June 1, 2004 / Vol. 29, No. 11 / OPTICS LETTERS 1197

Schematic eye with a gradient-index lensand aspheric surfaces

Damian Siedlecki and Henryk Kasprzak

Institute of Physics, Wroclaw University of Technology,Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland

Barbara K. Pierscionek

School of Biomedical Sciences, University of Ulster at Coleraine,County Londonderry, Northern Ireland BT52 1SA, UK

Received October 23, 2003

A new schematic eye with aspheric surfaces and a radially varying refractive-index distribution lens is pro-posed. Image quality and spherical aberration are determined by use of ray tracing, and the results arepresented as spot diagrams and compared with five existing model eyes. The proposed model provides thebest image quality and lowest spherical aberration. © 2004 Optical Society of America

OCIS codes: 330.5370, 110.2760, 170.4460.

The optical system of the eye is complex and ele-gant but aberrations that reduce optical quality exist.Spherical aberration (SA), astigmatism, and chromaticaberration have the greatest effect on retinal imagery;other off-axis aberrations play a smaller role. Thiswork reviews current models of the eye and presentsa new model that provides the sharpest focus in theretinal plane and minimizes SA.

The simplest schematic eye models, such as those ofListing1 and Emsley,2 have a single refractive surface.Gullstrand3 proposed two models and was the first toacknowledge the inhomogeneous nature of the lenticu-lar refractive index. A slightly modified version ofGullstrand’s model eye was introduced by Le Grandand Hage,4 but it used a homogeneous index lens.Koojiman5 extended the Le Grand and Hage modeland included aspheric surfaces to minimize aberra-tions and in particular to reduce SA. There are alsoa number of models of the individual optical elementsof the eye: the lens and cornea.6,7

The relative paucity of models of the gradient-indexlens results from difficulties in measuring the lenticu-lar index distribution. Ellipsoidal, isoindicial sur-faces have been used to describe the three-dimensionaldistribution of the refractive index in the lenses ofrabbits8 and cats.9 However, in the human lens thismodel does not appear to be appropriate.10 In thisstudy we introduce a radial model of the lens in whichthe refractive index varies only with distance from theoptic axis (Fig. 1). This model has great potential asa model for the design of intraocular implants becauseit significantly minimizes SA and enhances retinalimage quality.

A ray-tracing program was developed and used toproduce the resulting image and calculate the SA forthe proposed model eye. The point of entry in themodel was determined from surface parameters givenby Koojiman.5 The lenticular refractive index fol-lowed a radial distribution, decreasing exponentially

0146-9592/04/111197-03$15.00/0

with distance from the optic axis. This variation canbe expressed by

n�R� � n1 2 �n1 2 n2�∑

1 2 exp�2bR�1 2 exp�2bRmax�

∏, (1)

where n1 is the refractive index along the optic axis, n2is the refractive index at the equator, b is the exponen-tial coefficient, R is the distance from the optic axis,and Rmax is the equatorial radius of the lens (maxi-mum distance from the optic axis to the equator). Inthe calculations n1 � 1.406, n2 � 1.386 (as used in theGullstrand No. 1 model eye), and Rmax � 4.5 mm. Alldistances, surface curvatures, and eccentricities werethose of Koojiman.5 We traced 308 parallel, incidentrays spaced 0.2 mm apart through a 4-mm pupil. Themethod of Sharma et al.11 was used to describe ray pas-sage through the lens.

Results are shown as spot diagrams in two planes:the plane of the retina (24.2-mm axial length in ac-cordance with Koojiman) and the plane in which thespread of all points in the spot diagram is minimal.

Fig. 1. Diagrammatic representation of the refractive-index contours in the (a) equatorial and (b) sagittal planesof the new model. The radial lines show the directions ofincreasing refractive-index magnitude.

© 2004 Optical Society of America

1198 OPTICS LETTERS / Vol. 29, No. 11 / June 1, 2004

The spread is quantified as a statistical variance by

V �NXi

�di 2 d�2, (2)

where di is the distance of each point from the opticaxis, d is the mean distance of all points from the opticaxis, and N is the number of rays (N � 308). Addi-tional parameters are the distance from the cornealapex to the plane in which the spread of image points isminimal (i.e., the plane of best focus), the value of thevariance in this plane, and the value of the variance inthe plane of the retina. The model was assessed andoptimized with the difference between the varianceobtained in the retinal plane and that obtained in theplane of best focus. This difference is referred to asthe difference in the values of variances. The changein the value of the minimal variance as a function ofcoeff icient b was used to describe the refractive-indexdistribution in the lens. The ray-tracing procedurewas repeated, with the same conditions for ray entry,on the eye models of Emsley,2 Gullstrand,3 Le Grandand Hage,4 and Koojiman.5 Calculations from raytracing through the proposed model were made forvalues of b from 22 to 11 in increments of 0.1. Asthe distance between the points in the retinal planeand those in the plane of best focus approached aminimum, calculations were made in smaller incre-mental steps. The results are shown in Fig. 2, fromwhich it can be seen that the minimum value occurswhen b � 20.91. In such a case the plane of bestfocus lies at a distance of 24.20 mm from the cor-neal apex.

The spot diagrams for all the model eyes, includingthe proposed model, are shown in Fig. 3 (relevantvalues are given in Table 1). It is clear that, of allthe models, the proposed model yields the best focusand the lowest SA: The diameter of the retinal imagedoes not exceed 0.024 mm, and most of the light is con-centrated in an area of 0.0076 mm in diameter. Thepoorest image and the highest level of SA is found forEmsley’s eye,2 which is not surprising given that thismodel incorporates a single spherical surface witha relatively small radius of curvature. In this casethe diameter of the image is almost 0.24 mm. Thenext-best image quality is found in the Gullstrand3

and Le Grand and Hage4 model eyes; image sizes forthese are comparable. The model eye of Koojiman5

provides the best image quality of all the existing

Fig. 2. Difference in the second-order moments (in squaremillimeters) as a function of index distribution coeff i-cient b.

Fig. 3. Spot diagrams calculated for various model eyes: (a) Emsley, (b) Gullstrand No. 1, (c) Gullstrand No. 2,(d) Le Grand and Hage, (e) Koojiman, (f ) proposed (b � 0.91). The empty squares indicate the spot diagram in theretinal plane; the f illed squares indicate the spot diagram in the plane of best focus. Dimensions are in millimeters.

June 1, 2004 / Vol. 29, No. 11 / OPTICS LETTERS 1199

Table 1. Distance of the Plane of Best Focus, Value of the Minimal Variance, Axial Length, and Value of theVariance in the Retinal Plane

Gullstrand Gullstrand Le GrandEmsley Eye Eye Model Eye Model and Hage Koojiman Proposed

Parameter Model (No. 1) (No. 2) Eye Model Eye Model Model

Distance of the plane of best focusfrom corneal apex (mm) 21.43 24.01 23.52 23.83 24.05 24.20

Value of the minimal variance 0.105 0.023 0.024 0.022 0.004 0.004Axial length (mm) 23.22 24.38 23.90 24.20 24.20 24.20Value of the variance

at the retinal plane 0.937 0.198 0.211 0.192 0.033 0.004

models tested because it uses aspheric surfaces.However, when compared with the proposed model,the Koojiman model eye5 yields variance that is almostten times greater in the plane of the retina (Table 1).The diameter of the image from Koojiman’s modelis 0.043 mm, compared with 0.024 mm from theproposed model.

In summary, the proposed eye model provides a bet-ter image and lower SA than all the other eye modelstested.12 Although this model is not a complete rep-resentation of the in vivo lens, it can serve as a modelfor intraocular implants, providing better image qual-ity than that produced through a homogeneous indeximplant, without requiring the complexity found in theliving lens.

The authors are grateful for financial assis-tance from the Komitet Badan Naukowych and theBritish Council. D. Siedlecki’s e-mail address isdamian.siedlecki@pwr.wroc.pl.

References

1. G. Smith and D. A. Atchison, The Eye and Visual Opti-cal Instruments (Cambridge U. Press, Cambridge, Eng-land, 1997).

2. H. H. Emsley, Visual Optics (Butterworth, London,1952), Vol. 1.

3. J. P. C. Southall, Helmholtz’s Treatise on Physiologi-cal Optics (Optical Society of America, Rochester, N.Y.,1924), Vol. 1.

4. Y. Le Grand and S. G. Hage, Physiological Optics(Springer-Verlag, Berlin, 1981).

5. A. C. Koojiman, J. Opt. Soc. Am. 73, 1544 (1983).6. G. Smith, B. K. Pierscionek, and D. A. Atchison, Oph-

thalmic Physiol. Opt. 11, 359 (1991).7. H. T. Kasprzak and E. Jankowska-Kuchta, J. Mod. Opt.

43, 1135 (1996).8. S. Nakao and S. Fujimoto, J. Opt. Soc. Am. 58, 1125

(1968).9. W. S. Jagger, Vision Res. 30, 723 (1990).

10. B. K. Pierscionek, Exp. Eye Res. 64, 887 (1997).11. A. Sharma, D. Vizia Kumar, and K. A. Ghatak, Appl.

Opt. 21, 984 (1982).12. The parameters of the eye models used are available

from the authors.