JOURNAL OF TIHE OPTICAL SOCIETY OF AMERICA
Theoretical Eye Model with Aspherics*
W. LOTMAR
Swiss Office of Weiglits and Measures, CII 3084 Wabern, Switzerland
(Received 2 April 1971)
A model for the human eye is proposed, similar to Gullstrand's well-known 4-radius model, however with
the front surface of the cornea and the back surface of the crystalline lens taken to be rotationally symmetric
aspherics. Whereas for the cornea a polynomial is used based on experimental data of Bonnet, a
second-order parabola was tentatively adopted for the back surface of the lens. This model results in slight
spherical undercorrection, in agreement with experimental findings. On the other hand, the sine condition
is not well satisfied, probably due to neglect of the shell structure of the lens. By ray tracing, astigmatism
and coma as well as the meridional and sagittal focal lengths were computed up to a visual angle of 900.
Calculations were also made for the same model preceded by a plano-concave contact lens (Goldmann
3-mirror contact glass), showing that this combination results in considerably reduced astigmatism.
INDEX HEADINGS: Aberrations; Image formation; Ray tracing.
In preventive treatment of human retinal detachment
by photocoagulation, the aberrations of the dioptric
apparatus prevent exact focusing in the far periphery
of the fundus oculi, because they increase rapidly with
visual angle; in particular, they impair both observation
and coagulation efficiency. We have shown that
these difficulties can be overcome to a large extent by
use of a plano-concave contact lens (mirror contact
glass of Goldmann.)" 2
Hence it was desirable to gain
some insight into the optical properties of the system,
eye plus contact glass, as compared to those of the
unaided eye.
In perimetry of the peripheral parts of the eye, socalled
refractional scotomas are known to occur.', 4
These are areas of reduced contrast sensitivity due not
to a disturbance of the neuro-visual system but to abnormal
blur of the target image. Such blur may be
caused by local bulging of the retina or variation of the
focal length of the optical media. Because correct
diagnosis of sensitivity depressions is of course highly
important, knowledge about the aberrations of the
normal eye could be expected to be useful.
10 mm
FIG. 1. Theoretical eye model of Gullstrand-Le Grand. a, visual
angle; e,internal angle; oy,angle of acceptance.
Another problem that arises in biomicroscopy of the
eye is the determination of the absolute dimensions of
objects and structures in the periphery of the retina.
Little is known about the dependence of the effective
focal length of the eye on visual angle.
Ferree and Rand5
and recently Rempt et al.6
have
shown that the astigmatic difference (Sturm's interval)
in the periphery of emmetropic human eyes varies considerably
within the population. For a visual angle of
600, values between 2 and 10 diopters have been reported,
the higher figures being preponderant. From
theoretical considerations, such a wide spread in a system
as simple as that of the eye appeared rather strange,
because for its radii, spacings, and refractive indices no
large variations are known. Experimentally, Sturm's
interval is measured in diopters, whereas values computed
from a model are in millimeters. To convert the
latter to the former both meridional and sagittal focal
lengths of the eye must be known as a function of visual
angle.
Finally, it is well known that the use of an indentor
considerably improves the observation of the far
periphery and the pars plana. However, quantitative
predictions of the possible improvement were not
available before for lack of optical data.7
Ray tracing through a theoretical eye model, based
on recent data about the cornea profile and the
correction on axis, should yield some pertinent information,
although this should serve as a first approximation
only. Lee el al.8 reported recently on ray
TABLE I. The eye model of Gullstrand-Le Grand.
Radius Spacing Refractive
(mm) (mm) index 1ld Medium
ri= + 7.8 1.0 air
r2=+ 6.5 d2= 0.55 1.3771 cornea
r3= +10.2 d3= 3.05 1.3374 aqueous
r4=- 6.0 d4= 4.0 1.420 lens
r5=-12.3 d5= 16.60 1.336 vitreous
f = 22.29 mm
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VOLUME 61, NUMBER 11 NOVEMBER 1971
THEORETICAL EYE MODEL WITH ASPHERICS
tracing through the eye preceded by a meniscus contact
lens, but gave no quantitative results.
EYE MODEL
As in previous theoretical eye models, we assume the
optical system of the eye to consist of four centered
refracting surfaces of rotational symmetry. The retina,
too, is approximated by a spherical surface centered on
the same axis. Our first model is that of ListingGullstrand,
slightly improved by newer data of Le
Grand (Fig. 1).9 The data are given in Table I. The
spacing between the lens and the retina, d5, equals the
intersection length of the preretinal media for an object
at infinity, in order to make the model eye paraxially
emmetropic.
Like Gullstrand, we assume a homogeneous lens,
neglecting the shell structure for lack of exact data;
this provides us with a relatively simple system for the
calculations.
A detailed analysis of the shell structure of the
rabbit-eye lens was made by Nakao et al.'0
; by ray
tracing, they showed that this structure has a pronounced
effect on the optical correction of the lens.
Similar data for the human lens would also be desirable
for derivation of a better model, but the calculations
would have to be much more involved.
As is well known, the Gullstrand model results in
strong spherical undercorrection on axis. This is quite
at variance with recent experimental findings. These
show that the normal unaccommodated eye, broadly
speaking, is spherically corrected."-" Undercorrection
is of course expected for such a system when the refracting
surfaces are spherical.
As a first refinement of the Gullstrand model, we assume
an aspherical, rotationally symmetric shape for
the front surface of the cornea, with the same radius of
curvature at the vertex. When this shape closely approximates
the actual geometry, we expect calculated
corrections that approach reality better than the
Gullstrand model not only on-axis but also for off-axis
visual angles. Our results seem to confirm this.
THE FORM OF THE CORNEA
To obtain the true shape of the cornea, several experimental
methods have been used. One of the most
precise appears to be stereophotogrammetry as applied
by Bonnet.' 4
"5
1He was able to fit his measurements to
the analytical expression (see Fig. 2),
logo0f =ka+b, (1)
where a is the angle between the normal to the profile
curve and the optical axis, j3 is the angle between this
normal and the radius of the osculating circle drawn
from the intersection point of the normal with this circle,
and k and b are constants for the individual cornea,
FIG. 2. Definition of angles in Eq. (1). B, profile curve
of Bonnet; C, osculating circle.
which may vary somewhat for different meridians of
a given cornea.
For 200 emmetropic eyes measured with a special
keratometer, Bonnet found that these constants are
distributed around average values of +3 and -3,
respectively, with a near-gaussian distribution and a
mean deviation of about 1.1. With these average values,
h
0 2 mm 4
FIG. 3. B, Bonnet's profile curve for the cornea according to Eq.
(3); C, osculating circle; E, ellipse; P, parabola.
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November 1971
W. LOTMAR
the formula for the corneal profile Eq. (1) becomes
logio$= 3(a- 1), (2)
with a and f3 in radians.
For ray tracing by computer, we have approximated
this relation by a power series,
I 2 5 J\2 1 Z \4]
2r0L 28 \r0 /12\ r/'
Ai
(3)
x being the coordinate along the optical axis, It the
distance from this axis, and r0 the radius of curvature at
It=0 (osculating circle). The curve described by Eq. (3)
is denoted by B in Fig. 3. a and 3are calculated from
Eq. (3),
tana = 1/x' =dI/dx
sinj3= (x+htx'-r0 ) sina/r0 .
(4)
(5)
Values of $ obtained from Eq. (5) agree with those
from Eq. (2) to better than 0.003 rad up to k=7 mm
and deviate by 0.03 rad for It=8 mm.
For comparison, three other profile curves are shown
in Fig. 3, all having the same radius of curvature for
k=0. Le Grand's value of 7.8 mm was taken for this
radius (r0) in all cases. The curves are (a) a circle, C,
(b) the ellipse that intersects the polynomial Eq. (3) at
9
rmm
h
mm
- 4
3
2
rIG. 5. Spherical aberration of eye
model according to data of Table I
with aspherics. As, aberration of intersection
length; Af, aberration of focal
length.
-1 1mm
I= 7
, E, and (c) a second-order parabola, P. In Fig. 4,
the principal radii of curvature of the surface of revolution
described by Eq. (3) are shown as a function of h
(note r0 =7.8). The meridional radius of curvature Pm
can be approximated by
pmcVro[il+8(h/j0)4] (6)
and the sagittal radius by
p3-ro0 (I+ 1.66X lo/rhM) (7)
Ray tracing through this model showed an over-all
spherical undercorrection of about one half that of the
Gullstrand model, whereas the spherical aberration of
the corneal front surface alone was practically fully
corrected. This was taken to mean that in reality the
balance of undercorrection is compensated by the lens.
The simplest modification of the model that would re-
30
20
10
2 4 6 mm h
FIG. 4. Meridional and sagittal radii of curvature p,,, and pt
respectively, of body of revolution corresponding to polynomial of
Eq. (3).
FIG. 6.Le Grand's eye model. Course of principal rays.
R, schematized retina (r5 of Table I).
1524 Vol. 61
1
THEORETICAL EYE MODEL
C
700
600 - --- ,/
so,1-- .1 -f-.
300 -
200 -I
100 - _ _ _ _ _ _ __ _ _ _ _ _ _
100 200 300 40O 500 600 700 800
mer
FIG. 9. Astigmatism
of eye model
of Table I with
aspherics. Ordinate
axis represents reference
image sphere r,.
R, position of retina
according to Le
Grand.
FIG. 7. Plot of internal angle evs visual angle a.
sult in the same effect was the introduction of a second
aspherical surface. We adopted the form of a parabola
for the back surface of the lens, again retaining curvature
at the vertex as given in Table I. For this surface,
the exact shape is less critical than for the cornea because
the difference of the refractive indices is much
smaller (0.08 as compared to 0.38). It is well known that
the backsurface of the lens is not spherical, but no
exact data are available at present.
The axial spherical aberration of this model is shown
in Fig. 5. It is small and of the order found experimentally.
Therefore, this set of data was used for more
extensive ray tracing.
600
400
200
RESULTS OF RAY TRACING
We traced parallel bundles of rays in the meridional
plane for visual angle" a up to 900 in steps of 100, assuming
an apparent diameter of 8 mm of the eye pupil.
The latter was taken to be a diaphragm of zero thickness
adjacent to the front surface of the lens. Principal
rays were made to intersect that surface at its intersection
with the optical axis.
Figure 6 shows the cross section of the eye according
to Le Grand, with traces of the principal rays. In Fig.
7 the internal angle e of principal rays is plotted vs
visual angle, where e is the angle between the ray in the
vitreous and the optical axis (see Fig. 1). The angle of
acceptance -y; i.e., the angle between the principal ray
and the retina, is plotted vs visual angle in Fig. 8, (a)
for the model retina of radius 12.3 mm, and (b) as determined
graphically for Le Grand's cross section (Fig.
6), however, averaging over the temporal and nasal
side. Astigmatism is plotted in Fig. 9 with the sagittal
focus calculated from known formulas. As is customary
in ophthalmology, the distances are measured along the
a
b
200 400 600 800 a
FIG. 8. Angle of acceptance y vs visual angle a. (a) for model
retina, (b) for Le Grand's cross section (Fig. 6).
FIG. 10. Definition of focal length for oblique incidence
of a parallel meridional bundle, fm=d/u.
a
\sag
-1 l mm
November 1971 WITH ASPHERICS 1525
so,
W. LOTMAR
25
20
20° 40 60o t
FIG. 11. Meridional and sagittal focal lengths, fm and fs, respectively,
vs visual angle for eye model of Table I with aspherics.
principal rays, not by projection on the axis as in lens
design. Moreover, we have to keep in mind that the
medium is the vitreous humor (not air). The foci are
plotted by reference to a spherical retina. The true
position of the retina is also indicated in the figure.
For an evaluation of image size and energy density on
the retina we have to know not only the positions of the
meridional and sagittal focus on the principal rays,
but also the corresponding focal lengths. We can obtain
these quantities by applying first-order optical imagery
to the principal rays, taking the finite angles of incidence
at the refracting surfaces into account. The meridional
focal length fm is defined, as for paraxial rays, by the
ratio of a (infinitesimally small) distance d between the
h
80° 400 200
mm
4
3
2mm
FIG. 13. Longitudinal comatic aberration of eye model with
aspherics. Reference image sphere is r5 of Table I. Plotted curves
are projections of true curves on It axis along the principal rays of
the corresponding bundles.
incident principal ray and another parallel ray in the
meridional plane, to the angle u that these rays form in
the image space' (see Fig. 10). An analogous definition
applies to the sagittal focal length f,. Values of fm and
f. for the model of Table I with aspherics are shown in
Fig. 11 as a function of visual angle.
The meridional focal length Im decreases considerably
towards the periphery. To account for this result it is
h
4
0.5
3
2
20° 40° 60' 80' a
FIG. 12. Convergence ratio of individual refracting surfaces of
eye model with aspherics. r, (sph), curve if the front surface of
cornea is spherical.
mm
' TAf IAs
FIG. 14. Spherical aberration of model
with lens consisting of seven shells.
-1 1mm
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THEORETICAL EYE MODEL WITH ASPHERICS
a
400
200
FIG. 17. Spherical aberration of ECG
system. Scale of sine-condition aberration
is comparable to that of Fig. 5.
-1 1 dptr
(b)
FIG. 15. Astigmatism of emmetropic human eyes according
to Ferree and Rand. Classes (a) and (b).
interesting to study the contribution to focusing by the
individual refracting surfaces (Fig. 12). The ordinate
values in this figure are relative, that is, fm'/fo for the
front surface, and y0/,yi for r2, r3, and r4, with yh the
convergence ratio, (or angular magnification), whereas
fo, lyo are the corresponding values on axis. For a given
visual angle
fmf= o- (fm'/fO) - (1/72) - (1/Y3) *(1/Y4). (8)
Finally, longitudinal meridional comatic aberration
along the principal rays, as in Fig. 9 for astigmatism,
is shown in Fig. 13 for a series of visual angles.
DISCUSSION
A. Spherical Correction
Figure 5 shows that spherical aberration of the proposed
eye model agrees to order of magnitude with experimental
findings. On the other hand, the sine condition
is not well satisfied, which leads to asymmetry of
the coma curves (Fig. 13). It is likely that this shortcoming
of the model is due to its neglect of the shell
-1 1 mm
structure of the lens. For the rabbit lens, the computations
of Nakao et al.10
show that the heavy spherical
undercorrection of a fictitious homogeneous lens is reduced
to practically zero by the actual shell structure.
At the same time, the sine condition is better satisfied."8
By analogy to his findings, a rough and purely hypothetical
shell model for the human lens was constructed,
consisting of seven shells, the thicknesses of which decrease
and their radii increase with distance from the
center, according to square-root laws, with refractive
indices varying from 1.38 to 1.41 in steps of 0.005. When
this model was substituted for the homogeneous lens
of Table I, a similar result was obtained. Spherical
aberration and the sine condition of this system are
shown in Fig. 14. However, we felt it would be pre-
mer
E
o00
/
FIG. 16. Eye model with aspherics in combination with planoconcave
contact glass (ECG system). For ray tracing, the angle
between the principal ray and the mirror was assumed to be e/2,
so that incidence on the front surface of the contact glass is at
right angles.
- 1 1
FIG. 18. Astigmatism of ECG system.
mer
h
4,
31I
As
-1 1 dptr
(a)
2
mm
November 1971 1527
1.528 W. LOTMAR
h
Imm
90
600
3
2
2mm
FIG. 19. Longitudinal comatic aberration of ECG
system. See remarks to Fig. 13.
mature to do extended ray tracing through this system
before more experimental data on the human lens become
available.
Furthermore, the advantages of a relatively simple
model would be lost.
FIG. 20. Meridional and sagittal convergence
ratios of ECG system.
Vol. 61
mm
6
4
2
0
200 400 60" 80" cx.
FIG. 21. Sturm's interval, A, for eye model with
aspherics (a), and ECG system (b).
B. Astigmatism
Figure 9 shows that the location of the retina, up to
a visual angle of 900, is roughly half the distance between
the meridional and sagittal foci, i.e., the locus of
the circle of least confusion. Strictly speaking this applies
only to bundles of infinitesimal numerical aperture,
but the coma curves of Fig. 13 indicate that for the
present model the circle of least confusion of larger
meridional bundles lies not very far from the meridional
focus in the stricter sense.
Ferree and Rand' concluded from a relatively restricted
number of cases that, with respect to astigmatism
in the periphery, emmetropic human eyes fall
into two distinct classes A and B, which are shown in
Fig. 15. Rempt et al.,6
having measured nearly 900 eyes,
found however that intermediate cases exist as well.
Types A and B of Ferree and Rand may be-considered
to correspond roughly to the limiting cases. Of 217
emmetropic eyes, 62% were of type A, which therefore
is taken to be the "normal type" by Rempt et al. From
comparison of Figs. 9 and 15, it is evident that the
present model comes close to this type. Sturm's interval
at 500 visual angle in Fig. 15(a) is 5.5 diopters. The
model, for the same visual angle, results in -0.55-mm
deviation for the meridional and +1.05 mm for the
sagittal focus. When these figures are converted to
diopters, taking the respective focal lengths (Fig. 11)
into account, we obtain a value of 6.25 diopters for
Sturm's interval. In view of the simplifications made in
our model, the agreement must be considered good.
For Gullstrand's eye model with spherical surfaces,
on the other hand, Sturm's interval at 500 visual angle
was found to be 15.1 diopters. Thus, the shape of the
cornea has a strong influence on astigmatism in the
periphery. Curves like those of class B of Ferree and
THEORETICAL EYE MODEL WITH ASPHERICS
Rand might therefore be explained by assuming a shape
for the profile curve somewhat more flattened than
that corresponding to Eq. (3).
C.Variation of Focal Length with Incidence Angle
From Fig. 12, the decrease of meridional focal length
at high visual angle is preponderantly due to the front
surface of the lens (4), and not to the cornea. This can
be understood because ray tracing gives angles of incidence
of the principal rays at the lens surface that are
larger than at the cornea. The cosines of these angles
govern the variation of the convergence ratio.
D. Coma
It is not clear at this time to what extent the calculated
curves approximate those of a more refined model
that takes the shell structure of the lens into account;
nor, owing to lack of experimental data on coma, do we
know how they agree with reality.
THE SYSTEM EYE PLUS CONTACT
GLASS (ECG)
Rays were traced through the same eye model preceded
by a plano-concave plastic lens (Goldmann
3-mirror contact glass,19
Fig. 16). This lens is characterized
by r1 = oo, r2 =+7.8 mm, d(=22 mm, nd = 1.4 9 2
.
It is separated from the cornea by 0.2 mm of water
(nd = 1.336).
As the compound system is nearly afocal (f=462.4
mm), we used convergent incident bundles. Their foci
were adjusted to a distance behind the first surface such
that the circle of least confusion, after passage of the
bundles through the system, approximately coincides
with the retina. The difference of this distance for
bundles of 00 and 800 visual angle was found to be 2
mm.
Spherical, astigmatic, and comatic aberrations are
shown in Figs. 17-19, and the dependence of the convergence
ratio on visual angle in Fig. 20. Comparison
of Figs. 18 and 9 shows that astigmatism is less for the
ECG system up to a visual angle of 800 (Fig. 21). This
explains the superior image quality provided by use of
the Goldmann contact glass. On the other hand, the
difference vanishes at the highest visual angles, where
the influence of the front surface of the crystalline lens
evidently predominates. The steep rise of astigmatism
in the case of the ECG system explains also the high
efficiency of scleral indentation in regard to image
quality.
The coma curves of Fig. 19 undergo a reversal of
asymmetry with increasing visual angle. This, too,
must be ascribed to the contact glass, because it is not
noted in Fig. 13.
Figures 11 and 20 show that in the ECG system the
decrease of the meridional convergence ratio is shifted
to higher visual angles, reflecting the behavior of
astigmatism. This result may be expected to hold also
for a model with a more refined lens structure.
ACKNOWLEDGMENTS
Drs. A. Dalcher of Kern & Co., Aarau, and A.
Roulier did the programming and ray tracing. The
drawings were made by H. R. Dappen.
REFERENCES
*This work has been supported by a grant to the University of
Bern, No. NB-03638-05 from the NINDB, National Institutes of
Health, Bethesda, Md. 20014.
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(1967).
2 W. Lotmar, F. Fankhauser, and A. Roulier, Arch. Ophthalmol.
82, 314 (1969).
3 T. Schmidt, Ophthalmologica 129, 303 (1955).
4F. Fankhauser and J. M. Enoch, Arch. Ophthalmol. 68, 240
(1962).
1C. E. Ferree and G. Rand, in Report of a Joint Discussion on
Vision (Cambridge U. P., New York, 1932), p. 244.
6 F. Rempt, J. Hoogerheide, and W. P. H. Hoogenboom,
Ophthalmologica 162, 1 (1971).
7F. Fankhauser and W. Lotmar, Acta Ophthalmol. 48, 253
(1970).
8 P. Lee, 0. Pomerantzeff, and C. L. Schepens, Arch. Ophthalmol.
84, 650 (1970).
9Y. Le Grand, Optique PiysiologiqueI (Ed. Rev. Opt., Paris,
1953), p. 52.
10S. Nakao, S. Fujimoto, R. Nagata, and K. Iwata, J. Opt. Soc.
Am. 58, 1125 (1968).
1 G. van den Brink, Vision Res. 2, 233 (1962).
12 H. Schober, H. Munker, and F. Zolleis, Opt. Acta 15, 47
(1968)
3F. Berny, Vision Res. 9, 977 (1969).
14R. Bonnet, La Topographie Corneenne (Desroches, Paris,
1964).
1'R. Bonnet and P. Cochet, Bull. Soc. Fran§. Ophtalmol. 73,
688 (1960).
18Visual angle at (Fig. 1)is different from the angle a in Fig. 2and
Eq. (1) as used by Bonnet.
17 W. Mert6, in HandbuchderPhtysik XVIII, edited by H. Geiger
and K. Scheel (Springer, Berlin, 1927), p. 73.
18S. Nakao, personal communication.
19H. Goldmann and T. Schmidt, Ophthalmologica 149, 481
(1965).
November 1971 1529