al

nd

nive

eiv

n acandis ar ofn thIt 1

errations, van Meeteren concluded that chromatic differ-

1684 J. Opt. Soc. Am. A/Vol. 14, No. 8 /August 1997 H.-L. Liou and N. A. Brennanence of focus and spherical aberration are the most impor-tant aberrations. A literature search to find a model eyethat estimates both spherical aberration and chromaticaberration of the eye within the tolerances of empiricalresults was unsuccessful. A review of the spherical ab-erration predicted by model eyes2 revealed that presentschematic models estimated spherical aberration valuesthat were higher than empirical results. Of these, onlythe model of Navarro et al.3 provides estimates of chro-matic aberration in addition to spherical aberration. Theonly model eye that has both aspheric surfaces and agradient-index lens (Smith et al.4) gives estimates ofspherical aberration that are higher than empirical re-sults and does not provide values for chromatic aberra-tion.There is one model eye that accurately predicts spheri-

cal and chromatic aberration of the eye (provided that ty-pographical errors in the paper are rectified8), and thiswas proposed by Thibos and co-workers.57 However, it

2. METHODWe developed a new finite model eye by adopting empiri-cal values of ocular parameters to produce a model struc-turally similar to the human eye. Then we modeledthose parameters that lack experimental results or accu-racy to give values for spherical and chromatic aberrationthat are close to empirical data. We reviewed the litera-ture on the biometry of the eye and examined parameterssuch as axial length, anterior chamber depth, curvatureand asphericity of refracting surfaces, gradient-indexstructure of the lens, and dispersion of the ocular media.Data adopted in our new finite schematic model were se-lected according to the following criteria:

1. Data used were obtained from healthy emmetropiceyes.2. An average of 45 years was used if the parameter

was age dependent. This choice of age was based on anAnatomically accurfor optica

Hwey-Lan Liou a

Department of Optometry and Vision Sciences, U

Received July 19, 1996; revised manuscript rec

There is a need for a schematic eye that models visiosurgical procedures, contact lens and spectacle wear,to anatomical, biometric, and optical realities. Thisand a gradient-index lens. It has an equivalent powemodel eye provides spherical aberration values withiaberration for wavelengths between 380 and 750 nm.tions and predicting optical performance of the eye.[S0740-3232(97)01407-5]

1. INTRODUCTIONRecent advances in biometric measurement of the eye andin computerization to expedite extensive, complex opticalcalculations have made it possible to model the opticalperformance of the eye accurately. Predicting optical im-age quality at various levels of defocus, contrast, and pu-pil size can be achieved by calculation of modulationtransfer functions from a model eye. This can be usefulin estimating visual performance of those eyes undergo-ing refractive surgical procedures, such as photorefractivekeratectomy and automated lamellar keratoplasty to cor-rect ocular refractive errors. However, the choice of asuitable model eye for such calculation relies on its closeconformity to the anatomical and optical characteristics ofthe human eye.In calculating the modulation transfer function of the

human eye in white light from experimental data on ab-1is a reduced eye model and therefore not suitable for thepurposes of modeling vision where one or more refracting

0740-3232/97/0801684-12$10.00 te, finite model eyemodeling

Noel A. Brennan

rsity of Melbourne, Parkville VIC 3052, Australia

ed January 9, 1997; accepted January 23, 1997

curately under various conditions such as refractivenear vision. Here we propose a new model eye closefinite model with four aspheric refracting surfaces60.35 D and an axial length of 23.95 mm. The newe limits of empirical results and predicts chromaticprovides a model for calculating optical transfer func-997 Optical Society of America

surfaces of the eye is altered, as in the case of refractivesurgical procedures. Nevertheless, it is a simple and ef-ficient model for predicting spherical and chromatic aber-ration of the eye and should be used if there is no addi-tional requirement.As current model eyes are limited in their application

for vision modeling, our aim in this study is to develop amodel eye that can be used to predict visual performanceunder normal and altered conditions of the eye. The spe-cific aims of the project are to develop a new finite modeleye that

1. Represents the ocular anatomy as closely as pos-sible, so that it can be used in future modeling of visualperformance in which one or more surfaces are altered;2. Predicts spherical and chromatic aberrations as

close to empirical results as possible, so that it can beused to predict optically dependent functions such asmodulation transfer functions of the eye.adult life extending from approximately 20 to 70 years ofage and the onset of presbyopia at about 45 years of age.

1997 Optical Society of America

rameters were modeled to produce spherical aberration

ts

H.-L. Liou and N. A. Brennan Vol. 14, No. 8 /August 1997 /J. Opt. Soc. Am. A 1685values close to those found experimentally. The changein refractive index with wavelength was modeled to pre-dict chromatic aberration of the eye.

3. SELECTION OF BIOMETRIC DATAA. Intraocular DistancesIntraocular distances refer to the various distances be-tween optical refracting surfaces within an eye; these in-clude anterior chamber depth, thickness of the cornea andlens, vitreous depth, and the overall axial length.Consistent values of axial length of the eye have been

determined by various studies. In 1948 Stenstrom9 re-ported mean values of axial length of 24.04 mm for malesand 23.89 mm for females. The difference in gender hasbeen confirmed by various other investigators,1012 withmean axial length values of approximately 24 mm formales and 2324 mm for females. Table 1 lists the ex-perimental results found by Stenstrom,9 Jansson,10 Yuet al.,11 and Koretz et al.12

Anterior chamber depth and lens thickness appear tobe age dependent. Jansson10 found that the mean ante-rior chamber depth for males decreases from 3.86 mm atage 2029 years to 3.58 mm at age 4049 years. Similarfindings were obtained by Leighton and Tomlinson13 andKoretz et al.12 The reverse was noted for lens thickness,which was found to increase in thickness with age.12,13

Numerous other studies have been done on the anteriorchamber depth and lens thickness. Fontana andBrubaker,14 Leighton and Tomlinson,13 Larsen,15 Week-ers and Grieten,16 and many others have measured thedepth of the anterior chamber. Clemmensen andLuntz,17 Jansson,10 Leighton and Tomlinson,13 Koretzet al.,12 and others have measured the lens thickness.

Table 1. Axial Length Measuremen

Author Year Method

Stenstrom9 1948 Roentgenology

Jansson10 1963 Ultrasonography

Yu et al.11 1979 Ultrasonography

Koretz et al.12 1989 Ultrasonographythickness alone found results equal to 0.49 mm orlarger.1821

The thickness of the peripheral cornea is usually largerthan that of the central cornea. However, measurementtechniques and distance from center at which it has beenmeasured are variable, and results vary from 0.6 to 0.7mm at 15 from the center to the limbal periphery.20,2224

Therefore these were not used in the modeling but ratherwere used to check the order of magnitude of the modeledperipheral corneal thickness.

B. CorneaThere are many ways of describing an aspheric surface.A common method is to use a conicoid in the form

x2 1 y2 1 ~1 1 Q !z2 2 2zR 5 0, (1)

where the origin is chosen at the surfaces apex, x is thehorizontal meridian, y is the vertical meridian, z is theaxis of revolution, R is the radius at the apex, and Q isthe asphericity parameter that specifies the type of coni-coid.This representation of the cornea is rotationally sym-

metric and does not allow meridional variations in thevalues of R (i.e., astigmatism) or Q unless otherwisemodified. In a study of ocular refraction in young menfor the British National Service, Sorsby et al.25 found thatsome 80% of 1680 eyes showed spherical refractions withless than 60.5 D of astigmatism. Given that the corneaaccounts for the majority of the refracting power of theeye, it is reasonable to assume that we can represent thecornea as a rotationally symmetric conicoid for the pur-pose of an eye model.As shown in Table 2, early investigations9,26 measured

anterior central corneal radius only, and some studies27,28

Reported by Various Investigators

Gender Number of EyesAxial Length

Measurements (mm)

Male 685 24.04Female 315 23.89Male 113 24.00Female 71 23.14Male 1749 23.76Female 40 23.74Male 32 24.08Female 68 23.423. Results from in vivo experiments were chosen inpreference to in vitro studies.4. All other factors being the same, data from the most

recent study and/or with the larger sample size wereused.

Once the necessary biometric data were compiled, weidentified those parameters that lacked empirical resultsor accuracy and used them as modeling variables. Forparaxial calculations, curvature and gradient-index pa-rameters were modeled with the aim of producing an eyewith an equivalent power of 60 diopters (D) and an axiallength of 24 mm. For finite calculations, asphericity pa-

We chose to adopt results of intraocular distances fromthe investigation by Koretz et al.,12 following the criteriadetailed above and because their single study provided allthe various intraocular distances of the eye. They mademeasurements on 32 males and 68 females. To giveequal weights to gender, the average of male and femaleresults were used rather than the mean value reportedfor all 100 subjects. The thickness of the central corneawas taken to be 0.50 mm as found by subtracting the vit-reous depth, lens thickness, and anterior chamber depthfrom the length of the globe. This value was used ratherthan the value of 0.47 mm directly measured in the Ko-retz et al.12 study, as various other studies on corneal

on

of E

08

0460

nt

e 23an A

frac5 2

1686 J. Opt. Soc. Am. A/Vol. 14, No. 8 /August 1997 H.-L. Liou and N. A. Brennanterior corneal surface without reference to the apical cor-neal radius. In 1982 Kiely et al.29 calculated values forboth corneal radius and asphericity, using the conicoidrepresentation. Using an autocollimating keratoscopedeveloped by Clark,30 Kiely et al.29 measured the anteriorcorneal shape of 176 healthy eyes. They found a meanvalue of 7.72 6 0.27 mm for the central radius R and20.26 6 0.18 for the asphericity Q. In 1986 Guillonet al.31 repeated a similar experiment but used akeratometer to measure the central cornea and a commer-cially available photokeratoscope for the peripheral cor-nea. Two hundred and twenty eyes representative of anormal population were studied and gave a mean value of7.77 6 0.25 mm for the central corneal radius R, Q5 20.17 6 0.13 for the flat meridian and Q 5 20.196 0.16 for the steep meridian.In a summary of nine papers dealing with central cor-

neal keratometric values, Clark30 reported a mean of 7.80mm and a range from 7.0 to 9.0 mm for a healthy Cauca-sian population. Given that Guillon et al.31 found a cen-tral corneal value (7.77 mm) closer to this average of 7.80mm and that the study is more recent and has a largersample size, the mean values of radius and asphericity re-ported in their study have been adopted for the newmodel eye.Data on the curvature of the posterior corneal surface

are scarce, but because of the much smaller refractivestep, the curvatures effect is not very great. Table 3 is asummary of investigations on both anterior and posteriorcorneal radii and their comparison. In a series of studies

and posterior corneal radii and their correlation. Theymeasured mean values of 7.65 and 6.46 mm for the ante-rior and posterior corneal radii, respectively, and foundthe two parameters to be related as follows: posterior ra-dius 5 0.791 3 anterior radius 1 0.409.In 1990 Royston et al.33 investigated a new method of

measuring the posterior radius using Purkinje imagesand compared it with the slit-lamp method used by Loweand Clark.32 They obtained similar results from the twomethods; the average anterior corneal radius was foundto be 7.77 mm with both methods, and the average poste-rior corneal radii were found to be 6.35 mm and 6.40 mmwith the slit-lamp and the Purkinje image methods, re-spectively. The same authors then repeated a similar ex-periment on 80 eyes34 and determined the ratio of ante-rior corneal radius to posterior corneal radius to be1:0.823. This ratio is adopted in the new model eye togive a posterior corneal radius of 6.40 mm, given an an-terior corneal radius of 7.77 mm.To our knowledge, there are no direct measurements of

the shape of the posterior corneal surface. Rivett andHo35 attempted to determine the shape of the posteriorcorneal surface by using data of anterior corneal surfaceand thickness. They found a Q value of 21.14 and21.52 for the right and the left eyes, respectively. Patelet al.36 tried similar calculations in 1993 and found Q val-ues of 20.36 and 20.48 for the vertical (y) and horizontal(x) meridians of the posterior corneal surface. However,they measured the anterior corneal surface to be nearlyspherical, and this differed from the usually acceptedin the 1970s concentrated purely on asphericity of the an-

Table 2. Summary of Experimental Results

Author Year Number

Townsley27 1970 35Mandell and St. Helen28 1971

Stenstrom9 1948 100Sorsby et al.26 1957 19Kiely et al.29 1982 17Guillon et al.31 1986 22

Table 3. Summary and Comparison of Results on A

Author YearNumberof Eyes Method

Lowe and Clark32 1973 92 Slit lamp AgMe

Royston et al.33 1990 15 Slit lampPurkinje image

Dunne et al.34 1992 80 Purkinje image Re

Patel et al.36 1993 20 Calculation usinganterior corneal surfacedone in 1973, Lowe and Clark32 investigated the anterior

Anterior Corneal Radius and Asphericity

yes Radius (mm) Asphericity (Q)

20.30 20.23

(20.40 to 20.72)7.86 (7.00 to 8.65)

7.79 6 0.27 7.72 6 0.27 20.26 6 0.187.77 6 0.25 20.19 6 0.16

(steep meridian)20.17 6 0.15(flat meridian)

erior and Posterior Corneal Radii of the Same Eyes

Comments

Anteriorradius (A)(mm)

Posteriorradius (P)(mm) Ratio of A/P

77 7.65 6 0.27 6.46 6 0.26 P 5 0.791Age561.4 10.409

7.77 6.35 1:0.8177.77 6.40 1:0.824

tive error range 7.98 (male) 6.44 1:0.82316.5 D to 14.7 D 7.84 (female) 6.36

(Sphericalcomponent only)

5.8

of 92 normal eyes under cycloplegia from subjects aged r 5 (4)

nte

me

61.as

l651

51.

H.-L. Liou and N. A. Brennan Vol. 14, No. 8 /August 1997 /J. Opt. Soc. Am. A 16872377 (mean age 5 61.4), they found that the lens radiushad a mean value of 10.29 mm with a range from 7.50 to15.38 mm. The anterior lens radius correlated nega-tively with age, and the mean value was approximately11.26 mm for 45 years of age. Similar results were ob-tained by Brown39 in a series of 200 emmetropic eyes un-der cycloplegia in subjects aged 382 years. He foundthe mean central anterior lens radius to be 12.4 mm andthe mean central posterior lens radius to be 8.1 mm. Healso provided measurements of peripheral anterior andposterior lens radius. These are 13.3 mm and 7.1 mm ata distance of 2.8 mm and 2.4 mm from the center, respec-tively. Measurements of lens asphericity were made byHowcroft and Parker.40 They found values of 7.3 mmand 5.35 mm for the anterior and posterior lens radii, re-spectively, and Q values of 23.13 and 21.0 for the ante-rior and the posterior lens asphericity, respectively. Thelens radii measured by these two researchers are consid-erably less than those found by other authors, probablybecause they used cadaver lenses, which are likely tohave undergone changes in lens shape after death.In order to adopt a full set of in vivo results for both

anterior and posterior surfaces and both radius and as-phericity values, Browns39 measurements were chosen.The asphericity values for the anterior and posterior lenssurfaces were obtained using the central and peripheral

Table 4. Summary of Experimental Results on A

Author YearNumberof Eyes Method Com

Lowe37 1972 92 in vivo Age 2377

Lowe and Clark38 1973 92 in vivo Mean age 5Same series

Brown39 1974 200 in vivo Age 382Emmetropia, 60.75DS, 60.50 cy

Parker70 1972 100 in vitro Mean age 5Range 129

Howcroft 1977 120 in vitro Age 187and Parker40 Mean age 5R2

or

Q 5R2 2 ~rR2!2/3

y2. (5)

These agree with equations obtained by Roberts41 if Q isreplaced by Q 5 2e2, where e is eccentricity of the conicsection. Inserting measurements of Brown39 into Eq. (5)gives anterior and posterior lens asphericities of 20.94and 10.96, respectively.Measurements of the gradient refractive index of the

lens are scarce. Nakao et al.42 investigated the distribu-tion of the refractive index in five lenses, using an inter-ference technique on lens sections. Highest values of thelens refractive index were found at the center of the lensand varied from 1.403 to 1.409. There was intersubjectvariability in the distribution of the gradient index.Campbell43 developed a nondestructive method for mea-suring the refractive index of intact crystalline lenses,and this method was used by Pierscionek and Chan44 tostudy the gradient-index profiles in three human lensesaged 16, 56, and 84 years. Results of their measure-ments for the 16-year-old human lens were used in themodeling of the new model eye. Pierscionek and Chan44

stated that their method of calculation necessitated an ac-

rior and Posterior Lens Radius and Asphericity

ntsAnterior Lens Radius

(mm)Posterior Lens Radius

(mm)

Mean 5 10.29 6 1.78

Range 5 7.50 2 15.384 Mean 5 11.26

Lowe 197237 for 46 years of age

Central 12.4 6 2.6 Central 8.1 6 1.6Peripheral 13.3 6 3.2 Peripheral 7.1 6 1.4

Radius 5 5 Radius 5 3.3Q 5 21.5 Q 5 21.0Radius 5 7.3 6 0.3 Radius 5 5.35 6 0.14

4 Q 5 23.13 Q 5 21.0value of asphericity, Q 5 20.18 to 20.26, which probablyaffected their calculated value of posterior corneal asphe-ricity.Given that the posterior corneal asphericity is one of

the more inadequately defined optical parameters of theeye, we entered it as a variable in the modeling of ourmodel eye.

C. LensThere are fewer studies on lens shape than on cornealshape because of the difficulties of measurement. Re-sults obtained by various investigators vary considerablydepending on the method used and whether measure-ments were taken in vivo or in vitro (see Table 4). Loweand Clark37,38 studied the anterior lens curvature inpeople with healthy and glaucomatous eyes. In a series

radius values provided by Brown, assuming a rotationallysymmetric conicoid representation for the anterior andposterior surfaces.The relation between the instantaneous tangential ra-

dius of curvature r( y) and asphericity Q can be derivedas follows. In the vertical ( y) meridian the conic sectionis expressed by

y2 1 ~1 1 Q !z2 2 2zR 5 0. (2)

In Cartesian coordinates, r can be derived with the gen-eral expression

r 5@1 1 ~y8!2#3/2

y9. (3)

This yields

~R2 2 Qy2!3/2

1688 J. Opt. Soc. Am. A/Vol. 14, No. 8 /August 1997 H.-L. Liou and N. A. Brennancurate estimate of the refractive-index value at the lenssurface. Errors in this value will affect the profile valuesnear the lens capsule but not those at the center. Thismeans that their results are valid near the center buthave increasing uncertainty at greater distances from thecenter.

D. Refractive Indices and Dispersion of the OcularMediaWe adopted a refractive-index value of 1.376 for the cor-nea following Gullstrands45 model eyes, because to ourknowledge there are no recent studies on the corneal re-fractive index. Similarly, we adopted Gullstrands re-fractive index of 1.336 for the aqueous and vitreous hu-mors.Experimental data on dispersion of the ocular media

are scarce. Le Grand46 used the Cornu formula to com-pute the refractive indices. Palmer and Sivak47 and Si-vak and Mandelman48 attempted to provide some basicdispersion data of the ocular media for humans and othervertebrates. However, measurements of the human ocu-lar media included only values for the lens capsule, lenscore, and lens periphery, and the standard deviations ofmean findings were significantly large. They found thatthe ocular refractive indices all increase rapidly at theviolet end of the spectrum, and Cornus formula cannotadequately represent this change in refractive index.In experimental studies of the chromatic dispersion of

the ocular media, Sivak and Mandelman48 investigatedthe dispersion of the ocular media of various vertebrates,including humans. They found that, in general, theaqueous humor and the cornea have a dispersion similarto that of water, whereas the lens is significantly moredispersive than water. Given the lack of experimentaldata on dispersion of the human ocular media, these willbe assumed to have dispersive properties similar to thoseof water in the modeling of the new model eye. The re-fractive indices presented above will be taken for thewavelength of 555 nm, which is the peak of the photopicVl curve.

E. PupilAnatomically, the pupil is positioned directly in front ofthe crystalline lens. It is not exactly centered with re-spect to the rest of the eye and is often displaced slightlynasally by ;0.5 mm.49 Because of the various aberra-tions of the eye, this displacement has some effect on im-age quality and visual performance.Recent research into pupil centration has revealed that

it changes with change in illumination and pupil size.50,51

Wilson et al.50 measured the centration of the pupil withrespect to the achromatic axis of the eye as a function ofpupil size. They found significant shifts of the pupil cen-ter (up to 0.6 mm) with pupil dilation in both nasal andtemporal directions. The effect was usually symmetricalbetween the two eyes, and the shift was linear in half ofthe subjects. Walsh51 found changes in pupil centrationof up to 0.4 mm in both natural and drug-induced pupildilations. For the 39 subjects tested, there was no sig-nificant relationship between the direction of the changein centration occurring and the degree of natural pupil di-lation in the dark. The mean (6SD) shift was very small(0.09 6 0.17 mm temporal and 0.03 6 0.15 mm inferior),although the mean absolute centration change (regard-less of direction) was significant (0.19 6 0.12 mm).Walsh and Charman52 investigated the effect of pupil

centration and diameter on ocular performance. Theyfound that while decentration caused relatively little dif-ference in modulation transfer function at the smaller pu-pil size, it could produce marked degradations for thelarger pupils.Given the importance of the position of the pupil, the

pupil was modeled as a circular aperture stop on the frontsurface of the lens with its center decentered 0.5 mm na-sally from the optical axis. We did not model a change ofpupil center with changing pupil size, in the absence ofempirical data that specifies a consistent change.50,51

F. Angle AlphaThe visual axis of the human eye does not coincide withthe optical axis, as the fovea is normally displaced tempo-rally from the optical axis. We have adopted the defini-tion of visual axis used by Bennett and Rabbetts53 andothers. In view of the meaning of visual axis, it is de-fined as the axis or chief ray of the actual pencil of raysthat enters the pupil and is converged to the fovea. Ittherefore denotes the incident ray path directed towardthe center E of the entrance pupil such that the conjugaterefracted ray falls on the fovea. The angle between theoptical axis and the visual axis is termed the angle alphaand is considered positive when the visual axis in objectspace lies on the nasal side of the optical axis. A positivevalue of approximately 5 is commonly found.53 To ourknowledge, no previous schematic eye has incorporatedangle alpha into its model. We have included a 5 anglealpha in this new model, and this is used in image-qualitycalculations such as the modulation transfer function.

4. MODELING OF THE EYEAll optical calculations were performed with the CodeVoptical design and analysis software from Optical Re-search Associates, Pasadena, Calif. Exact ray tracingwas involved in all calculations.From the examination and selection of biometric data,

two parameters were identified as being inadequately de-fined by experimental results, namely, the posterior cor-neal asphericity and the gradient-index distribution ofthe lens. The latter affects paraxial parameters, whilethe former has an effect on the prediction of spherical ab-erration. We first modeled the distribution of the gradi-ent index of the lens to produce an eye with the normallyaccepted equivalent power of 60 D (Ref. 45) and an axiallength of 24 mm.The circles in Fig. 1 represent the experimental data of

Pierscionek and Chan.44 The lens index is plotted as afunction of r, which is the normalized distance from thelens center to the periphery. Because these data pointshave increasing uncertainty at greater distances from thelens center, we weighted the individual data points differ-ently in the curve-fitting procedure. Weighting equals 1at r 5 0 (i.e., in the core of the lens), and weighting 5 0at r 5 1 (i.e., at the lens surface). Different weightingswere investigated for data points between 0 , r , 1;

H.-L. Liou and N. A. Brennan Vol. 14, No. 8 /August 1997 /J. Opt. Soc. Am. A 1689these took the forms of 1 2 r, A1 2 r, A4 1 2 r, and unity(i.e., not weighting), as they produced equivalent powersof the eye ranging from 58 to 62 D. The shape of the in-dex distribution was assumed to be parabolic, as other-wise there exist an infinite number of higher-order possi-bilities for the mathematical form of a nonparabolicdistribution, and there are not enough data to ascertainany particular higher-order form. Figure 1 shows the fit-ted parabolic curves for different weightings comparedwith the original experimental results. It is assumedthat the variation of the lens index as a function of nor-malized distances is the same in the equatorial and sag-ittal planes. For simplicity of calculation, the lens is di-vided into anterior and posterior sections. The method ofSmith et al.4 is used to derive the thicknesses of the an-terior and posterior sections and to convert the parabolicgradient-index distribution, expressed in normalized dis-tance r, to one expressed in actual lens thicknesses. Thedistribution of the gradient index is represented in theform

n~w, z ! 5 n00 1 n01z 1 n02z2 1 n10w

2, (6)

where z is along the optical axis, w is the radial distanceperpendicular to the z axis (w2 5 x2 1 y2), and n00 ,n01 , n02 , and n10 are the index coefficients for a parabolicgradient-index distribution in an unaccommodated lens.Because of the partition of the lens, we have introduced

an imaginary plane surface between the anterior and pos-terior parts to demarcate the start of the posterior sec-tion. This surface has zero radius of curvature and noasphericity.We have previously reviewed empirical results of the

spherical aberration of the eye from various studies2 andfound that the relationship between longitudinal spheri-cal aberration SA and ray height h can be best describedby a linear model where SA 5 0.20h and the range is ap-proximately 60.25 D from the value of spherical aberra-tion for ray heights of up to 4 mm from the optical axis.

Fig. 1. Refractive-index distribution data of Pierscionek andChan44 (circles). The four parabolic curves represent the vari-ous smoothed-lens refractive-index distributions with differentweightings given to the experimental data points. A, weight5 1 2 r; B, weight 5 A1 2 r; C, weight 5 A4 1 2 r; D, noweighting (i.e., weighting 5 1 for all data points), where r is thenormalized distance from the lens center.The posterior corneal asphericity was modeled to predictspherical aberration of the eye by the above relationship.We investigated asphericity values between Q 5 0 and21.0 in steps of 0.2 for the posterior corneal surface. Q5 0 represents a spherical surface, Q 5 21 is a parabo-loid with the axis along the optical axis, and 21 , Q, 0 represent ellipsoids with the major axis along theoptical axis.To predict the chromatic aberration of the eye, we as-

sumed the ocular media to have dispersive propertiessimilar to those of water. Sivak and Mandelman48 mea-sured the variation of the refractive index of water withwavelength. We adopted their findings and fitted a curveto express the results as follows:

n ~water! 5 1.3847 2 0.1455l 1 0.0961l2, (7)

where l is in micrometers. Given that the refractive in-dex of water is 1.3335 at 0.555 mm, Eq. (7) can be rewrit-ten as

n ~water at l mm! 5 n ~water at 0.555 mm!

1 0.0512 2 0.1455l

1 0.0961l2. (8)

The various ocular media are assumed to have disper-sive properties similar to those of water, and their refrac-tive indices at various wavelengths l can be found by re-placing water by media as follows:

n ~mediaa at l mm! 5 n ~media at 0.555 mm!

1 0.0512 2 0.1455l

1 0.0961l2. (9)

For calculations of the modulation transfer-function(MTF), angle alpha was incorporated by using a pencil ofrays directed toward the center of the entrance pupil atan angle of 5 nasally from the optical axis. We have de-noted the nasal side of the optical axis to be positive, soangle alpha 5 15 and pupil decentration 5 10.5 mm(i.e., nasally, too).

5. RESULTSTable 5 lists the equivalent power and the axial length ofthe model eye with different weightings. The A4 1 2 rweighting of the gradient-index lens distribution gives anequivalent power of 60.35 D and an axial length of 23.95mm, and this is used in the final eye model.

Table 5. Equivalent Power and Axial Length of theNew Schematic Eye When Different Weightings

of Data Points Are Used in Curve Fittingthe Gradient-Index Distribution of the Lens

Weighting Equivalent Power (D) Axial Length (mm)

None (unity) 61.68 23.55A4 1 2 r 60.35 23.95A1 2 r 59.19 24.311 2 r 57.87 24.74

1690 J. Opt. Soc. Am. A/Vol. 14, No. 8 /August 1997 H.-L. Liou and N. A. BrennanFigure 2 shows the spherical aberration predicted fromthe new model eye, with asphericity values of Q 5 0 de-creasing to 21.0 in steps of 0.2 for the posterior corneal

Fig. 2. Spherical-aberration values predicted by the new sche-matic eye, with various asphericity values Q for the posteriorcorneal surface (dashed curves) compared with the line of bestlinear fit (solid line) and range (shaded region) of empirical re-sults.

Fig. 3. Diagram showing the variation of corneal thicknessgiven aspheric front and back surfaces.

Table 6. Structural Parameters of the NewSchematic Eye

Surface Radius Asphericity Thickness n at 555 nm

1 7.77 20.18 0.50 1.3762 6.40 20.60 3.16 1.3363 (pupil) 12.40 20.94 1.59 Grad A4 Infinity 2.43 Grad P5 28.10 10.96 16.27 1.336surface. The best linear fit and range of spherical aber-ration obtained from compiling various experimentalinvestigations2 are shown for comparison. When the as-phericity of the posterior corneal surface is Q > 20.4, thespherical aberration estimated from the model eye is lessthan the mean experimental data at all ray heights. Pos-terior corneal asphericity values of Q , 20.6 result inpredicted values greater than the mean experimentaldata at 4 mm ray height, close to the mean experimentalvalue at 23 mm ray height and less than this at 1 mmray height. Of these, the curve describing spherical ab-erration using Q 5 20.6 is the only one that lies entirelywithin the range of empirical results. We took this to bethe curve best approximating the mean experimentalspherical aberration, and the value of Q 5 20.6 isadopted as the posterior corneal asphericity in the newmodel eye.The aspheric shape of the posterior corneal surface in

combination with the aspheric front surface gives a thick-ness of 0.50 mm to the central cornea and 0.91 mm at aperipheral point 6 mm from the center (i.e., at the edge ofa 12-mm cornea), the thickness gradually increasing fromthe center to the periphery (see Fig. 3). Had the poste-rior cornea been spherical, as assumed by most previousfinite schematic models,3,4,5456 the peripheral corneawould have been thicker, with values as high as 1.97 mmat 6 mm from the center.Table 6 lists all the parameters of the new proposed

model eye, whose format can be readily used in any opti-cal design and analysis software, and Fig. 4 illustratesthe schematic features of the new proposed model eye andshows the directions of angle alpha and pupil decentra-tion. Details regarding Table 6 are as follows. Surface 3contains the pupil aperture whose center is decentered by0.5 mm nasally. The aspheric surface 3 is still centeredaround the optical axis like all other surfaces, eventhough the pupil aperture is not centered. Surface 4 isan imaginary plane dividing the lens into anterior andposterior sections. The gradient index of the lens is de-scribed by n(w,z) 5 n00 1 n01z 1 n02z

2 1 n10w2, where

z is along the optical axis, w is the radial distance per-pendicular to the z axis (w 5 x2 1 y2), and n00 , n01 ,n02 , and n10 are the index coefficients for a parabolicgradient-index distribution in an unaccommodated lens.In grad A, n00 5 1.368, n01 5 0.049057, n025 20.015427, n10 5 20.001978; and in grad P, n005 1.407, n01 5 0.000000, n02 5 20.006605, n105 20.001978. The different ocular refractive indices atvarious wavelengths l can be found as n(media at l mm) 5 n (media at 0.555 mm) 1 0.05122 0.1455l 1 0.0961l2.In Fig. 5 the spherical aberration predicted by the new

model eye is compared with the mean experimental re-sults for spherical aberration. The spherical aberrationestimated from the Gullstrand no. 245 and Le Grand46

paraxial schematic eyes and the Kooijman55 and Navarroet al.3 finite schematic eyes are shown for comparison.The contribution of the gradient-index lens to the spheri-cal aberration of the eye is shown in Fig. 6.Applying Eq. (9) to obtain the refractive indices of dif-

ferent ocular media at various wavelengths resulted in achromatic aberration close to experimental results

ma

H.-L. Liou and N. A. Brennan Vol. 14, No. 8 /August 1997 /J. Opt. Soc. Am. A 1691recent experimental results of Artal et al.62 and Navarroet al.63

6. DISCUSSIONMany finite schematic eyes proposed in the past werebased on a previous paraxial schematic model and incor-porated various refracting surfaces as required3,54,55 ormodified the lens into a gradient-index structure.4,56 Forthese models, much of the original biometric data incor-porated into the earlier paraxial model were retained.Given the vast amount of recently published biometricdata, it is possible to develop a new model eye that incor-porates data specifically collected with the most modernand accurate techniques. This means that errors due tolimitations of the accuracy of outdated instruments arenot propagated, and the new model accounts for changingtrends in the population with respect to biometric data.

results.

Fig. 6. Contribution of the gradient-index lens (GRIN) to thespherical aberration of the eye. Here 1ve SA and 2ve SA meanpositive and negative spherical aberration, respectively.Fig. 4. Schematic drawing of the new sche

Fig. 5. Comparison of the spherical aberration predicted by thenew schematic eye with spherical aberrations estimated by vari-ous paraxial and finite model eyes and the mean experimental(squared multiple correlation R2 5 0.98). Data fromWald and Griffin,57 Bedford and Wyszecki,58 Howarthand Bradley,59 and Cooper and Pease60 are plotted in Fig.7 for comparison with the longitudinal chromatic aberra-tion predicted from the new model eye.Sine-wave polychromatic modulation transfer functions

of the new model eye were calculated for a 4-mm pupiland are shown in Fig. 8. The StilesCrawford effect isincorporated into the calculation by introducing a two-dimensional Gaussian apodization filter for the amplitudeof the effect, as was done by van Meeteren.1 This has theform I 5 102(a/2)w

2where I is the intensity, a 5 0.05,

and w is the radial distance from the pupil center inmillimeters.61 Polychromatic MTFs for an equienergy il-luminant were calculated, taking into account the photo-pic spectral sensitivity function of retinal receptors (Vlcurve). The latter was simulated by using wavelengthsof 510, 555, and 610 nm with weightings of 1, 2, and 1 re-spectively. Figure 8 compares the modulation transferfunction calculated from the new model eye with the most

tic eye (see Table 6 for parameter values).

1692 J. Opt. Soc. Am. A/Vol. 14, No. 8 /August 1997 H.-L. Liou and N. A. BrennanA. Anatomical AccuracyThe anatomical accuracy of a schematic eye is very impor-tant for the purpose of modeling vision in refractive pro-cedures. We produced a model eye with all of the refract-ing surfaces represented, so that each individualparameter could be varied according to the refractive pro-cedure applied (e.g., photorefractive keratectomy modifiesthe front surface of the cornea only, and an intraocularlens replaces the entire crystalline lens).The equivalent power of the new model eye is 60.35 D.

This is close to the currently accepted value of 60 D forthe eye and is similar to those of other modeleyes.35,45,46,5456 The axial length of the new model eyeis 23.95 mm, which compares favorably with the averagevalue of 24 mm for males and slightly less than 24 mm forfemales. Finite eyes such as those proposed by Lotmar,54

Kooijman,55 and Navarro et al.3 were based on LeGrands46 paraxial model eye and had axial lengths of ap-proximately 24.2 mm, slightly longer than values deter-mined experimentally.

Fig. 7. Comparison of the chromatic aberration predicted by thenew schematic eye with various experimental results.

Fig. 8. Sine-wave polychromatic MTF for a 4-mm pupil calcu-lated with horizontal gratings, compared with the most recentexperimental data of Navarro et al.63 and Artal et al.62In comparison with other finite eyes, our new model eyeattempts to be anatomically accurate while providing op-tical information about the human eye that is as close aspossible to experimental results. Lotmar,54 Kooijman,55

and Navarro et al.3 used aspheric refracting surfaces butdid not incorporate a gradient-index lens. Blaker56

adopted a gradient-index structure for the lens to describeaccommodative changes but did not introduce asphericityin any refracting surfaces. Smith et al.4 included bothaspheric surfaces and a gradient-index lens. However,they based their model on the Gullstrand no. 1 eye. Theequivalent power of their eye is therefore 58.64 D, lessthan the accepted 60 D; and the axial length for the eye is24.35 mm, slightly longer than the average experimentalvalue. Furthermore, they adopted positive asphericityvalues for the lens (7.9 and 2.6 for the anterior and pos-terior surfaces, respectively), which led to estimates of thespherical aberration of the eye that were higher than em-pirical values.The aspheric shape of the posterior corneal surface in

combination with the aspheric front surface provides athickness of 0.50 mm to the central cornea and 0.91 mmat a peripheral point 6 mm from the center, the thicknessgradually increasing from the center to the periphery.At 3 mm from the center, the thickness value is 0.62 mm,which compares favorably with measurements of periph-eral corneal thickness by Hirji and Larke,24 who found athickness of 0.60 6 0.05 mm and 0.57 6 0.03 mm at 15(approximately 3 mm) nasal and temporal to the center,respectively. This feature of the new model eye is morerealistic than thickness values in previous finite sche-matic models,3,4,5456 as the peripheral cornea in thesemodels are much thicker than that in empirical data ow-ing to the spherical back surface adopted for the cornea.As far as we know, the new proposed model eye is the

first to incorporate both angle alpha and decentration ofthe pupil, thus distinguishing it from all previously pub-lished schematic eyes. Failure to incorporate angle al-pha assumes coincidence of the visual and optical axes,which is anatomically inaccurate and may lead to opticalinaccuracy.Pupil decentration was included for anatomical accu-

racy of the representation of the eye. The decentration ofthe human pupil is taken to be 0.5 mm on average andtherefore leads to slight transverse chromatic aberration(TCA). According to Thibos et al.64 the magnitude ofTCA present at the fovea can be approximated by theequation

f 5 hDRx, (10)

where h is the pupil decentration in meters, DRx is theLCA in diopters, and f is TCA in radians.For the present model, this translates to a foveal TCA

of approximately 1.2 arcmin for red and blue lights ofwavelengths 605 and 497 nm, respectively. This com-pares favorably with results of Rynders et al.,65 whofound that the absolute magnitude of foveal TCA for anygiven individual was from 0.05 to 2.67 arcmin for red andblue lights of 605 and 497 nm, respectively. The meanfoveal TCA of 85 young adults from their study was 0.8arcmin.

H.-L. Liou and N. A. Brennan Vol. 14, No. 8 /August 1997 /J. Opt. Soc. Am. A 1693B. Importance of the Gradient-Index LensA lens with a gradient-index (GRIN) structure was intro-duced in this new model to represent the anatomicalstructure of the human crystalline lens. Most previousfinite schematic eyes either had aspheric refracting sur-faces and a homogeneous lens3,54,55 or a GRIN lens struc-ture with spherical surfaces,56 but not both.The added complexity of GRIN is required in order to

provide the functional power of the lens while keeping therealistic values of the refractive index of the lens. Previ-ous schematic models with a homogeneous lens had toadopt a refractive as high as 1.42 for the lens in order toproduce the functional power of the lens (approximately20 D). However, the refractive index of the lens has beenmeasured to vary from 1.386 in the cortex to 1.404 in thenucleus,44 and these values can produce the power of thelens only if they are in the form of GRIN. This consti-tutes a major difference from previous schematic eyemodels (Smith et al.4 used GRIN, but it does not comparefavorably in modeling vision). It is also because of theGRIN lens that we were able to obtain a more empiricallyaccurate value of the axial length for the new model eyewithout increasing the equivalent power.

C. Optical PerformanceSpherical aberration predicted by the new model eye com-pares well with experimental results (see Fig. 5). Thissimilarity is achieved by incorporating into the model eyea combination of aspheric surfaces for the posterior cor-nea and both the anterior and the posterior surfaces ofthe lens. Also, the gradient distribution of the lens indexprovides some negative spherical aberration to offset thepositive spherical aberration that is due to the cornea (seeFig. 6). This matches the general view that the lensplays an important role in reducing the aberrations of theeye. El Hage and Berny66 measured the aberration ofone whole eye and the shape of its cornea and deducedthat the lens must play a large compensatory role result-ing in diminishing corneal aberration. This opinion wasshared by Hartridge.67 Jenkins68 found that the corneahad about the same amount of aberration as the entireeye, thus implying that the lens is almost free of aberra-tion. Results by Millodot and Sivak69 showed that theaberration of the lens does not systematically neutralizethat of the cornea. Of the 17 subjects for whom theymeasured lenticular spherical aberration, two had slightnegative values, four showed no measurable amount, andthe rest had positive values.Most previous finite models3,4,5456 assumed the poste-

rior cornea to be spherical. Although the posterior cor-neal surface does not play an important role in refractingrays, an aspheric surface has the effect of fine-tuning thespherical aberration of the eye at large ray heights fromthe optical axis in our model. Modifications of the pa-rameters of the GRIN lens would also be very effective inaltering the estimates of spherical aberration. However,we did not consider manipulating the parameters of theGRIN lens to match the spherical-aberration data fromthe literature, as this would defeat the goal of anatomicalaccuracy. The asphericity of the back surface of the cor-nea was chosen as the mechanism for fine-tuning the av-erage spherical aberration of the eye instead, because webelieve that it is the least well defined parameter of theeye. Individual changes in the parameters of the GRINlens would most likely account for the large intersubjectvariability encountered in spherical-aberration measure-ments, as described above.To account for the chromatic aberration of the eye, val-

ues for the dispersion of the ocular media were incorpo-rated into the new model. The dispersion of the ocularmedia can be approximated by that of water, since resultsof chromatic aberration for wavelengths from 380 to 780nm for the new model eye predict experimental measure-ments accurately, as seen from Fig. 4 (R2 5 0.98). Wedid not choose to adopt dispersion measurements of thehuman ocular media,47,48 because of the large errors in-volved in the measurements. In comparison, water wasmuch better defined.The sine-wave MTF calculated from our model eye com-

pares favorably with the most recent experimentalresults62,63 (see Fig. 8). The remarkably close agreementbetween our computed polychromatic MTF and the mono-chromatic MTFs measured by Artal et al.62 and Navarroet al.63 is surprising. One would expect the polychro-matic MTF to be worse than the monochromatic one be-cause of the effect of chromatic aberration on image qual-ity. In this case, it is probable that the computed MTFresults are better than expected because irregular aber-rations of the eye have not been included in the model.The incorporation of angle alpha and pupil decentra-

tion affects calculations of image quality. Angle alphadecreases the modulation sensitivity of a centered system,but the decentration of the pupil in the same direction off-sets some of this effect, especially at low spatial frequen-cies. Modulation transfer functions estimated from otherschematic eyes were not significantly higher than empiri-cal results, even though they did not include angle alpha.This is probably because the spherical aberration in thesemodel eyes was higher than in experimental results andtherefore contributed to lower modulation sensitivities.

D. Future ResearchWhile our new model eye predicts spherical and chro-matic aberration of the eye, it is limited in its opticalstructures to those refracting surfaces that can be de-scribed mathematically. Further improvements of themodel would include incorporation of irregular or nonro-tationally symmetric surfaces. Such surfaces may not berepresentative of the general population but might bemore useful for the purpose of assessing image-qualityfunctions in individual eyes. This would be necessary toachieve the best result in modeling vision in individualeyes for purposes such as refractive surgery, where thecharacteristics of each person are unique. Additionalbiometric knowledge such as the curvature of the retina isalso required for analysis of off-axis aberrations.The present model does not yet account for changes

with age, accommodation, or individual variability. Fur-ther research on the changes of ocular biometry with ageis required for incorporation of this factor into the eyemodel. To extend the model further, we need to acquiremore knowledge about the surface structure and distribu-tion of the refractive index of the lens as well as how thesechange during accommodation. Although a schematic

1694 J. Opt. Soc. Am. A/Vol. 14, No. 8 /August 1997 H.-L. Liou and N. A. Brennaneye represents an average model, it is possible to incorpo-rate individual characteristics for the exact prediction ofthe visual performance of a given subject. Further re-search is necessary to improve and expand this model eye.The spherical-aberration curve predicted by the new

model eye has a somewhat parabolic shape, as expectedfrom aberration theory, rather than the linear relation-ship indicated by empirical data. More detailed biomet-ric data and empirical results for spherical aberration arenecessary to investigate the deviations between experi-mental and calculated curves.The new model eye presented here is based on experi-

mental biometric data except for the posterior corneal as-phericity and dispersion values due to lack of data.Careful compilation of experimental measurements pro-duced quite successfully a finite schematic eye model.This shows that the various biometric measurementsavailable are fairly accurate when carefully screened forthe desired purpose. The posterior corneal asphericityand the gradient-index distribution of the lens were mod-eled for the new model eye to have an equivalent power of60 D and an axial length of 24 mm and to predict spheri-cal aberration. Future investigation into the empiricalvalues of the above two parameters need to be performedin order to confirm the values obtained from the presentmodeling.

7. CONCLUSIONTo the best of our knowledge, the new finite model eyepresented in this paper is the first schematic eye modelthat predicts both chromatic and spherical aberrations ofthe eye within the tolerances of empirical results. Apartfrom providing optical parameters and serving as a ve-hicle for image-quality calculations, the new model can beconsidered to be a quasi-true anatomical representationof an average emmetropic human eye. Its close anatomi-cal representation to the human eye makes it suitable forstudies investigating visual performance under normal oraltered conditions, such as reshaping of the cornea in re-fractive surgery, estimating vision where the cornea isdistorted, changing the ocular refraction by application ofcontact lenses or spectacle lenses, and even implantationof intraocular lenses.

ACKNOWLEDGMENTWe thank M. C. Rynders and J. J. Vos for helpful com-ments on the manuscript.

REFERENCES AND NOTES1. A. van Meeteren, Calculations on the optical modulation

transfer function of the human eye for white light, Opt.Acta 21, 395412 (1974).

2. H. L. Liou and N. A. Brennan, The prediction of sphericalaberration with schematic eyes, Ophthalmic. Physiol. Opt.16, 348354 (1996).

3. R. Navarro, J. Santamara, and J. Bescos, Accommodation-dependent model of the human eye with aspherics, J. Opt.Soc. Am. A 2, 12731281 (1985).

4. G. Smith, B. K. Pierscionek, and D. A. Atchison, The opti-cal modelling of the human lens, Ophthalmic. Physiol.Opt. 11, 359369 (1991).5. M. Ye, X. X. Zhang, L. N. Thibos, and A. Bradley, A newsingle-surface model eye that accurately predicts chromaticand spherical aberrations of the human eye, Invest. Oph-thalmol. Visual Sci. (Suppl.) 34, 777 (1993).

6. L. N. Thibos, M. Ye, X. X. Zhang, and A. Bradley, The chro-matic eye: a new reduced-eye model of ocular chromaticaberration in humans, Appl. Opt. 31, 35943600 (1992).

7. L. N. Thibos, M. Ye, X. X. Zhang, and A. Bradley, A newoptical model of the human eye, Opt. Photon. News 4, 12(1993).

8. A. Bradley, School of Optometry, Indiana University,Bloomington, Ind. 47405-3201 (personal communication,1995): corrections to the Ye et al. abstract (Ref. 5): Therefracting surface defined as Y 5 0.0899X2 1 0.0006X4

should be Y 5 0.0899X2 1 0.0005X4 instead.9. S. Stenstrom, Investigation of the variation and the corre-

lation of the optical elements of human eyes, Am. J. Op-tom. 25, 340350 (1948).

10. F. Jansson, Measurements of intraocular distances by ul-trasound, Acta Ophthalmol. Suppl. 74, 149 (1963).

11. C. S. Yu, D. Kao, and C. T. Chang, Measurement of thelength of the visual axis by ultrasonography in 1789 eyes,Chin. J. Ophthalmol. 15, 4547 (1979).

12. J. F. Koretz, P. L. Kaufman, M. W. Neider, and P. A. Go-eckner, Accommodation and presbyopia in the humaneyeaging of the anterior segment, Vision Res. 29, 16851692 (1989).

13. D. A. Leighton and A. Tomlinson, Changes in axial lengthand other dimensions of the eyeball with increasing age,Acta Ophthalmol. 50, 815826 (1972).

14. S. T. Fontana and R. F. Brubaker, Volume and depth ofthe anterior chamber in the normal aging human eye,Arch. Ophthalmol. 98, 18031808 (1980).

15. J. Larsen, The sagittal growth of the eye, Acta Ophthal-mol. 49, 239262 (1971).

16. R. Weekers and J. Grieten, Mesure de la profondeur de lachambre anterieure en clinique, Soc. Belg. Ophthalmol.129, 361381 (1961).

17. V. Clemmensen and M. H. Luntz, Lens thickness andangle-closure glaucoma, Acta Ophthalmol. 54, 193197(1976).

18. R. F. Lowe, Central corneal thickness, Br. J. Ophthalmol.53, 824826 (1969).

19. F. K. Hansen, A clinical study of the normal human cen-tral corneal thickness, Acta Ophthalmol. 49, 8289 (1971).

20. E. L. Martola and J. L. Baum, Central and peripheral cor-neal thickness, Arch. Opththalmol. 79, 2830 (1968).

21. P. S. Soni and I. M. Borish, A report on central and periph-eral corneal thickness, Int. Contact Lens Clin. 6, 6670(1979).

22. D. M. Maurice and A. A. Giardini, A simple optical appa-ratus for measuring the corneal thickness and the averagethickness of the human cornea, Br. J. Ophthalmol. 35,169177 (1951).

23. A. Tomlinson, A clinical study of the central and periph-eral thickness and curvature of the human cornea, ActaOphthalmol. 50, 7382 (1972).

24. N. K. Hirji and J. R. Larke, Thickness of human corneameasured by topographic tachometry, Am. J. Optom. Arch.Am. Acad. Optom. 55, 97100 (1978).

25. A. Sorsby, M. Sheridan, A. G. Leary, and B. Benjamin, Vi-sion, visual acuity and ocular refraction in young men,Brit. Med. J. 1, 13941398 (1960).

26. A. Sorsby, B. Benjamin, J. B. Davey, M. Sheridan, and J. M.Tanner, Emmetropia and Its Aberrations. A Study in theCorrelation of the Optical Components of the Eye. MedicalResearch Council Special Report Series No. 293 (Her Maj-estys Stationary Office, London, 1957).

27. M. Townsley, New knowledge of the corneal contour, Con-tacto 14, 3843 (1970).

28. R. B. Mandell and R. St. Helen, Mathematical model forthe corneal contour, Br. J. Physiol. Opt. 26, 183197(1971).

29. P. H. Kiely, G. Smith, and G. Carney, The mean shape ofthe human cornea, Opt. Acta 29, 10271040 (1982).

30. B. A. J. Clark, Variations in corneal topography, Aust. J.Optom. 56, 399413 (1973).

31. M. Guillon, P. M. Lydon, and C. Wilson, Corneal topogra-phy: a clinical model, Ophthalmic. Physiol. Opt. 6, 4756(1986).

32. R. F. Lowe and B. A. Clark, Posterior corneal curvature,Br. J. Ophthalmol. 57, 464470 (1973).

33. J. M. Royston, M. C. M. Dunne, and D. A. Barnes, Mea-surement of the posterior corneal radius using slit lampand Purkinje image techniques, Ophthalmic. Physiol. Opt.10, 385388 (1990).

34. M. C. M. Dunne, J. M. Royston, and D. A. Barnes, Normalvariations of the posterior corneal surface, Acta Ophthal-mol. 70, 255261 (1992).

35. A. G. Rivett and A. Ho, The posterior corneal topography,Invest. Ophthalmol. Visual Sci. (Suppl.) 32, 10011001(1991).

36. S. Patel, J. Marshall, and F. W. Fitzke, Shape and radiusof posterior corneal surface, Refract. Corneal Surg. 9, 173181 (1993).

tion of the human eye, Ophthalmic. Physiol. Opt. 8, 178182 (1988).

52. G. Walsh and W. N. Charman, The effect of pupil centra-tion and diameter on ocular performance, Vision Res. 28,659665 (1988).

53. A. G. Bennett and R. B. Rabbetts, Clinical Visual Optics,2nd ed. (Butterworth-Heinemann, Oxford, 1989), pp. 1718.

54. W. Lotmar, Theoretical eye model with aspherics, J. Opt.Soc. Am. 61, 15221529 (1971).

55. A. C. Kooijman, Light distribution on the retina of a wide-angle theoretical eye, J. Opt. Soc. Am. 73, 15441550(1983).

56. J. W. Blaker, Toward an adaptive model of the humaneye, J. Opt. Soc. Am. 70, 220223 (1980).

57. G. Wald and D. R. Griffin, The change in refractive powerof the human eye in dim and bright light, J. Opt. Soc. Am.37, 321336 (1947).

58. R. E. Bedford and G. Wyszecki, Axial chromatic aberrationof the human eye, J. Opt. Soc. Am. 47, 564565 (1957).

59. P. A. Howarth and A. Bradley, The longitudinal chromatic

H.-L. Liou and N. A. Brennan Vol. 14, No. 8 /August 1997 /J. Opt. Soc. Am. A 169537. R. F. Lowe, Anterior lens curvature, Br. J. Ophthalmol.56, 409413 (1972).

38. R. F. Lowe and B. A. J. Clark, Radius of curvature of theanterior lens surface, Br. J. Ophthalmol. 57, 471474(1973).

39. N. Brown, The change in lens curvature with age, Exp.Eye Res. 19, 175183 (1974).

40. M. J. Howcroft and J. A. Parker, Aspheric curvatures forthe human lens, Vision Res. 17, 12171223 (1977).

41. C. Roberts, The accuracy of power maps to display curva-ture data in corneal topography systems, Invest. Ophthal-mol. Visual Sci. 35, 35253532 (1994).

42. S. Nakao, T. Ono, R. Nagata, and K. Iwata, The distribu-tion of refractive index in the human crystalline lens, Jpn.J. Clin. Ophthalmol. 23, 903906 (1969).

43. M. C. W. Campbell, Measurement of refractive index in anintact crystalline lens, Vision Res. 24, 409415 (1984).

44. B. K. Pierscionek and D. Y. C. Chan, Refractive index gra-dient of human lenses, Optom. Vis. Sci. 66, 822829(1989).

45. A. Gullstrand, Helmholtzs Physiological Optics (Optical So-ciety of America, New York, 1924), Appendix, pp. 350358.

46. Y. Le Grand, Physiological Optics (Springer-Verlag, NewYork, 1980), pp. 5455.

47. D. A. Palmer and J. Sivak, Crystalline lens dispersion, J.Opt. Soc. Am. 71, 780782 (1981).

48. J. G. Sivak and T. Mandelman, Chromatic dispersion ofthe ocular media, Vision Res. 22, 9971003 (1982).

49. G. Westheimer, Image quality in the human eye, Opt.Acta 17, 641658 (1970).

50. M. A. Wilson, M. C. W. Campbell, and P. Simonet, Changeof pupil centration with change of illumination and pupilsize, Optom. Vis. Sci. 69, 129136 (1992).

51. G. Walsh, The effect of mydriasis on the pupillary centra-aberration of the human eye, and its correction, VisionRes. 26, 361366 (1986).

60. D. P. Cooper and P. L. Pease, Longitudinal chromatic ab-erration of the human eye and wavelength in focus, Am. J.Optom. Physiol. Opt. 65, 99107 (1988).

61. R. A. Applegate and V. Lakshminarayanan, Parametricrepresentation of StilesCrawford functions: normalvariation of peak location and directionality, J. Opt. Soc.Am. A 10, 16111623 (1993).

62. P. Artal, M. Ferro, I. Miranda, and R. Navarro, Effects ofaging in retinal image quality, J. Opt. Soc. Am. A 10,16561662 (1993).

63. R. Navarro, P. Artal, and D. R. Williams, Modulationtransfer of the human eye as a function of retinal eccentric-ity, J. Opt. Soc. Am. A 10, 201212 (1993).

64. L. N. Thibos, A. Bradley, and X. X. Zhang, Effect of ocularchromatic aberration on monocular visual performance,Optom. Visual Sci. 68, 599607 (1991).

65. M. Rynders, B. Lidkea, W. Chisholm, and L. N. Thibos,Statistical distribution of foveal transverse chromatic ab-erration, pupil centration, and angle c in a population ofyoung adult eyes, J. Opt. Soc. Am. A 12, 23482357 (1995).

66. S. G. El Hage and F. Berny, Contribution of the crystallinelens to the spherical aberration of the eye, J. Opt. Soc. Am.63, 205211 (1973).

67. H. Hartridge, Recent Advances in the Physiology of Vision(Churchill, London, 1950), pp. 7884.

68. T. C. A. Jenkins, Aberrations of the eye and their effects onvision: part 1, Br. J. Physiol. Opt. 20, 5991 (1963).

69. M. Millodot and J. Sivak, Contribution of the cornea andlens to the spherical aberration of the eye, Vision Res. 19,685687 (1979).

70. J. A. Parker, Aspheric optics of the human lens, Can. J.Ophthalmol. 7, 168175 (1972).