Vol. 2, No. 8/August 1985/J. Opt. Soc. Am. A 1273
Accommodation-dependent model of the human eye with
aspherics
R. Navarro, J.Santamaria, and J.Besc6s
Instituto de Optica, Serrano 121, 28006 Madrid, Spain
Received February 24, 1984; accepted February 28, 1985
We consider a schematic human eye with four centered aspheric surfaces. We show that by introducing recent experimental
average measurements of cornea and lens into the Gullstrand-Le Grand model,the average spherical
aberration of the actual eye ispredicted without any shape fitting. The chromatic dispersions are adjusted to fit
the experimentally observed chromatic aberration of the eye. The polychromaticpoint-spread function and modulation
transfer function are calculated for several pupil diameters and show good agreement with previous experimental
results. Finally, from this schematic eye an accommodation-dependent model is proposedthat reproduces
the increment of refractive power of the eye during accommodation. The variation of asphericity with accommodation
is alsointroduced in the model and the resulting optical performance studied.
INTRODUCTION
Many experiments in physiological optics as well as in other
fields, such as optical design, digital image processing, and
robotics, must take into account the behavior and contribution
of the optical system of the eye as a part of the whole system.
In some cases a generalized behavior characterized by the
optical transfer function (OTF) is sufficient, but a moredetailed
model of the optical system is often needed. Since the
beginning of this century, many modelshave been proposed,
progressively developed, and improved. Among them we
shall point out those of Helmholtz1
and Gullstrand.2
The
latter, after being revised by Le Grand,3
has been widely used
for first-order calculations, even though this model does not
agree with the measured values of eye aberrations. To obtain
better agreement, aspheric surfaces and a graded-index lens
have been incorporated in theoretical eye models. In this way
Lotmar4
introduced asphericitiesforthe corneaand the back
surface of the lens in the Gullstrand-Le Grand model. He
found a spherical aberration ofthe sameorder as that of the
experimental findings, but the shell structure (or graded
index) of the lens was required for off-axis aberrations to be
predicted. El Hage and Berny 5
computed the aspheric shape
of a two-surfacelens that fitted their experimental measurements
of the spherical aberration of the eye. Nakao et al.6
showedthat the spherical aberration of the lens can be predicted
by the actual shell structure and proposed a theoretical
eye model in accordance with it. Other eye models7
'8
also take
into account the shell structure of the lens. However, because
of the increasing complexity of these models (a large number
of parameters are involved), attempts to model the optics of
the eye by using a schematic two-surface lens are still of great
interest. In this way Kooijman9
has recently used a schematic
model, similar to the one proposed in the present paper, to
compute the retinal illumination corresponding to a Ganzfeld
illuminance field.
It is not expected that a schematic model could predict the
aberrations corresponding to the actual eye if the shell
structure of the lens is not considered. It is generally agreed
that such a model cannot predict off-axis aberrations accurately,
but whether it canpredict spherical abberation is not
so clear. The research of Lotmar 4
and El Hage and Berny 5
indicatesthat, instead ofthe shellstructure (orgraded index)
of the lens, an effective refractive index could be used to
predict axial-spherical aberration. The resulting model,although
not anatomical, isstrongly simplifiedwith respect to
the shell of graded-index models, and the ray tracing becomes
easier.
In the present paper a simple schematic eye, to be used in
on-axis calculations, is proposed and tested. (See Fig. 1.) The
model considers an optical system formed by four centered
quadric refracting surfaces with rotational symmetry; each
surface is defined by two parameters, the radius and the
asphericity., The variations of the refractive indices with
wavelength havebeen computed to fit the chromatic aberration
of the eye. The optical performance of the resulting
model has beentested by computingspherical aberration and
the polychromatic modulation transfer function (MTF) and
the point-spread function (PSF), and it shows good agreement
with experimental findings. This fact indicates that the onaxis
optical performance of the eye can be modeled without
considering the shell structure of the lens.
Attempts to model the accommodationdependence of the
eye have also been made by a small number of authors.
Gullstrand 2
and Le Grand 3
give different lens parameters for
the unaccommodated and forthe fully accommodated theoreticaleyebut
not forthe moreinterestingintermediate states.
A paraxial model that varies with accommodation wasproposed
by Blaker.10
He considers the continuous variation of
the shape and graded-index structure of the lens with accommodation.
In the present paper an accommodationdependent
model based on our unaccommodated schematic
eye isalso proposed. Finally, a variation of the lens asphericity
isalso introduced in the model and tested.
SCHEMATIC EYE
The optimized version of the schematic-eye model has been
obtained by introducing some recent average experimental
data about the cornea and the lens into the Gullstrand-Le
0740-3232/85/081273-09$02.00 © 1985 Optical Society of America
Navarroet al.
1274 J. Opt. Soc. Am. A/Vol. 2, No. 8/August 1985 Navarro et al.
difficult. First, there is a lack of experimental data about
dispersions (or constringences) on human eyes,17
and second,
the dispersion of the effective refractive index assumed for the
lens cannot be compared with the dispersions of the actual
graded-index lens.'8
On the other hand, the Herzberger
formula,19
as a better description of optical media, has been
applied here instead ofthe Cornu formula used by Le Grand
tocompute the refractive indices.16
The Herzbergerformula
is expressed as follows:
n(X) = al(X)n** + a2(X)nF + a3 (X)n, + a4(X)n*, (2)
where n** = n(0.365 ,m), nF = n(0.4861 Am), n,
1m), n* = n(1.014,um), and
= (0.6563
Fig. 1. Schematic eye.
Grand theoretical eyeand by computingrefractive indices to
adjust the chromatic aberration ofthe eye. Here an aspheric
surface is substituted for the spherical cornea of the Gullstrand-Le
Grand eye. The radius and the asphericity correspond
to an average human cornea obtained from measurements
made in vivo by Kiely et al. "- Aspherics are also
used for the crystalline-lens surface. The asphericity parameters
have been taken from the average data obtained in vitro by
Howcroftand Parker.'2
Here,the Gullstrand-Le Grand radii
have been conserved for two main reasons. First, excised
lenses show smaller radii than those measuired in vivo,-perhaps
because they are in an intermediate accommodation state.
Second, the Gullstrand-Le Grand model does predict refractive
power of the lens' and hence there is no a priori reason
to change its radii of curvature. The effective refractive index
of the lens has been also kept because it implies a strong
simplification of the complex internal structure of the lens,
and, as we will see, it allows the on-axis optical performance
of the eye to be represented within a good approximation.
The schematic-eye parameters are. shown in Table 1.
Those of Gullstrand-Le Grand3
and Kooijman9
are alsorepresented
for comparisonpurposes. Note that our refractive
index of the cornea is different. It has been taken from the
original Gullstrand eye2
to compensate partially for the
greater refracting power introduced by a smaller radius (our
corneal radius is also closeto the original from Gullstrand).
Compensationis not total, and the resulting refractive power
isslightly greater than that of the Gullstrand-Le Grand eye.
The image plane (retina) is then shifted 0.2mm backward
with respect to the Gullstrand-Le Grand eye, and thus the
vitreous thickness is assumed to be 16.4mm. The Kooijman
model coincides with that of Gullstrand-Le Grand except for
the asphericities. Our anterior lens surface asphericity differs
slightly from that of Kooijman,although both of them have
been taken from the same source.1 2
Each quadric surface of
the model isrepresented by the formula
x2 +y2 + (1+Q)z2-2Rz =0, (1)
where R is the radius of curvature and Q the asphericity parameter;
x, y, and z are spatial coordinates, the z axis being
the optical axis.
The refractive indicesof the ocularmedia forthe different
wavelengths have been determined to fit the experimental
chromatic aberration' 3
"14
taking as the departure point the
constringences of Polack,15
cited by Le Grand.' 6
Here there
are two problems that make comparison with experiments
al(X) = 0.66147196 - 0.040352796\2
0.2804679 0.03385979
_~~
2- Xd
2
(X
2
- X02)2
a2(X) - -4.20146383 + 2.73508956X2
1.50543784 0.11593235
X2 - X2 (X2
- Xo2
)2
a3W = 6.29834237 - 4.69409935X2
1.5750865 0.10293038
2 - X22 (X2- X02)2
a4(X) = 1.75835059 + 2.36253794X2
0.35011657 0.02085782
+\2- 2 (X2-X02)2
where Xo2
= 0.028 AM
2
.
Table 1. Schematic Eye Parameters Comparedwith
Those of Gullstrand-Le Grand and Kooijman
Schematic GullstrandParameters
Eye Le Grand Kooijmana
Radius of curvature (mm)
Anterior surface of 7.72 7.8 7.8
cornea
Posterior surface of 6.5 6.5 6.5
cornea
Anterior surface lens 10.2 10.2 10.2
Posterior surface lens -6 -6 -6
Asphericity.
Cornea anterior -0.26 -0.25(+1)
Cornea posterior 0 -0.25(+1)
Lens antbrior -3.1316 -3.06(+1)
Lens posterior -1 -1(+1)
Thickness (mm)
Cornea 0.55 0.55 0.55
Aqueous 3.05 3.05 3.05
Lens 4 4 4
Vitreous 16.4 16.6 16.6
Refractive index
Cornea 1.367 1.3771 1.3771
Aqueous 1.3374 1.3374 1.3374
Lens 1.42 1.42 1.42
Vitreous 1.336 1.336 1.336
Resulting refractive 60.4 59.94 59.94
power (diopters)
a Kooijman1 0
givesconicalparameters p instead of asphericity Q;p = 1+
Q.
Vol. 2, No. 8/August 1985/J. Opt. Soc. Am. A 1275
Table 2. Refractive Indices of the Schematic Eye
n** nF nc n*
(X= (X= (X= (X=
Ocular 0.365 0.4861 0.6563 1.014
Medium gm) Am) gim) ptm)
Cornea 1.3975 1.3807 1.37405 1.3668
Aqueous 1.3593 1.3422 1.3354 1.3278
Lens 1.4492 1.42625 1.4175 1.4097
Vitreous 1.3565 1.3407 1.3341 1.3273
Table 3. Constringences of the Schematic Eye
Comparedwith Data of Polack and Sivak and
Mandelman
Ocular Theoretical
Medium Eye Polacka Sivak and Mandelmanb
Cornea 56.5 56 54 (cow) 61 (cat)
Aqueous 49.61 53.3 56 (water) 61 (cat)
Lens 48 52 29 (periphery) 35 (core)
Vitreous 50.9 53.3 56 (water) 60 (cat)
a Ref. 15.
bRef.17.
Table 2showsthe refractive indices n**,nF, nc, and n* of
the ocular media of the schematic eye that we finally obtained,
and Table 3showsthe constringences,definedas (nD - 1)/(nF
- nc ) compared with those of Polack15
and those of Sivak and
Mandelman17
(from cat, cow,water, and the human lens).
These indices agree only in their order of magnitude. As
mentioned above, our refractive indices and constringences
have been computed to fit chromaticaberration, and thus we
have not tried to make them anatomical.
SPHERICAL AND CHROMATIC ABERRATIONS
In order to test the model, the on-axis aberrations of the
schematic eye have been computed by ray tracing. The resulting
longitudinal-spherical aberration (LSA)of the schematic
eye as well as the contributions of the corneaand the
lens are shown in Fig. 2. The parabolic adjustment made by
Van Meeteren 20
from experimental data of several au-
thors13
'2 1
-2 3
and the spherical aberration of the Gullstrand-Le
Grand eye are also included in the figure for comparison
purposes. A good agreement with the Van Meeteren adjustment
isobtained, whereas the Gullstrand-Le Grand eye
predicts larger aberration. With respect to the contribution
of the cornea and the lens tothe spherical aberration, our results
are in agreement with those of El Hage and Berny5
; the
lens has a spherical aberration of opposite sign from and
smaller than the cornea, playing a compensatory role. There
is agreement in average aberration values,but disagreement
can be found in individuals' eyes because monochromatic
aberrations show wide variations among subjects.'3
'2 1
-
2 4
Chromatic aberration, the most important in fovealvision,
has moresimilar values for different eyes than monochromatic
aberration. This is probably because the refractive indices
and chromatic dispersions, which depend on the composition
of the media, are practically the same for all subjects (only
Millodot and Newton2 5
reported a possiblechange of refractive
indices with age), whereas the shapes of the refracting
surfaces have bigger variations. As has been pointed out
above, the refractive indices of our schematic eye have been
computed to fit the experimentally measured chromatic aberration,
using the Herzberger formula. The chromatic aberration
was computed by paraxial-ray tracing, and the refractive
indices wereprogressively changed until the experimental
aberration was obtained. Refractive indices are shown
in Table 2,and the goodagreement of chromatic aberration
with the experimental findings of Ivanoff'3
and Wald and
Griffin' 4
can be seen in Fig. 3.
On the other hand, night myopia also appears in this eye
model. With an 8-mm pupil diameter, the best image was
found in a plane shifted 0.77 D from the paraxial-focus plane.
This value, added to 0.6-0.8 D, which corresponds to the state
of accommodation in darkness,2 6
'27
givesthe mean valueestimated
for night myopia (about 1.5D).
HEIGHT (mm)
-3 -2 - I LSA (dp)
Fig. 2. LSAof the cornea (- - - -),'of the lens (- -), and of the
wholetheoretical eye (-). The results corresponding to the Gullstrand-Le
Grand eye (....) and to Van Meeteren's parabolic adjustment
(-- - - -) are also represented. (dp, diopters.)
C A.
2,-
0
-I
-2
, P F, . 9 I A
400 500 Goo X(nm) 700
Fig. 3. Longitudinal chromatic aberration (C.A., in diopters) of the
theoretical eye (-). Experimental results ofWald and Griffin (A)
and of Ivanoff (X ) are also represented.
Navarro-et al.
1276 J. Opt. Soc. Am. A/Vol. 2, No. 8/August 1985
DIFFRACTION PATTERNS AND THE
MODULATION TRANSFER FUNCTION
The polychromatic PSF and the MTF corresponding to the
schematic eyehave been computed inthe plane ofmaximum
illuminance for several pupil diameters. The computing
method is described in what follows:
The pupil function for each wavelength is obtained by
polynomialfitting of the LSA,the waveaberration being obtained
by integration. The chromatic aberration (treated as
a defocusing)and the Stiles-Crawford appodizing effect are
also introduced, resulting in the pupil function
P(r) = exp(-0.05Rp 2
r
2
In 10)[exp ike(6OW2 0 r
2
+ W4or4
+ W60r6
+ W80r8
+...)],
a
0
C
(3)
where 0 • r < 1 isthe normalizedpupilar radial coordinate;
Rp is the pupil radius (in millimeters); BOW20 is a defocusing
that accountsfor the chromaticaberration and eventuallyfor
an additional shift of focus; and W4o,W60, W80, ... are the
resulting coefficientsof the spherical-waveaberration. The
first factor exp(-0.05Rp2
r2
In 10) is the effective transmittance
produced by the Stiles-Crawford effect.2 0
The monochromatic PSF and MTF are computed by
Fourier transform and autocorrelation of the pupil function,
respectively. Then the polychromatic functions are computed
by integration of the monochromatic ones along the
visible spectrum sampled in 40 intervals. Integration is made
for the D65 spectral distribution and for the spectral-sensitivity
functions of the eyeto obtain the tristimulus values.2 8
The maximum-illuminance plane is found by progressive
shifts of focus.
The considerable influence of the Stiles-Crawford appodization
is shown in Fig. 4. The figure represents the
monochromatic MTF for a 4-mm pupil diameter with and
withoutconsideringthe Stiles-Crawfordeffect. Theresulting
distribution of illuminance and chromaticity along a radius
of the polychromatic PSF and the polychromatic MTF are
represented in Fig. 5 for several pupil diameters.
C
Z 1
0.5
Normalizedspatialfreauencv
Fig. 4. Influence of Stiles-Crawford effect on the monochromatic
MTF of a 4-mm pupil-diameter schematic eye: aberration-free
system (--), eye model without apodization (-) and with apodi-
zation(---).
0.6,
y
04
0.21
a)
025 05 075
Normalizedspatial coordinate
p sina' (jm)
F~iI \
I / '//
II /7'
/I7
0.2 0.4
b)
06 X
C)
C
00.5
2
*0
Normalizedspatialfrequency
Fig. 5. Distribution of a) illuminance and b) chromaticity coordinates
(x,A)of the polychromaticPSF and c)the polychromaticMTF
for the schematic eye with the following pupil diameters: 2mm (O),
3mm (0), 4mm (X),5mm (A),6mm (13),and8mm (A). v=1corresponds
to the cutoff frequencyat A= 600nm, p is the radial coordinate
onthe imageplane,and a' isthe semiaperture angle. Dashed
curves correspondto the aberration-free system.
Navarroet al.
Vol. 2, No. 8/August 1985/J. Opt. Soc. Am. A 1277
Acomparisonwith experimental measurements ismade in
Figs, 6 and 7. Figure 6 shows the theoretical MTF for a 4-mm
pupil diameter compared with the experimental findings of
Arnulf,29
who measured the MTF directly, and those of
Campbelland Gubisch,3 0
who computed it by Fourier transform
from their experimental line-spread function. The
F
U
0.5
-J
ARNULF
N
N
N.
0,5
NORMALIZEDSFATIALFRECUENCY
Fig. 6. Polychromatic MTF of the schematic eye (theoretical)
computed for 4-mm pupil diameter, compared with experimental
curves from Arnulf (o)29 and Campbell and Gubisch (V).
3 0
(The last
corresponds to 3.8-mmpupil diameter.)
0.8.
0,6. 7\0.
0.4
6 7 8
. PUPIL DIAMETER
Fig. 7. Strehl ratio versus pupil diameter for the schematic eye
(theoretical) compared with experimental findings from Gubisch
(0).31 They differby an almost constant factor of the order of 0.7.
theoretical MTF lies between the two experimental curves.
The Strehl ratio, defined as the ratio of the maximum illuminancein
the PSF formedby a real systemto the maximum
illuminance in the PSF formed by an ideal optical system
working at the same aperture and with the same spectral
composition of light, is represented versuspupil diameter in
Fig. 7. The figure shows the Strehl ratio for the schematic eye
(theoretical) and for the experimental data of Gubisch.3
'
The theoretical Strehl ratio is higherthan the experimental
one, and both of them are related by an almost constant factor
(about 0.7). This discrepancy can be explained in the following
way: The experiment was made by a double pass
through the optical system ofthe eye30
'3
' that includes a diffusereflection
onthe retina, and thus the actual retinal image
is expected to have a little better quality. Moreover, irregular
aberrations32
'33
have not been considered in the model. According
to Van Meeteren,2 0
their influence on the image is not
greater than a defocusing of 0.15 D. We should remember
that the PSF and the MTF have been obtained in the plane
of maximum illuminance (maximum Strehl ratio), and the
inclusion of a small defocusing of about 0.15 D in the computations
would be enough to produce a better fitting.
ACCOMMODATION-DEPENDENT MODEL
As we have shown above, a model with four aspheric surfaces
can predict the on-axis performance of the average eye. Thus,
if a paraxial accommodation-dependent model with four
surfaces is available, the shapes (asphericities) of both anterior
and posterior lens surfaces could be adjusted to fit spherical
aberration for each accommodation state in the samewayas
wasdone by El Hage and Berny5
to fit the aberration of the
unaccommodated lens. In what follows, a paraxial model that
varies continuously with accommodation is proposed first.
Then, instead of computing asphericities for each accommodation
state, a functional variation of the asphericities with
accommodation is introduced in the model and tested.
It iswellknownthat the changeinthe anterior radius ofthe
lens is the major factor in the increment of refracting power
of the lens with accommodation,the posterior radius and the
thickness having a secondaryrole. This continuous change
of the anterior radius can be represented by a mathematical
function expressed by Blaker,10
who used a linear function.
Other authors have used a variation of the radius inversely
proportional to the accommodation in theoretical computations.3
4
However, experimental measurements'3
' 26
' 35
'3 6
show
wide variation of the form of the dependence. We looked for
a simple mathematical function that would connect both
unaccommodated and fully accommodated anterior radii of
the Gullstrand-Le Grand schematic eye and would have a
similar shape to the Ivanoff experimental average curve.13
The function finally adopted waslogarithmic. Comparison
of Ivanoff's mean curve and the adopted function ismade in
Fig.8,showingthe similarity of the shapes. The samefunctionhas
alsobeenused forthe variationof the posteriorradius
and the thickness of the lens by fitting averaged valuesfrom
Brown.36
The aqueous thickness is decreased by half ofthe
increase of the lens thickness. The resulting functions are
given in Table 4.
These changes during accommodation do not predict the
total increment in refractive power of the lens, which is given
by the change of the internal graded-index structure of the
Navarro et al.
1278 J. Opt. Soc. Am. A/Vol. 2, No. 8/August 1985 Navarro et al.
E \
I0
C,,
z \ \ IVANOFF'SMEANCURVE
JW8 z-.
THEORETICALEz
6
2 4 6 8 10
ACCOMODATION (dp)
Fig. 8. Variation of the anterior lens radius with accommodation.
The figure shows the experimental mean curve from Ivanoff13
and
that adopted for the schematic theoretical eye. They have similar
shapes, but the theoretical eye curve differs somewhat from that of
Ivanoff because our schematic eye starts from the Gullstrand-Le
Grand eye,whoseanterior unaccommodatedradius differs fromthat
of Ivanoff. (dp, diopters.)
Table 4. Accommodation Dependence of the Lens
Parameters on Accommodation A (in diopters)
Accommodation
Lens Parameter Dependence
Anterior lens radius R3 (A) = 10.2 - 1.75 *ln(A + 1)
Posterior lens radius R4 (A) = -6 + 0.2294 *ln(A + 1)
Aqueous thickness D2 (A) = 3.05 - 0.05 *ln(A + 1)
Lens thickness D3 (A) = 4 + 0.1 - ln(A + 1)
Lens refractive index n3(A) = 1.42 + 9 X 10-5 - (10 - A + A
2
)
lens with accommodation. However,an effective refractive
index instead of the graded-index structure has been used in
the model. In order to account forthe additional necessary
increment in refractive power, the intracapsular mechanism
of accommodation (postulated by Gullstrand 2
), by which the
effective refractive index of the lens also changes with accommodation,
has been introduced in the model. This
mechanism isfictitious and must be considered not as an actual
fact but as a mathematical artifice that permits a strong
simplification of the modeling. As a first approximation, a
parabolic adjustment has been made to fit the refractive
powers for 5 and 10 D of accommodation. The resulting expression
is also shown in Table 4. Figure 9 shows the resulting
refractive power of the eye, computed by paraxial-ray tracing,
versus accommodation. Starting from this paraxial model
and fromthe asphericities ofthe unaccommodatedschematic
eye, the case in which the asphericities also have a logarithmic
dependence on accommodation, with an amount of change of
the same order of magnitude than experimental findings,36
has been considered. The adopted functions are inTable 5.
With these functions the spherical aberration has been computed
for several accommodations, and the results are represented
in Fig. 10. Agreement with published experimental
results has been found in an individual eye37
(0 D and 3.4D
of accommodation).
The effective refractive indices have been computed in
order to keep the chromatic aberration for each accommodation
state (in longitudinal units) equal to that ofthe unaccommodatedeye.
This impliesthat the chromaticaberration,
expressed in diopters, increases slightly with accommodation,
a7 0
2i68
Uj 66 -
64-
62
60
0 2 4 6 8 IO
ACCOMODATION (dp)
Fig. 9. Refractive power of the schematic eye versus accommodation
(@). The straight line corresponds to the ideal fitting. (dp, diop-
ters.)
Table 5. Test Parameters Used in Optical
Performance Computations
Asphericities
Anterior lens asphericity Q3(A) = -3.1316 - 0.34 - ln(A + 1)
Posterior lens asphericity Q4 (A) = -1 - 0.125 - ln(A + 1)
Refractive Indices of the Lens
Accommodation
(diopters) n** nF nc n*
3 1.4511 1.4298 1.421 1.4134
5 1.4533 1.43313 1.42423 1.4162
HEIGHT (mm)
....-r4
-3 -2 - I
LSA (dp)
Fig. 10. LSA obtained for several accommodation states: unaccommodated
(-), 3D (----), 5 D (--), and 10D (...). (dp,
diopters.)
Vol. 2, No. 8/August 1985/J. Opt. Soc. Am. A 1279
,0
.2
0.5
0.61
y
0.4
.a
0
C
0
0
a)
025 05
Normalized spatialcoordinate
p sin Oa(/AM)
F
I
I
I
I
I
I
I
0.75
b)
0.2 0.4 0.6 X
C)
0 0.5
Normalized spatialfrequency
Fig. 11. Distribution of a) illuminance and b) chromaticity coordinates
(x, y) of the polychiomatic PSF and the c) polychromatic MTF
obtained for the 3 D accommodated eye with the following pupil diameters:
2mm (@),3mm (O),4 mm (X),5mm (A),6 mm (o), and
8mm (A). v = 1corresponds to the cutoff frequency at X = 500 nm,
p is the radial coordinate on the image plane, and a' is the semiaperture
angle. Dashed curves correspond to the aberration-free
system.
in accordance (in a first approximation) with the effect observed
experimentally. 38
The resulting refractive indices for
3 and 5 D are shown in Table 5. Finally, the polychromatic
PSF and MTF have also been computed forseveral pupil diameters
and accommodations, and the results for.the 3-D
accommodated eye are shown in Fig. 11 (the results for 4 and
5 D are very similar). This simple model, although it does not
accurately predict average spherical aberration, does predict
the small improvement in optical performance for intermediate
accommodation states (near vision)that isusually ob-
served.
DISCUSSION AND CONCLUSIONS
Unaccommodated Schematic Eye
A schematic eye with four aspheric surfaces is proposed in this
paper. It is shown that, by introducing recent average experimental
measurements about the corneaand the lens into
the Gullstrand-Le Grand eye, the experimentally observed
average spherical aberration is predicted. The main difference
between our model and the previous work of El Hage and
Berny5
is that they computed the shape of the lens surfaces
to fit the experimental spherical aberration, whereas the
asphericities introduced in our schematic eye have been taken
from anatomical measurements.12
The resulting modeldoes
predict averagesphericalaberrationwithoutany shapefitting.
This fact seems to be in disagreement with previous findings 6
that show that the graded-index structure of the lens, not
considered in our schematic eye, has an important role on the
state of spherical correction of the lens. In this waya schematic
eyethat does not include the graded-index lens is not
expected to reproduce the spherical aberration of the actual
eye. 'However, Lotmar4
pointed out that not only the
graded-index structure of the lens but the asphericities
(shapes) have an important role in the spherical aberration
of the eye. In fact, he found order-of-magnitude agreement
in spherical aberration just by introducing asphericities In the
cornea and in the back surface of the lens. He also pointed
out that, on the contrary, off-axisaberrations cannot be predicted
without consideringthe graded-indexlens. Therefore
it appears that asphericities play the major rolein spherical
aberration, whereas off-axis behavior is managed by the
graded-indexstructure ofthe lens. Onthe otherhand, itmust
be taken into account that the asphericities of the lens surfaces
used in our model were measured from excised lenses,1
2 the
corresponding radii of curvature being smaller than those
measured in vivo. Perhaps both radii and asphericities
should correspond to an intermediate state of accommodation,
and consequently the actual asphericities of the unaccommodated
eye could differ somewhat from those used in the
model. Such a difference (expected to be small) could explain
the apparent contradiction of predicted spherical aberration
with anatomically determined asphericities and without
considering the graded-index lens.
The schematic eyeproposed in this paper is similar to the
one described by Kooijman in Ref. 9, which was published
when our workwas almost finished, and therefore a similar
monochromatic optical performance must also be expected.
He used his model as a wide-angle eyeto compute retinal illumination
produced by uniform field but not to reproduce
the optical performance of the eye. Nevertheless he found
Navarro et al.
I
1280 J. Opt. Soc. Am. A/Vol. 2, No. 8/August 1985
better results with his aspheric model than with the Gullstrand-Le
Grand model.
It must be pointed out that the refractive indices proposed
(Table 2) differ from those of Le Grand16
and are not anatomical.
They have been computed to fit the chromatic aberration
of the whole eye, and so the model could fail to predict
chromatic aberration ofthe different components of the eye.
The contribution of each ocular medium to the chromatic
aberration has been calculated by Millodot and Newton25
using Le Grand's model.
The model predicts the on-axis optical performance of an
average eye. Disagreements could be found if it is compared
with individual eyes because of the wide variation in the
spherical aberration from different eyes. Somemultilayered
or graded-index models can fit individual spherical aberra-
tion,7
but they need a large number of parameters (a greater
number if the polychromatic performance also has to be
modeled). In spite of their complexity they do not predict
irregular aberrations or intraocular scattering. Therefore
simplified eye models are still useful, mainly in those cases in
which on-axis performance is required. Irregular aberrations
are caused by the lack of rotational symmetry, decentering of
the surfaces, etc., and they have an influence on the image no
greater than a defocusingof 0.15D,according to Van Meet-
eren.20
However, more recent work on monochromatic aberrations
of the eye33
attributes a more important roleto the
irregular aberrations, even more important than the spherical
aberration. This seems reasonable for individual subjects,
because the eye is not a rotationally symmetric system and,
on the other hand, the fovea is placed off the optical axis of the
eye. However, when an average eye is modeled, the average
aberration is then composed mainly of a rotationally symmetric
aberration (spherical) and a residual aberration (irregular).
Also,the Stiles-Crawford effectand the chromatic
aberration soften the relative influence of the monochromatic
irregular aberrations.
Accommodation-Dependent Model
Based on the unaccommodated eye model, a schematic eye
that varies continuously with accommodation is also proposed.
A logarithmic variation for the radius and the thickness of the
lens has been adopted instead of the linear variation proposed
by Blaker.10
Both kinds of variation are approaches to the
actual changes during accommodation. Linear variation has
been found in a few individual cases, while logarithmic variation
appears to be a better approach to average variation.
The intracapsular mechanism of accommodation has been
introduced in the model to account for the increment of refractive
power produced by the change in the graded-index
structure of the lens during accommodation. The results
obtained inthe accommodation-dependent case showthat it
is a better approach than previous paraxial models. In fact,
although spherical aberration ofthe accommodatedeyeisnot
accurately predicted, the resulting polychromatic optical
performance seems reasonable. Further improvements on
the accommodation-dependent model can be made by computing
more adequate asphericities and constringences, although
more experimental data are needed to permit effective
modeling.
Navarro et al.
REFERENCES
1. H. von Helmholtz, Physiologische Optik, 3rd ed. (Voss, Hamburg,
1909).
2. A. Gullstrand, appendix in H. von Helmholtz, Physiologische
Optic, 3rd ed. (Voss, Hamburg, 1909), Bd. 1, p. 299.
3. Y. Le Grand and S. G. El Hage, Physiological Optics (SpringerVerlag,
Berlin, 1980).
4. W. Lotmar, "Theoretical eye model with aspherics," J. Opt. Soc.
Am. 61, 1522-1529 (1971).
5. S. G. El Hage and F. Berny, "Contribution of the crystalline lens
to the spherical aberration of the eye," J. Opt. Soc. Am. 63,
205-211 (1973).
6. S. N. Nakao, K. Mine, K. Nishioka, and S. Kamiya, "New schematic
eye and its clinical applications," presented at the
Twenty-First International Congress of Ophthalmology, M6xico
DF, Mexico,March 8-14, 1970.
7. 0. Pomerantzeff, H. Fish, J. Govignon, and C. L. Schepens,
"Wide-angle optical model of the eye," Opt. Acta 19, 387-388
(1972).
8. F. W. Fitzke,"A newschematic eye and its applications to psychophysics,"
presented at the Optical Society of America Meeting
on Recent Advances in Vision, Sarasota, Florida, April 30-May
3, 1980.
9. A. C. Kooijman, "Light distribution on the retina of a wide-angle
theoretical eye," J. Opt. Soc. Am. 73, 1544-1550 (1983).
10. J. W.Blaker, "Toward and adaptive model of the human eye,"
J. Opt. Soc. Am. 70, 220-223 (1980).
11. P. H. Kiely, G. Smith, and G.Carney, "The mean shape of the
human cornea," Opt. Acta 29, 1027-1040 (1982).
12. M. J. Howcroft and J. A.Parker, "Aspheric curvatures for the
human lens," Vision Res. 17, 1217-1223 (1977).
13. A.Ivanoff,Les Aberrations de l'Oeil (Masson,Paris, 1953).
14. G. Wald and D. T. Griffin, "The change in refractive power of the
human eye in dim and bright light," J. Opt. Soc. Am. 37,321-329
(1947).
15. A. Polack, "Le chromatisme de l'oeil," Bull. Soc. Ophthalmol.
Paris 9, 498 (1923).
16. Y.Le Grand,L'Espace Visuel,Vol.III of Optique Physiologique
(Masson, Paris, 1956).
17. J. G. Sivak and T. Mandelman, "Chromatic dispersion of the
ocular media," Vision Res. 22, 997-1003 (1982).
18. D.A.Palmer and J. Sivak,"Crystalline lens dispersion," J. Opt.
Soc. Am. 7, 780-782 (1981).
19. M. Herzberger, "Colour correction in optical systems and a new
dispersion formula," Opt. Acta 6, 197-215 (1959).
20. A. Van Meeteren, "Calculations on the optical modulation
transfer function of the human eye," Opt. Acta 21, 395-412
(1974).
21. M. Koomen, R. Tousey, and R. Scolnik, "The spherical aberration
of the eye," J. Opt. Soc. Am. 39, 370-376 (1949).
22. M. Francon, "Aberration sperique chromatisme et puvoir sbparateur
de l'oeil," Rev. Opt. Theor. Instrum. 30, 71-86 (1951).
23. H. Schober, H. Nunker, and F. Zolleis, "Die Aberration des
menschlichen Auges und ihre Messung," Opt. Acta 15,47-55
(1968).
24. M. Millodot and J. G. Sivak, "Contribution of the cornea and lens
to the spherical aberration of the eye," Vision. Res. 19, 685-687
(1979).
25. M. Millodot and J. A. Newton, "A possible change of refractive
index with age and its relevance to chromatic aberration," Albrecht
V. Graefes Arch. Ophthalmol. 201, 159-167 (1976).
26. M. Gomez, "Estudio sobre la acomodaci6n del ojo en presencia
de estimulos exteriores pr6ximos a los umbrales de percepci6n,"
Ph.D. dissertation (Universidad Complutense de Madrid, Madrid,
1965).
27. F. W.Campbell and J. A.E. Primrose, "The state of accommodation
ofthe human eye in darkness," Trans. Ophthalmol. Soc.
UK 73, 353-361 (1953).
28. J. Besc6s and J. Santamaria, "Colourbased quality parameters
for white light imagery," Opt. Acta 28, 43-55 (1981).
29. A.Arnulf,"Le systemeoptique de l'oeil en visionphotopique et
Vol. 2, No. 8/August 1985/J. Opt. Soc. Am. A 1281
mesopique," Excerpta Med. Int. Congr. Ser. 125, 135-151
(1965).
30. F. W. Campbell and R. W. Gubisch, "Optical quality of the human
eye," J. Physiol. 186, 558-578 (1966).
31. R. W. Gubisch, "Optical performance of the human eye," J. Opt.
Soc. Am. 57, 407-415 (1967).
32. M. S. Smirnov, "Measurement of the wave aberrations of the
human eye," Biophysics (USSR) 6, 776-795 (1961).
33. M. C. Howland and B. Howland, "A subjective method for the
measurement of monochromatic aberrations of the eye," J. Opt.
Soc. Am. 67, 1508-1518 (1977).
34. R. F. Fisher, "The significanceof the shape of the lens and capsular
energy changes in accommodation," J. Physiol. 201, 21-47
(1969).
35. E. F. Fincham, The Mechanism of Accommodation (Pullman,
London, 1937).
36. N. Brown, "The change in shape and internal form of the lens of
the eye on accommodation," Exp. Eye Res. 15, 441-459 (1973).
37. B. Patnaik, "A photographic study of accommodative mechanisms:
changes in the lens nucleus during accommodation,"
Invest. Ophthalmol. 6, 601-611 (1967).
38. F. Berny, "Etude de la formation des images retiniennes et determination
de l'aberration de sphericite de l'oeil humain," Vision
Res. 9,977-990 (1969).
39. K. Ukai and H. Ohzu, "Dynamic laser speckle pattern used to
determine eye refraction. I. Changes in chromatic aberration
ofthe eye with accommodation," presented at ICO-11,Madrid,
September 10-17, 1978.
Navarro et al.