1544 J. Opt. Soc. Am./Vol. 73, No. 11/November 1983 Aart C. Kooijman
Light distribution on the retina of a wide-angle theoretical
eye
Aart C.Kooijman
Kliniek voor Oogheelkunde, Rijks-Universiteit Groningen, Postbus 30001, 9700 RB Groningen,the Netherlands
Received November 24,1982; July 18, 1983
In a theoretical eye with spherical and aspheric surfaces, the retinal illumination is calculated if a Ganzfeld luminance
field is used. The resulting retinal light distribution is nearly homogeneous over the whole retina. The homogeneity
is not much influenced by the size of the pupil or the shape of the optical surfaces. The corresponding
retinal area and the luminous flux entering the eye are calculated as functions of the size of the visual field. The
values of the length of the light path through the crystalline lens and of the angle of incidence on the retina are described
as functions of the angle in the visual field.
INTRODUCTION
Ganzfeld illumination is used in electroretinography, darkadaptation
tests, and perimetry. Usually an internally illuminated
globe with a homogeneous luminance is used,'- 3
but
light-diffusing contact lenses are also suitable.4
'5
The following
questions arise:
(1) What retinal illumination results from Ganzfeld illumination
as a function of the angle of incident light relative
to the optic axis?
(2) How does the light distribution on the retina depend
on the size of the pupil?
(3) How strong is the influence of modifications in the
shapes of the eye surfaces on the light distribution?
Although these questions look classical, it has not been possible
to find all the answers in the literature. Recently, Bedell
and Katz6
deduced from the experimental results on pupillary
area7
and the calculated values of retinal area per solid angle
of visual field8
that the retinal illuminance of a target should
remain essentially constant up to about 800. Gouras' mentions
that the pupillary aperture places restrictions on the
homogeneity of the retinal illumination. Hoffman et al.9
conclude that the pupil acts as a field stop that limits the visual
angle for large fields, such as a Ganzfeld source. Gross'0
and Young" measured the luminance of the image of an object
from the back of the excised rat eye as a function of eccentricity.
None of these authors refers to calculations of the
retinal light distribution. In this study a theoretical model
is used to calculate retinal light distribution and the influence
of some model parameters on the relative retinal illumina-
tion.
METHODS
Calculations are done on a schematic eye with optical surfaces
that are surfaces of rotation of a spherical or aspheric conic
section (Table 1and Fig. 1). The dimensions ofthe schematic
eye are based on the theoretical eye of Le Grand.12
A ray-tracing program in BASIC on a Commodore 3032
microcomputer was used to calculate the light rays in the
meridional plane of the eye. The steps of the calculation are
as follows:
(1) The apparent pupil diameter PD(sp) in the meridional
plane is calculated as seen from oblique directions. The apparent
pupil is the image of the physical pupil focused by the
cornea in air. The physical pupil is assumed to be coincident
with the anterior lens surface. The thickness of the iris edge
is assumed to be zero in this study. A real thickness of the iris
edge causes vignetting at all peripheral angles of incidence of
10° and greater but has an appreciable effect on the apparent
pupil area only at angles of view greater than 700.7 The apparent
pupil, as seen along the optic axis, has a circular shape.
This shape changes if the eye is viewed obliquely. The apparent
shape, seen from an angle so with the optic axis, is defined
as the shape of the cross section of the light pencil of
parallel rays incident upon the eye at an angle s° with the optic
axis and filling up the whole pupil after refraction by the
cornea. This is the pupil as seen from the outside ofthe eye.
An apparent pupil diameter observed along the optic axis
corresponds to a smaller physical pupil diameter at the anterior
surface of the lens. For apparent pupil diameters of 2
and 8 mm on the optic axis, these physical pupil diameters are
calculated and are used to calculate the apparent pupil diameters
for oblique directions. It is supposed that the apparent
pupil diameter perpendicular to the meridional plane
is independent of the angle so. The apparent pupil area PA(so)
is calculated with the area equation of an ellipse:
PA(sp) = 7r X PD(sp) XPD(0)/4.
The principal ray of the pencil of parallel rays with an angle
(pwith the optic axis passes through the center ofthe entrance
pupil related to that pencil of rays (Ref. 13, p. 186). As a
consequence of aberrations, the principal ray does not necessarily
pass through the center of the physical pupil and can
be different for large and small pupils.
(2) The retinal-area-per-steradian visual field is calculated
as a function of p. For that purpose the position and
the length of the retinal arc that correspond to an increment
of +0.25' of the angle of the principal ray in the meridional
plane are calculated as a function of (p. A ring-shaped retinal
0030-3941/83/111544-07$01.00 ©1983 Optical Society of America
Vol. 73, No. 11/November 1983/J. Opt. Soc. Am. 1545
Table 1. Schematic Eyea
Refractive Index Apical Radius (mm) Conical Value p Distance Corneal Vertex (mm)
Cornea 1.3771 7.8 (anterior) 0
6.5 (posterior) 1 or0.75 0.55
Aqueous 1.3374 Lens
1.42 10.2 (anterior) 1 or -2.06 3.6
-6 (posterior) 1or 0 7.6
Vitreous 1.336
Retina - -10.8 or -14.1 1 or 1.346 24.2
a The theoretical eye ofLe Grand has spherical surfaces (p = 1). In this studysurfaces with an aspheric conical cross section are also used (p #- 1). Calculations
are done on the basic eye model with only spherical surfaces (model A) and on the basic eye model with only one of the following modifications: (model B) aspheric
anterior and posterior corneal surfaces with an elliptical curvature, (model C) aspheric retinal surface with an elliptical curvature, (model D) aspheric anterior lens
surface with a hyperbolic curvature, or (model E) aspheric posterior lens surface with a parabolic curvature.
1 cm
Fig. 1. Schematic eye. Solid lines are spherical surfaces (theoretical
eye of Le Grand). Dashed lines are aspheric surfaces as used in this
study.
area dRA(qo) around the optic axis is determined by this arc
(Fig. 2). On this area fall the light rays that emerge from a
ring-shaped solid angle dSA(o) around the optic axis with a
width of ±0.250 at a peripheral angle s°. The quotient RS(<p)
= dRA(so)/dSA(p) is the retinal area per solid angle at eccentricity
s°. For small and large pupils the principal rays are
not identical to each other. Therefore dRA(yp) and RS(<p) are
calculated for two pupil sizes.
(3) The retinal illumination RI(p) with Ganzfeld luminance
A in the object space is calculated as function of sp:
RIp) = SPA(s)X A,
RS(~o)
in which R((p) is the retinal illumination in lumens per square
meter (lm m2
), PA(so) is the apparent pupil area in square
millimeters (mm2
), RS(so) is the retinal area per solid angle
in square millimeters per steradian (mm2
sr-'), and A is the
Ganzfeld luminance in lumens per square meter X steradian
(lm m2
sr-1
).
Light losses in the eye media (from scattering, absorption,
reflection, etc.) are not taken into account. Parallel incident
light rays are assumed to focus perfectly at the retina at the
point of incidence of the principal ray. This last assumption
will overestimate inhomogeneities in the light distribution
over the retina. Light spreading by spherical and comatic
aberration will result in smaller differences in retinal illumination
at various angles of incidence.
(4) The luminous flux L(<p) entering the eye is calculated
for a luminance field with a limited size. The light source is
assumed to be a homogenous field at infinity centered on the
optic axis, subtending the solid angle SA(so) bounded by the
principal rays having an angle of incidence s° with the optic
axis. The apparent pupil area PA(so) (mm2
) multiplied by
the solid angle dSA(sp) (sr2
) is the luminous flux dL(sp) (lm)
that enters the eye in the ring-shaped solid angle dSA(so) if
the light source has a luminance of 1 lm/m-2
sr-1
. The luminous
flux L((p) is calculated with a discrete Simpson integration
routine on dL(so).
(5) The retinal area RA(so) corresponding to the solid
angle SA(sp) is calculated with the discrete Simpson integration
routine on dRA(qo).
(6) The angle of incidence PI(qp) of the principal ray on
the retina is obtained by these calculations. Because it may
be relevant to discussions of photoreceptor orientation it is
plotted in Fig. 3.14,15
(7) The path length PL(sp) of the principal ray through
the crystalline lens can be used to estimate the light absorption.
Light absorption depends on the age ofthe subject16
and
on the wavelength ofthe light.17
Neither variable is considered
in this paper. Therefore the absorption is not calculated,
and only the path length in the lens is plotted (Fig. 4).
The calculations in points (1)-(3), (6), and (7) are done at
angles of 00 to 1000 with steps of 100. In points (4) and (5),
the Simpson integration routine is used on values calculated
in points (1)-(3). Hence the values of the luminous flux L(ep)
and the retinal area RA(<p) are obtained at 20°-interval steps.
The curves in Figs. 3-9 are drawn by eye through the calculated
values.
All calculations are done on a schematic eye with spherical
surfaces only12
and on schematic eyes with surfaces with
lip)
dSA(Y)
Fig. 2. Retinal area dRA(ep) corresponding to an annular solid angle
of visual field dSA(sp) calculated for various values of qoand with an
annular width of 0.50.
Aart C. Kooijman
1546 J. Opt. Soc. Am./Vol. 73, No. 11/November 1983
. ........ A2zo
A2----B2
L)Oo
T. C2
°I~o 801
WV'
0
0 20 40 60 80 100
ANGLE WITH OPTIC AXIS IN
VISUAL FIELD ( °)
Fig. 3. Angle of incidence on the retina ofthe principal ray ofa light
pencil ofparallel rays as a function of angle of incidence in the visual
field. A, B, and C refer to the eye models A, B, and C in the caption
of Table 1. The numbers 2 and 8 indicate that the eye model had a
pupil diameter of 2 or 8 mm.
I
IM
11-E
tDZ:
Z: D
zL-J
-J W
M
0 20 40 60 80
ANGLE WITH OPTIC AXIS IN
VISUAL FIELD ( °)
with the x axis along the optic axis and the apex in the origin.
The shape ofthe cross section depends on r and p. The apical
radius of curvature r determines the paraxial optical power
and is not influenced by the value of the conic constant p.
The value of p is less than zero for a hyperbola, equal to zero
for a parabola, between zero and one for an ellipse with the
longer axis along the x axis, one for a circle, and larger than
one for an ellipse with the shorter axis along the x axis. The
relation between p and the elliptic eccentricity e (Refs. 8, 19,
20) is given by p = 1 - e2
. A typographical error in formula
(4) of Ref. 19 confused Drasdo and Fowler8
at this point but
was corrected by Drasdo and Peaston.2 0
The expression I(b/a)- 1 21, with a and b the semiaxis in
the x and the y direction, respectively, is a little bit confusing
r
0a:
r
-J
4
a:
Wi-
W
w
F-
a:
-J
w
100
Fig. 4. Path length ofthe principal light ray through the lens. A and
B refer to the eye models A and B in the caption of Table 1. L is
calculated in an eye with aspheric anterior and posterior lens surfaces.
It is the combination of models D and E in the caption of Table 1.
spherical and aspheric conic-section profiles (Table 1). The
cross sections of these surfaces can be described by the conic
equation (Ref. 18, p. 392)
( 2r +Y =r2
P P P
*p - 2xr + y2 = o,
0 20 40 60 80 10 0
ANGLE WITH OPTIC AXIS IN
VISUAL FIELD ( )
Fig. 5. Retinal illumination. A, B, and C are calculated in eye
models A, B, and C as defined in the caption of Table 1.
A
Aart C. Kooijman
Aart C. Kooijman
0 20 40 60 80 100
ECCENTRICITY OF THE BORDER OF
THE FIELD OF UIEW ( 0)
Fig. 6. Relative light flux into the eye witha homogeneous luminance
field oflimited size. All eye models (irrespective of whether the pupil
diameter is 2 or 8 mm) in this study show the same relation between
visual field and light flux. With a Ganzfeld with a luminance of 1
cd/M2
the absolute light flux is 212 tlm with an 8-mm pupil and 13
ttlm with a 2-mm pupil.
Co
CD
A8 X
-- A2 /
n---B2 G
- o
1,III,
E
ECETIITCFTECODRO
W/
0:
WLI
0 20 40 60 80 100
ECCENTRICITY OF THE BORDER OF
THE FIELD OF UIEW ( 0)
Fig. 7. Retinal area corresponding with angular field of view. A, B,
and C refer to the eye models A, B, and Cin the caption of Table 1.
The numbers 2 and 8indicate that the eye model had a pupil diameter
of 2 or 8 mm.
Vol. 73, No. 11/November 1983/J. Opt. Soc. Am. 1547
because it holds only for the conic functions with real values
of b < a (half of the family of elliptical curves) and imaginary
values of b (hyperbolic curves). The relation between
I(b/a) - II and the conic constant p is
la I
P = (a)2 = (I_1-a 11l)2
~LJ
-<I
M-.
J
a-
10wc
D
A:
IW
"
MLa
ifa and b are real and b <a,
0 20 40 60 80 100
ANGLE WITH OPTIC AXIS IN
UISUAL FIELD (°0)
Fig. 8. Retinal area per solid degree in the visual field. B2: Calculated
relation in the eye model with aspheric (conic constant p =
0.75) anterior and posterior corneal surfaces and spherical lens and
retinal surfaces. Pupil diameter, 2 mm. D&F: Relative values
calculated by Drasdo and Fowler8
in an eye model with a single
aspheric (conic constant p = 0.5) corneal surface and spherical lens
and retinal surfaces.
CD
0
E o
E04
-J
I-
I
0:c
7'o
0 20 40 60 80
ANGLE WITH OPTIC AXIS IN
UISUAL FIELD ( 0)
100
Fig. 9. Length of retinal arc between the optic axis and the point of
incidence ofthe principal ray with eccentricity so. A, B, and C refer
tothe eye models A, B, and C in the caption ofTable 1. The numbers
2 and 8 indicate that the eye model had a pupil diameter of 2 or 8
mm.
11 1 1
LOX
U IV.
- ---- B2 \. .D&F
t
- I I I I I I I I I
1548 J. Opt. Soc. Am./Vol. 73, No. 11/November 1983
b
|--1l >1
a if b is imaginary and a is real.
p = (bI = 1 _ |- - 112
aa
The value of p for the aspherical surfaces was taken from
papers on measurements of the curvature of the cornea,19
the
lens,21
and the retinal surface.22
An ellipsoid anterior and posterior corneal surface (p =
0.75), a hyperboloid anterior lens surface (p = -2.06), and a
paraboloid posterior lens surface (p = 0) were used in the
calculations. The apical radii of curvature were the same as
in the calculations with spherical surfaces. Only for the retinal
surface was the apical radius of curvature changed with
a change in the value of p. In the front-to-back cross section
ofthe eye the retinal surface is considered to be either a part
of a circle with a radius of 10.8 mm or a part of an ellipse with
p = 1.346 and an apical radius of curvature of 14.1 mm. The
value of p is taken from the data of Krause.2 2
The axial
length values (19.72 and 22.88 mm) of Krause are too short to
cross the posterior corneal surface near the position of the
corneoscleral junction. Therefore slightly larger values are
used in these calculations. The lengths of the short and long
axes ofthe retinal surface are taken as 20.96 and 24.30 mm and
are, respectively, along and perpendicular to the optic axis.
RESULTS
The results of the calculations on apparent pupil size and
retinal area per solid angle are tabulated (Table 2). The apparent
pupil area and the retinal area per solid angle are used
to calculate retinal illumination (Table 2 and Fig. 5). The
apparent pupil area is also used to calculate the luminous flux
into the eye in a solid angle centered on the optic axis with
various values of stimulus field size (Fig. 6). The retinal area
dRA(0) is used to calculate the retinal area RA(ep) corresponding
to a solid angle around the optic axis and bounded
Aart C. Kooijman
by the principal rays with an angle s° with the optic axis (Fig.
7). The calculations with aspheric anterior or posterior lens
surfaces yielded data on apparent pupil diameter, retinal area
per solid angle, and retinal illumination, which were nearly
identical to the data obtained with the spherical eye model (all
differences <2%).
In the calculations no limitations were set on the diameter
of-the cornea. It appeared from the calculation that the extreme
peripheral light rays (so = 1000) hit the corneal surface
at 6.91 mm (spherical cornea) and 7.32 mm (ellipsoid cornea)
perpendicularly from the optic axis.
DISCUSSION
Spherical Eye Model
Calculations are done on an eye model with an iris without
thickness. The apparent pupil areas calculated with 2- and
8-mm pupils for oblique directions are in good agreement with
the experimentally measured values of Spring and Stiles.2 3
In a model with an iris without thickness, vignetting occurs
at angles of incidence of 900 and 1000. The incident light rays
cannot reach the edge of the pupil at the other side of the
optical axis. At that side the entrance pupil is limited by the
light rays that are just refracted by the convex crystalline lens
surface.
In a model with an iris with some thickness, vignetting occurs
at all angles of incidence of 100 and above by the posterior
and anterior edge of the iris. Jay7
measured a smaller pupil
area at angles of over 700 than did Spring and Stiles,23
and he
assumed that it was an effect of the thickness of the iris. It
has a limited influence on the calculations on the retinal light
distribution because the retinal illumination is nearly homogeneous
up to 700, and only at higher angles of incidence
will the calculated retinal illumination be changed.
Drasdo and Fowler8
calculated the retinal area per solid
degree. The absolute values in the present study differ from
the values of Drasdo and Fowler because of a difference in the
axial lengths of the theoretical eye models. The relative
Table 2. Apparent Pupil Diameter, Retinal Area per Solid Angle, and Retinal Illumination Resulting from a
Ganzfeld Light Source with a Luminance of 1cd/M2
.O SPHERIC EYE MODEL
cr, o
rN.
-J
- E- E
Iix
w
p..-.
CC e
WE
ax
ON
- E
E
, .
ASPHERIC CORNEARQ 8SPHERIC RETINR
I.-
A-E
IE
p. '..-
WI---W
c:W
cc r
.¢cln
at!.-J
IM
crt-
Z:'%
~_E
E
a!
-J E
~c'E
a! -
-J.-,
-LLE
D E
G _
crr
p.R.
¢C$
¢r S
- E
wI E
ax
-J
J E
-a E
ax
0 ABSOL. 2 .085 .011 8 .085 .18 2 .085 .011 2 .085 .011
0 1 1 1 1 1 1 1 1 1 1 1
10 S .99 .99 1 .99 .99 1 .99 .99 1 .99 1
20 v .96 .96 1 .96 .96 1 .95 .96 1 .97 .99
30 r .90 .91 .99 .91 .91 1 .90 .90 .99 .93 .97
40 v .83 .84 .99 .84 .84 1 .82 .83 .99 .89 .94
50 W .74 .76 .98 .75 .76 1 .73 .74 .98 .83 .89
60 v .64 .66 .97 .66 .66 1 .62 .64 .97 .76 .83
20 - .52 .55 .95 .54 .55 .98 .50 .53 .96 .68 .76
80 5 .39 .43 .92 .40 .44 .90 .38 .41 .93 .57 .684
90 w .24 .29 .85 .23 .34 .68 .25 .28 .87 .41 .59
100 X .07 .19 .37 .11 .25 .45 .12 .18 ,55 .28 .25
XC,
- X3¢s-
_J
Z.a
a:o
-JW
: Ew
Lr-Z W
W¢
Ofn
w-
-
1 11
J-1E
t-- E
W
JX %.'
Vol. 73, No. 11/November 1983/J. Opt. Soc. Am. 1549
retinal area per solid degree as a function ofperipheral angle
is in accord with the results of Drasdo and Fowler8
(Fig. 8),
who used an ellipsoid corneal surface with a conic constant p
= 0.5. The use ofp = 0.5 or p = 0.75 in the present study does
change the curve less than the line width in the figure.
The apparent pupil area determines the light flux entering
the eye. The distribution of this flux over the retina depends
on the retinal area per solid degree. The relative values
(Table 2) show that both change by nearly the same factor as
a function of angle of incidence. The relative retinal illumination
is the ratio between apparent pupil area and the retinal
area per solid degree. The result is that the illumination is
surprisingly flat over the retinal surface (Fig. 5A).
The relative light distribution over the retina is barely influenced
by the size of the pupil. [One has to realize that a
distinct angle of incidence does not necessarily correspond
with the same retinal locus if the size ofthe pupil or the shape
of any surface is changed (Fig. 9)].
Aspheric Eye Model
The schematic eye with spherical surfaces is a reasonable
representation of a real emmetropic eye for the on-axis optical
properties. The off-axis dimensions of real eyes can deviate
considerably from the spherical shape. The cross sections of
the cornea and the retina can be better described by ellip-
ses.'9
'22
The anterior and the posterior human lens surfaces
have cross sections that can be described by a hyperbola and
a parabola.2
1 The sensitivity of the retinal light distribution
to modifications in the model of the eye is calculated by using
these surfaces in the model. To separate the influences of the
surface modifications the calculations are done on the
spherical eye model with only one or two surfaces at a time
changed into an aspheric shape.
A realistic value of asphericity of the cornea results in a
computed light distribution over the retina that is nearly as
homogeneous as that calculated for the spherical eye model
(Fig. 5B). A change of the anterior or the posterior lens surface
into an aspherical shape has only a very small influence
on the retinal light distribution (<1.5%). The modification
of the anterior lens surface changes the position of the real
pupil because it is assumed to be coincident with the anterior
lens surface. This results in a small change of the apparent
pupil area (<1%). The retinal area per solid angle is influenced
(<2%) because the principal rays are a little displaced
and cross the anterior lens surface at another point. Modification
of the posterior lens surface leaves the apparent pupil
size undisturbed and has only a slight influence on the retinal
area per solid angle (<1%).
Modification of the shape of the retinal surface has the
largest effect on the light distribution on the retina (Fig.5C).
Because at peripheral angles of incidence a larger retinal area
is illuminated per solid angle of visual field in comparison with
the area illuminated on a spherical retina, the resulting illumination
decreases much more with eccentricity than in the
models with a spherical retina.
The above theoretical results do not agree with experimental
studies on retinal illumination with peripheral
angle10
"' and retinal light distribution with small and large
pupil diameters. 9
Both Gross'0
and Young" measured the
luminance of the retinal image of a small light source as a
function of peripheral angle. But the quality ofthe imaging
system deteriorates with peripheral angle, and therefore the
wU
LA.
WU)
LU
0
C-)
Z_ELvr_
y/ff//L////2 mm
'0 20 40 60 80 100
ANGLE OF INCIDENCE WITH
THE OPTIC AXIS < 0)
Fig. 10. Corneal arc in the meridional plane, covered by the light rays
that pass through the pupil, depends on the angle of incidence and
on the size of the pupil. These data are collected with the spherical
eye model.
luminance ofa small image decreases as the lightis spread over
a larger area.24
I have calculated the retinal illumination with
Ganzfeld illumination whereby image quality influences
retinal illumination less than in the case in which the image
of a small light source is being used. If the image of a small
light source deteriorates, then the retinal illumination decreases
as the light spreads. A Ganzfeld light source can be
represented by many small light sources side by side, and the
decrease of the illumination in each image will probably be
compensated for in the most part by light spread from surrounding
images. Bedell and Katz6
recently combined the
experimental results of Jay7
on pupillary area with the calculations
ofDrasdo and Fowler8
on retinal area per solid angle
of visual field to calculate the retinal illumination. The
conclusion of Bedell and Katz that retinal illumination is
nearly constant up to about 800 in the peripheral field is
identical to the calculated results in this study (Fig. 5).
Hoffman et al.9
measured electroretinogram (ERG) responses
on Ganzfeld stimulation with large and small pupil diameters
and concluded from their results that the pupil acts not only
as an aperture but also as a field stop that influences the distribution
of light on the retina. However, they used an ERG
contact lens with a clear corneal lens (diameter 9 mm) and a
black scleral rim. This configuration particularly influences
the peripheral retinal illumination because light rays cannot
enter the eye through the peripheral parts of the corneal
surface (Fig.10). The influence of this corneal diaphragm on
the light input from peripheral angles is different for small and
large pupils. This can be the cause of differences in the ERG
responses obtained with small and large pupils.
Aart C. Kooij'man
1550 J. Opt. Soc. Am./Vol. 73, No. 11/November 1983
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Aart C. Kooijman
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