, magnetic fields, and plasma dimensions of many laboratory plasma experiments require the use of microwave frequencies in the region of 3 to 90 Gc for free-space (beamed) transmission experiments. Fortunately, this frequency interval includes several of the bands that have been developed extensively. Some experiments call for higher frequencies, the hardware for which is still largely experimental. Table 9.1 lists some of the bands and the present standard waveguides and flanges of each. Plasma experiments involving spacecharge wave transmission and hybrid ion resonance effects are conveniently done at lower frequencies, for example, 500 to 5000 Mc, using coaxial cable components. Extensive lines of components are available, employing type N, C, BNC, TNC, and other well-matched fittings and covering octave band widths. 9.1.1 Waveguide considerations. For millimeter wavelengths, the nominal skin-depth in metals is given by 1 meters, (9.1.1) where/is the frequency, fj.0 is the permeability, and a is the conductivity, mhos/meter; S is of the order of 2.5 to 5- 10"s cm (0.25 to 0.5 micron). The wave attenuation in a waveguide therefore is influenced by surface roughness, oxidation and chemical deposits, and work-hardening (Thorp, 1954). A quantity that describes the surface condition is the surface resistivity, ohms/unit square. (9.1.2) 306 Microwave hardware and techniques Chap. 9 Table 9.1 Standard waveguide bands 9.1 Transmission lines and fittings 307 Frequency Band-center Designations Waveguide Flanges range, Gc wavelength JAN EIA 2.6 3.95 10 cm S RG-48 WR284 UG 53, 54A 8.2-12.4 3 cm X, Xs RG-52 WR 90 UG 39,40A 12.4-18 20 mm Ku, P RG-91 WR 62 UG419, 541 18-26.5 12 mm K RG-53 WR 42 UG425, 595, 596 26.5-40 8.6 mm Ka, R, V, V RG-96 WR 28 UG381,599, 600 33-50 6.8 mm Q RG-97 WR 22 UG383 50 -75 4.3 mm M, V, W RG-98 WR 15 UG385 60-90 3 mm E RG-99 WR 12 UG387 75-110 3 mm — — WR 10' 90-140 2.2 mm F RG-138 WR 8 EIA ► standard flanges* 110-170 140-220 170-260 2 mm 1.5 mm 1.2 mm G RG-136 RG-135 RG-137 WR WR WR 7 5 4 220-325 1 mm — RG-139 WR 3 j * A U.N. commission, the International Electro-Technical Commission, is currently studying waveguide standards problems and is making recommendations for standard flanges in the millimeter bands. These will become EIA standard flanges. R5 has the same value as the d-c resistivity of a plane conductor of thickness S and conductivity a. Surface roughness and porosity increase Rs. For example, a microscopic roughness of depth and spacing equal to 2S will increase Rs about 50% (Lending, 1955; Morgan, 1949). High-conductivity, nonoxidizing materials commonly are used for waveguide fabrication, including pure copper, coin silver (90% silver, 10% copper) and pure silver laminated on copper or bronze. At the shorter wavelengths, gold, chromium, or iridium often are plated on exposed surfaces to impede oxidation. Figure 9.1 shows skin depths for several materials at various frequencies. Waveguide attenuation is dependent on the frequency, the dimensions of the waveguide, and the type of transmission mode. For example, the attenuation due to wall losses for the TE10 mode in rectangular waveguide of width a, height b is •Hm r{1+v(é)1 dB<'meter 0-L3> 20 30 40 50 60 80 100 Frequency [GcJ 200 300 FIG. 9.1 Skin depth in metals of various conductivities as a function of frequency, d-c Conductivities of common metals in units of 10" mhos/meter: silver 64; copper 59; gold 41; chromium 38; aluminum 35; magnesium 22; iridium 16; brass 15; platinum 9; soft solder 7; mercury I.I. where A is the free-space wavelength. The attenuations of the common RG-type waveguides in the TE10 mode are shown in Fig. 9.2. In addition, the values for some high-mode waveguides are shown, including some RG waveguides of extreme oversize (Valenzuala, 1963). In oversize or high-mode waveguides, mode purity is sometimes difficult to maintain (Lewin, 1959). Usually, bends, junctions, or obstructions lead to mode conversion, with attendant frequency sensitivity and attenuation. A common practice is to use conventional-waveguide-size components for circuits, short runs, sharp bends, etc., and taper up to the oversize or high-mode guide with a long, electroformed transition for long straight runs, where excessive attenuation in the conventional waveguide would normally occur. Round copper tubing that has been slightly flattened 310 Microivave hardware and techniques Chap. 9 9.1 Transmission lines and fittings 311 between rollers to prevent polarization rotation is a convenient method to cover distances up to about 50 feet, cheaply, with propagation in the oversize TEn mode. Bends having radii of curvature of about 20 wavelengths or more can be tolerated with little mode conversion. Typical 50-foot runs of flattened £-inch tubing at 90 Gc have a loss of 7 to 10 dB, including the two transitions at the ends, as compared to 70 to 80 dB for RG-99/U. Even lower losses areobtainable withTE01 round waveguide (Miller and Beck, 1953), at the expense of complicated mode transitions and the necessity for inserting mode filters at frequent intervals along the run. Figure 9.3 shows some TE0i-modc circular waveguide components (Lanciani, 1954). Figure 9.4 shows views of several other mode transitions, filters, and bends, as used at the University of California Lawrence Radiation Laboratory. Highly-oversize, special-mode components are available commercially, although they tend to be somewhat costly. 9.1.2 Open waveguide transmission lines. Other interesting types of wave-guiding structures utilize surface waves on metallic (Sobel et al., 1961), dielectric-coated metallic (Diament et al., 1961), or purely dielectric E-Field lines H-Field lines VI -Metal plates _ Dielectric center strip FIG. 9.5 Sketches of field patterns of open waveguide transmission lines. The fields are seen lo fringe some distance outside the line, (a) Dielectric rod, HEU dipole mode, (b) Dielectric rod, TM0i mode, (c) Trough, (d) //-Guide. surfaces and rods (Chandler and Elsasser, 1949; Weiss and Gyorgy, 1954). Dielectric rod waveguides of teflon, polystyrene, etc., are particularly interesting, since they are nonconductors and can be used to bridge across high voltage environments or to operate in a corrosive atmosphere. The fields extend some distance outside the rod, however, as sketched in Fig. 9.5, and coupling can occur between rods coming close to each other or from the rod to support structures. Also, radiation will occur at sharp bends or regions of nonuniform cross sections. Dielectric rod radiators, discussed in Section 9.3.3, are made by tapering the end of a rod. Various open waveguide or surface-wave propagation structures have been devised. Among the more useful ones are H-line, V-line, trough-line (Tischer, 1956, 1958), and dielectric-coated wire or G-line (Goubau and Schwering, 1961). The attenuation at millimeter wavelengths of these lines can be several orders of magnitude lower than that of conventional waveguide, the attenuation decreasing with frequency. Wave launching FIG. 9.6 Free space transmission path utilizing dielectric lenses. Path length of 4 feet has a loss of I dB al 70 Gc. (Courtesy TRG, Inc.) 312 Microwave hardware and techniques Chap. 9 is generally d i Ilk u It and the structure tolerances are often extreme. They are useful Tor very long runs, but have little advantage for short ones. 9,1.3 Free-space radiation. Free-space radiation links are possible at millimeter wavelengths because large-aperture antennas (in wavelengths) are easily realizable. The minimum insertion loss is limited in practice by diffraction and interference conditions, as pointed out in Section 4.9. Large apertures can be obtained with lenses, Fresnel zone plates (van Buskirk and Hendrix, 1961) and parabolic reflectors. An example of a transmission link using lenses is shown in Fig. 9.6. Transmission over distances of 15 to 20 meters between zone-plate radiators with attenuation as low as 10 d B at 140 Gc is obtainable. Even lower loss is obtained if the cross-sectional phase distribution is corrected at periodic intervals (Goubau and Schwering, 1961) by inserting long-focal-length lenses in the transmission path. Attenuations as low as 2 dB over path lengths of a kilometer have been achieved. 30 40 50 60 80 100 Frequency [Gc] 300 400 FÍG. 9.7 Approximate atmospheric attenuation of electromagnetic waves for horizontal propagation as a function of frequency. The water vapor density was 7.5 grams/meter3 for the upper curve and 1.0 gram meter3 for the low^cr curve, (Compiled from data of S'.raiton and Tolberg, 1960; Dieke et a'.., 1946; and Thcissing and Caplan, 1956.) 9.2 Special components 313 In free-space transmission, atmospheric attenuation becomes important. Except over very long paths, however, the air attenuation is practically negligible. Figure 9.7 shows the attenuation due to molecular resonances and the atmospheric "windows" between (Dicke et al., 1946; Theissing and Caplan, 1956; Straiten and Tolbert, 1960). 9.2 Special components Many techniques and component designs can be carried over from the highly perfected waveguide bands into millimeter wavelengths by simple scaling laws. There are, of course, some problems: (1) Dimensional tolerances become exacting, (2) Losses increase, due to small skin depth and small component size. (3) Oscillators and amplifiers become less efficient (and more expensive) because of greater beam density and heat dissipation requirements. (4) Crystal detectors become less efficient because of increased internal impedance, and have lower power handling capabilities. The cost of waveguide systems, in general, increases with frequency in a nearly linear fashion above JC-band. For example, a 4-mm interferometer costs about twice as much as an 8-mm one, and a 2-mm interferometer FIG. 9.8 Simple waveguide bending tool and samples of sonic elbows fabricated in RG-96/U waveguide. (Courtesy General Atomic, San Diego, Calif.) 314 Microwave hardware and techniques Chap. 9 9.2 Special components 315 about twice as much again. The choice of the frequency at which to do diagnostics, then, clearly contains an economic factor as well as physical factors. Nearly always, some kind of compromise is necessary. Often, the budget can be eased by fabrication of components, both waveguide and electronic, within the laboratory shops. Elbows, twists, phase shifters, terminations, etc., are particularly easy and inexpensive to make. As an example, a simple waveguide bending tool and some elaborate bends made in RG-96/U waveguide are shown in Fig. 9.8. No reflections or attenuation due to these bends were measurable in the systems where they were employed. The flange faces were finished off in a lathe after soldering. The same techniques have been used successfully up to 90 Gc. 9.2.1 Phase shifters. Waveguide phase shifters change the electrical length of a section of waveguide, either physically (as with a line stretcher) or by varying the cut-off frequency and thus the phase velocity. Commonly, phase shifters are made with a dielectric vane oriented in the E-plane and either lowered through a slot in the top wall or moved back and forth across the broad dimension of the guide. Both or these types must be used with considerable caution. The first type tends to radiate badly, unless the vane is well shielded and, even then, may introduce standing waves. Both types can support higher modes when the vanes are well in, or when the frequency is near the top of the band; frequency FIG. 9.9 Squeeze-section phase shifter in R.G-96/U waveguide. Maximum phase shift fti 360" at 35 Gc. (Courtesy General Atomic, San Diego, Calif.) sensitivity and standing waves then result, leading to significant errors in phase measurements. A phase shifter whose cut-off frequency is decreased by operation (no higher modes arc then possible) is the squeeze section. A longitudinal slot is cut through both top and bottom walls of the waveguide along the center line for about ten guide wavelengths. A captive screw is then soldered to one side wall, and a manipulator attached that can both spread and squeeze together the slit. Phase shifts of 2-rr to 4?r are easily obtained, writh essentially no radiation or reflections. A squeeze-section phase shifter is shown in Fig. 9.9. A more elegant type of phase shifter is formed by rotating a half-wave plate, in circular waveguide carrying a circularly polarized wave, between two quarter-wave plates that transform from circular to linear polarization (Fox, 1947). The phase is shifted Itr with each rotation. If the center section is rotated at constant velocity with a motor, a frequency adder (or single-side-band modulator) thus results. The rotation velocity can be made very large if it is performed electrically by means of a ferrite section driven by a varying or rotating magnetic field (Cacheris, 1954; and Fox et al., 1954). Such devices are useful for serrodync detection systems. The line-stretcher types of phase shifters include a short-slot hybrid with ganged shorting plungers (Barnett. 1955) and a circulator with a shorting plunger. 9.2,2 Hybrid junctions. Hybrid junctions are used as power dividers, phase comparators, and in balanced mixers (Jones. 1961). The matching posts and irises used in the conventional "magic tee1' at frequencies up to A!-band become difficult to fabricate in the millimeter range. The hybrid ring or "rat-race" is often used, since no matching is required. It may be milled, dye-cast or electroformed, even in small waveguide sizes. Typical band width, determined by the geometry of the ring, is about 6%. The basic design, shown in Fig, 9.10, is straightforward. The isolation is obtained by arranging the coupling arms A/4 apart, so that a signal entering arm I divides equally between arms 2 and 4, the waves that travel around the ring in opposite directions reinforcing. The waves reaching arm 3 just cancel. The junction ports are reciprocal, of course. Isolation between opposite arms is usually about 20 dB within the 6% band. Short-slot hybrids and multihole directional couplers having 3 dB coupling can also be used as hybrid junctions over fairly wide band widths, although directional couplers tend to be somewhat larger and not as well balanced as rat-races (Riblet, 1952). Fin-line couplers and turnstile junctions (Sections 9.3.3 and 9.2.5) also are useful for hybrid junctions and balanced mixers (Loth, 1956). 316 Microwave hardware and techniques Chap. 9 E 1 X \ '4 Jj FIG. 9.10 Plan of rat-race hybrid junction, showing the lengths of the ring segments. The E-plane dimension of the ring groove is typically 1/V2 that of the side arm grooves. Coaxial-line hybrid junctions use hybrid rings, A/4 couplers, etc., and may have octave band width. 9.2.3 Polarization and frequency diplexers. A simple and useful polariza-tion diplexer is the fin-line coupler (Robertson, 1956), sketched in Fig. 9.11 and shown in Fig. 9.4. By installing two side arms at 45° on either side of the plane of symmetry in the cylindrical form, a 3-dB coupler is obtained. Two arms, installed at 90° to each other, make a polarization diplexer, with the arms decoupled from each other. By properly phasing the drive to the two arms, a circularly polarized wave results. A square-cross-section coupler is made either by elcctroforming walls around a copper plate that has had a slot etched in it (see Fig. 9.4) or by machining and soldering. One way of fabrication is to mill off the top walls of two pieces of waveguide, fit them together with the fins sandwiched between and with the side arm positioned by a jig. The assembly is then soldered. The cross-coupling between two waves having their electric vectors oriented at 90" to one another in either type of fin-line coupler is typically down by 20 to 40 dB. The coupling loss to each of the polarizations is typically 0.5 to 1.5 dB, most of the loss being in reflections and in excitation of resonances in the structure. Resonances also lead to cross-coupling. Fin-line couplers are useful at frequencies between about 3 and 70 Gc. 9.2 Special components 317 Resistive sheet to damp spurious mode responses FIG. 9.11 Sketch of a Jin-line coupler in cylindrical waveguide in a cut-away view, to show the internal construction of the coupling vanes or fins and the resistive mode suppressor. The fin-line coupler can also be used as a frequency diplexer if the frequencies are not too widely spaced. For example, a 25-Gc wave can be propagated in one polarization and a 35-Gc wave in the other, with little cross-talk. An electroformed waveguide having a square cross section for carrying the two polarizations is also shown in Fig. 9.4. A square-cross-scction vacuum window and horn antenna complete the dual-polarization system. Some measurements utilizing such a system are described in Section 6.5. Frequency diplexers or multiplexers ordinarily use filters to separate the channels. A waveguide itself is a high-pass filter; a squeeze section of waveguide then becomes a variable-cutoff high-pass filter. A simple diplexer can be made using a series tee junction, containing a squeeze section in one arm for the high frequency signal and a phase shifter and low-pass filter or an impedance transformer in the other arm for the low frequency vvavc. TiTc phase shifter and squeeze section arc adjusted for matching at the junction at the frequencies in use. Isolation in excess of 318 Microwave hardware and techniques Chap. 9 9.2 Special components 319 Detected low-frequency signal Series T junction v Sä •'n-f*"** \ r, ' V:-rJ. frequency^|jg input Detected high-frequency signal Impedance transformer section to reject the high frequency Tapered squeeze section having variable cut-off position for the low frequency FIG. 9.12 Dual frequency diplexer-detector for frequencies 30% to 80% apart. 30 dB between 25 Gc and 35 Gc signals propagating in RG-96/U, with little signal loss and reflection, have been obtained with such a diplexer. An advantage of this type of diplexer over that using fixed-frequency filters is that it is easy to adjust for changes in operating frequency, and not very sensitive to incidental frequency changes. The unit is sketched in Fig. 9.12. 9.2.4 Filters. Waveguide filters commonly are made up of inductive or capacitive iris-coupled sections. The susccptance of the iris and the spacing between them determines the cutoff characteristics (Conn, 1957). Low-pass, high-pass, and band-pass configurations can be made, the design following standard transmission line theory (Guillemin, 1948). Typical six-section filters have an increase in insertion loss beyond cutoff of 36 dB per octave. High-Q cavity filters of the band-pass or band-rejection type may have a 30-dB change for a 1% frequency change (Riblet, 1958b). For example, the band-rejection filter used with the scattering experiment described in Section 6.7, consisting of two orifice-coupled tunable waveguide cavities, has an insertion attenuation of ~50 dB over a 20-Mc band and >20 dB over a 50-Mc band, but falls to 1.5 dB outside FIG. 9.13 Band rejection filter. Two-section orifice-coupled cavity filter, tunable between 30 and 40 Gc with peak rejection >50 dB. (Courtesy General Atomic, San Diego, Calif.) /o ±100 Mb, A photograph of the filter is shown in Fig. 9.13. The orifice-coupled waveguides are A„/2 cavities, tuned by conventional shorting plungers. Low-pass filters find application in rejecting harmonics from mixers and from magnetrons or klystrons. Band-pass and high-pass filters remove unwanted noise or image frequencies in superhet receivers. High-f), band-pass filters are usually a transmission cavity, that is, two waveguides orifice-coupled through a resonant cavity. Minimum insertion loss is usually at least 2 to 6 dB. A common construction method for iris-coupled waveguide filters is to saw thin slots part way through the waveguide, insert copper shim stock to the depth giving desired performance, and solder the shims in place. Alternatively, the iris vanes (shim stock) may be inserted in slots in an aluminum mandrel and the walls then electroformed around them. Either method gives satisfactory performance, using graphical design data to obtain the spacings (sec Microwave Engineers Handbook and Cohn, 1957). 9.2.5 Circular polarizers. A basic arrangement for transforming between linear and circular "polarization is a quarter-wave plate oriented at 45° with respeel to the linear polarization. In waveguide terms, a quarter- 320 Microwave hardware and techniques Chap. 9 2,2 2.1 11 2.0 1.9 1.8 1 Odd CjiT Ever 0 0.1 0.2 0.3 0.4 0.5 0.6 Eccentricity e FIG 9.14 Mathieu-function roots for TEn-Iike modes in elliptical waveguide, as functions of eccentricity. wave plate is a section of waveguide capable of supporting two orthogonal modes having different phase velocities such that the respective electrical lengths differ by one-quarter wavelength. A simple technique is to deform circular waveguide (TEU mode) into an elliptical cross section. The cut-off wavelength in elliptical waveguide is given by lira (9.2.1) where a is the semimajor axis and rlm is a Mathieu function root. These roots are given for TEn-like modes in Fig. 9.14 as functions of eccentricity. The length of a quarter-wave section is given in Fig. 9.15 as a function of wavelength, perimeter, and eccentricity (Blau and Heald, 1959). In practice, quarter-wave sections of this type can be made by electroforming on a precut mandrel or by mechanically deforming a section of circular waveguide, with gradually tapered transitions from circular to elliptical cross section to suppress reflections. End effects from the transitions usually require a linal empirical adjustment by further deformation. The useful frequency band width is about 4%. 9.2 Special components 321 1Q0 50 20 10 0M 0.2 1.6 0.1 e 0.2 03 04, / 2.0 2.5 3.0 3.5 4.0 FIG. 9.15 Dependence of length L on wavelength A for waveguide quarter-wave sections in elliptic waveguide of eccentricity e and perimeter p. Waveguide quarter-wave plates may also be made by dielectric or ferrite loading. Wide-band designs have been developed (Ayres et al., 1957), as well as designs in which the quarter-wave plate can be rotated 45° to provide the option of circular or linear polarization. Such a polarizer, used as a horn feed, is shown in Fig. 9.16. A system for transmitting simultaneously in the two counterrotating circular polarizations can be made by using a polarization diplexer with the quarter-wave section. Figure 9.17 illustrates such a system in which both modes arc fed from a single power source. The transmitting antenna launches a linearly polarized wave which is equivalent to two counter-rotating circularly" polarized waves of equal amplitude. The receiving antenna analyzes the clfcct 6l the plasma region between the horns in 322 Microwave hardware and techniques Chap. 9 FIG. 9.16 Components used to launch circularly polarized waves. Bottom, rectangular to circular transition; center, rotatable quarter-wave section; top, circular horn antenna. (Courtesy DeMornay-Bonardi Corp.) changing the relative phases, amplitudes, and polarizations of these waves. Another interesting circular polarizer is the turnstile junction (Montgomery et al., 1948; Meyer and Goldberg, 1955). The basic junction, sketched in Fig. 9.18«, is a six-terminal network, consisting of four Transmitting horn Rectangular guide Receiving horn S Quarter-wave section Circular (or square) guide Fin-line polarization diplexer Transistious back to rectangular guide FIG. 9.17 System for diplcxing circularly polarized waves from a common source, using a polarizer and lin-line coupler. 9.2 Special components 323 rectangular waveguide arms and one circular waveguide arm excited in two orthogonal TEn modes. Two pins in the junction, concentric with the circular waveguide axis, are used to match the junction. Tn the matched condition, and with all arms properly terminated, power entering arm 1, denoted by Ex, will divide, half of it creating EA in the circular arm and half of it splitting between E3 and Et. No power reaches arm 2. Likewise, power entering arm 3 splits between arms 1 and 2 and creates EB in the circular guide. The junction is thus useful as a polarization diplexer, as well as for a number of other applications; we shall discuss, here, its use as a circular-polarization analyzer. To generate a right-hand circularly polarized wave, we short-circuit arms 3 and 4 at 5Ag/8 and 7AJ8, respectively, and drive the junction from input 1, as shown in Fig. 9.18ft. The wave emitted will have a clockwise rotation sense. As a receiver, a circularly polarized wave having a counterclockwise rotation sense will be coupled into the side arm 2 and one having a clockwise sense into arm 1. If matched detectors are put on the two arms, we now have a direct measurement of an elliptically polarized wave entering the junction. Matching posts (a) FIG. 9.18a Basic turnstile junction, showing the four rectangular waveguide arms and the circular waveguide. The matching post is used to isolate opposite arms and hi mutch impedances. 324 Microwave hardware and techniques Chap. 9 9.2 Special components 325 Ili^.Jv,',V^k; Arm 3 shorted | FIG. 9.18b Turnstile junction connected to transmit or receive right-hand or left-hand circularly polarized waves. The addition of a conical horn to the circular junction completes the instrument for polarization analysis (Allen and Tompkins, 1959). 9.2.6 Resonant cavities. Cavities are used in a number of applications. Calibrated wave meters measure the frequency; coupled cavities are used in filters; low-g cavities are employed in some waveguide high-mode couplers; and evacuated resonant cavities are used to heat and measure density of plasmas, as discussed in Section 5.1. A common configuration is the right circular cylinder, sketched in Fig. 9.19. The wavelength of the resonance frequency for TM modes is (Moreno, 1948) 2h A„ = [S2 + {hja)\P„J-nf\' (9.2.2) where h is the height, a the radius, S is the integral number of nodes along h, Pm„ is the nth root of Jm(P) = 0, in is the number of circumferential maxima, and n is the number of radial maxima. 1.9-, 1.8- 1.7- 1.6- 1.5- "yTMn ■vTH, 1.4- oTM02i lt411 o TM 022 oTE122 °TM212 °TE3 ° TEn TM023 §TElM, TE4]3 TM213 0te313 °TEoi3,TMii3 °TE2i3 8TM013 .TMii2E"3 1- 0 0.5 1.0 °TE3n °TE0]1,TMm oTE °TE2i2 oTM012 6TEm 1.5 1.3- 1.2 \.\-_ 1.0-0.9 0.8-1 0.7 o.e-oh I'M 1 211 oTMon oTEm 03 Jf'TMn la Example: 2a/X h 1.08 TM Za/k 1.5 -2.0 Given: lajh Find: 2o/X 1.5 1.08 ■2.5 PIG, 9.19 Mode chart for resonant cavity modes in a right circular cylinder. i< 'i.iificsy K. N. Ilraccwell, Stanford University, Stanford, Calif.) FIG. 9.21 Gain parameters Cm and G8 for linear rectangular horn antennas. (From SchelkunofT and Friis, 1952, courtesy of Bell 5 Telephone Laboratories.) u, IS 328 Microwave hardware and techniques Chap. 9 32 FIG. 9.22 Gain of conical horn antennas. (From King, 1950, courtesy of the Proceedings of the Institute of Radio Engineers.) The Q of a cavity depends upon the ratio of skin depth 8 to wavelength (see Fig. 9.1). For TE modes, the Q is generally maximum when the cavity height and diameter are about equal, as shown in Fig. 9.20. Further extensive data on resonant frequencies, mode configurations, Q, re-entrant end walls, and coupling techniques can be found in several texts (Montgomery, 1947; Wilson et al, 1946-1947; and Moreno, 1948). Cavities that must be evacuated require vacuum seals for electrode leads and waveguides. These considerations are discussed in Section 9.6. 9.3 Antennas and radiators 329 9.3 Antennas and radiators 9.3.1 Horns. One of the most useful microwave antennas is the pyramidal horn. A horn, gradually flared over several wavelengths, matches the wave impedance of a waveguide (typically 450 to 500 ohms) to that of free space (377 ohms) and provides directivity to the radiated wave. The directivity, expressed as gain over an omnidirectional antenna, depends primarily upon the height and width (in wavelengths) of the final aperture, but also secondarily upon the length. The gain of a horn flared in both the E-plane and H-plane is given as (Schelkunoff and Friis, 1952) ? GmÁ GCX :32 h a (9.3.1) where Gm and Ge are gain factors given in Fig. 9.21. Gain data for circular (conical) horns are given in Fig. 9.22 (King, 1950). For both types of horns, at a particular length, there is a particular aperture for which maximum gain is achieved. A horn having these dimensions is called an optimum horn. Conversely, for given aperture dimensions, the gain does not increase appreciably with increasing axial length beyond the long horn length of abjX. In most plasma diagnostics experiments, where aperture diameters must be restricted, the long horn is preferred in order to achieve reasonably good gain and minimum radiation field curvature. The curvature can be further corrected by using a lens in front of the horn (see Section 4.9). The long horn also has a larger effective area for radiation, the area S being about 80% of the geometrical area. The optimum horn has S about 50% the geometrical area. The angle off-axis at which the radiation intensity has dropped 3 dB is related to the gain /3 x 104\ % %s«Fi^| degrees. (9.3.2) Several of the horn-design parameters are summarized in Fig. 9.23. We note in passing that, at millimeter wavelengths, horns having modest gains (for example, 10 to 15 dB) are not physically large. As pointed out in Section 4.9, the maximum permissible coupling between two horns is limited by Fresnel interferences to about —6 dB. The presence of the plasma, because of its refractive index, may cause refraction effects that change the effective coupling. Often, the coupling is seen to improve by a decibel or so when a dilute plasma is present in the transmission path. The effect is not signal amplification, but is due simply to ehhance"c1 coupling or better impedance matching, unless there be some amplifying mechanism also present. FIG. 9.23 Summary of horn-design parameters. 9.3 Antennas and radiators 331 FIG. 9.24 Metal plate (artificial dielectric) microwave lens. The electric Held is parallel to the metal plates. '1.3.2 Lenses. Two modified forms of the ordinary horn find applications in many experiments: the horn-lens and the horn-fed dielectric rod. Microwave lenses are classed as dielectric, artificial-dielectric, or metal-plate. The metal-plate or "vcnetian-blind" lenses are made or parallel strips of metal of varying width, supported parallel to the incident electric field, as shown in Fig. 9.24. The phase velocity of the wave between the plates is greater than c (equivalent refractive index less than unity), so that a concave lens focuses (Kunz, 1954). Short-focal-length, stigmatic lenses that bring the waves to a line focus on the axis of a plasma column are straightforward to make by this method. The properties, including focal length, are frequency-sensitive, since it is the plate spacing that determines the equivalent refractive index (Brown, 1953). Dielectric lenses may be simply convex, as the common optical lens, or made up of zones (van Buskirk and Hendrix, 1961). The relation between a lens and Fresnel zone plates is shown in Fig. 9.25. The simple /one plate rejects half of the incident power because alternate zones are clear and opaque, as shown in Fig. 9.26. The phase-reversing zone plate is made by arranging the thickness of dielectric material in adjacent zones to rive a phase difference of n, The full transmission then results. 332 Microwave hardware and techniques Chap. 9 15 Simple zone plate Phase-reversing zone plate Quarter-period zone plate Fresnel lens Sim pie lens FIG. 9.25 Relationship between a Fresnel zone plate and a microwave lens. (Courtesy Electronic Communications, Inc.) FIG. 9,26 Fresnel zone plate for use as a microwave lens. (Courtesy Electronic Communications, Inc.) 9.3 Antennas and radiators 333 FIG. 9.27 Dielectric microwave lens, using impedance matching grooves. (Courtesy TRG, Inc.) Wave launching horn 7-/-rr f I V^. ^-Dielectric lens O-Ring vacuum seal" I ^Plasma •r" Vacuum chamber Wave receiving system PIGi 9.2K Microwave lens used as a vacuum window in a plasma experiment (see also Fig. 6.23). (Gardner, t962.) 334 Microwave hardware and techniques Chap. 9 9.3 Antennas and radiators 335 Horn-lens combinations ordinarily require far-field illumination, or at least as long a horn as possible. They are useful for either focussing the waves at some point in the plasma experiment, or to correct for wave-front curvature. The horn-lens shown in Fig. 9.27 has lineal grooves, which are A/4 deep, but evenly spaced, to reduce reflections from the surface of the dielectric convex lens. The lenses may serve as vacuum seals, as shown in Fig. 9.28. In vacuum systems that do not require baking, the flat side of the lens is sealed with an O-ring, the convex side being illuminated by a horn. At 4 mm wavelength, a lens of 4 inches diameter is found to be quite satisfactory (Gardner, 1962). Some reflections, due to the dielectric interface, are present; a matching layer (analogous to coated optics) could be added to correct for these at the expense of band width, or the grooves shown in the lens of Fig. 9.27 could be used. 9,3.3 Dielectric rod antennas. The waves propagating in a dielectric rod (see Section 9.1.2) tend to excite radiation fields when discontinuities are present in the line. To intentionally make a radiator, then, one shapes the dielectric radiator cross section to achieve both the desired radiation pattern and a good impedance match to the feed waveguide (Mueller and Tyrell, 1947). Dielectric rods can support various modes, as shown in Rectangular 'waveguide (a) (b) (c) FIG. 9.29 Dielectric rod antennas, showing launching mechanisms. Dimensions arc discussed in the text in Section 9.3.3. Fig. 9.5. One mode that closely approaches the circular TEU waveguide mode is the dipole or HEn mode (Brown and Spector, 1957). The mode may be launched by simply inserting a tapered dielectric rod into an open waveguide (Harvey, 1963), as shown in Fig. 9.29«. A circularly polarized wave at 35 Gc, launched by a teflon Tod having íí= 1.0 inch, b = 1.0 inch, = 1.125 inch, > ±1 £ ps 3 -a *0 o o pí 9 -ii — a JO ■ 5 -? Ó 3. i—i I I I 348 Microwave hardware and techniques Chap. 9 sensitivity vs. receiver band width for various crystal figure-of-merit values. The figure of merit (FM) is obtainable from the manufacturer. Typical sensitivities vary from -57 dBm at 2.5 Gc (FM=200) to -40 dBm at 300 Gc (FM=4), for 10 Mc video band width. For 100-kc video band width, the sensitivities are improved to -67 dBm and -50 dBm, respectively. Further improvement can be realized at the lower microwave frequencies by placing a low-noise traveling-wave amplifier before the detector (Taussing, 1962; Wade, 1961). The crystal detector often is mounted directly on the video preamplifier-chassis to provide minimum capacitive shunting and to lessen the likelihood of stray pickup. If the preamplifier uses vacuum tubes, the detector must be mounted well away from them to avoid heating the crystal; temperature increases both degrade the noise figure and lower the power handling capabilities (typically <5 milliwatts CW). Video preamplifiers should be designed to have minimum input capacitance and a smaller noise contribution than the detector itself. Circuits, such as the cascode, both for vacuum tubes and transistors, are ideal from both standpoints. FIG. 9.41 Crystal video detector and video preamplifier. Amplifier voltage gain is 400 over a broad video band of 100 cps to 2 Mc or 1000 to 2000 over narrow, tuned bands at 10 kc, 100 kc, and 1 Mc (see circuit diagram in Fig. 9.49). (Photograph courtesy of General Atomic, San Diego, Calif.) 9.5 Signal detection 349 D-c heaters for vacuum lubes, and ripple-free, well-regulated power supplies are necessary, since the signals being amplified are as small as a microvolt An example of a crystal video receiver is shown in Fig. 9.41. The cascode triodes, followed by a pentode and a cathode follower, have an over-all voltage gain of 400, over a band from 100 cps to 2 Mc. The output can then be fed directly to an oscilloscope. To obtain the full video band width, the crystal detector may have to be shunted with a 5000-ohm or smaller resistor to reduce the RC time constant. This compromises the figure of merit, hut improves the relative response above ;i megacycle or so. <>.5.2 Superheterodyne receivers. The microwave superheterodyne has considerably increased sensitivity over the simple crystal video detector (Harvey, 1963). The threshhold signal magnitude is related to the noise figure of the receiver, as shown in the curves of Fig. 9.42. Typical noise figures for balanced mixers at centimeter to millimeter wavelengths lie between 10 and 18 dB, yielding sensitivities of -90 dBm to -100 dlím for usual video band widths, and -120 dBm for very narrow band widths and coherent detection (Smith, 1951; also see Section 9.5.5). The minimum detectable signal power may also be expressed as an equivalent noise temperature Tn of the receiver, the conversion factor being derived from (7.2.8) p . rp _J mm (9.5.1) For i°raI„=l picowatt (—90 dBm) and 4/= 10 Mc hand width, the equivalent noise temperature is r„ = 7250 °K = 0.625 eV. The greater sensitivity of the superhet over the simple video detector is iluc mainly to its higher intermediate frequency. Since, for a given video band width, the shot noise is roughly proportional to 1//, the choice of a high intermediate frequency reduces the over-all noise figure, l.f. amplifiers, however, have worsening performance above 20 Mc or so, requiring it compromise in intermediate frequency; the optimum value, for normal hand widths from 2 to 10 Mc, is between 30 and 60 Mc (Rennie, 1957). The over-all receiver noise figure depends also on the mixer conversion efficiency, which improves with higher crystal current (Pound, 1948). Since the shot noise becomes worse with higher crystal currents, a compromise again is required; the optimum crystal current, usually, is found h) lie between 0.2 and 0.6 milliamperes, corresponding to a local oscillator power of about a milliwatt for a balanced mixer (2 crystals). The current (lowing in mixer crystals raises their operating point on the l-V characteristic ctirve well out of the square-law region. For small llgndlSi the output voltage is Huts nearly linear with input signal, rather 9.5 Signal detection 351 than quadratic as in direct video detection. For this reason, superheterodyne systems have much greater dynamic range than crystal video systems, besides having considerably better sensitivity (Tauband Giordano, 1954). Harmonic mixing provides a useful detection method for millimeter wavelengths where local oscillators are not available. The local oscillator is operated at a subharmonic of the frequency at which detection is desired. The pulses in the rectified crystal current contain harmonics that mix with the incoming signal to produce beats at the intermediate frequency (Johnson, 1954). Silicon and gallium arsenide crystals seem to give best results, yielding over-all noise figures intermediate between straight superhct receivers and crystal video receivers. Harmonics as high as the fifth and sixth give useful mixing action. The superhet has some disadvantages, however, the chier one being the need for a local oscillator. For transmission and scattering experiments, in order to keep the receiver tuned to the transmitter, it is usually necessary to provide automatic frequency control or AFC (Jenks, 1947), requiring still further circuit and operational complication. The reflex klystron is a convenient local oscillator for use with AFC because it is voltage-tunable over a small band. Changing the transmitter frequency over a large excursion requires retiming the local oscillator. A backward-wave oscillator is voltage-tunable over a wide range and, with proper adjustment, can be made to track a transmitter over its entire tuning range. Ordinarily, the mixer assembly requires peaking for each particular frequency, although some mixers retain low noise figures over a 6% band. A swept-frequency superhet can be made by sweeping the local oscillator electrically or mechanically. A well-matched, broad-band mixer, such as a short-slot hybrid or rat-race, is necessary for this type of service. An application requiring a swept receiver was mentioned in Section 8.3 for examining the electron gyro-frequency harmonics. 9.5.3 Parametric mixer-amplifiers. The parametric amplifier (Bloom and Chang, 1957), in some respects, resembles a superheterodyne receiver; the pump is analogous to the local oscillator, and the nonlinear reactive element is analogous to the mixer. The idler frequency appears at the sum and difference frequencies, for down-mixing, just as does the intermediate frequency of a superhet. In its component form, the microwave parametric amplifier uses a nonlinear capacitor, often a silicon p-n junction Detractor with reverse bias so that it forms a voltage-variable, capacitive depletion layer, but draws no current. The resistive component of the "mixer" is thus practically absent, and the shot and Johnson noise contributions are very'small (van der Ziel, 1959). Equivalent noise temperatures of (mils operated under ambient conditions may be as low as 300 "K 352 Microwave hardware and techniques Chap. 9 and, when cooled to liquid nitrogen temperature (77 °K), the noise temperature may be reduced to about 80 °K. In addition, since the nonlinear element has little loss, the over-all conductance can be made negative by proper choice of operating conditions. Instead of "conversion loss," as in a diode mixer superhet, the unit then exhibits "conversion gain" over a narrow frequency spectrum. The varactor usually is mounted in a resonant structure, which makes tuning difficult. A typical A"-band parametric amplifier, pumped at 10.15 Gc, exhibited a gain of 50 dB over a band width of 2.4 Mc at an input frequency of 8.5 Ge, with an equivalent noise temperature of 600 °K (Bossard et al., 1960). Generally speaking, parametric amplifiers, including traveling-wave and distributed-line1 units (Mount and Begg, 1960), have the widest application at the UHF and low microwave frequencies in lixed-tuned operation. Typical output powers are between I and 500 milliwatts. As components and techniques improve, the upper cut-off frequencies are being raised, and amplification at frequencies as high as A>band or the 8-mm band (35 Gc) has now become possible (de Loach, 1960). Other nonlinear Teactive elements, such as ferrites, electron beams, garnets, and various solid-state, crystalline substances, are also finding application (Mount and Begg, 1960). 9.5.4 Miscellaneous receiver systems. Quantum-mechanical amplifiers or masers arc useful for very low-level amplification at particular frequencies corresponding to the energy gaps in the molecules of various solids, liquids, and gases. To allow the levels to be populated to high enough concentrations to give spontaneous emission, the element must be cooled, often to liquid helium temperatures. In. principle, at 0 °K, the maser should be capable of detecting individual microwave photons (Weber, 1959). To achieve level-splitting or paramagnetic resonance, a magnetic field is required. The maser, thus, is not a simple amplifier to operate but, for some critical applications, such as radio astronomy, it provides amplification at noise temperatures of a few degrees Kelvin and frequency stability unmatched by other devices (Gordon et al., 1955). Tunnel diodes (Esaki, 1958) will oscillate if placed across a tuned circuit because they possess negative resistance. When isolated by nonreciprocal elements, such as a circulator, however, they act as amplifiers, having noise figures as low as 7 dB. Gallium arsenide diodes have been used as Esaki amplifiers as high as 26 Gc (Holonyak and Lesk, I960). The diodes are low-power elements, operating at levels up to a milliwatt or so. Thermionic diodes are useful as low-level detectors up to about S-bancl (3 Gc) before transit-time effects degrade the efficiency. At higher power levels (100 watts to 100 kW pulsed power), coaxial diodes, in which the 9.5 Signal detection 353 thermionic cathode is the center conductor, have been constructed with waveguide inputs for frequencies from 3 to 35 Gc (Hawkins et al., 1958). Commercially available units have large enough voltage outputs to deflect a cathode ray tube directly, with 0.1 microsecond response times and, thus, may be used directly for power oscillator monitoring. 9.5.5 Microwave radiometers. Receivers intended for reception of thermal or quasi thermal radiation in the microwave spectrum are called radiometers (Harris, I960). The received signals have the characteristics of noise, generally over a broad frequency band, leading to the name white noise. The signals may be steady, slowly fluctuating, or transient. Receiving antenna ^ X^Receivi 'irS^Vacnum T ^ window Calibrated A AAfAf attenuator >vVV Balanced miser Fertile chopping switch i - i i ľ? Low noise balanced i.f. amplifier '-- 3D Mc Noise source (nur. discharge tube et lint lead in : ■ ■ ■ I: I-.- j Chopping frequency generator Local oscillator (swept or CW) Second detector, video amplifier Synchronous detector-amplifier Narrow-hand filter-amplifier V Integrating circuit r = D.Ol sec Direct . wide-hand video output Medium lime-constant video outpul Long time-constant d-c output lUi. 22 dB decrease) are obtained by making the lattice members with random widths or by letting points protrude a wavelength or so. Still other types of absorbers consist of an array of absorbing spears, set side by side, or bundles of short lengths of glass tubing, on which metal has been evaporated or colloidal graphite painted. 9.7 Circuitry considerations 9.7.1 Electronic circuits. Most of the electronic circuitry used in the transmitters, receivers, control and timing chassis, and power supplies required for microwave diagnostics are more or less standard. Power 364 Microwave hardware arid techniques Chap. 9 supplies, generally, must be well regulated and completely free of ripple and noise (. The circuit sketched in Fig. 9.49 includes a switch to select one of several tuned frequencies or broad-band video amplification. Power supplies for klystrons and for crystal video receivers often require regulator on top of regulator to eliminate the hum and noise. D-c heater supplies for the microwave sources and preamplifier tubes are generally desirable. 9.7.2 Circuit interference and stray pickup. Stray pickup around plasma experiments is a constant source of concern. The large pulsed currents (kiloamperes to megamperes) used in controlled fusion experiments make ground-current loops a major problem. Fortunately, a waveguide run can be broken up by inserting insulating films between choke llanges or by using short lengths of dielectric rod. Video crystal mounts especially must have insulation breaks, since even small currents flowing in the low level video-current return paths can induce serious interference. An example of a voltage isolation section in WR-28 (RG-96/U) waveguide is shown in Fig. 9.50. Mylar film 0.002 inches thick separates the flanges, which are fastened with nylon screws. The assembly holds off 1500 volts dry and 3000 volts when the holes through the mylar are filled with a silicone grease. The VSWR is 1.05 at 35 Gc, and the loss is less than 0.5 dEt. To hold off higher voltages requires external clamps (no holes in the insulating sheet) and a thicker sheet. Flange alignment then becomes 9.7 Circuitry considerations 365 FIG. 9.50 Waveguide voltage isolation section. Loss is 0.5 dB and VSWR is 1.05 at 35 Gc. Hold-off voltage is 1500 V. (Courtesy of General Atomic, San Hiego, Calif.) a problem and with sheets thicker than about 0.005 inches there is some leakage as well as reflection. Screen-cage enclosures are generally helpful for the diagnostics system tilthough the main requirement to eliminate pickup is simply to make all ground returns have low inductance paths. Running all waveguides, trigger cables, etc., so that they are well grounded as they pass through a large metal sheet or a metal wall usually suffices. In addition, the power lines, oscilloscope trigger lines, and timing signal lines may need decoupling filter networks or transformers. Coordination with the other diagnostic instrument leads is also necessary. Direct radio-frequency interference hy electromagnetic radiation is seldom encountered, but occasionally spark gaps radiate "hash" (broad-band rf noise) and power tubes used in modulators, etc., may have parasitic or Barkhausen oscillations that radiate to sensitive circuits. CHAPTER 10 General plasma diagnostic techniques 10.1 Tabulation of some useful diagnostic techniques Several diagnostic techniques yield information similar to that obtained by microwaves, that is, the plasma electron properties. Many of the same experimental requirements also apply, such as shielding against stray pickup, multiple-channel simultaneous measurements, and the need for simplified, automatic data presentation. Some measurements involve internal probing, such as Langmuir and magnetic probes, which may perturb the plasma, while other external measurements, such as optical diagnostics, are nonperturbing. Some techniques are useful only in dense plasmas or plasmas containing a strong magnetic field. To sort out the measurements useful for a given kind of determination, brief tabulations of several techniques, grouped according to applications, will be helpful. Techniques useful for ion diagnostics and other plasma measurements are also included. The ranges of applicability shown are not necessarily exclusive, but are intended only as qualitative guides. Elaboration on a few of the techniques will be given in subsequent sections. A much more extensive discussion may be found in Huddle-stone and Leonard, ed., Plasma Diagnostic Techniques (Academic Press, New York, 1965). ELECTRON DENSITY AND DISTRIBUTION (a) Microwave interferometer; 1010<«,,< JO14 cm-3. (b) Microwave cavity perturbation; 108<«e<1012 cm-3. (c) Rf-conductivity probes; 1081016 cm"3 for 10,000 gauss. (g) Optical spectroscopic intensities; ne>W2 cm-3; equilibrium plasmas. (h) Optical scattering; 1014<«e< 1019 cm-3. (i) Optical Balmer series limit; I0130.1 eV; stable plasmas. (/;) Doppler broadening of cyclotron radiation line; 7^50 eV. (c) Infrared and optical intensities; rc>10 eV; equilibrium plasmas. (d) X-ray intensities; Te>6 keV; wall problems. (e) Relative intensities of spectral lines; 1 < T..< 50 eV. (/) Relative intensities of bremsstrahlung and recombination radiation. (g) Doppler broadening of optical (Thomson) scattering; Te>5 eV. (/?) Langmuir probes; 0.1 < Te< 1000 eV; moderate densities. li )N DENSITY AND DISTRIBUTION (a) Stark broadening of spectral lines; Atj>1015 cm-3. (b) Langmuir probes, single and double. (c) Electron, ion, neutral atom, or neutron beam probes; n{> 1014 cm-3. (d) Diamagnetic effect (requires knowledge of temperature). {<■) Alfven and sound wave propagation; dense plasmas. (/) Calorimetry (requires knowledge of temperature). (g) Radioactive gas tracers and collimatcd detectors. (It) Charge-exchange neutral detectors. ION TEMPERATURE AND ENERGY (a) Calorimetry; total energy and momentum. (b) Doppler broadening of spectral lines; r;>5 eV. (c) External energy-momentum analyzer; samples escaping ions. ( I ■. 111 iniliiri/alion prism. (Ctmrlcsy of W. E, Quitin, G. A. Sawyer, and F. L. Ribe, I . n Alamos Scientific Laboratory] University of California, Los Alamos, N.M.) 372 Generat plasma diagnostic techniques Chap. 10 FIG. 10.2b Mach-Zehntler interferometer presentation as in Fig. 10.2«, but showing the turbulence due to a flute instability, developing at high compression ratios. (Courtesy of W. E. Quinn, G, A. Sawyer, and F. L. Ribe, Los Alamos Scientific Laboratory, University of California, Los Alamos, N.M.) A similar optical arrangement is used for Schlieren photography, except that changes in refractive index are recorded as modulations of light intensity, rather than interference fringes. An experimental arrangement for studying shock waves is shown in Fig. 10.4 (Lovberg, 1963). The light source is a spark between tungsten electrodes in nitrogen. The accurate timing necessary to follow the fast front is obtained by a Kerr-ceil light shutter. A typical photograph is shown in Fig. 10.5, showing the sharp electron density gradient in the "snow plow" front. From Table 10.1, it is apparent that greater sensitivity for plasmas of medium density is obtained in the far infrared. Golay cell and bolometer 10.2 Optical and infrared prohing 373 FIG, 10.3 Streak intcrfcrogram of the refractive index changes vs. time in an electromagnetic shock tube. The plasma is generated by a conical pinch driver (Scott and Josephson, 1957) in hydrogen at 2 torr. The Mach number was 20, at 18 inches down stream from the driver. The initial positive deflection of fringes ( —4u) is due to an increase in neutral density by compression and dissociation, followed by the negative deflection ( —<4f) due to the electrons in the plasma. Peak electron density was 5.7x 10lc cm-3. (Courtesy of A, F. Klein, Aerospace Corp., I (is Angeles, Calif.) defectors are useful at wavelengths all the way across the infrared to the microwave band, although their response times arc ofthe order of a second. Nevertheless, some plasma experiments have been done at infrared wavelengths (Brown, 1962; Harding et al., 1961) with results that compare favorably with microwave results. Optical Faraday rotation can be used to study dense plasmas in strong magnetic fields (Dougal, 1963). The total rotation is given by (6.5.7), where the collisionless linear approximation is easily justified for these frequencies. Equation (6.5.7) then reduces to ^=180° L n B\BU (10.2.1) where B is the magnetic field and BK is the magnetic field necessary to give |>.yroresonance, Blt = (omle. As an example, the plasma density in a ID cm path with 20 kilogauss applied, necessary to give 90° of rotation at A 2.6 microns is 4,36xlOlit cm"11. The sensitivity increases in direct 374 Generat plasma diagnostic techniques Chap. 10 Plasma acceleration chamber Entrance lens No. 1 Knife edge Acceleration plates Mo, 2 Knife edge vFront-surface mirror Light source Camera FIG. 10.4 Experimental arrangement for studying fast current sheets in a parallel-plate plasma accelerator by Schlieren photography. (Courtesy of R. Lovberg, General Atomic, San Diego, Calif.) proportion to the density, the path length, and the magnetic field applied. Crossed polarization plates, sensitive to rotations as small as 5°, are experimentally feasible. 10.3 Conductivity probes Plasmas having high collision rates have appreciable real components of conductivity, as can be seen from (1.3.14). A small rf coil immersed in such a plasma will induce currents in the plasma. The current is complex, the real part extracting power, and the imaginary part changing the coil's effective inductance by diamagnetic effects. Figure 10.6 shows two methods to measure the effect of the plasma on the coil. In Fig. 10.6a, the voltage across the coil is a measure of its impedance, and the phase angle in respect to the input voltage gives the relative magnitudes of /( to Jr. In Fig. 10.66, the unloaded coil is resonated at the drive frequency f0. The plasma current then both detunes and de-g's the circuit, much as in the resonant cavity cases in Section 5.1. Equation (5.1.2) may he used to calculate the loading as a function of conductivity, When 10.3 Conductivity probes 375 PIG. 10.5 Schlieren photographs of a current sheet, traveling at 7 cm/psec between .....III Id plates in a plasma accelerator. (Courtesy of R. Lovberg, General Atomic, Nan Diego, Calif.) 376 General plasma diagnostic techniques Chap. 10 FJG. 10.6 Conductivity probes for high collision rate plasmas. In (a) the changes in coil impedance are measured. In (b) the changes in Q and f0 of a resonant circuit arc measured. C2 > lOd.; R X u>L. r»to and a>„»jD, the major effect of the plasma on the coil is in lowering its Q. If the coil is immersed in the plasma so that a is uniform in space, (5.1.2) may be approximated as 1 QM Q0~ ."JA £0OJ (10.3.1) In most experiments, however, o- is not uniform, since the coil is mounted on a form or otherwise obstructs the plasma. Calibration is then conveniently carried out with ionic solutions (van der Pol, 1920), whose conductivities can be measured directly with a platinum electrode conductivity cell. For example, the data of Fig. 10.7 were obtained in an experiment at Livermore, Calif. (Wharton and Hawke, 1962) from a coil wrapped around a section of pyrex plasma chamber, which had been removed and filled with a conducting solution. The Q was measured by observing the width 8f at half-height of the frequency response, where Mr (10.3.2) 10.3 Conductivity probes 377 150 0.5 1.0 Conductivity [mhos/meter] III.. 10.7 Typical rf conductivity probe calibration data. The changes of Q of n Domini circuits due to the conductivity of enclosed ionic solutions, for frequencies ..I I, 5, and 14 Mc are shown. (After Wharton and Hawke, 1962.) I In- center frequency f0 was held constant by trimming Cj slightly as the conductivity was varied. Q0 was 105 for the 5-Mc coil and 85 for the II Mc coil, with distilled water in the chamber. A Faraday screen inside ihc coil helped reduce electrostatic effects between the coil and the solution, n in I the coil and flTc plasma, when later the assembly was used to study liil'li density plasmas. 378 General plasma diagnostic techniques Chap. 10 10.4 Langmuir probes A conducting probe immersed in a plasma will emit or collect current, depending upon the voltage impressed (Tonks and Langmuir, 1929; Chen, 1965). The technique has been extensively used and in many cases the quantities measured compare very well with those obtained by other means (Schulz and Brown, 1955; Talbot et al., 1963). 10.4.1 Single Langmuir probe, no magnetic field. Typical characteristics for single probes, whose dimensions are small compared to electron and ion mean-free-paths, are shown for a plasma, in the absence of magnetic 10.4 Langmuir probes 379 Reference probe (or wall) Glass Probe *'P Movable vacuum seal 1 Straight probe rJ (Current density) (J + [J+I> + 10 Volts FIG. 10.8 Langmuir probe V-l characteristics, (a) Linear characteristic, (b) Logarithmic characteristic, (c) Double floating probe characteristic. Glass 90" (bent) probe Disc- Disc probe Double floating probe ■ FIG. J0.9 Some typical Langmuir probes. Glass capillary tube Bold, in Fig. 10.8. Sketches of some typical probes and their geometry hi- given in Fig. 10.9. The probe potential is measured in respect to some convenient, fixed-pOtential point, such as the anode or walls of a discharge tube or a floating "wall probe," which has an area at least 50 times as large as the probe Itielf. The requirement that the potential difference between the plasma and I he reference point remain fixed often excludes the use of the anode or ■ dumber walls for this purpose because of current fluctuations. We shall presume, for our fTurposes, that the probe potential can be specified in icpecl lo the plasma potential, and potential V refers to that value. 380 General plasma diagnostic techniques Chap. 10 When V\s made very negative, all electrons are repelled and only ions collected. The random ion current passing through an area A in the plasma is related to the ion density and the velocity _/+_« + evth_n+e l2kT\Vl (10.4.1) where /., is the random ion current, amp, y.,. is the random ion current density, amp/m2, Ap is the area of the probe, m2, m+ is the ion mass, 1.67 x 10~27 kg for protons, n+ is the ion density, no./m3, Dm = (2/r77m+)1'/2 = mean kinetic ion velocity, m/sec. Equation (10.4.1) would be valid also for ion current collected by a probe if the presence of the probe caused no perturbation in the surrounding random plasma currents. The probe does perturb the plasma, however. The volume which the probe occupies provides an energy sink for all particles which strike it and the fringing fields extend for a considerable distance into the surroundings (Genevalov, 1959; Bernstein and Rabinowitz, 1959). As a result, the collected ion current density seems to be more a function of the electron temperature than of the ion temperature. This effect is due to the formation of a positive sheath around the probe. The extent of the sheath's influence is determined by the electron temperature. For TtV) + AtJ0 exp(gAK) (10.4.8) where Ali2 are the areas of the respective probes, JQ is the random electron space current density, yi:2 are the plasma-to-probe potentials, * k% re[eV]" 11,600 n[°K] I he logarithm of (10.4.8) is ■4>VD+ln (10.4.9) The slope of (10.4.9) plotted against I ,; yields the electron temperature, just as in Section 10.4.1. I(1.4.6 Double probes in a magnetic field. Double probes are not influenced as much by magnetic fields as single probes. The collected i in rent is governed by ion mobility, and it is not until the magnetic field Itrength is very large (several thousand gauss) that the ion gyroradius is as miiiiII as the probe size. When the negative voltage is sufficiently large that the sheath thickness [| Luge compared to the probe diameter, the theory of Langmuir and Motl-Smith (1924) applies, even in magnetic fields up to several thousand Uiiuss. The current-voltage relationship then follows a square law leV F+ TT [kT .1, is linear with ion density, as shown in (10.4.1). (10.4.10) The ion density, far out in region I, then is found to be (Langmuir and Mott-Smith, 1924) (2irm A'''* J+ kT, j e = 3.32x 101 A {in,,) (10.4.11a) (10.4.11b) 386 General plasma diagnostic techniques Chap. 10 where S is the slope of I2 vs. V expressed in amperes and volts, and A is the effective collecting area of the probe sheath. Gardner (1962) expresses the relationship in terms of current density, not specifying^, but assuming that it remains constant as V varies 1 d(J+2) 7rsm. dV 2e* (10.4.12) The circuit of Fig. 10.10, in the switch position marked "Square," gives direct readings ofn + 2, when the cursor slope has been calibrated. 10.4.7 Double probes: miscellaneous effects. Although floating probes represent less of a perturbation than a single probe in an equilibrium discharge, when the ion temperature is well above the neutral gas temperature any kind of an object in the plasma leads to severe energy loss. The probe element suffers from bombardment and, even though it floats, it possesses capacitance, which tends to hold its potential fixed long enough to expose it to arcing, which of course melts the elements away. 10.5 Plasma wave and resonant probes When plasma waves or oscillations are present, they may be detected with probes. Spacecharge waves may also be launched with probes, but this method of launching tends to excite all modes. Langmuir-typc probes, having coaxial shields brought up near the collecting surface, are adequate for many measurements up to frequencies of 1000 Mc (Bailey and Emeleus, 1955). A pair of small disc probes has been used successfully to measure Co,, in dilute plasmas (Yeung and Sayers, 1957) and, in fact, wire and disc probes were used in experiments that probably were the first microwave diagnostic measurements (van der Pol, 1920). When a current-collecting probe simultaneously has a large rf voltage applied to it, the nonlinear sheath characteristics cause rectification of the rf signal. The d-c current is thus altered a small amount. The rf electric field around the probe is maximum at a frequency near the plasma frequency, leading to an increase in the rectified direct current as the applied frequency is swept through the local plasma frequency (Takayama et al., 1960; Ikegami and Takayama, 1963). The probe, of course, perturbs the plasma and thus the frequency measured is slightly below the true plasma frequency. Resonant probes, fed by transmission lines (Levitskii and Shashurin, 1961), permit measurement of the plasma impedance, plasma density, and guide wavelength (see Sections 5.5 and 5.6). Probes loosely coupled to a tunable filter, such as a motor-driven coaxial resonator (Malmberg et al., 10.7 Ballistic probes 387 1963), permit rapid analysis of the frequency spectrum of oscillations picked up in the plasma, or as a means to filter received signals. The movable probes shown in Fig. 9.47 were used for these purposes. 10.6 Magnetic probes Small inductive probes immersed in the plasma will have voltages induced in them by changes in the local magnetic field, dBjdt (Glasstone and Lovberg, 1960; Colgate et al., 1958). Field sensitive elements, such as I lall current probes, measure the instantaneous magnitude of the magnetic field B. Magnetic probes may be made as small as 1 mm in diameter and grouped in x-y-z arrays to measure three-dimensional field configurations (Pollock et al., 1960). Current density contours and the presence of hydromagnetic instabilities in dense plasmas are measured by a linear array across current channels. The data can be displayed by rapid sampling. The output voltage of the coil-type probe may be integrated to yield the magnitude of field. The resulting signals are very small (depending on the integration time), and care must be used to avoid stray pickup. Hall probes have outputs of a volt or so, response times up into the megacycles, and are easily calibrated with a standard magnet. They are somewhat temperature sensitive. Another kind of coil assembly that measures rates of change in enclosed current channels is the Rogowsky loop or girdle (Golovin et al., 1958; < 'ooper, 1963). The assembly consists of two sets of coils, one around the in lire experimental region and the other around only the current channel or a part of it. The difference in induced voltage represents the currents not enclosed, such as wall currents. The coils may be segmented, with leads brought out separately, to indicate current profiles. Low inductance coils can also be used to pick up high frequency llucluations, such as those associated with ion-wave instabilities or ion-cyclotron-frequency instabilities. These frequencies are typically from li) kc to 10 Mc. 10.7 Ballistic probes I lie measurement of thrust, shock intensity, and momentum transfer by plasmas is possible by using ballistic probes (Marshall, 1958). Considerable care is required to avoid electrostatic force deflections and thermal [hock deflections. Piezo-clectric transducers (Stern and Dacus, 1961) are pailictilarly useful, since they can be driven by insulated pickup probes and produce an output voltage having very fast time response. These transducers are also sensitive to thermal shock produced by intense light pulses (lui example, laser beams) or microwave pulses (While, 1962). 388 General plasma diagnostic techniques Chap. JO 10.8 Optical spectroscopy Spectroscopy is a very large subject in itself, having wide and well-developed applications to nearly all categories of plasma research. At best, we can only hope to give a few of the highlights in this section, indicating where fuller treatments of the various spectroscopic techniques may be found. 10.8.1 Constituent identity and state. Identification of radiating species has been studied perhaps longer than any other of the spectroscopic techniques, and numerous tables of line wavelengths, intensities, and related information exist (Chemical Rubber Handbook, 1962; M.I.T Tables, 1955; Kelly, 1959; and A.I.P. Handbook, 1963). A photographic plate exposed in a calibrated spectrograph can reveal immediately a great deal of qualitative information about the degree of ionization, whether molecular dissociation is occurring, what impurities are present, and even some knowledge of the electron temperature by observing the highest ionization potentials excited. The one-dimensional extent of the various species is obtained from the lengths of the lines. The presence of a continuum between the lines indicates the presence of recombination or bremsstrahlung. A quantitative examination of the lines, such as with a line splitter (Scott et al., 1962) or microdensitometer, then reveals detailed information about plasma properties (Wulff, 1959). Of the three types of line broadening, only the Stark and Doppler broadening give measurable effects in most normal plasmas. Collisional or pressure broadening is generally a very small effect. Zeeman splitting of lines, if observable in laboratory plasmas, gives information on the local magnetic field strength. Field strengths of 50 to 100 kilogauss are required to be able to see the effect at all, in most cases. 10.8.2 Stark broadening. In high-density, low-temperature plasmas («> 1015cm"3, Te <4 eV) the spectral lines are shifted and broadened by the electric fields due to ions and electrons. In the Holtsmark theory (Griem, 1960), the line profile is described by a slowly varying function S(a) S(a) d*= 1 (10.8.1) where a=AXjF0 F0 — 2.6\ en54 = the Holtsmark normal field strength n = electron and ion density, number/cm3 A A = displacement from unperturbed line, A e = electron charge J0.8 Optical spectroscopy 389 S(a) is a complicated function, and it is convenient to use either approximations or tabulated values (Griem, Kolb, and Shen, 1959; Underhill and Waddell, 1959). Several modifications to the Holtsmark theory have been made to account for correlations among ions (Margenau, 1932, 1951) and for nonadiabatic effects. The density dependence, nevertheless, does not depart much from the n%, especially in the wings of the Balmcr and Lyman series lines. Some lines exhibit linear Stark effect and others quadratic effects; the total broadening is calculated by summing over all of the Stark coefficients. Typical measurements of Stark broadening range from a fraction of an angstrom for the N IV 3479 A line due to a 5% N2 impurity in a 1017 cm-3 D+ plasma (Lukyanov and Sinitsin, 1958) to tens of angstroms in dense H2 plasmas having n X 10ie cm-3 (Wilcox et al., 1961). The Stark shift, which is generally quite small, has a slight temperature dependence und a density dependence ~n^22 (Margenau, 1951). 10.8.3 Doppler broadening. The relative velocity, v, of a radiating atom in respect to the observer leads to a doppler shift of the radiation frequency Av — v — va = v0 v/c. (10.8.2) II the emitting atoms are in kinetic equilibrium, this leads to a gaussian line shape for the doppler broadening (10.8.3) The doppler half-width in frequency or wavelength is _J^ = J^=(|n2)W?*B* (10.8.4) (ln2)^0 (In2)'-*A0 \Mcz) v where M is the ion mass. For example, the doppler broadening of the Bftlmer HB line is approximately ^«4-10-5(7,H+[eV])^. An (10.8.5) In forms of the intensity at the line center, /„, the intensity I(y) is typical line half-widths are between 0.3 to 30 A for hydrogen ion ttmperatures between 105 to 109 °K. Transient plasmaSfoften arc not reproducible from pulse to pulse, and it desirable to observe the line profile completely with each shot. A line 390 General plasma diagnostic techniques Chap. 10 FIG. 10.12 Optical line splitter assembly, showing the exit slit with the cylindrical lens A, the deflection mirror B, the photomultiplier array C, and the light-tight entrance tube D. (Courtesy of F. R. Scott, General Atomic, San Diego, Calif.) splitter (Scott et a],, 1962) permits a sampling of a broadened line at six intervals over a narrow wavelength band. A cylindrical lens, attached to a 25(V-wide slit on a 500-mm JACO monochromator, allows a variation in the dispersed band from 0.04 A/mm to 18 A/mm. Seven 1P21 photo-multipliers are used as detectors. The instrument is shown in Fig. 10.12. Besides giving the profile shape, if the detector outputs are observed for signal time correlation, one can determine if the broadening is true d op pier (thermal) broadening or due to mass motions in the plasma. 10.9 Bremsstrahlung and recombination continuum 391 10.9 Bremsstrahlung and recombination continuum The radiation from free-free and free-bound transitions (see Section 7.3) also may be analyzed in the X-ray, ultraviolet, optical, or infrared wavelengths as a diagnostic technique. In the optical spectrum, this radiation appears as a continuum between the emission lines. The continuum is made up of bremsstrahlung radiation (due to the acceleration of electrons during collisions), from high, overlapping harmonics of synchrotron radiation (due to acceleration of electrons between collisions), and from recombination radiation (due to the radiative capture of electrons into the hound states of atoms). The contributions of these three depend, in different manners, on the electron density and temperature and on the wavelength at which observations are made. For example, in equilibrium hydrogen plasmas (electrons having a Maxwellian velocity distribution), the dilference in background intensities of the Balmer continuum (that is, I he background intensity between Balmer series lines) and that just beyond l he series limit (A < 3642 A) is strongly dependent on the electron tempera-lure, but not very sensitive to density (McWhirter, 1965). The absolute intensity of the continuum, however, is proportional to with a square-root dependence on temperature. At short wavelengths, then, where cllects of impurity radiation are not too serious, the electron density can be measured by absolute intensity measurements of the continuum (< ilasstone and Lovberg, 1960; McWhirter et al., 1959). The frequency dependence of the continuum intensity (that is, the shape Of the spectral intensity curve) is a strong function of temperature in other portions of the spectrum away from the Balmer series limit as well, especially if there is self absorption. For example, in the far infrared the bremsstrahlung continuum merges into blackbody radiation. In experiments at Harwell, England, by Harding and Roberts (1961), the gap between upiical and microwave radiation was bridged by the use of a special far infrared spectrometer, covering wavelengths of 0.1 to 2.0 mm. The radiation intensities were found to follow blackbody curves quite well up in about the plasma frequency and then to saturate, becoming insensitive lo further changes in frequency. The experimental curves had the general shape of Fig. 7.7«. 10.8.4 Doppler shift. If the plasma as a whole is moving (mass motion), the emission lines will be shifted in wavelength. In a rotating plasma, the spectral lines originating in that part of the plasma moving toward the observer will be shifted to shorter wavelengths and vice versa. If the entrance slit to the spectrometer spans the plasma diameter, the lines will be tilted, leading to the slant effect (Dickerman and Morris, 1961), I The purpose of this appendix is to review the well-known arguments of electromagnetism leading to the concepts of complex conductivity and dielectric constant and to wave propagation in a general lossy medium. The final sections consider microscopic fields and the Lorentz-tcrm paradox, and propagation in anisotropic media. For a more extensive discussion of these matters, see von Hippel's Dielectrics and Waves (Technology Press/Wiley, New York, 1954), and also the standard treatises on electromagnetism by Panofsky and Phillips, Jackson, Stratton, and others. A.l Basic relations for a linear medium Maxwell's equations for a macroscopic medium are, in rationalized mks units, V-D = P (A.1) V-B=0 (A.2) (A.3) (A.4) VXE=-^ tit VxH=J+ ÖD 8t where p is the volume density of free charge and J the explicit current density. The field vectors E, D, B, and H are related by the constitutive relations D = *0E + P (A.5) Mo (A.6) APPEN DIX A Review of electromagnetic wave propagation A.l Basic relations for a linear medium 393 where P and M are the polarization and magnetization (net dipole moments per unit volume) and e0, p0 are the usual dimensional constants of mks units.1 In the case of a linear, isotropic medium (A.7) (A.8) where % and fm, the electric and magnetic susceptibilities, are constants characteristic of the medium.2 For this linear case, the constituitive relations (A.5) and (A.6) reduce to D = eE= KTEnE (A.9) (A. 10) where e and p. are the permittivity and permeability of the medium and K=l + Ve (A. 11) «„,= 1 + ^ (A-12) are the dielectric constant (or relative permittivity) and relative permeability. Likewise a linear, isotropic medium obeys Ohm's law, J = aE (A.13) Where a is the conductivity? Thus the electromagnetic properties of a linear, isotropic medium are specified by the three independent constants k, xm, and c\ 1 The mks units of these various quantities are E volt/meter D, P coulomb/meter2 It weber/meter2= 104 gauss H, M ampere/meter ji coulomb/meter1 .1 ampere/meter2 I/4^c0 = 10~7e2 K9• 10° meter/farad (c = velocity of light) /i,p/4m= 10"7 henry/meter. 1 We follow convention here in writing M proportional to H, rather than to the Average microscopic field B. ' Many authors use mobility, defined as the ratio of average electron velocity to electric li. M. rather than conductivity in discussing ionized gases. Thus, mobility equals t)nt, II more than one species of charged particle contributes to the current, the conductivity lS«7|i-2 «»?»«*>*i where nk, qk, and .fLk are the density, charge, and mobility of the Ath species. 392 394 Review of electromagnetic wave propagation Appendix A From an energy point of view, the constants k and iV(1)H V XH=(lt+jwKe0)E. (A. 14) (A. 15) (A. 16) If the medium exhibits dispersion, a, k, and km are functions of frequency. The form of (A. 16) makes it convenient to incorporate one of the two constants a, k in the other by means of complex notation. This technique is directly analogous to the familiar representation of the resistive and reactive processes of circuit theory by a complex impedance. In the case of ordinary dielectrics and wave propagation, it is customary to define a complex dielectric constant such that a+jcoKe0 —> j(M