Abstract

We introduce a model of the human eye for which we take into consideration the laminated nature of lens fibers. The thickness of each lamina is 5.6 μm; thus the lens comprises 300 eccentric lenses of minute dimensions. The index gradient of the lens is such that the index of refraction increases exponentially from the lens core to its peripheral zone. A vector ray-tracing technique is employed to study the optical characteristics of the system. Both paraxial and marginal rays are simulated, and the angles of incidence vary from 0° to ±20°. Special attention is given to the meridional caustic surfaces as well as the wave-front distortion of the refracted rays. A quasi-Newton optimization technique is employed to obtain the best parameters for the system. A computer modeling program, written in fortran 77, is used to simulate a ray’s refraction through the multisurfaces of the eye. The results show full agreement with previous data and that the cornea is responsible for eliminating possible spherical aberration of the system.

© 1995 Optical Society of America

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References

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  1. A. Gullstrand, Y. Le Grand, “Schematic eye,” in Treatise on Physiological Optics, 3rd ed., H. von Helmholtz, ed., translated by J. P. C. Southall (Optical Society of America, Washington, D.C., 1924), pp. 320–322.
  2. R. Navarro, J. Santamaria, J. Bescós, “Accommodation-dependent model of the human eye with aspherics,” J. Opt. Soc. Am. A 2, 1273–1281 (1985).
    [CrossRef] [PubMed]
  3. O. Pomerantzeff, H. Fish, J. Govignon, C. L. Schepens, “Wide-angle optical model of the eye,” Opt. Acta 19, 387–388 (1972).
    [CrossRef]
  4. O. Pomerantzeff, M. Pankratov, G. Wang, P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61(3), 166–176 (1984).
    [CrossRef] [PubMed]
  5. K. P. Pflibsen, O. Pomerantzeff, R. N. Ross, “Retinal illuminance using a wide-angle model of the eye,” J. Opt. Soc. Am. A 5, 146–150 (1988).
    [CrossRef] [PubMed]
  6. O. Pomerantzeff, M. Pankratov, G. Wang, “Calculation of an IOL from the wide angle optical model of the eye,” Am. Inter-Ocular Implant Soc. J. 11, 37–43 (1985).
  7. J. W. Blaker, “Toward an adaptive model of the human eye,” J. Opt. Soc. Am. 70, 220–223 (1980).
    [CrossRef] [PubMed]
  8. W. M. Rosenblum, J. W. Blaker, M. G. Block, “Matrix method for the evaluation of lens systems with radial gradient-index elements,” Am. J. Optomet. Physiol. Opt. 65, 661–665 (1988).
    [CrossRef]
  9. I. H. Al-Ahdali, “Optimization of three and four-element lens systems by minimizing the caustic merit function,” Ph.D. dissertation (University of Alabama at Birmingham, Birmingham, Ala., 1989).
  10. I. H. Al-Ahdali, D. L. Shealy, “Optimization of three- and four-element lens systems by minimizing the caustic surfaces,” Appl. Opt. 29, 4551–4559 (1990).
    [CrossRef] [PubMed]
  11. W. M. Rosenblum, D. L. Shealy, “Caustic analysis of intraocular lens implants in humans,” Contact Intraocular Lens Med. J. 5, 136–140 (1979).
  12. W. M. Rosenblum, D. L. Shealy, “Caustic and illuminance calculations for a model of the human eye,” in Fiber Optics I, C. T. Holliday, ed., Proc. Soc. Photo-Opt. Instrum. Eng.14, 237–240 (1968).
  13. J. C. Mann, “The organ of the senses,” J. Anat. London 7, 1270–1271 (1924).
  14. M. W. Charles, “Dimensions of the human eye relevant to radiation protection,” Phys. Med. Biol. 20, 202–218 (1975).
    [CrossRef] [PubMed]
  15. E. F. Finchman, “The mechanism of accommodation,” Br. J. Ophthalmol. Monogr. 8, 417–419 (1975).
  16. S. Duke-elders, A. David, Opthalmamic Optics and Refraction (Henry & Kimpton, London, 1970), pp. 115–130.
  17. W. H. Press, S. A. V. Teukolsky, W. T. Etterling, B. P. Flannery, “Minimization and maximization of functions,” in Numerical Recipes in fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, London, 1992), pp. 406–413.
  18. Military Standardization Handbook: Optical Design, MILH-DBK-141.5 (US. GPO, Washington, D.C., 1962).
  19. J. F. Koretz, G. H. Handelman, N. P. Brown, “Analysis of human crystalline lens curvature as a function of accommodative state and age,” Vision Res. 24, 1141–1151 (1984).
    [CrossRef] [PubMed]

1990 (1)

1988 (2)

W. M. Rosenblum, J. W. Blaker, M. G. Block, “Matrix method for the evaluation of lens systems with radial gradient-index elements,” Am. J. Optomet. Physiol. Opt. 65, 661–665 (1988).
[CrossRef]

K. P. Pflibsen, O. Pomerantzeff, R. N. Ross, “Retinal illuminance using a wide-angle model of the eye,” J. Opt. Soc. Am. A 5, 146–150 (1988).
[CrossRef] [PubMed]

1985 (2)

O. Pomerantzeff, M. Pankratov, G. Wang, “Calculation of an IOL from the wide angle optical model of the eye,” Am. Inter-Ocular Implant Soc. J. 11, 37–43 (1985).

R. Navarro, J. Santamaria, J. Bescós, “Accommodation-dependent model of the human eye with aspherics,” J. Opt. Soc. Am. A 2, 1273–1281 (1985).
[CrossRef] [PubMed]

1984 (2)

O. Pomerantzeff, M. Pankratov, G. Wang, P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61(3), 166–176 (1984).
[CrossRef] [PubMed]

J. F. Koretz, G. H. Handelman, N. P. Brown, “Analysis of human crystalline lens curvature as a function of accommodative state and age,” Vision Res. 24, 1141–1151 (1984).
[CrossRef] [PubMed]

1980 (1)

1979 (1)

W. M. Rosenblum, D. L. Shealy, “Caustic analysis of intraocular lens implants in humans,” Contact Intraocular Lens Med. J. 5, 136–140 (1979).

1975 (2)

M. W. Charles, “Dimensions of the human eye relevant to radiation protection,” Phys. Med. Biol. 20, 202–218 (1975).
[CrossRef] [PubMed]

E. F. Finchman, “The mechanism of accommodation,” Br. J. Ophthalmol. Monogr. 8, 417–419 (1975).

1972 (1)

O. Pomerantzeff, H. Fish, J. Govignon, C. L. Schepens, “Wide-angle optical model of the eye,” Opt. Acta 19, 387–388 (1972).
[CrossRef]

1924 (1)

J. C. Mann, “The organ of the senses,” J. Anat. London 7, 1270–1271 (1924).

Al-Ahdali, I. H.

I. H. Al-Ahdali, D. L. Shealy, “Optimization of three- and four-element lens systems by minimizing the caustic surfaces,” Appl. Opt. 29, 4551–4559 (1990).
[CrossRef] [PubMed]

I. H. Al-Ahdali, “Optimization of three and four-element lens systems by minimizing the caustic merit function,” Ph.D. dissertation (University of Alabama at Birmingham, Birmingham, Ala., 1989).

Bescós, J.

Blaker, J. W.

W. M. Rosenblum, J. W. Blaker, M. G. Block, “Matrix method for the evaluation of lens systems with radial gradient-index elements,” Am. J. Optomet. Physiol. Opt. 65, 661–665 (1988).
[CrossRef]

J. W. Blaker, “Toward an adaptive model of the human eye,” J. Opt. Soc. Am. 70, 220–223 (1980).
[CrossRef] [PubMed]

Block, M. G.

W. M. Rosenblum, J. W. Blaker, M. G. Block, “Matrix method for the evaluation of lens systems with radial gradient-index elements,” Am. J. Optomet. Physiol. Opt. 65, 661–665 (1988).
[CrossRef]

Brown, N. P.

J. F. Koretz, G. H. Handelman, N. P. Brown, “Analysis of human crystalline lens curvature as a function of accommodative state and age,” Vision Res. 24, 1141–1151 (1984).
[CrossRef] [PubMed]

Charles, M. W.

M. W. Charles, “Dimensions of the human eye relevant to radiation protection,” Phys. Med. Biol. 20, 202–218 (1975).
[CrossRef] [PubMed]

David, A.

S. Duke-elders, A. David, Opthalmamic Optics and Refraction (Henry & Kimpton, London, 1970), pp. 115–130.

Dufault, P.

O. Pomerantzeff, M. Pankratov, G. Wang, P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61(3), 166–176 (1984).
[CrossRef] [PubMed]

Duke-elders, S.

S. Duke-elders, A. David, Opthalmamic Optics and Refraction (Henry & Kimpton, London, 1970), pp. 115–130.

Etterling, W. T.

W. H. Press, S. A. V. Teukolsky, W. T. Etterling, B. P. Flannery, “Minimization and maximization of functions,” in Numerical Recipes in fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, London, 1992), pp. 406–413.

Finchman, E. F.

E. F. Finchman, “The mechanism of accommodation,” Br. J. Ophthalmol. Monogr. 8, 417–419 (1975).

Fish, H.

O. Pomerantzeff, H. Fish, J. Govignon, C. L. Schepens, “Wide-angle optical model of the eye,” Opt. Acta 19, 387–388 (1972).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. V. Teukolsky, W. T. Etterling, B. P. Flannery, “Minimization and maximization of functions,” in Numerical Recipes in fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, London, 1992), pp. 406–413.

Govignon, J.

O. Pomerantzeff, H. Fish, J. Govignon, C. L. Schepens, “Wide-angle optical model of the eye,” Opt. Acta 19, 387–388 (1972).
[CrossRef]

Gullstrand, A.

A. Gullstrand, Y. Le Grand, “Schematic eye,” in Treatise on Physiological Optics, 3rd ed., H. von Helmholtz, ed., translated by J. P. C. Southall (Optical Society of America, Washington, D.C., 1924), pp. 320–322.

Handelman, G. H.

J. F. Koretz, G. H. Handelman, N. P. Brown, “Analysis of human crystalline lens curvature as a function of accommodative state and age,” Vision Res. 24, 1141–1151 (1984).
[CrossRef] [PubMed]

Koretz, J. F.

J. F. Koretz, G. H. Handelman, N. P. Brown, “Analysis of human crystalline lens curvature as a function of accommodative state and age,” Vision Res. 24, 1141–1151 (1984).
[CrossRef] [PubMed]

Le Grand, Y.

A. Gullstrand, Y. Le Grand, “Schematic eye,” in Treatise on Physiological Optics, 3rd ed., H. von Helmholtz, ed., translated by J. P. C. Southall (Optical Society of America, Washington, D.C., 1924), pp. 320–322.

Mann, J. C.

J. C. Mann, “The organ of the senses,” J. Anat. London 7, 1270–1271 (1924).

Navarro, R.

Pankratov, M.

O. Pomerantzeff, M. Pankratov, G. Wang, “Calculation of an IOL from the wide angle optical model of the eye,” Am. Inter-Ocular Implant Soc. J. 11, 37–43 (1985).

O. Pomerantzeff, M. Pankratov, G. Wang, P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61(3), 166–176 (1984).
[CrossRef] [PubMed]

Pflibsen, K. P.

Pomerantzeff, O.

K. P. Pflibsen, O. Pomerantzeff, R. N. Ross, “Retinal illuminance using a wide-angle model of the eye,” J. Opt. Soc. Am. A 5, 146–150 (1988).
[CrossRef] [PubMed]

O. Pomerantzeff, M. Pankratov, G. Wang, “Calculation of an IOL from the wide angle optical model of the eye,” Am. Inter-Ocular Implant Soc. J. 11, 37–43 (1985).

O. Pomerantzeff, M. Pankratov, G. Wang, P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61(3), 166–176 (1984).
[CrossRef] [PubMed]

O. Pomerantzeff, H. Fish, J. Govignon, C. L. Schepens, “Wide-angle optical model of the eye,” Opt. Acta 19, 387–388 (1972).
[CrossRef]

Press, W. H.

W. H. Press, S. A. V. Teukolsky, W. T. Etterling, B. P. Flannery, “Minimization and maximization of functions,” in Numerical Recipes in fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, London, 1992), pp. 406–413.

Rosenblum, W. M.

W. M. Rosenblum, J. W. Blaker, M. G. Block, “Matrix method for the evaluation of lens systems with radial gradient-index elements,” Am. J. Optomet. Physiol. Opt. 65, 661–665 (1988).
[CrossRef]

W. M. Rosenblum, D. L. Shealy, “Caustic analysis of intraocular lens implants in humans,” Contact Intraocular Lens Med. J. 5, 136–140 (1979).

W. M. Rosenblum, D. L. Shealy, “Caustic and illuminance calculations for a model of the human eye,” in Fiber Optics I, C. T. Holliday, ed., Proc. Soc. Photo-Opt. Instrum. Eng.14, 237–240 (1968).

Ross, R. N.

Santamaria, J.

Schepens, C. L.

O. Pomerantzeff, H. Fish, J. Govignon, C. L. Schepens, “Wide-angle optical model of the eye,” Opt. Acta 19, 387–388 (1972).
[CrossRef]

Shealy, D. L.

I. H. Al-Ahdali, D. L. Shealy, “Optimization of three- and four-element lens systems by minimizing the caustic surfaces,” Appl. Opt. 29, 4551–4559 (1990).
[CrossRef] [PubMed]

W. M. Rosenblum, D. L. Shealy, “Caustic analysis of intraocular lens implants in humans,” Contact Intraocular Lens Med. J. 5, 136–140 (1979).

W. M. Rosenblum, D. L. Shealy, “Caustic and illuminance calculations for a model of the human eye,” in Fiber Optics I, C. T. Holliday, ed., Proc. Soc. Photo-Opt. Instrum. Eng.14, 237–240 (1968).

Teukolsky, S. A. V.

W. H. Press, S. A. V. Teukolsky, W. T. Etterling, B. P. Flannery, “Minimization and maximization of functions,” in Numerical Recipes in fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, London, 1992), pp. 406–413.

Wang, G.

O. Pomerantzeff, M. Pankratov, G. Wang, “Calculation of an IOL from the wide angle optical model of the eye,” Am. Inter-Ocular Implant Soc. J. 11, 37–43 (1985).

O. Pomerantzeff, M. Pankratov, G. Wang, P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61(3), 166–176 (1984).
[CrossRef] [PubMed]

Am. Inter-Ocular Implant Soc. J. (1)

O. Pomerantzeff, M. Pankratov, G. Wang, “Calculation of an IOL from the wide angle optical model of the eye,” Am. Inter-Ocular Implant Soc. J. 11, 37–43 (1985).

Am. J. Optom. Physiol. Opt. (1)

O. Pomerantzeff, M. Pankratov, G. Wang, P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61(3), 166–176 (1984).
[CrossRef] [PubMed]

Am. J. Optomet. Physiol. Opt. (1)

W. M. Rosenblum, J. W. Blaker, M. G. Block, “Matrix method for the evaluation of lens systems with radial gradient-index elements,” Am. J. Optomet. Physiol. Opt. 65, 661–665 (1988).
[CrossRef]

Appl. Opt. (1)

Br. J. Ophthalmol. Monogr. (1)

E. F. Finchman, “The mechanism of accommodation,” Br. J. Ophthalmol. Monogr. 8, 417–419 (1975).

Contact Intraocular Lens Med. J. (1)

W. M. Rosenblum, D. L. Shealy, “Caustic analysis of intraocular lens implants in humans,” Contact Intraocular Lens Med. J. 5, 136–140 (1979).

J. Anat. London (1)

J. C. Mann, “The organ of the senses,” J. Anat. London 7, 1270–1271 (1924).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

O. Pomerantzeff, H. Fish, J. Govignon, C. L. Schepens, “Wide-angle optical model of the eye,” Opt. Acta 19, 387–388 (1972).
[CrossRef]

Phys. Med. Biol. (1)

M. W. Charles, “Dimensions of the human eye relevant to radiation protection,” Phys. Med. Biol. 20, 202–218 (1975).
[CrossRef] [PubMed]

Vision Res. (1)

J. F. Koretz, G. H. Handelman, N. P. Brown, “Analysis of human crystalline lens curvature as a function of accommodative state and age,” Vision Res. 24, 1141–1151 (1984).
[CrossRef] [PubMed]

Other (6)

A. Gullstrand, Y. Le Grand, “Schematic eye,” in Treatise on Physiological Optics, 3rd ed., H. von Helmholtz, ed., translated by J. P. C. Southall (Optical Society of America, Washington, D.C., 1924), pp. 320–322.

W. M. Rosenblum, D. L. Shealy, “Caustic and illuminance calculations for a model of the human eye,” in Fiber Optics I, C. T. Holliday, ed., Proc. Soc. Photo-Opt. Instrum. Eng.14, 237–240 (1968).

S. Duke-elders, A. David, Opthalmamic Optics and Refraction (Henry & Kimpton, London, 1970), pp. 115–130.

W. H. Press, S. A. V. Teukolsky, W. T. Etterling, B. P. Flannery, “Minimization and maximization of functions,” in Numerical Recipes in fortran: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, London, 1992), pp. 406–413.

Military Standardization Handbook: Optical Design, MILH-DBK-141.5 (US. GPO, Washington, D.C., 1962).

I. H. Al-Ahdali, “Optimization of three and four-element lens systems by minimizing the caustic merit function,” Ph.D. dissertation (University of Alabama at Birmingham, Birmingham, Ala., 1989).

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Figures (11)

Fig. 1
Fig. 1

Anatomical section through the margin of a human lens.

Fig. 2
Fig. 2

Schematic representation of the variation of the index of refraction distribution (laminated lens model).

Fig. 3
Fig. 3

Schematic representation of the ray trace in a laminated lens.

Fig. 4
Fig. 4

Dependence of the focal length on the β factor at different shell thicknesses (σ = Dt/300).

Fig. 5
Fig. 5

Refractive-index distribution inside a laminated lens at different β.

Fig. 6
Fig. 6

Dependence of the focal length on α at different values of β (Dt = 1.7 mm).

Fig. 7
Fig. 7

Variation of the focal length with ray height for schematic, laminated, and parabolic cornea models.

Fig. 8
Fig. 8

Variation of the focal length with ray height at different degrees of corneal curvatures.

Fig. 9
Fig. 9

Comparison of the caustic surfaces for schematic, laminated, and parabolic cornea models. The z coordinate was measured from the anterior pole of the cornea.

Fig. 10
Fig. 10

Caustic surfaces for a parabolic cornea model at different angles of incidence. The z coordinate was measured from the anterior pole of the cornea.

Fig. 11
Fig. 11

Emerged wave fronts of schematic, laminated, and parabolic cornea models. The z coordinate was measured from the anterior pole of the cornea.

Tables (1)

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Table 1 Dimensions and Refractive Indices of the Unaccommodated Eye According to the Gullstrand Modela

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

P ( j ) = a j 3 + b j 2 + c j + d .
n ( y , z ) = i = 0 ( j = 0 n j 0 z 2 j ) y 2 i .
n ( y ) = n 0 ( 1 ± A 2 2 y 2 ) .
D t = σ N ,
R a ( I ) = R o a ( 1 - { 1 - exp [ α Z ( I ) ] } [ 1 - exp ( α D t ) ] ) ,
Z ( I ) = σ I ,             0 I 300 , hence 0 Z ( I ) ω .
R P ( I ) = R o p [ 1 - ( 1 - exp { α [ Z ( I ) - D t ] } ) { 1 - exp [ α ( ω - D t ) ] } ] ,
n ( I ) = n o + ( n c - n o ) { 1 - exp [ - β ( I - 1 ) ] 1 - exp [ - β ( N - 1 ) ] } ,
Y 2 = ( δ R ) Z ,

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