Chapter 7   Plane Electromagnetic Waves and Wave Propagation—SI
10   Causality in the Connection Between D and E; Kramers-Kronig Relations
A. Nonlocality in Time
Another consequence of the frequency dependence of e(oj) is a temporally nonlocal connection between the displacement D(x, t) and the electric field E(x, t). If the monochromatic components of frequency co are related by
D(x, <u) = e(w)E(x, at) (7.103)
the dependence on time can be constructed by Fourier superposition. Treating the spatial coordinate as a parameter, the Fourier integrals in time and frequency
can be written
1 r
D(x, t) = —i=      D(x, (o)e-lM doi
'2tt
and (7.104)
If"
D(x, cS) = —=      D(x, t')etwt dt'
\Z2lT J-oo
with corresponding equations for e. The substitution of (7.103) for D(x, cS) gives
i r
D(x, t) = ——       e(w)e(x, (o)e~loa doi We now insert the Fourier representation of e(x, co) into the integral and obtain
if"00 f °°
D(x, t) = —      da> e(co)e-iwt      dt' eicot'E(x, t')
2.7t j —<xi J —o=
With the assumption that the orders of integration can be interchanged, the last expression can be written as
D(x, 0 - c0|e(x, 0 + J    G(r)e(x, t - r) drj (7.105) where G{r) is the Fourier transform of \e - ^(^)/eo _ 1:
G(t) = J-       [e(cS)/e0 - l]e"'WT da> (7.106)
att j -«>
Equations (7.105) and (7.106) give a nonlocal connection between D and E, in which D at time t depends on the electric field at times other than t* If e(<o) is
♦Equations (7.103) and (7.105) are recognizable as an example of the faltung theorem of Fourier integrals: if A(i), B(t), C(t) and a(w), b(w), c(w) are two sets of functions related in pairs by the Fourier inversion formulas (7.104), and
c(w) = a((0)b(a>) then, under suitable restrictions concerning integrability,
C(f) = —L= I    A(t')B(t - f) dt'
V2-7T J-=o
Sect. 7.10   Causality in the Connection Between D and E; Kramers-Kronig Relations 331
independent of co for all co, (7.106) yields G(r) a: d(r) and the instantaneous connection is obtained, but if e(co) varies with co, G(t) is nonvanishing for some values of r different from zero.
B. Simple Model for G(t), Limitations
To illustrate the character of the connection implied by (7.105) and (7.106) we consider a one-resonance version of the index of refraction (7.51):
e((o)/e0 - 1 = co2p(co20 - co2 - iyco)^ (7.107)
The susceptibility kernel G(r) for this model of e(co) is
col f00 e~ioiT
G(t) = ~       —7-5-:— dco (7.108)
277 J-oo a>0 — co — lyco
The integral can be evaluated by contour integration. The integrand has poles in the lower half-w-plane at
2
ly y
coh2 = -— ± vQ,      where v\ = co\ - — (7.109)
For r < 0 the contour can be closed in the upper half-plane without affecting the value of the integral. Since the integrand is regular inside the closed contour, the integral vanishes. For r > 0, the contour is closed in the lower half-plane and the integral is given by —277? times the residues at the two poles. The kernel (7.108) is therefore
G(r) = cole^12 d(r) (7.110)
where 6»(r) is the step function [0(t) = 0 for r < 0; 0(t) = 1 for r > 0]. For the dielectric constant (7.51) the kernel G(t) is just a linear superposition of terms like (7.110). The kernel G(r) is oscillatory with the characteristic frequency of the medium and damped in time with the damping constant of the electronic oscillators. The nonlocality in time of the connection between D and E is thus confined to times of the order of y-1. Since y is the width in frequency of spectral lines and these are typically 107-109 s-1, the departure from simultaneity is of the order of 10~7-10~9 s. For frequencies above the microwave region many cycles of the electric field oscillations contribute an average weighed by G(r) to the displacement D at a given instant of time.
Equation (7.105) is nonlocal in time, but not in space. This approximation is valid provided the spatial variation of the applied fields has a scale that is large compared with the dimensions involved in the creation of the atomic or molecular polarization. For bound charges the latter scale is of the order of atomic dimensions or less, and so the concept of a dielectric constant that is a function only of co can be expected to hold for frequencies well beyond the visible range. For conductors, however, the presence of free charges with macroscopic mean free paths makes the assumption of a simple e(co) or tr(co) break down at much lower frequencies. For a good conductor like copper we have seen that the damping constant (corresponding to a collision frequency) is of the order of y0 ~ 3 X 1013 s-1 at room temperature. At liquid helium temperatures, the damping constant may be 10~3 times the room temperature value. Taking the Bohr velocity in
332    Chapter 7   Plane Electromagnetic Waves and Wave Propagation—SI
hydrogen (c/137) as typical of electron velocities in metals, we find mean free paths of the order L ~ c/(137y0) ~ 10 4 ni at liquid helium temperatures. On the other hand, the conventional skin depth 5 (5.165) can be much smaller, of the order of 10~7 or 10~8 m at microwave frequencies. In such circumstances, Ohm's law must be replaced by a nonlocal expression. The conductivity becomes a tensorial quantity depending on wave number k and frequency co. The associated departures from the standard behavior are known collectively as the anomalous skin effect. They can be utilized to map out the Fermi surfaces in metals.* Similar nonlocal effects occur in superconductors where the electromagnetic properties involve a coherence length of the order of 10~6 m.f With this brief mention of the limitations of (7.105) and the areas where generalizations have been fruitful we return to the discussion of the physical content of (7.105).
C. Causality and Analydeity Domain of e(w)
The most obvious and fundamental feature of the kernel (7.110) is that it vanishes for t < 0. This means that at time t only values of the electric field prior to that time enter in determining the displacement, in accord with our fundamental ideas of causality in physical phenomena. Equation (7.105) can thus be written
D(x, t) = £0|e(x, t) + Jo G(t)E(x, t - r) drj (7.111)
This is, in fact, the most general spatially local, linear, and causal relation that can be written between D and E in a uniform isotropic medium. Its validity transcends any specific model of e(oj). From (7.106) the dielectric constant can be expressed in terms of G(t) as
f* CO
e(<u)/e0 = 1 +      G(r)elcOTdr (7.112) Jo
This relation has several interesting consequences. From the reality of D, E, and therefore G(t) in (7.111) we can deduce from (7.112) that for complex co,
e(-<o)/e0 = e*(co*)/e0 (7.113)
Furthermore, if (7.112) is viewed as a representation of e(oo)/e0 in the complex co plane, it shows that e(co)/e0 is an analytic function of co in the upper half-plane, provided G(r) is finite for all t. On the real axis it is necessary to invoke the "physically reasonable" requirement that G(t) —» 0 as r^> °° to assure that e(co)/e0 is also analytic there. This is true for dielectrics, but not for conductors, where G(r) -» o7e0 as °o and e(<y)/e0 has a simple pole at co = 0 (e —» icr/oo as co —> 0). Apart, then, from a possible pole at co = 0, the dielectric constant e(co)/e0 is analytic in co for Im co > 0 as a direct result of the causal relation (7.111)
*A. B. Pippard, in Reports on Progress in Physics 23, 176 (1960), and the article entitled "The Dynamics of Conduction Electrons," by the same author in Low-Temperature Physics, Les Houches Summer School (1961), eds. C. de Witt. B. Dreyfus, and P. G. de Gennes, Gordon and Breach, New York (1962). The latter article has been issued separately by the same publisher.
tSee, for example, the article "Superconductivity" by M. Tinkham in Low Temperature Physics, op. cit.
Sect. 7.10   Causality in the Connection Between D and E; Kramers-Kronig Relations 333
between D and E. These properties can be verified, of course, for the models discussed in Sections 7.5.A and 7.5.C.
The behavior of e(co)/e0 — 1 for large co can be related to the behavior of G(r) at small times. Integration by parts in (7.112) leads to the asymptotic series,
iG(0) G'(0)
e(co)/e0 - 1 =---— + • • •
co co
where the argument of G and its derivatives is r = 0+. It is unphysical to have G(0~) = 0, but G(0+) ¥= 0. Thus the first term in the series is absent, and e(co)/e0 - 1 falls off at high frequencies as co'2, just as was found in (7.59) for the oscillator model. The asymptotic series shows, in fact, that the real and imaginary parts of e((o)/e0 - 1 behave for large real co as
Re[e(<u)/e0 - 1] = o(j-j,      Im e(co)/e0 = (7.114)
These asymptotic forms depend only upon the existence of the derivatives of G(t) around r = 0+.
D. Kramers-Kronig Relations
The analyticity of e(a>)/e0 in the upper half-w-plane permits the use of Cau-chy's theorem to relate the real and imaginary part of e(co)/e0 on the real axis. For any point z inside a closed contour C in the upper half-co-plane, Cauchy's theorem gives
If [6(a/)/60 - 1] ,
e(z)/e0 = 1 + — <P -;-dco
2m Jc      co z
The contour C is now chosen to consist of the real co axis and a great semicircle at infinity in the upper half-plane. From the asymptotic expansion just discussed or the specific results of Section 7.5.D, we see that e/e0 - 1 vanishes sufficiently rapidly at infinity so that there is no contribution to the integral from the great semicircle. Thus the Cauchy integral can be written
e(z)/e0 = 1 + — -;-da (7.115)
Ztti j-°°      co — z
where z is now any point in the upper half-plane and the integral is taken along the real axis. Taking the limit as the complex frequency approaches the real axis from above, we write z = co + id in (7.115):
e(co)/e0 = 1 + —      —-— dco (7.116)
2m j-°° co — co — id
For real co the presence of the i8 in the denominator is a mnemonic for the distortion of the contour along the real axis by giving it an infinitesimal semicircular detour below the point co' = co. The denominator can be written formally as
1 1       + m8(co' - co) (7.117)
co   — CO
id       \ co' co
334    Chapter 7   Plane Electromagnetic Waves and Wave Propagation—SI
where P means principal part. The delta function serves to pick up the contribution from the small semicircle going in a positive sense halfway around the pole at co' = co. Use of (7.117) and a simple rearrangement turns (7.116) into
r \i       1  ,   1 p f" [e(q/)/e0 - 1] ^ , , e(co)/e0 = 1 H--: P \ -;-dco (7.118)
777      J-o°        CO   — CO '
The real and imaginary parts of this equation are
1     f" Im e(co')/e0 , , Re e(o»)/e0 = 1 + - P \ -, dco'
77      j-*,      (o   — co T       ,  V 1 P f 6(0)0/66 - 1] ,
Im e(co)/e0 =--/J I---dco
77 O)  — co
(7.119)
These relations, or the ones recorded immediately below, are called Kramers-Kronig relations or dispersion relations. They were first derived by H. A. Kramers (1927) and R. de L. Kronig (1926) independently. The symmetry property (7.113) shows that Re e(co) is even in co, while Im e(co) is odd. The integrals in (7.119) can thus be transformed to span only positive frequencies:
2     T     Im e(o)')/e0 , ,
Re e(co)/e0 = 1 + - P \ -v  i dco'
W 77    Jo       a/2 - a;2 (7.120)
Im e(w)/e0 =--"--?-«to
77       JO CO     - OT
In writing (7.119) and (7.120) we have tacitly assumed that e(co)/e0 was regular at co = 0. For conductors the simple pole at co = 0 can be exhibited separately with little further complication.
The Kramers-Kronig relations are of very general validity, following from little more than the assumption of the causal connection (7.111) between the polarization and the electric field. Empirical knowledge of Im e(co) from absorption studies allows the calculation of Re e(co) from the first equation in (7.120). The connection between absorption and anomalous dispersion, shown in Fig. 7.8, is contained in the relations. The presence of a very narrow absorption line or band at co = co0 can be approximated by taking
77#
Im e(co') —-8(co' — co0) + • • ■
2oj0
where K is a constant and the dots indicate the other (smoothly varying) contributions to Im e. The first equation in (7.120) then yields
Re e(co) = e + -=-5 (7.121)
COq — CO
for the behavior of Re e(co) near, but not exactly at, co = co0. The term e represents the slowly varying part of Re e resulting from the more remote contributions to Im e. The approximation (7.121) exhibits the rapid variation of Re e(co) in the neighborhood of an absorption line, shown in Fig. 7.8 for lines of finite width. A more realistic description for Im e would lead to an expression for Re e in complete accord with the behavior shown in Fig. 7.8. The demonstration of this is left to the problems at the end of the chapter.
Sect. 7.11   Arrival of a Signal After Propagation Through a Dispersive Medium 335
Relations of the general type (7.119) or (7.120) connecting the dispersive and absorptive aspects of a process are extremely useful in all areas of physics. Their widespread application stems from the very small number of physically well-founded assumptions necessary for their derivation. References to their application in particle physics, as well as solid-state physics, are given at the end of the chapter. We end with mention of two sum rules obtainable from (7.120). It was shown in Section 7.5.D, within the context of a specific model, that the dielectric constant is given at high frequencies by (7.59). The form of (7.59) is, in fact, quite general, as shown above (Section 7.10.C). The plasma frequency can therefore be defined by means of (7.59) as
co2p ~ lim{a>2[l — e(co)/e0]}
Provided the falloff of Im e(co) at high frequencies is given by (7.114), the first Kramers-Kronig relation yields a sum rule for co2p:
2 r
col = — I   oj Im e(co)/e0 dco (7.122)
77 J 0
This relation is sometimes known as the sum rule for oscillator strengths. It can be shown to be equivalent to (7.52) for the dielectric constant (7.51), but is obviously more general.
The second sum rule concerns the integral over the real part of e(co) and follows from the second relation (7.120). With the assumption that [Re e(co')/e0 - 1] = -co2plco'2 + 0(1/go'4) for all co' > N, it is straightforward to show that for co > N
Im e(co)/e0 = — (-— + f   [Re e(co')le0 - 1] dco'\ + o(—?
ttco  I     N       jo j \c0~
It was shown in Section 7.10.C that, excluding conductors and barring the un-physical happening that G(0+) # 0, Im e(co) behaves at large frequencies as co~3. It therefore follows that the expression in curly brackets must vanish. We are thus led to a second sum rule,
1   CN co2
- I Re e(co)/e0 dco = 1 + ^ (7.123)
which, for N ^> °°, states that the average value of Re e(co)/e0 over all frequencies is equal to unity. For conductors, the plasma frequency sum rule (7.122) still holds, but the second sum rule (sometimes called a super convergence relation) has an added term — 77o72e0jV, on the right hand side (see Problem 7.23). These optical sum rules and several others are discussed by Altarelli et al.*
7.11  Arrival of a Signal After Propagation Through a Dispersive Medium
Some of the effects of dispersion have been considered in the preceding sections. There remains one important aspect, the actual arrival at a remote point of a
*M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, Phys. Rev. B6, 4502 (1972).