A. V. Goncharov and C. Dainty
Vol. 24, No. 8 / August 2007 / J. Opt. Soc. Am. A
2157
Wide-field schematic eye models with
gradient-index lens
Alexander V. Goncharov* and Chris Dainty
Applied Optics Group, Department of Experimental Physics, National University of Ireland, Galway, Ireland
*Corresponding author: alexander.goncharov@nuigalway.ie.
Received October 20, 2006; revised February 18, 2007; accepted March 18, 2007;
posted April 2, 2007 (Doc. ID 76289); published July 11, 2007
We propose a wide-field schematic eye model, which provides a more realistic description of the optical system
of the eye in relation to its anatomical structure. The wide-field model incorporates a gradient-index (GRIN)
lens, which enables it to fulfill properties of two well-known schematic eye models, namely, Navarro’s model for
off-axis aberrations and Thibos’s chromatic on-axis model (the Indiana eye). These two models are based on
extensive experimental data, which makes the derived wide-field eye model also consistent with that data. A
mathematical method to construct a GRIN lens with its iso-indicial contours following the optical surfaces of
given asphericity is presented. The efficiency of the method is demonstrated with three variants related to
different age groups. The role of the GRIN structure in relation to the lens paradox is analyzed. The wide-field
model with a GRIN lens can be used as a starting design for the eye inverse problem, i.e., reconstructing the
optical structure of the eye from off-axis wavefront measurements. Anatomically more accurate age-dependent
optical models of the eye could ultimately help an optical designer to improve wide-field retinal imaging.
© 2007 Optical Society of America
OCIS codes: 330.4060, 110.2760, 330.4460, 080.3620, 010.1080.
1. INTRODUCTION
The earliest schematic eye model involving a high refractive index core and a lower index cortex for crystalline
lens was proposed nearly a century ago by Gullstrand [1].
This model with five spherical surfaces, usually referred
to as Gullstrand’s “No. 1” model, was intended to describe
aberrational properties of the human eye on axis. After
being revised by Le Grand and El Hage [2] (using a homogeneous index lens), this model has been widely used
as a first-order approximation in spite of poor agreement
with the measured values of the ocular aberrations. To
make theoretical eye models more consistent with experimental data, aspheric surfaces and a lens with a varying
refractive index have been considered. In these schematic
models, the lens is approximated either by a finite number of concentric “shells” with a slightly different index of
refraction or by gradient-refractive-index, or “gradientindex” (GRIN) elements with a smooth index decrease
from the lens center to its periphery. In the first group,
the index of the lens changes stepwise. For example, in
the theoretical eye model proposed by Lotmar [3], the lens
was constructed of seven shells with refractive indices
varying from 1.38 to 1.41 in steps of 0.005 (at wavelength
␭ = 0.543 nm). Pomerantzeff et al. [4] constructed a shell–
lens consisting of 398 layers with different indices, radii
of curvature, and thickness varying as a function of a
high-order polynomial. Al-Ahdali and El-Messiery [5] proposed an eye model incorporating 300 spherical shells in
the lens. In the eye model proposed by Popiolek-Masajada
[6], the anterior and posterior surfaces of the shell–lens
were made hyperboloidal with a rather high value of asphericity compared with that of recent population studies.
1084-7529/07/082157-18/$15.00
Liu et al. [7] established an anatomically more accurate
eye model with 602 concentric ellipsoidal shells. Unfortunately, even for such a large number of shells, the noncontinuous structure of the lens produces multiple foci [6];
i.e., the longitudinal spherical aberration (SA) becomes a
discontinuous function when the ray enters the lens at
certain critical heights. The second group of lens models
is free of this effect, since the GRIN lens has a continuous
index gradient usually described by a set of equations.
Well-known schematic eye models with a GRIN lens are
those of Gullstrand [1], Blaker [8], Smith et al. [9], and
Liou and Brennan [10]. A good review on the optical properties of the crystalline lens and their significance to image formation was given by Smith [11]. The two kinds of
crystalline lens models featuring the continuous GRIN
and shell structures are well described in [12].
Both groups of these eye models are applicable only to
describing foveal vision (on visual axis), except for Lotmar’s [3] and Pomerantzeff ’s [4] models with a shell lens,
which are of high complexity owing to a large number of
parameters involved compared with a GRIN lens model.
Therefore, we believe there is a need for a simpler wideangle model with a GRIN lens to describe aberrational
properties of the eye at oblique angles, which is of great
importance for peripheral vision and imaging of the peripheral fundus.
Together with Lotmar’s and Pomerantzeff ’s models,
there have been several attempts to develop wide-field
models using a lens of constant refractive index. The
Gullstrand–Le Grand model [2] was modified by Kooijman [13], who introduced a moderate asphericity 共k =
−0.25兲 on both surfaces of the cornea and a relatively
© 2007 Optical Society of America
2158
J. Opt. Soc. Am. A / Vol. 24, No. 8 / August 2007
large asphericity on the anterior and posterior surfaces of
the homogenous lens (k = −3.06 and k = −1, respectively) to
obtain a wide-angle model. Shortly after, Navarro et al.
[14] proposed a schematic eye model with similar asphericities of the corneal surfaces (k = −0.26 and 0) and surfaces of the lens 共k = −3.1316 and −1.0) derived from averaged anatomical measurements obtained in vitro by
Howcroft and Parker [15]. Navarro’s eye model had a lens
with a homogenous index changing with accommodation.
The off-axis aberrations of the unaccommodated form of
this model was extensively analyzed later by EscuderoSanz and Navarro [16]. Navarro’s model agrees well with
experimental findings for off-axis aberrational properties
of the real eye, yet it does not model the graded-index
structure in the crystalline lens. Therefore it would be
very appropriate to include such anatomical features of
the lens into the eye model, especially if one needs to use
it for solving the eye inverse problem, i.e., reconstructing
the optical structure of the eye from off-axis wavefront
measurements.
In the present paper, a simplified wide-field eye model
with a GRIN lens, to be used as a starting point for solving the eye inverse problem, is proposed and is optimized
to fit both chromatic aberrations and overall root-meansquare (RMS) wavefront error observed at different visual
field angles. The experimental data for chromatic aberrations is taken from a simple chromatic eye model developed by Thibos and colleagues [17]. This model being
valid on axis, after a minor modification of the refractive
surface shape, its asphericity also provides the correct
amount of SA [18]. We shall refer to this modified model
as the Indiana eye, which we use on axis, since Navarro’s
model shows a slightly excessive amount of SA.
The experimental data for RMS wavefront error as a
function of visual field angle is taken from averaged measurements of two different studies by Navarro et al. [19]
and Atchison and Scott [20,21] carried out with a laser
ray-tracing technique [22,23] and the Shack–Hartmann
(SH) wavefront sensor [24–26], respectively. These studies seem to indicate a similar amount of off-axis aberrations of the order of those predicted by Navarro’s schematic eye, which suggests that Navarro’s model is a good
starting point for deriving a new wide-field mode of the
eye with a GRIN lens.
Recent advances in the understanding of the refractive
index distribution [27] in the crystalline lens make it possible to narrow down our search for promising solutions to
fit simultaneously the aberrations of the Indiana eye on
axis and Navarro’s model off axis in one single wide-field
model with a GRIN lens matching the typical distribution
of the refractive index. The ultimate goal is to find a relatively simple function of a few physical parameters representing the index distribution within the lens, which can
be easily adjusted while reconstructing the structure of
the real eye from off-axis wavefront measurements. The
success of such a reconstruction will depend on the initial
schematic eye model used and the correctness of the index
distribution in the lens. Therefore it is important to develop a new eye model closely resembling the anatomical
structure of the real eye and providing just a few efficient
parameters to account for intersubject variability and age
effects in the GRIN lens [28].
A. V. Goncharov and C. Dainty
2. ANALYTICAL MODEL OF THE GRIN
LENS
Ideally, a new schematic eye model should predict accurately the aberrations arising at each ocular component
as well as the overall ocular aberrations. As pointed out
by Smith [11], even using current sparse data on the optical structure of the lens, one could use mathematical
modeling to study the effects of surface shapes and gradient refractive index structure on the Gaussian and aberration properties of the lens. We believe that using a
GRIN lens in the schematic eye could facilitate our search
for a more accurate model of the optical system matching
the experimental data for on-axis and off-axis measurements of ocular aberrations. In our case, the averaged experimental data are implicitly presented in the Indiana
and Navarro eye models. The GRIN lens should have a realistic anatomical structure of the crystalline lens consistent with the characteristic distribution of the refractive
index reported by many researchers [27,29]. As we mentioned in the introduction, there have been several attempts to employ a GRIN lens for modeling of ocular aberrations on axis (at one field point) [1,8–10,30], but to
our knowledge off-axis aberrations have been modeled
only with a homogeneous index lens [13,14,16] or a shell–
lens [3,4]. Interestingly, wide-angle eye models of a rainbow trout [31] and octopus [32] have been constructed using a GRIN lens and analyzed for off-axis performance.
These studies suggest that a spherical symmetry of the
GRIN lens allows maintenance of a well-corrected retinal
image far into the peripheral field. Both lens models have
a strong refractive index gradient over nearly the same
range, increasing from 1.38 at the lens cortex to 1.50 at
the center. To describe this index gradient, a polynomial
of the tenth order (as a function of radial distance from
the lens center) has been used. The human crystalline
lens does not have spherical symmetry, yet it is assumed
to be rotationally symmetrical about its optical axis.
Early studies of the refractive index distribution in the
human lens by Nakao et al. [33] using an interference
technique showed that the index profile, in both sagittal
and equatorial sections, could be approximated by an
even polynomial of second order. However, experimental
studies by Pierscionek et al. [34,35], based on an
equatorial-to-sagittal transposition method developed by
Chan et al. [36], showed that the distribution of refractive
index in the human lens has a nonparabolic profile; it is
relatively flat over the inner two thirds of the lens with a
steep falloff in the cortical region. A similar type of distribution was found along the optical axis of the human lens
for protein concentration [37]. More recent studies by
Peirscionek [29,38] using a reflectometric fiber optic sensor [39] for direct measurements of refractive index along
the equatorial and sagittal planes showed nearly flat index profile in the aged lenses (for both planes), whereas
young lenses exhibited a rapid decrease of refractive index at the periphery in the equatorial plane and relatively
low gradient in the sagittal plane. The latest studies by
Moffat et al. [40] and Jones et al. [27] using a noninvasive
magnetic resonance imaging technique obtained refractive index maps through crystalline lenses. It was confirmed that with increasing age, index profiles become
flatter in the central region and steeper at the periphery,
A. V. Goncharov and C. Dainty
Vol. 24, No. 8 / August 2007 / J. Opt. Soc. Am. A
especially in the equatorial plane. This fact explains the
earlier difficulty in fitting the equatorial index profile to a
second-order polynomial. Therefore we consider modeling
the lens structure of young adults only, leaving a more
complex case of irregular, nonsymmetric lenses with a flat
central index profile for future investigation.
A. General Model of the GRIN Lens
In our analytical model for a rotationally symmetric
GRIN lens of axial thickness d, we assume that it consists
of two parts. For the first, anterior part of the lens, the
refractive index n is described by a fourth-order even
polynomial of radial distance r from the optical axis in the
equatorial plane and by a fourth-order polynomial of longitudinal distance z measured from the lens anterior surface in the sagittal plane:
na共z,r兲 = n0 + n1r2 + n2r4 + n3z + n4z2 + n5z3 + n6z4 , 共1兲
where n0 is the refractive index at the anterior surface of
the lens and n1, n2, n3, n4, n5, and n6 are GRIN lens coefficients. The refractive index distribution along the optical axis na共z兲 is an increasing function ranging from n0
to nmax. For the second, posterior part of the lens, the refractive index is also described by a fourth-order polynomial:
np共z,r兲 = nmax + n1r2 + n2r4 + n3,2z + n4,2z2 + n5,2z3 + n6,2z4 ,
共2兲
where nmax is the refractive index at the intermediate
plane, where the refractive index reaches its maximum
value, and n1, n2, n3,2, n4,2, n5,2, and n6,2 are GRIN lens
coefficients. Here the longitudinal distance z is calculated
from the plane of maximum refractive index (peak plane)
so that the axial distribution np 共z兲 is a decreasing function ranging from nmax to n0.
Our task is to find such a refractive index distribution
within the GRIN lens, which corresponds to the right
amount of SA and at the same time has an anatomically
sound structure; that is, iso-indicial lines of constant refractive index should follow the optical surfaces of the
lens. The latter condition can be fulfilled in the rotationally symmetric GRIN lens of the form given by Eqs. (1)
and (2) if the optical surfaces of the lens are conicoids (a
conicoid is formed by rotating a conic section about its
axis of symmetry).
To derive formulas for the GRIN lens coefficients, we
assume that the anterior lens surface has a radius of curvature ra and conic constant ka; therefore the iso-indicial
contour at the anterior vertex of the lens has to follow the
shape of the lens surface, given by the conic section equation
r2 = 2raz − 共1 + ka兲z2 .
共3兲
Using Eq. (3) for r2 in Eq. (1) and regrouping the sum as a
fourth-order polynomial of z, we obtain four linear relationships for GRIN lens coefficients by equating to zero
the coefficients of the polynomial. For our derivation, we
denote by zm the distance from the anterior surface to the
peak plane. To reach the maximum value nmax in this
plane, one has to fulfill two conditions: na共zm兲 = nmax and
na⬘ 共zm兲 = 0, where na⬘ 共zm兲 is the derivative of na with re-
2159
spect to z evaluated at z = zm. Similarly, for the posterior
lens surface we have
r2 = 2rpt − 共1 + kp兲t2 ,
共4兲
where rp and kp are the radius of curvature and conic constant of the surface, respectively, and t is a parameter defining the axial coordinate at the posterior vertex of the
lens such that
共5兲
z = t + d − zm .
Using Eqs. (4) and (5) in Eq. (2) and regrouping the sum
as a fourth-order polynomial of t, we obtain four additional relations for GRIN lens coefficients by equating to
zero the coefficients of the polynomial. For the second part
of the lens, the condition for reaching the maximum refractive index in the peak plane becomes simply n3,2 = 0.
The derivation of the formulas for GRIN lens coefficients
from these relations is straightforward; therefore we
present here only the final expressions. Introducing two
auxiliary parameters,
⌬n = 共nmax − n0兲,
2
m = zm
共1 + ka兲 − 2razm ,
the index coefficients can be expressed in the following
forms:
n1 = 2⌬n/m,
n2 = ⌬n/m2,
n3 = − 4⌬nra/m,
n4 = − 2⌬n关3ra2 − 共ra − zm共1 + ka兲兲2兴/m2 ,
n5 = 4⌬n共1 + ka兲ra/m2,
n6 = − ⌬n共1 + ka兲2/m2 ,
n3,2 = 0,
n5,2 = 4⌬n共1 + kp兲关rp + 共d − zm兲共1 + kp兲兴/m2 ,
n6,2 = − ⌬n共1 + kp兲2/m2 ,
n4,2 = 共1 + kp兲n1 − 4n2rp2 − 3共d − zm兲n5,2 − 6共d − zm兲2n6,2 .
共6兲
The location of the peak plane in the GRIN lens is found
from a quadratic equation
2
共kp − ka兲zm
− 2关d共1 + kp兲 + rp − ra兴zm + d关d共1 + kp兲 + 2rp兴
= 0.
共7兲
It is worth noticing that for a given aspheric shape of the
lens (defined by its geometrical parameters ra, rp, ka, kp,
and d) and chosen optical parameter ⌬n, the refractive index distribution in the GRIN lens is uniquely defined by
Eqs. (6) and (7). The refractive index at the anterior and
posterior surfaces of the lens is equal to n0. This represents the case of a balanced GRIN lens; that is, all isoindicial contours of the lens complete each other on both
sides of the peak plane. We should also mention one interesting feature of this model. If kp + 1 ⬎ −rp / d, then
there exists an additional solution for choosing the peak
plane location zm,
2160
J. Opt. Soc. Am. A / Vol. 24, No. 8 / August 2007
A. V. Goncharov and C. Dainty
zm = d + rp/共1 + kp兲.
共8兲
In this case, the refractive index np共d − zm , 0兲 at the posterior lens surface is higher than the anterior surface refractive index n0, and thus we shall call this case an unbalanced lens. Note that inserting zm from Eq. (8) into Eq.
(6) for the expression of n5,2 leads to n5,2 = 0 and that from
Eq. (2) we find np共d − zm , 0兲 = nmax + n4,2共d − zm兲2 + n6,2共d
− z m兲 4.
B. Simplified Model of the GRIN Lens
It is possible to simplify the model of the GRIN lens if we
relax the condition of iso-indicial contours being coincident with the external surfaces of the lens. In such a case,
the refractive index distribution in the GRIN lens is described by Eq. (1) as a single element with index coefficients defined by three parameters of the lens shape ra,
rp, and d; index range ⌬n; and two free parameters zm
and n2, which are used to set the refractive index n0 and
amount of SA in the lens within the expected range, respectively. Similar to the previous derivation of a balanced GRIN lens, we find the index coefficients in the following explicit forms:
n1 = − ⌬nzmd 共d − 2zm兲共d − zm兲/m ,
2
*
*
2
n4 = − ⌬nd关d3ra − 3d共3ra + rp兲zm
3
+ 4共2ra + rp兲zm
兴/m* ,
n5 = 2⌬n关d3ra − d2共3ra + rp兲zm
2
+ 3共ra + rp兲zm
兴/m* ,
共9兲
2
2
共d − zm兲2关ra共d − zm兲2 + rpzm
兴. This simplified
where m* = zm
model of the GRIN lens has its iso-indicial contours coincident with the optical surfaces only near their vertices,
given that the asphericity of the surfaces was excluded
from the parameters of the lens shape. The index coefficient n2 and location of the peak plane zm are responsible
for the aspheric shape of the marginal iso-indicial contours, which have a refractive index of n0 and pass
through the vertices of the lens. In spite of the highly aspheric shape of the marginal contours, which cannot be
described as a conicoid, we can approximate their equivalent conic constant by using a second term in the Taylor
series expansion. The asphericities of anterior and posterior marginal iso-indicial contours are
− zm兲共d − 2zm兲兴,
kb = − 1 + 关n4 + dn5 − 4rp共n2rp − n1/d兲兴m*/关⌬nzmd2
⫻共d − zm兲共d − 2zm兲兴.
F = − 6n0n1d/共3n0 − 2n1d2兲.
共11兲
It can be seen that the lens power depends on refractive
index n0 and coefficient n1, which is a function of zm;
therefore we may adjust the power of the lens and the
whole eye by simply choosing the appropriate position of
the peak plane while retaining the value of n0 at the expected level, which is about 1.37 according to a recent
study by Jones and colleagues [27].
Following Smith and Atchison [42], we calculate the
primary SA coefficient W4,0 of the simplified GRIN lens
using the Seidel aberration coefficient SI, which corresponds to third-order SA. These coefficients are related as
follows:
W4,0 = SI/共8h04兲.
共12兲
where the individual contributions are related to the
GRIN lens coefficients by the equations
n6 = − ⌬n关d2ra − 2d共2ra + rp兲zm
ka = − 1 − 共n4 +
where g = n1 / 共n0 − 2n1d2 / 3兲 for our GRIN lens models. Assuming the entrance height h0 = 1, the equivalent power of
the GRIN lens bulk can be expressed as follows:
W4,0 = Wa + Wb1 + Wb2 + Wp ,
3
+ 2共ra + rp兲zm
兴/m* ,
4n2ra2兲m*/关⌬nzmd2共d
h共z兲 = h0共1 + gz2兲,
For the eye, the wavefront aberration is usually of the order of 10−3 mm; hence it is more practical to use the units
of the primary SA in micrometers, while the ray height is
expressed in millimeters. With this in mind, we derive an
expression for the primary SA of the GRIN lens as a sum
of the anterior surface refractive contribution Wa, two
transfer contributions of the GRIN lens bulk Wb1 and
Wb2, and the posterior surface refractive contribution Wp:
n3 = 2⌬nzmrad 共d − 2zm兲共d − zm兲/m ,
2
that is, the height of the ray h 共z兲 above the optical axis is
given by
共10兲
The equivalent optical power of the GRIN lens can be approximately calculated assuming a parabolic ray path
through the lens as suggested by Smith and Atchison [41];
Wa = 500共n1 + n3/4ra兲/ra ,
Wb1 = 25F4共− 6300n02 + 588n0n1d2
+ n1d3共1260n3 + 1080n4d + 945n5d2
+ 840n6d3 − 184n1d兲兲/1512n04n1d,
Wb2 = 25F4n2共3780n03n1d2 − 3402n02n12d4
+ 1236n0n13d6 − 187n14d8
− 2835n04兲/1134n04n14d3 ,
Wp = − 500关n1 + 共n3 + 2n4d + 3n5d2
+ 4n6d3兲/4rp兴/rp .
共13兲
These equations are derived from a basic expression of
Sands [43], who established the formula for third-order
SA (Seidel sum SI) of a GRIN medium. Since we assumed
again a parabolic ray path through the medium, Eq. (12)
is an approximation. It can be seen that the second transfer contribution Wb2 depends on the index coefficient n2,
and as we show later it plays a major role in balancing the
amount of SA in the GRIN lens.
A. V. Goncharov and C. Dainty
Vol. 24, No. 8 / August 2007 / J. Opt. Soc. Am. A
Modeling older eyes with a steeper refractive index profile in the equatorial plane might require polynomials of
higher order of r in Eqs. (1) and (2); for example, the University of Rochester representation includes two extra
terms r6 and r8, a modified version of which has been used
for modeling relaxed and accommodated states of the
crystalline lens [44]. Introducing these additional terms
of even power of the radial coordinate r will affect the
amount of high-order SA of the lens, but not its optical
power.
3. MODELING THE SPHERICAL
ABERRATION OF THE EYE
The SA of the eye has been extensively studied by many
researchers [45–47], yet its relative contribution to ocular
2161
aberrations is a subject of debate. For comparison, we
have collected from the literature the experimental data
on ocular SA in Table 1. Various psychophysical methods
(PM) for direct measurements of the longitudinal SA
(LSA) show a relatively large amount of aberration
[45–47] except for a study by Millodot and Sivak [48]. The
latter work was chosen as the basis for the linear model of
LSA proposed by Liou and Brennan [49], for which the
value of LSA was converted into dioptric power changes
⌬F = n/共f⬘ + LSA兲 − n/f⬘ ,
where f⬘ is the equivalent focal length of the eye in meters
and n is the refractive index of the last refractive medium, leading to a linear relationship between LSA and
ray height h (in millimeters) at the entrance pupil of the
eye: ⌬F = 0.2 h. This LSA linear model was used by Liou
Table 1. Summary of Experimental Data for the Ocular and Corneal Spherical Aberrations of the Human
Eye and Different Eye Modelsa
Ocular SA
Corneal SA
Number
of Eyes
Mean
Age
(years)
⌬F (D)
W40 共␮m / mm−4兲
Z40 共␮m兲
W40
Z40
Pupil
Diameter
20
10
12
3
218
200
13
13
2
15
15
7
10
75
90
228
72
30
5
27
16
16
—
—
—
33
42
26
35
66
39
25
68
35
60
44
18
50
45
21
30
25
45
65
0.5
0.4
0.7
1.5
—
—
—
—
0.8
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
0.005
0.013
—
—
—
—
—
—
—
—
—
—
0.014
—
—
—
—
—
—
—
0.138
0.120
—
—
—
0.095
0.175
0.110
0.303
0.175
0.060
—
0.160
0.132
—
—
—
—
—
—
—
—
—
—
0.032
0.029
—
—
—
—
—
—
—
—
—
—
0.074
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
0.255
0.300
0.281
0.260
0.207
—
0.192
0.260
0.265
6.0
4.0
6.0
6.2
5.7
6.0
5.4
5.4
5.6
6.0
6.0
5.9
5.9
6.0
6.3
6.0
6.0
6.0
4.0
6.0
6.0
6.0
Eye models
SA1, Liou and Brennan [10]
SA2, Indiana eye [18,50]
Navarro [14,16]
45
—
—
0.6
1.5
1.7
0.016
0.031
0.033
0.095
0.187
0.199
0.035
—
0.027
0.211
—
0.166
6.0
6.0
6.0
20U
20B
20S
20
20
20
0.7
0.6
0.6
0.013
0.012
0.013
0.078
0.071
0.075
0.040
0.040
0.040
0.241
0.241
0.241
6.0
6.0
6.0
30U
30B
30S
30
30
30
1.3
1.1
1.2
0.021
0.020
0.022
0.128
0.121
0.132
0.043
0.043
0.043
0.260
0.260
0.260
6.0
6.0
6.0
40U
40B
40S
40
40
40
1.5
1.3
1.4
0.029
0.027
0.030
0.174
0.162
0.179
0.047
0.047
0.047
0.283
0.283
0.283
6.0
6.0
6.0
Study/Method
Millodot and Sivak [48]/PM
Ivanoff [46]/PM
Jenkins [47]/PM
Koomen et al. [45]/PM
Porter et al. [52]/SH
Thibos et al. [53]/SH
Smith et al. [55]/CA, VK
Salmon et al. [26]/SH, PM
Calver et al. [56]/CA
Artal et al. [57]/SH
Amano et al. [58]/SH
He et al. [54]/PM, VK
Wang et al. [70]/VK
Alió et al. [59]/SH, VK
Kelly et al. [71]/SH, VK
Artal and Guirao [68]/SH, VK
Guirao et al. [62]/VK
a
PM, psychophysical methods; SH, Shack–Hartmann; CA, crossed-cylinder aberroscope; VK, videokeratographic system; U, unbalanced; B, balanced; S, simplified.
2162
J. Opt. Soc. Am. A / Vol. 24, No. 8 / August 2007
and Brennan in their subsequent finite model of the eye
with a GRIN lens [10]. As pointed out by Thibos et al. [50],
before using raw psychophysical data to estimate the
amount of ocular SA, one has to account for odd aberrations, such as coma, which arise on axis due to the lack of
rotational symmetry in real eyes. They illustrated the
concept for eliminating the coma contribution from measurements of transverse aberration of the eye [51]. The
estimated amount of the SA was modeled by a reduced
schematic eye with an elliptical refracting surface [50],
for which k = −0.43. The latter model was consistent with
their earlier single-surface chromatic eye model [18].
In that respect only psychophysical measurements by
Koomen et al. [45], utilizing an annular pupil mask with
different diameters, provided averaging of the coma contribution, and their results are in good agreement with
the Indiana eye model.
On the other hand, wavefront measurements with modern wavefront sensing techniques [24] do not always
agree with the Indiana eye model. Several studies [52,53]
of ocular wavefront aberrations in large populations using
the SH sensor showed that in a fourth-order Zernike expansion the mean value of coefficient Z40, responsible for
primary SA, is significantly different from zero. In a large
study by Porter et al. [52], the mean value of Z40 was found
to be somewhere in between the value predicted by the Indiana eye model and that of Liou and Brennan listed in
Table 1. However, two large studies of young subjects by
Thibos et al. [53] and He et al. [54] clearly supported Liou
and Brennan’s model. Similarly, analysis of the data obtained by Smith et al. [55] using a crossed-cylinder aberroscope (CA) to measure the mean value of the primary
SA coefficient W4,0 supports Liou and Brennan’s model. It
is worth mentioning that for a given pupil size, one can
use a simple relation
W4,0 = 6冑5Z40 ,
provided that there is no defocus error in the eye. Another
recent study [26] comparing a psychophysical method
with the SH wavefront sensor demonstrated consistency
of both techniques on two subjects. The estimated amount
of the LSA in dioptric power was somewhere in the middle
of these two models.
Due to a relatively small number of measurements currently available for ocular SA, and in view of its large intersubject variability, it is reasonable to keep both options
for representing ocular SA: Liou and Brennan’s model
[10], hereafter simply called the SA1 case, and the Indiana eye model [18,50] with a single ellipsoidal refracting
surface 共kSA2 = −0.43兲, referred to as the SA2 case. The SA
of the SA1 case can be accurately reproduced by the Indiana eye model with a modified ellipsoidal surface 共kSA1 =
−0.495兲. Therefore we shall use only the Indiana eye
model with an appropriate conic constant k to mimic both
SA1 and SA2 cases. Figure 1 depicts the ocular SA for
four cases, including Navarro’s wide-angle eye model.
A comparative analysis of the wavefront measurements
of the Zernike coefficient Z40 for primary SA sorted into
two different age groups of young adults and middle-aged
people as given by Porter et al. [52] and Calver et al. [56]
indicates that positive value of Z40 increases with age,
A. V. Goncharov and C. Dainty
which is consistent with findings of other studies by
Smith et al. [55], Artal et al. [57], Amano et al. [58], and
Alió et al. [59]. In spite of intersubject variability, we can
deduce from these studies that the primary SA of young
adults seems to favor the SA1 case, whereas the SA2 case
is more appropriate for middle-aged people.
4. EFFECT OF AGING ON THE
ANATOMICAL STRUCTURE OF THE EYE
Constructing a new schematic eye model that is structurally similar to the human eye demands a thorough consideration of the biometric data. Empirical values of ocular parameters available in the literature display a mixed
effect of intersubject variability and restructuring of the
eye due to aging. Averaging such diverse biometric data
without taking into account the effect of aging on anatomical structure is more likely to result in some unrealistic eye model, which might not be applicable even for a
specific age group. Therefore, we consider here three age
groups and their corresponding schematic models representing 20-, 30-, and 40-year-old eyes. Since averaging
biometric data specifically for each group is not always
possible, we used data from various experimental studies
describing the changes in shape and internal geometry of
the human eye as a function of age. The resulting geometrical parameters of our three models are listed in
Table 2, with references to the original data given below.
The optical layout of the eye is shown on the right-hand
side in Fig. 2.
A. Anterior Cornea
According to the data analysis by Lam and Douthwaite
[60], the correlation between the cornea anterior radius
rca and posterior radius rcp derived from a regression line
for horizontal meridian gives
rcp = 0.87rca − 0.24.
On the other hand, estimating the average ratio rcp / rcp
for our age groups gives the mean value of 0.84, which is
comparable to Edmund’s study [61]. In vertical meridian,
the cornea shape is less curved and the average ratio
rcp / rcp is about 0.83. Since data are available only for offaxis aberration in the eye measured in the horizontal field
[19–21], we have adopted here the averaged values for the
corneal anterior radius of curvature in horizontal meridian from the data of Lam and Douthwaite [60] and more
recent data by Guirao et al. [62] derived from measurements with a videokeratographic system (VK) [63]. We
also use Guirao’s data to account for the effect of aging on
the conic constant kca of the anterior cornea surface,
which becomes less prolate with increasing age. Our chosen values for kca in Table 2 agree well with findings by
Sheridan et al. [64] 共kca = −0.11兲 and Aoshima et al. [65]
共kca = −0.08兲 but are slightly higher than the mean value
of− 0.18± 0.18 reported in other studies [66,67]. We are
aware of this discrepancy and are ready to support our
choice by additional analysis of experimental data for corneal SA converted for a 6 mm pupil. Table 1 contains SA
coefficients W40 and Z40 of the anterior corneal surface estimated for three age groups (20, 30, and 40 years) with
the parameters rca and kca given in Table 2. We assumed
A. V. Goncharov and C. Dainty
Vol. 24, No. 8 / August 2007 / J. Opt. Soc. Am. A
2163
Fig. 1. (Color online) Longitudinal spherical aberration of the eye predicted by different models: 0, linear model based on Millodot and
Sivak’s data [48]; 1, Liou and Brennan’s eye model with a GRIN lens [10]; 2, Indiana eye model with a single ellipsoidal refracting surface
[50] 共k = −0.43兲; 3, Navarro’s wide-angle eye model [16]; 20U, 30U, and 40U models with an unbalanced GRIN lens; 20B, 30B, and 40B
models with a balanced GRIN lens; 20S, 30S, and 40S models with a simplified GRIN lens.
obtained by Amano et al. [58] gave Z40 = 0.26 ␮m. These
two large studies showed no statistically significant correlation between age and corneal SA. On the other hand,
in a more recent study by Alió et al. [59], the corneal SA
Zernike sum 共Z40 + Z60兲 showed a weak increase with age,
starting at 0.26 ␮m (for 20-year age) and reaching
0.28 ␮m at the age of 40 years. For comparison, we
that the corneal refractive index is 1.375 at ␭ = 589 nm,
which is our reference wavelength.
Artal and Guirao [68] estimated Seidel aberration coefficients; the mean value was W40 = 0.04 ␮m / mm4. Two
studies of young subjects by He et al. [54,69] showed the
mean value Z40 = 0.3 ␮m. In the large study by Wang et al.
[70], the mean value was Z40 = 0.28 ␮m, whereas the data
Table 2. Effect of Aging on the Anatomical Structure of the Human Eye
Average Model Age (Years)
Anatomical Structure
Cornea anterior radius, rca (horizontal)
Cornea posterior radius, rcp (horizontal)
Cornea anterior conic constant, kca
Cornea posterior conic constant, kcp
Central cornea thickness
Anterior chamber depth (ACD)
Lens thickness, d
Lens anterior radius, ra
Lens posterior radius, rp
Vitreous chamber depth (VCD)
20
30
40
7.85
6.59
−0.12
−0.23
0.55
3.28
3.49
12.28
−7.87
16.58
7.76
6.52
−0.10
−0.30
0.55
3.06
3.69
11.51
−7.67
16.60
7.67
6.44
−0.08
−0.37
0.55
2.84
3.89
10.74
−7.47
16.62
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J. Opt. Soc. Am. A / Vol. 24, No. 8 / August 2007
A. V. Goncharov and C. Dainty
ridional profile of the posterior corneal surface was approximated by an eighth-order aspheric, which after refitting to a conicoid (1 ␮m fitting error) gave us an average
kpc = −0.22. Therefore we chose the conic constant kpc to be
in the range of these two studies. The choice of this parameter is not critical, because the contributions to ocular
SA from the posterior corneal surface is not significant
due to the small difference between corneal and aqueous
refractive indices, the latter being 1.3374 as in Navarro’s
eye model at ␭ = 589 nm.
Fig. 2. (Color online) Optimization of the eye model by reverse
ray tracing. For clarity, optical systems are shown on both sides
of the reference plane (vertical dashed line), where rays start to
traverse backward.
should also give typical values of SA for eyes with more
negative values of kca. Assuming corneal radius of curvature ra = 7.85 mm and kca = −0.18, we get less SA from the
cornea as Z40 = 0.20 ␮m and W40 = 0.033 ␮m / mm4 due to its
gradual decrease to zero at kca = −0.53.
The majority of experimental data in Table 1 (except for
data by Kelly et al. [71]) indicates the significance of the
corneal contribution to the ocular SA, and hence we
choose a relatively small absolute value of its conic constant. However, a more recent study by Navarro et al. [72]
of the mean shape of the anterior cornea showed a more
negative conic constant k ⬍ −0.4, which would predict
much lower corneal SA. The reasons for such a striking
difference might be due partly to a discovered 2.5 deg tilt
of the corneal axis with respect to the optical axis of the
lens, whereas other studies assumed rotational symmetry
of the eye.
It is clear that different authors would tend to choose
somewhat different anatomical parameters for their models according to their current knowledge. However, due to
rapid evolution in the field, some biometric data may soon
become outdated. For this reason we will focus on the
methodology of the construction of the GRIN lens for a
given set of optical parameters, assuming rotational symmetry of the eye.
B. Posterior Cornea
In a large study of 500 eyes (500 subjects, mean age
= 31 years) by Lleó et al. [73], there was no significant correlation found between central corneal thickness and age.
The mean corneal thickness for a young group (283 subjects, range of 18– 30 years) was 0.545 mm, for the second
group (155 subjects, range of 31– 40 years) 0.551 mm, and
for the third group (62 subjects, range of 41– 67 years)
0.549 mm. A comparable mean value of 0.546 mm was
found in a study of corneal thickness for 92 eyes (46 subjects, mean age= 31 years) by Lam and Chan [74]. We
chose the average value of 0.55 mm for corneal thickness
in all our models. The radius of curvature for the posterior corneal surface is found as rcp = 0.84rca. The asphericity of the posterior corneal surface is known with less confidence; according to Dubbelman et al. [67], the mean
value of kpc is −0.38± 0.27 and the surface becomes more
prolate (kpc becomes more negative) with increasing age,
showing a trend of −0.007 per year, which we adopted in
our models. In a study by Aoshima et al. [65], based on
corneal topography with the Orbscan II system, the me-
C. Anterior Segment and Crystalline Lens
The accuracy of estimating the curvature and especially
the asphericity of the lens surfaces is limited by the correction technique used for interpreting the Scheimpflug
slit images of the eye [75]. The raw images of the lens
shape are distorted due to the refractive properties of the
cornea and more significantly due to the GRIN structure
of the lens itself. To take into account these factors one
needs some additional knowledge of the corneal shape
and the GRIN structure. The latter aspect presents a real
challenge (Dubbelman et al. [76]). This uncertainty creates a large degree of diversity in the literature results
obtained from data analysis of the age-related changes in
the lens shape. Koretz et al. [77] supported their data
analysis with independent magnetic resonance imaging,
which we use in our models; however, the eye stability
over the scanning period and the finite size of the pixels
limit the resolution of this technique, and therefore other
correction methods might provide more accurate results
(see Dubbelman and Van der Heijde [78]).
The geometry of the anterior segment of the human eye
as a function of age is based on averaged data from highresolution magnetic resonance images and Scheimpflug
slit-lamp images by Koretz et al. [77,79]. For the anterior
chamber depth (ACD) and lens thickness d expressed in
millimeters, we adopted the following age functions:
ACD= 4.27− 0.022A and d = 3.09+ 0.02A, where A is a parameter of age in years. Using these expressions we can
estimate dependence of the anterior segment length
(ASL) on age: ASL= ACD+ d = 7.36− 0.002A, which is in
good agreement with the data obtained with Scheimpflug
imaging [79]. The age function for lens thickness is comparable with another study by Alió et al. [59]. We define
the radius of curvature for the anterior and posterior surfaces of the crystalline lens as ra = 13.82− 0.077A and rp
= −8.27+ 0.02A, respectively; the latter expression is comparable with earlier findings by Brown [80]. We should
note that the posterior surface of the lens is one of the
most challenging objects to characterize because its location is the least accessible for imaging; as a consequence,
the age function chosen for rp may contain larger uncertainty than other parameters of the eye.
Ideally, for a constant length of the globe (axial distance between cornea and retina), assuming its average
value of 23.9 mm from two large studies [81,82], the vitreous chamber depth (VCD) has to slightly increase with
age: VCD= 16.54+ 0.002A, which is reflected in Table 2.
However, during our optimization of the eye models, we
first set the optical power of the eye approximately to
60 D by adjusting the peak plane position defined by Eq.
(7) or (8), and then we remove defocus by slightly altering
A. V. Goncharov and C. Dainty
the VCD table values. The reason for keeping the power
of the eye constant is because it is relatively stable between the ages of 20 and 40 years [83], after which there
is a shift in the hypermetropic direction [84,85]; that is,
the optical power of the eye decreases with age.
5. OPTIMIZATION OF THE GRIN LENS
MODELS
A. Unbalanced GRIN Lens
We start this section with a description of three wide-field
schematic eye models having an unbalanced GRIN lens,
labeled as 20U, 30U, and 40U. In order to reach a certain
level of ocular SA comparable to SA1 or SA2 cases, we optimized all models with reverse ray tracing [86], a technique for duplicating aberrations of optical systems. A recent work on personalized eye models by Navarro et al.
[87] showed the reconstruction of the optical system of the
eye (including the GRIN lens) from on-axis wavefront
measurements using an optimization strategy, which has
a strong parallelism with reverse ray tracing. They used a
phase plate placed at the pupil plane instead of a reversed
eye model.
Figure 2 shows the optical system of the Indiana eye
(left-hand side) and the optical system under optimization
(right-hand side). The SA of the latter is matched to that
of the Indiana eye for a 7 mm beam. Off-axis image quality is compared with that of Navarro’s eye model after optimization. We use a conventional plot of RMS wavefront
aberration versus the field angle ranging from 0 to 40 deg
(see Fig. 3). Ray tracing and optimization were carried
out with Zemax-EE optical design software (Focus Software, Inc.).
As a result of our extensive search for optimal solutions, we found three models. Their image quality on axis
is presented in Fig. 1, which depicts the longitudinal SA
as a function of pupil semidiameter h. Table 3 contains
optical parameters of the GRIN lens n0 and ⌬n and parameters of its aspheric shape ka and kp. The final values
of the optimization parameters are marked by an asterisk. By adjusting n0, we could bring the focal length (or
VCD) in line with the table values (Table 2). The SA and
the optical power of the lens were set by optimizing parameters ka and kp; the latter defines the peak plane lo-
Fig. 3. (Color online) Off-axis wavefront aberrations versus field
angle ␻ for optimized models of the eye with an unbalanced
GRIN lens. Navarro’s eye model is shown for comparison 共␭
= 0.589 ␮m兲.
Vol. 24, No. 8 / August 2007 / J. Opt. Soc. Am. A
2165
cation zm in Eq. (8) and hence the optical power [see Eq.
(11)], provided that the value of ⌬n is chosen in advance.
We selected ⌬n = 0.035 for all three models. At the final
stage, the image defocus was removed by fine-tuning the
VCD and the field aberrations were minimized by adjusting the retinal curvature rim. In principle, one could
achieve an even better match to the off-axis performance
of Navarro’s model by reducing ⌬n below the 0.035 level;
nevertheless, smaller values of ⌬n would not be consistent with the latest findings with magnetic resonance imaging [27].
Using four parameters n0, ⌬n, ka, and kp from Table 3
and three parameters d, ra, and rp from Table 2, one can
estimate the GRIN lens coefficients from Eqs. (6) and (8),
which are listed in Table 4. We present the resulting refractive index variation as a function of distance from the
optical axis r, and the axial distance from the anterior
surface z in Figs. 4(a) and 4(b), respectively. Figure 5
shows sagittal maps of refractive index variation within
the GRIN lens for each model. The diameter of the lens
shown is approximately 8 mm; one can easily see a
gradual increase in the lens thickness from 3.49 mm
(model 20U) to 3.89 mm (model 40U).
One of the interesting features of these models is related to the peak plane position, which remains at the
same distance of about 5.28 mm from the anterior surface
of the cornea. This fact supports a hypothesis of unchanged position of the lens nucleus with age [88].
Comparing refractive index profiles for our models, we
can see that the maximum index value nmax in the core
and at the anterior surface decreases at a constant rate of
−0.004 per decade (a bit slower at the posterior surface)
and that the index profiles in the sagittal and equatorial
planes gradually flatten out (see Figs. 4 and 5). It is quite
likely that such a synchronized refractive index decrease
is not a real effect but a consequence of our assumption
that iso-indicial contours are coincident with the optical
surfaces of the lens.
In order to analyze the aberrational characteristics of
the lens, we estimate the lens contribution to the primary
SA by subtracting the corneal aberrations from the ocular
aberrations listed in Table 1; for models 20U, 30U, and
40U we have Z40 = −0.163, −0.132, and −0.109 ␮m, respectively. Our results support earlier findings in several
studies [68,71] indicating that the lens partly compensates for SA introduced by the anterior corneal surface
and that this compensatory mechanism becomes less efficient with age. Our values of internal SA of the eye
(6 mm pupil) are consistent with the empirical data by
Alió et al. [59] for the intraocular SA expressed as a function of age (regression line is in the form Z40 = 0.00287A
− 0.198).
B. Balanced GRIN Lens
An unbalanced GRIN lens demonstrates one possible refractive index distribution that provides a realistic
amount of ocular SA. Alternatively, we shall now investigate the case for a balanced GRIN lens, which has the
same refractive index for its anterior and posterior surfaces. A thorough exploration of multivariable space for
the optimal shape of the GRIN lens (with its iso-indicial
contours following the optical surfaces) revealed that
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J. Opt. Soc. Am. A / Vol. 24, No. 8 / August 2007
A. V. Goncharov and C. Dainty
Table 3. Optical Parameters of the Wide-Field Eye Models with Unbalanced (U), Balanced (B),
and Simplified (S) GRIN Lensesa
Age
n0
⌬n
ka
kp
zm
VCD
rpp (mm)
rim, (mm)
Power (D)
20U
30U
40U
20B
30B
40B
20S
30S
40S
1.369*
1.373*
1.377*
1.376
1.376
1.376
1.362
1.362
1.362
0.035
0.035
0.035
0.040
0.040
0.040
0.040
0.040
0.040
−2.8*
−2.9*
−3.0*
0.0*
−1.0*
−2.0*
−1.0
−1.0
−1.0
2.9*
2.8*
2.7*
2.0*
1.0*
0.0*
0.5
0.5
0.5
1.47
1.67
1.87
1.77
2.00
2.22
1.54*
1.68*
1.81*
16.76*
16.70*
16.62*
16.74*
16.69*
16.62*
16.73*
16.64*
16.53*
−7.87
−7.67
−7.47
−14.31*
−15.34*
−15.93*
−7.87
−7.67
−7.47
12.2*
12.0*
12.0*
12.0*
12.0*
12.0*
12.0*
12.0*
11.8*
60.06
60.06
60.14
60.00
59.98
60.02
59.91
60.01
60.19
a
Asterisks denote optimized values.
Table 4. Refractive Index Coefficients for the Wide-Field Eye Models with Unbalanced (U)
and Balanced (B) GRIN Lenses
Model
n1
n2
n3
n4
n5
n6
n4,2
n5,2
n6,2
20U
30U
40U
20B
30B
40B
−0.0017476
0.0000218
0.0429220
−0.0100135
−0.0019289
−0.0000707
−0.0041134
0.0
−0.0003318
−0.0015986
0.0000183
0.0367995
−0.0063555
−0.0015967
−0.0000659
−0.003927
0.0
−0.0002636
−0.0014833
0.0000157
0.0318609
−0.0042843
−0.0013502
−0.0000629
−0.0037343
0.0
−0.0002151
−0.0019829
0.0000246
0.0486997
−0.0168058
0.0012071
−0.0000246
−0.0082314
−0.0026985
−0.0002212
−0.0017358
0.0000188
0.0399587
−0.0099794
0.0
0.0
−0.0107824
−0.0018025
−0.0000753
−0.0015200
0.0000144
0.0326487
−0.0051421
−0.0006203
−0.0000144
−0.0128755
−0.0008624
−0.0000144
there are probably no satisfactory solutions for an index
variation range ⌬n ⬎ 0.02, owing to the unrealistically
small amount of ocular SA, which was even less than that
of the SA1 case.
On the other hand, according to recent measurements
of the refractive index distribution [27], the typical range
for index variation is ⌬n = 0.04, . . . , 0.05. This range is affected by neither age nor intersubject variability of the refractive index n0 at the surface. Consequently, we should
adhere to this refractive index range and bring the
amount of SA in line with expected values [59].
Interestingly, a study by Peirscionek [29] indicated that
earlier schematic eye models [4,9,36] with concentric isoindicial contours following the shape of the lens could not
be supported, especially for young lenses. According to
Brown [89], density contours obtained from biomicroscopic images do not exactly follow the external shape of
the lens. In light of that, we relax our condition for concentric iso-indicial contours for one of the lens surfaces.
We chose the posterior surface of the lens, as it has a
larger impact on image formation, especially at off-axis
angles, due to its distant location from the pupil. Restricting this surface to fulfill the condition of the concentric
iso-indicial contours might lead to no feasible solution.
The posterior surface is also more convex than the anterior surface (ra / rp is about 1.5), which makes it more sensitive to any changes in shape in terms of balancing the
SA of the eye, since the primary SA of a refractive surface
is inversely proportional to the cube of its radius of curvature.
We now present three wide-field schematic eye models
with a balanced GRIN lens, labeled as 20B, 30B, and 40B,
having iso-indicial contours less curved than those of the
posterior lens surface, which helped to adjust the amount
of ocular SA. For our chosen index range ⌬n = 0.04, the radius of curvature for the marginal iso-indicial contour at
the posterior vertex (pole) of the lens, denoted as rpp, is
about twice the radius of the lens surface. As a result, the
refractive index at the posterior surface increases gradually from 1.376 at the pole to 1.39 at 4 mm away from the
optical axis.
Table 3 lists the optical and geometrical parameters of
the GRIN lens n0, ⌬n, ka, kp, and zm and the radius of curvature rpp at the posterior pole used in Eq. (7) as a substitute for rp. The parameter rpp was adjusted so that the
VCD is consistent with values in Table 2. In contrast to
the optimization of the unbalanced GRIN lens, we selected the same value n0 = 1.376 for all models. By varying
the conic constants ka and kp, we could obtain the right
amount of SA and the desired optical power of 60 D. The
refractive index coefficients are calculated from Eqs. (6)
and (7) and are given in Table 4.
The LSA of our models with a balanced GRIN lens is
presented in Fig. 1. It is clearly seen that plotted curves of
the LSA for the 30B and 40B models show a less dramatic
change at the edge of the pupil compared with the curves
of the 30U and 40U models. This means that the balanced
GRIN lens has a smaller amount of high-order SA, which
allows us to achieve a better fit to the SA2 case. Table 1
gives the ocular SA. The contribution from the GRIN lens
A. V. Goncharov and C. Dainty
Vol. 24, No. 8 / August 2007 / J. Opt. Soc. Am. A
2167
Fig. 4. Refractive index profiles in the peak plane (a) and sagittal plane (b) for the 20U, 30U, 40U, 20B, 30B, 40B, 20S, 30S, and 40S
models.
Fig. 5. Iso-indicial contours following the shape of the crystalline lens. The refractive index values are in increments of 0.002, starting
from the anterior surface value n0 and reaching the central contour at 1.411, 1.407, and 1.403 for the 20U, 30U, and 40U models,
respectively.
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J. Opt. Soc. Am. A / Vol. 24, No. 8 / August 2007
is Z40 = −0.17, −0.14, and −0.12 ␮m for the 20B, 30B, and
40B models, respectively. They display an even larger
compensation of the corneal SA than that found in the
20U and 30U models.
The RMS wavefront aberrations (units in ␭) are presented in Fig. 6 together with the aberrations of the Navarro’s eye model for comparison. Similar to the models
with an unbalanced lens, wavefront aberrations at 40 deg
are slightly higher than that of Navarro’s model. Analyzing the impact of the index range ⌬n on off-axis aberrations, we found that reducing the range ⌬n by half helps
to lower the off-axis aberrations but is not sufficient to
reach the level of Navarro’s model at 40 deg.
Using Eqs. (6) and (7), we present the refractive index
variation as a function of distance from the optical axis r
and axial distance z in Figs. 4(a) and 4(b), respectively.
The sagittal maps of refractive index within the GRIN
lens for each model are shown in Fig. 7. The growth of the
lens and the internal shift of the core can be easily ob-
Fig. 6. (Color online) Off-axis wavefront aberrations versus field
angle ␻ for optimized 20B, 30B, and 40B models of the eye with
a balanced GRIN lens 共␭ = 0.589 ␮m兲.
A. V. Goncharov and C. Dainty
served. The diameter of the lens defined by the optical
surfaces is about 9.5 mm.
C. Simplified GRIN Lens
Finally, we investigate the usability of a GRIN lens model
with its marginal iso-indicial contours more curved than
the optical surfaces of the lens. In order to make the isoindicial contours grow steeper, while at the same time
maintaining SA at a realistic level for our chosen index
range ⌬n = 0.04, we need to reduce the index coefficient n2
so that it becomes negative. Setting the coefficient n2 to a
negative value and using Eqs. (6) leads to a more general
case, where the shape of the optical surfaces of the GRIN
lens changes form a conicoid to an aspheric of higher order. For that reason, we shall use a simplified single-core
GRIN lens with index coefficients defined by Eqs. (9). We
present here three models, 20S, 30S, and 40S, optimized
by varying the position of the peak plane zm and index coefficient n2 directly. Even though our initial concept of setting the asphericity of the lens surfaces is not applicable
for the simplified GRIN lens model, the index coefficient
n2 is helpful for adjusting the steepness of the iso-indicial
contours, thanks to its direct link to the refractive index
profile in the equatorial plane. This makes the coefficient
n2 an ideal parameter to regulate the amount of SA in the
lens without any changes in the paraxial properties of the
lens, since the radii of curvatures ra and rp remain the
same. This is also evident from analyzing the contribution
Wb2 in Eq. (13). Adjusting the index coefficient n2, we
brought the SA of the lens in good agreement with the
data by Alió et al. [59].
The optical and geometrical parameters of the GRIN
lens n0, ⌬n, ka, kp, and zm are presented in Table 3. The
refractive index coefficients are calculated from Eqs. (9)
and are listed in Table 5. The asphericity of the marginal
anterior and posterior iso-indicial contours calculated
from Eq. (10) are approximately 12.7 and 3.4 for the 20S
model, 9.0 and 1.3 for the S30 model, and 5.5 and −0.7 for
Fig. 7. Iso-indicial contours following the anterior surface of the crystalline lens. The refractive index values are in increments of 0.004,
starting from the surface value n0 = 1.376 and reaching the central contour at 1.412 for all models 共nmax = 1.416兲.
A. V. Goncharov and C. Dainty
Vol. 24, No. 8 / August 2007 / J. Opt. Soc. Am. A
the 40S model. For all three models, the marginal isoindicial contours have a steeper shape than the optical
surfaces of the lens. As a result, the refractive index at
the anterior and posterior surfaces decreases gradually
from the pole to the equator.
The longitudinal SA of our models with a simplified
GRIN lens is presented in Fig. 1. Similar to the models
with a balanced lens, the plotted curves of LSA for the
30S and 40S models appear less curved compared with
the 30U and 40U models, which indicates a better fit to
the SA2 case. The ocular and corneal SAs are presented
in Table 1, from which we can estimate the contribution of
the GRIN lens: Z40 = −0.166, −0.128, and −0.1048 ␮m for
the 20S, 30S, and 40S models, respectively. The RMS
wavefront aberrations of these simplified models shown
in Fig. 8 more closely resemble aberrations of Navarro’s
eye model, especially at 40 deg off axis, while the other
models show somewhat higher aberrations at oblique
angles.
D. Lens Paradox
The lens paradox is the phenomenon in which the external surfaces of the crystalline lens become steeper with
age [80] without producing any noticeable increase in optical power of the human eye. On the contrary, according
to Sounders [84,85], the ocular power decreases; that is,
the eye becomes hypermetropic with age, showing a mean
power reduction of 2 D between the ages of 30 and
60 years. Grosvenor [90] reanalyzed biometric data from
the late 1950s and found a reduction of 0.6 mm in the
mean axial length of the eye for a 50+ age group comTable 5. Refractive Index Coefficients for the
Wide-Field Eye Models with a Simplified (S)
GRIN Lens
Model
20S
30S
40S
n1
n2
n3
n4
n5
n6
−0.0023783
−0.0000110
0.0584122
−0.0258500
−0.0035000
−0.0002547
−0.0021490
−0.0000106
0.0494670
−0.0159580
0.0001715
0.0001410
−0.0019508
−0.0000090
0.0419020
−0.0085090
−0.0020070
0.0003663
Fig. 8. (Color online) Off-axis wavefront aberrations versus field
angle ␻ for optimized 20S, 30S, and 40S models of the eye with a
simplified GRIN lens 共␭ = 0.589 ␮m兲.
2169
pared with the third-decade age group. The effect of the
gradual decrease in axial length with age might resolve
the lens paradox. However, more recent data do not show
this effect [77,91]. To explain the lens paradox Koretz and
Handelman [92] suggested that with age the effect of increasing curvatures of the lens is precisely balanced by
the lens growth along the optical axis (gradual thickening). On the other hand, calculations by Dubbelman and
Van der Heijde [78] showed that the thickening of the lens
could only partly compensate for its more convex shape
and that some additional mechanism for stabilizing the
optical power is needed. They also pointed out that Scheimpflug slit images of the crystalline lens used to estimate
the lens shape could be distorted due to the refractive
properties of the cornea and more significantly due to the
GRIN structure of the lens itself; as a result, the effect of
lens steepening reported by Brown [80] might not be so
dramatic.
Since the corneal power does not become weaker with
age [60], one of the possible mechanisms resolving the
lens paradox is an age-related change in the refractive index distribution in the lens. The optical power of the lens
has two distinct components: namely, the refracting
power associated with the anterior and posterior surfaces
and the power due to the refractive index distribution
within the lens. Pierscionek [93] suggested that a slight
change in the slope of refractive index in the cortex might
compensate the increase in lens curvature and prevent
the eye from becoming myopic with age. This hypothesis
was shown to be feasible [94]. Using the Wood lens as an
approximation, Smith and Pierscionek [95] examined a
GRIN lens model with the inner refractive index distribution based on elliptical iso-indicial contours. Considering
the biochemistry of the lens, they assumed that the indices at the edge and center of the lens do not change with
age. The decrease of lens power was attributed to the
gradual steepening of the refractive index profile in the
cortex of the lens. An earlier study of biometric data for
two different age groups by Hemenger et al. [96] led to a
similar conclusion, that subtle changes in the distribution
of refractive index within the lens might compensate to a
large extent changes in surface curvatures.
Another hypothesis proposed to resolve the lens paradox includes a gradual reduction in the index difference
between the edge and the center of the lens. There are
two possibilities for this case: Either the edge refractive
index could increase [97] or the central refractive index
could decrease with age [98]. Theoretically, either of these
scenarios is possible [99]; however, in reality it is more
likely to find several different factors contributing to the
lens paradox.
The ultimate confirmation of the theoretical modeling
of the age-related changes in refractive index distribution
has been limited by practical difficulties in measuring the
index distribution in the sagittal plane of the lens.
The equivalent optical power of the crystalline lens can
be found approximately from Eq. (11) or by exact ray tracing. For these two methods, Table 6 gives the optical
power of the lens surrounded by the media with refractive
index of n0 identical to the surface value of the lens. We
can see that Eq. (11) predicts quite accurately the optical
power of the lens. Table 6 also shows results for more
2170
J. Opt. Soc. Am. A / Vol. 24, No. 8 / August 2007
A. V. Goncharov and C. Dainty
Table 6. Equivalent Optical Power of the GRIN Lens in Media with Refractive Index n = n0 and n = 1.336
(Natural Conditions)
Model
20U
30U
40U
20B
30B
40B
20S
30S
40S
Fapprox (D)
Fexact (D)
Fnatural (D)
12.1
12.5
22.1
11.7
12.0
21.5
11.4
11.6
21.0
13.7
13.8
21.9
12.7
12.8
21.3
11.7
11.8
20.6
16.4
16.5
21.8
15.6
15.8
21.3
14.9
15.1
20.8
Fig. 9. Iso-indicial contours within the crystalline lens. The refractive index values are in increments of 0.004, starting from the surface
value n0 = 1.362 and reaching the central contour at 1.398 for all models 共nmax = 1.402兲.
natural conditions in vivo with surrounding refractive index of 1.336 (vitreous), which mimics the experimental
study of isolated crystalline lenses by Jones et al. [27]. According to their average data, the optical power of isolated
(unstretched) lenses decreases with age from 25.7 D
(20-year age) down to 21.0 D (40-year age). Lenses in our
models show somewhat slower decrease of optical power
with age, which is likely to be due to their unaccommodated state, whereas isolated lenses lack stretching support from ciliary muscles and therefore assume more convex shape (accommodated state), especially young lenses.
The decrease of the equivalent power of the crystalline
lens with equatorial radius and thickness was pointed by
Perez et al. [100], who used the same data by Koretz et al.
[79] to account for age-related changes in the lens shape.
Similarly, in a study of 27 human lenses, Glasser and
Campbell [101] reported noticeable decrease of optical
power with age when no stretching was applied to the
lens. However, for artificially stretched isolated lenses,
the optical power showed insignificant increase with age
for young eyes, probably due to overstretching, since no
attempt was made to measure the magnitude of the
stretching force, and it was set to the maximum level that
produces no visible damage to the lens fibers.
All three groups of models of the GRIN lens predict decrease in optical power with age in spite of the fact that
the lens radius of curvature increases with age. The
changes in refractive index distribution within the lens
are responsible for this age effect. Our unbalanced eye
models demonstrate another possible compensatory
mechanism; namely, the refractive index difference between the edge and the center of the lens remains unchanged, while the maximum and minimum values of the
index slowly decrease with age (see Figs. 4 and 9). For the
other models, as one can see from Fig. 4(a), relatively
small increase in refractive index at the periphery of the
lens in the equatorial plane is sufficient to retain the overall optical power of the eye and to slightly reduce the lens
power with age. These results support Pierscionek’s hypothesis [93,94] of possible age-related changes in refractive index profile (flattening in the central part) that
could potentially prevent the eye from becoming myopic
with age.
6. DISCUSSION AND CONCLUSIONS
The optical system of the human eye is highly complex
due to the multilayered structure of the crystalline lens
with distributed refractive index. This feature plays an
important role in image formation. We made an attempt
to describe the refractive index distribution in the crystalline lens by using two analytical models, which can be
easily adapted for age-related changes in the shape of the
lens and its optical power.
The general model of the GRIN lens has two segments
joining at the peak plane. This model fulfills the condition
A. V. Goncharov and C. Dainty
that marginal iso-indicial contours of refractive index be
coincident with the external surfaces of the lens. The
GRIN structure is constructed using five geometrical parameters of the lens shape (axial thickness d, radii of curvature ra and rp, and conic constants ka and kp of the external surfaces) and one optical parameter, the refractive
index range ⌬n. All parameters of shape are age dependent; therefore we presented their typical values for three
age groups (20, 30, and 40 years) derived from various experimental data except for asphericity of the lens surfaces, which are not so well known. The anterior and posterior surface asphericities were kept as free variables
during optimization of the models to achieve a realistic
amount of ocular SA. In principle, one could vary geometrical parameters to form a particular accommodation
state of the lens, although we considered only emmetropic
eyes. We bring the optical power of the eye to 60 D by
regulating the position of the peak plane zm. The index
range ⌬n affects the amount of ocular SA. Therefore, the
first step is to select ⌬n and find the asphericity (conic
constants ka and kp) of the lens such that ocular SA is in
the range of the SA1 and SA2 cases.
The resulting models show a gradual change in asphericity of the lens with age. The anterior surface becomes
more hyperboloidal, whereas the oblate ellipsoidal posterior surface tends to reduce its asphericity by approaching a spherical shape. This process of restructuring of the
lens shape is more evident in models with a balanced
GRIN lens. Excessive restructuring might be an artifact
of the condition of concentricity of iso-indicial contours
with the external surfaces imposed on the models, since it
is quite unlikely that the lens undergoes drastic changes
during the third and fourth decades. Alternatively, moderate restructuring of the shape indicates that models are
more likely to give a plausible solution. In that respect,
considering a sequence of age-dependent models as the
restructuring process, we can avoid an unrealistic solution at an earlier stage. As a final test of plausibility of the
models, we performed ray tracing at oblique angles and
compared wavefront aberrations with that of Navarro’s
model. The general GRIN and Navarro models exhibit a
similar amount of wavefront aberrations in the midperiphery of the field (20 deg off axis). At the far periphery
(40 deg off axis), however, wavefront aberrations of the
general GRIN models are higher, reaching 9 ␭ RMS as
compared with 7 ␭ RMS for Navarro’s model.
To achieve a better agreement with wavefront aberrations of Navarro’s model at far periphery, we abandoned
the condition of concentricity of iso-indicial contours and
simplified the model of the GRIN lens by reducing it to a
single-segment model. This model is constructed using
three geometrical parameters d, ra, and rp and three optical parameters such as ⌬n, position of the peak plane
zm, and index coefficient n2. In the beginning, we select
⌬n and optimize both the peak plane location zm and surface index n0 to attain 60 D optical power for the whole
eye. The index coefficient n2 is used exclusively to control
the amount of SA in the lens at the final stage of optimization. As seen from Eq. (11), the optical power does not
depend on the index coefficient n2, which is the characteristic feature of the simplified GRIN lens model. It enables
Vol. 24, No. 8 / August 2007 / J. Opt. Soc. Am. A
2171
us to use conic constants ka and kp of the lens and to some
extent the parameter zm for simultaneous optimization of
the wavefront aberration at the field periphery.
In contrast to the simplified model, for a given ⌬n, ra,
and rp, SA of the general GRIN lens model is uniquely determined by the conic constants ka and kp, with the shape
of the iso-indicial contours automatically controlled. The
condition of concentricity of the iso-indicial contours
makes the general model particular suitable for studies of
aberrations at different accommodation states for the eye.
Explicit representation of the lens shape together with its
iso-indicial contours might add stability to the process of
reconstructing the optical system of the eye from experimental measurements of ocular aberrations.
The simplified model of the GRIN eye has an additional
independent parameter (coefficient n2) influencing the SA
of the eye at the expense of losing the direct link between
the asphericities of the lens and the marginal iso-indicial
contours. Reducing the general GRIN lens description to
a single equation [Eq. (1)] changes the shape of conicoid
surfaces of the constant refractive index to a more complex high-order aspheric form. The simplicity of using a
single equation and the flexibility in fitting aberration
both on axis and off axis are the main advantages of the
model. We were able to obtain a closer fit of the wavefront
aberration of Navarro’s model at the field periphery for all
three age-dependent models with no altering of the conic
constants, which were selected as ka = −1 and kp = 0.5 to
give a realistic shape of the lens. The coefficient n2 was
gradually increased (see Table 5) without significant
change, which indicates only moderate restructuring of
the GRIN lens. The refractive index profile slowly flattens
out in the equatorial plane with age, resulting in a reduced optical power of the lens despite its more convex
shape. This observation supports the hypotheses [93,94]
proposed for resolving the lens paradox through the possible mechanism of restructuring the refractive index distribution in the GRIN lens.
An ultimate test of the two GRIN lens models requires
real data for ocular aberrations across the field, yet there
is a strong evidence to believe that their inclusion in the
wide-field schematic models provides a more realistic optical system of the eye. The analysis of the experimental
data for ocular parameters is essential for a good starting
design prior to optimization of the models. The reverse
ray-tracing procedure can also be used for solving the inverse problem of the eye with a GRIN lens.
It is worth pointing out a few other features of the derived wide-field schematic eye models that add to the
credibility of the proposed method to construct the GRIN
lens. All nine schematic eye models are consistent with
available data on ocular and intraocular SAs showing
partial compensation of the corneal contribution by the
GRIN lens. Following the restructuring process of the
whole eye, we can also note that this compensatory
mechanism becomes less efficient with age, a fact that has
been confirmed experimentally [59]. The GRIN lens models allow us to mimick subtle changes in the gradientindex distribution. The refractive index ⌬n and surface
index n0 are in the expected range of the crystalline lens
[27]. Our thorough analysis [102] of field aberrations such
2172
J. Opt. Soc. Am. A / Vol. 24, No. 8 / August 2007
as coma, astigmatism, and field curvature in the 30S
model confirms that its overall performance is in good
agreement with experimental findings.
A. V. Goncharov and C. Dainty
22.
23.
ACKNOWLEDGMENTS
This research was supported by Science Foundation Ireland under grant SFI/01/PI.2/B039C. We are grateful to
R. Navarro, who is supported by the Spanish CICyT under grant FIS2005-05020-C03-01.
24.
25.
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(alexander.goncharov@nuigalway.ie)