Abstract

A dispersive model of a gradient refractive index (GRIN) lens is introduced based on the idea of iso-dispersive contours. These contours have constant Abbe number and their shape is related to iso-indicial contours of the monochromatic geometry-invariant GRIN lens (GIGL) model. The chromatic GIGL model predicts the dispersion throughout the GRIN structure by using the dispersion curves of the surface and the center of the lens. The analytical approach for paraxial ray tracing and the monochromatic aberration calculations used in the GIGL model is employed here to derive closed-form expressions for the axial and lateral color coefficients of the lens. Expressions for equivalent refractive index and the equivalent Abbe number of the homogeneous equivalent lens are also presented and new aspects of the chromatic aberration change due to aging are discussed. The key derivations and explanations of the GRIN lens optical properties are accompanied with numerical examples for the human and animal eye GRIN lenses.

© 2012 OSA

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References

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  1. H. L. Liou and N. A. Brennan, “Anatomically accurate, finite model eye for optical modeling,” J. Opt. Soc. Am. A14, 1684–1695 (1997).
    [CrossRef]
  2. J. A. Díaz, C. Pizarro, and J. Arasa, “Single dispersive gradient-index profile for the aging human lens,” J. Opt. Soc. Am. A25, 250–261 (2008).
    [CrossRef]
  3. M. Bahrami and A. V. Goncharov, “Geometry-invariant GRIN lens: analytical ray tracing” J. Biomed. Opt.17, 055001 (2012).
    [CrossRef] [PubMed]
  4. B. K. Pierscionek, “Presbyopia - effect of refractive index,” Clin. Exp. Optom.73, 23–30 (1990).
    [CrossRef]
  5. G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modeling the power of the aging human eye,” J. Opt. Soc. Am. A9, 2111–2117 (1992).
    [CrossRef] [PubMed]
  6. G. Smith and B. K. Pierscionek, “The optical structure of the lens and its contribution to the refractive status of the eye,” Ophthalmic Physiol. Opt.18, 21–29 (1998).
    [CrossRef] [PubMed]
  7. C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, “Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI),” Vision Res.45, 2352–2366 (2005).
    [CrossRef] [PubMed]
  8. R. Navarro, F. Palos, and L. González, “Adaptive model of the gradient index of the human lens. I. formulation and model of aging ex vivo lenses,” J. Opt. Soc. Am. A24, 2175–2185 (2007).
    [CrossRef]
  9. S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Vis. Sci.49, 2531–2540 (2008).
    [CrossRef] [PubMed]
  10. D. A. Palmer and J. Sivak, “Crystalline lens dispersion,” J. Opt. Soc. Am. A71, 780–782 (1981).
    [CrossRef]
  11. J. G. Sivak and T. Mandelman, “Chromatic dispersion of the ocular media,” Vision Res.22, 997–1003 (1982).
    [CrossRef] [PubMed]
  12. D. A. Atchison and G. Smith, “Chromatic dispersions of the ocular media of human eyes,” J. Opt. Soc. Am. A22, 29–37 (2005).
    [CrossRef]
  13. The geometry-Invariant lens computational code. This is a computable document format (CDF) for the equations presented in Ref. [3]. Our source CDF code can be accessed via Mathematica, the computational software developed by Wolfram Research (Oct. 2011), http://optics.nuigalway.ie/people/mehdiB/CDF.html .
  14. M. J. Kidger, Fundamental Optical Design (SPIE Press, 2002).
  15. B. Gilmartin and R. E. Hogan, “The magnitude of longitudinal chromatic aberration of the human eye between 458 and 633 nm,” Vision Res.25, 1747–1755 (1985).
    [CrossRef] [PubMed]
  16. R. I. Barraquer, R. Michael, R. Abreu, J. Lamarca, and F. Tresserra, “Human lens capsule thickness as a function of age and location along the sagittal lens perimeter,” Invest. Ophthalmol. Vis. Sci.47, 2053–2060 (2006).
    [CrossRef] [PubMed]
  17. Y. Le Grand, Form and Space Vision, rev. ed., translated by M. Millodot and G. Heath (Indiana University Press, 1967).
  18. R. Navarro, J. Santamaría, and J. Bescós, “Accommodation dependent model of the human eye with aspherics,” J. Opt. Soc. Am. A2, 1273–1281 (1985).
    [CrossRef] [PubMed]
  19. S. R. Uhlhorn, D. Borja, F. Manns, and J. M. Parel, “Refractive index measurement of the isolated crystalline lens using optical coherence tomography,” Vision Res.48, 2732–2738 (2008).
    [CrossRef] [PubMed]
  20. C. Ware, “Human axial chromatic aberration found not to decline with age,” A. Graefes Arch. Klin. Exp. Ophthalmol.218, 39–41 (1982).
    [CrossRef]
  21. P. A. Howarth, X. X. Zhang, D. L. Still, and L. N. Thibos, “Does the chromatic aberration of the eye vary with age?,” J. Opt. Soc. Am. A5, 2087–2096 (1988).
    [CrossRef] [PubMed]
  22. M. Millodot, “The influence of age on the chromatic aberration of the eye,” A. Graefes Arch. Klin. Exp. Ophthalmol.198, 235–243 (1976).
    [CrossRef]
  23. J. A. Mordi and W. K. Adrian, “Influence of age on chromatic aberration of the human eye,” A. Graefes Arch. Klin. Exp. Ophthalmol.198, 235–243 (1976).
    [CrossRef]
  24. N. Brown, “The change in lens curvature with age,” Exp. Eye Res.19, 175–183 (1974).
    [CrossRef] [PubMed]
  25. A. V. Goncharov and C. Dainty, “Wide-field schematic eye models with gradient-index lens,” J. Opt. Soc. Am. A24, 2157–2174 (2007).
    [CrossRef]
  26. R. H. H. Kröger and M. C. W. Campbell, “Dispersion and longitudinal chromatic aberration of the crystalline lens of the African cichlid fish Haplochromis burtoni,” J. Opt. Soc. Am. A13, 2341–2347 (1996).
    [CrossRef]
  27. W. S. Jagger and P. J. Standsl, “A wide-angle gradient index optical model of the crystalline lens and eye of the octopus,” Vision Res.39, 2841–2852 (1999).
    [CrossRef] [PubMed]

2012 (1)

M. Bahrami and A. V. Goncharov, “Geometry-invariant GRIN lens: analytical ray tracing” J. Biomed. Opt.17, 055001 (2012).
[CrossRef] [PubMed]

2008 (3)

J. A. Díaz, C. Pizarro, and J. Arasa, “Single dispersive gradient-index profile for the aging human lens,” J. Opt. Soc. Am. A25, 250–261 (2008).
[CrossRef]

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Vis. Sci.49, 2531–2540 (2008).
[CrossRef] [PubMed]

S. R. Uhlhorn, D. Borja, F. Manns, and J. M. Parel, “Refractive index measurement of the isolated crystalline lens using optical coherence tomography,” Vision Res.48, 2732–2738 (2008).
[CrossRef] [PubMed]

2007 (2)

2006 (1)

R. I. Barraquer, R. Michael, R. Abreu, J. Lamarca, and F. Tresserra, “Human lens capsule thickness as a function of age and location along the sagittal lens perimeter,” Invest. Ophthalmol. Vis. Sci.47, 2053–2060 (2006).
[CrossRef] [PubMed]

2005 (2)

D. A. Atchison and G. Smith, “Chromatic dispersions of the ocular media of human eyes,” J. Opt. Soc. Am. A22, 29–37 (2005).
[CrossRef]

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, “Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI),” Vision Res.45, 2352–2366 (2005).
[CrossRef] [PubMed]

1999 (1)

W. S. Jagger and P. J. Standsl, “A wide-angle gradient index optical model of the crystalline lens and eye of the octopus,” Vision Res.39, 2841–2852 (1999).
[CrossRef] [PubMed]

1998 (1)

G. Smith and B. K. Pierscionek, “The optical structure of the lens and its contribution to the refractive status of the eye,” Ophthalmic Physiol. Opt.18, 21–29 (1998).
[CrossRef] [PubMed]

1997 (1)

1996 (1)

1992 (1)

1990 (1)

B. K. Pierscionek, “Presbyopia - effect of refractive index,” Clin. Exp. Optom.73, 23–30 (1990).
[CrossRef]

1988 (1)

1985 (2)

B. Gilmartin and R. E. Hogan, “The magnitude of longitudinal chromatic aberration of the human eye between 458 and 633 nm,” Vision Res.25, 1747–1755 (1985).
[CrossRef] [PubMed]

R. Navarro, J. Santamaría, and J. Bescós, “Accommodation dependent model of the human eye with aspherics,” J. Opt. Soc. Am. A2, 1273–1281 (1985).
[CrossRef] [PubMed]

1982 (2)

C. Ware, “Human axial chromatic aberration found not to decline with age,” A. Graefes Arch. Klin. Exp. Ophthalmol.218, 39–41 (1982).
[CrossRef]

J. G. Sivak and T. Mandelman, “Chromatic dispersion of the ocular media,” Vision Res.22, 997–1003 (1982).
[CrossRef] [PubMed]

1981 (1)

D. A. Palmer and J. Sivak, “Crystalline lens dispersion,” J. Opt. Soc. Am. A71, 780–782 (1981).
[CrossRef]

1976 (2)

M. Millodot, “The influence of age on the chromatic aberration of the eye,” A. Graefes Arch. Klin. Exp. Ophthalmol.198, 235–243 (1976).
[CrossRef]

J. A. Mordi and W. K. Adrian, “Influence of age on chromatic aberration of the human eye,” A. Graefes Arch. Klin. Exp. Ophthalmol.198, 235–243 (1976).
[CrossRef]

1974 (1)

N. Brown, “The change in lens curvature with age,” Exp. Eye Res.19, 175–183 (1974).
[CrossRef] [PubMed]

Abreu, R.

R. I. Barraquer, R. Michael, R. Abreu, J. Lamarca, and F. Tresserra, “Human lens capsule thickness as a function of age and location along the sagittal lens perimeter,” Invest. Ophthalmol. Vis. Sci.47, 2053–2060 (2006).
[CrossRef] [PubMed]

Adrian, W. K.

J. A. Mordi and W. K. Adrian, “Influence of age on chromatic aberration of the human eye,” A. Graefes Arch. Klin. Exp. Ophthalmol.198, 235–243 (1976).
[CrossRef]

Arasa, J.

Atchison, D. A.

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Vis. Sci.49, 2531–2540 (2008).
[CrossRef] [PubMed]

D. A. Atchison and G. Smith, “Chromatic dispersions of the ocular media of human eyes,” J. Opt. Soc. Am. A22, 29–37 (2005).
[CrossRef]

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, “Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI),” Vision Res.45, 2352–2366 (2005).
[CrossRef] [PubMed]

G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modeling the power of the aging human eye,” J. Opt. Soc. Am. A9, 2111–2117 (1992).
[CrossRef] [PubMed]

Bahrami, M.

M. Bahrami and A. V. Goncharov, “Geometry-invariant GRIN lens: analytical ray tracing” J. Biomed. Opt.17, 055001 (2012).
[CrossRef] [PubMed]

Barraquer, R. I.

R. I. Barraquer, R. Michael, R. Abreu, J. Lamarca, and F. Tresserra, “Human lens capsule thickness as a function of age and location along the sagittal lens perimeter,” Invest. Ophthalmol. Vis. Sci.47, 2053–2060 (2006).
[CrossRef] [PubMed]

Bescós, J.

Borja, D.

S. R. Uhlhorn, D. Borja, F. Manns, and J. M. Parel, “Refractive index measurement of the isolated crystalline lens using optical coherence tomography,” Vision Res.48, 2732–2738 (2008).
[CrossRef] [PubMed]

Brennan, N. A.

Brown, N.

N. Brown, “The change in lens curvature with age,” Exp. Eye Res.19, 175–183 (1974).
[CrossRef] [PubMed]

Campbell, M. C. W.

Dainty, C.

Díaz, J. A.

Gilmartin, B.

B. Gilmartin and R. E. Hogan, “The magnitude of longitudinal chromatic aberration of the human eye between 458 and 633 nm,” Vision Res.25, 1747–1755 (1985).
[CrossRef] [PubMed]

Goncharov, A. V.

M. Bahrami and A. V. Goncharov, “Geometry-invariant GRIN lens: analytical ray tracing” J. Biomed. Opt.17, 055001 (2012).
[CrossRef] [PubMed]

A. V. Goncharov and C. Dainty, “Wide-field schematic eye models with gradient-index lens,” J. Opt. Soc. Am. A24, 2157–2174 (2007).
[CrossRef]

González, L.

Hogan, R. E.

B. Gilmartin and R. E. Hogan, “The magnitude of longitudinal chromatic aberration of the human eye between 458 and 633 nm,” Vision Res.25, 1747–1755 (1985).
[CrossRef] [PubMed]

Howarth, P. A.

Jagger, W. S.

W. S. Jagger and P. J. Standsl, “A wide-angle gradient index optical model of the crystalline lens and eye of the octopus,” Vision Res.39, 2841–2852 (1999).
[CrossRef] [PubMed]

Jones, C. E.

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, “Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI),” Vision Res.45, 2352–2366 (2005).
[CrossRef] [PubMed]

Kasthurirangan, S.

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Vis. Sci.49, 2531–2540 (2008).
[CrossRef] [PubMed]

Kidger, M. J.

M. J. Kidger, Fundamental Optical Design (SPIE Press, 2002).

Kröger, R. H. H.

Lamarca, J.

R. I. Barraquer, R. Michael, R. Abreu, J. Lamarca, and F. Tresserra, “Human lens capsule thickness as a function of age and location along the sagittal lens perimeter,” Invest. Ophthalmol. Vis. Sci.47, 2053–2060 (2006).
[CrossRef] [PubMed]

Le Grand, Y.

Y. Le Grand, Form and Space Vision, rev. ed., translated by M. Millodot and G. Heath (Indiana University Press, 1967).

Liou, H. L.

Mandelman, T.

J. G. Sivak and T. Mandelman, “Chromatic dispersion of the ocular media,” Vision Res.22, 997–1003 (1982).
[CrossRef] [PubMed]

Manns, F.

S. R. Uhlhorn, D. Borja, F. Manns, and J. M. Parel, “Refractive index measurement of the isolated crystalline lens using optical coherence tomography,” Vision Res.48, 2732–2738 (2008).
[CrossRef] [PubMed]

Markwell, E. L.

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Vis. Sci.49, 2531–2540 (2008).
[CrossRef] [PubMed]

Meder, R.

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, “Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI),” Vision Res.45, 2352–2366 (2005).
[CrossRef] [PubMed]

Michael, R.

R. I. Barraquer, R. Michael, R. Abreu, J. Lamarca, and F. Tresserra, “Human lens capsule thickness as a function of age and location along the sagittal lens perimeter,” Invest. Ophthalmol. Vis. Sci.47, 2053–2060 (2006).
[CrossRef] [PubMed]

Millodot, M.

M. Millodot, “The influence of age on the chromatic aberration of the eye,” A. Graefes Arch. Klin. Exp. Ophthalmol.198, 235–243 (1976).
[CrossRef]

Mordi, J. A.

J. A. Mordi and W. K. Adrian, “Influence of age on chromatic aberration of the human eye,” A. Graefes Arch. Klin. Exp. Ophthalmol.198, 235–243 (1976).
[CrossRef]

Navarro, R.

Palmer, D. A.

D. A. Palmer and J. Sivak, “Crystalline lens dispersion,” J. Opt. Soc. Am. A71, 780–782 (1981).
[CrossRef]

Palos, F.

Parel, J. M.

S. R. Uhlhorn, D. Borja, F. Manns, and J. M. Parel, “Refractive index measurement of the isolated crystalline lens using optical coherence tomography,” Vision Res.48, 2732–2738 (2008).
[CrossRef] [PubMed]

Pierscionek, B. K.

G. Smith and B. K. Pierscionek, “The optical structure of the lens and its contribution to the refractive status of the eye,” Ophthalmic Physiol. Opt.18, 21–29 (1998).
[CrossRef] [PubMed]

G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modeling the power of the aging human eye,” J. Opt. Soc. Am. A9, 2111–2117 (1992).
[CrossRef] [PubMed]

B. K. Pierscionek, “Presbyopia - effect of refractive index,” Clin. Exp. Optom.73, 23–30 (1990).
[CrossRef]

Pizarro, C.

Pope, J. M.

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Vis. Sci.49, 2531–2540 (2008).
[CrossRef] [PubMed]

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, “Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI),” Vision Res.45, 2352–2366 (2005).
[CrossRef] [PubMed]

Santamaría, J.

Sivak, J.

D. A. Palmer and J. Sivak, “Crystalline lens dispersion,” J. Opt. Soc. Am. A71, 780–782 (1981).
[CrossRef]

Sivak, J. G.

J. G. Sivak and T. Mandelman, “Chromatic dispersion of the ocular media,” Vision Res.22, 997–1003 (1982).
[CrossRef] [PubMed]

Smith, G.

Standsl, P. J.

W. S. Jagger and P. J. Standsl, “A wide-angle gradient index optical model of the crystalline lens and eye of the octopus,” Vision Res.39, 2841–2852 (1999).
[CrossRef] [PubMed]

Still, D. L.

Thibos, L. N.

Tresserra, F.

R. I. Barraquer, R. Michael, R. Abreu, J. Lamarca, and F. Tresserra, “Human lens capsule thickness as a function of age and location along the sagittal lens perimeter,” Invest. Ophthalmol. Vis. Sci.47, 2053–2060 (2006).
[CrossRef] [PubMed]

Uhlhorn, S. R.

S. R. Uhlhorn, D. Borja, F. Manns, and J. M. Parel, “Refractive index measurement of the isolated crystalline lens using optical coherence tomography,” Vision Res.48, 2732–2738 (2008).
[CrossRef] [PubMed]

Ware, C.

C. Ware, “Human axial chromatic aberration found not to decline with age,” A. Graefes Arch. Klin. Exp. Ophthalmol.218, 39–41 (1982).
[CrossRef]

Zhang, X. X.

A. Graefes Arch. Klin. Exp. Ophthalmol. (3)

M. Millodot, “The influence of age on the chromatic aberration of the eye,” A. Graefes Arch. Klin. Exp. Ophthalmol.198, 235–243 (1976).
[CrossRef]

J. A. Mordi and W. K. Adrian, “Influence of age on chromatic aberration of the human eye,” A. Graefes Arch. Klin. Exp. Ophthalmol.198, 235–243 (1976).
[CrossRef]

C. Ware, “Human axial chromatic aberration found not to decline with age,” A. Graefes Arch. Klin. Exp. Ophthalmol.218, 39–41 (1982).
[CrossRef]

Clin. Exp. Optom. (1)

B. K. Pierscionek, “Presbyopia - effect of refractive index,” Clin. Exp. Optom.73, 23–30 (1990).
[CrossRef]

Exp. Eye Res. (1)

N. Brown, “The change in lens curvature with age,” Exp. Eye Res.19, 175–183 (1974).
[CrossRef] [PubMed]

Invest. Ophthalmol. Vis. Sci. (2)

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Vis. Sci.49, 2531–2540 (2008).
[CrossRef] [PubMed]

R. I. Barraquer, R. Michael, R. Abreu, J. Lamarca, and F. Tresserra, “Human lens capsule thickness as a function of age and location along the sagittal lens perimeter,” Invest. Ophthalmol. Vis. Sci.47, 2053–2060 (2006).
[CrossRef] [PubMed]

J. Biomed. Opt. (1)

M. Bahrami and A. V. Goncharov, “Geometry-invariant GRIN lens: analytical ray tracing” J. Biomed. Opt.17, 055001 (2012).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (10)

R. Navarro, F. Palos, and L. González, “Adaptive model of the gradient index of the human lens. I. formulation and model of aging ex vivo lenses,” J. Opt. Soc. Am. A24, 2175–2185 (2007).
[CrossRef]

H. L. Liou and N. A. Brennan, “Anatomically accurate, finite model eye for optical modeling,” J. Opt. Soc. Am. A14, 1684–1695 (1997).
[CrossRef]

J. A. Díaz, C. Pizarro, and J. Arasa, “Single dispersive gradient-index profile for the aging human lens,” J. Opt. Soc. Am. A25, 250–261 (2008).
[CrossRef]

G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modeling the power of the aging human eye,” J. Opt. Soc. Am. A9, 2111–2117 (1992).
[CrossRef] [PubMed]

R. Navarro, J. Santamaría, and J. Bescós, “Accommodation dependent model of the human eye with aspherics,” J. Opt. Soc. Am. A2, 1273–1281 (1985).
[CrossRef] [PubMed]

D. A. Palmer and J. Sivak, “Crystalline lens dispersion,” J. Opt. Soc. Am. A71, 780–782 (1981).
[CrossRef]

D. A. Atchison and G. Smith, “Chromatic dispersions of the ocular media of human eyes,” J. Opt. Soc. Am. A22, 29–37 (2005).
[CrossRef]

A. V. Goncharov and C. Dainty, “Wide-field schematic eye models with gradient-index lens,” J. Opt. Soc. Am. A24, 2157–2174 (2007).
[CrossRef]

R. H. H. Kröger and M. C. W. Campbell, “Dispersion and longitudinal chromatic aberration of the crystalline lens of the African cichlid fish Haplochromis burtoni,” J. Opt. Soc. Am. A13, 2341–2347 (1996).
[CrossRef]

P. A. Howarth, X. X. Zhang, D. L. Still, and L. N. Thibos, “Does the chromatic aberration of the eye vary with age?,” J. Opt. Soc. Am. A5, 2087–2096 (1988).
[CrossRef] [PubMed]

Ophthalmic Physiol. Opt. (1)

G. Smith and B. K. Pierscionek, “The optical structure of the lens and its contribution to the refractive status of the eye,” Ophthalmic Physiol. Opt.18, 21–29 (1998).
[CrossRef] [PubMed]

Vision Res. (5)

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, “Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI),” Vision Res.45, 2352–2366 (2005).
[CrossRef] [PubMed]

J. G. Sivak and T. Mandelman, “Chromatic dispersion of the ocular media,” Vision Res.22, 997–1003 (1982).
[CrossRef] [PubMed]

S. R. Uhlhorn, D. Borja, F. Manns, and J. M. Parel, “Refractive index measurement of the isolated crystalline lens using optical coherence tomography,” Vision Res.48, 2732–2738 (2008).
[CrossRef] [PubMed]

B. Gilmartin and R. E. Hogan, “The magnitude of longitudinal chromatic aberration of the human eye between 458 and 633 nm,” Vision Res.25, 1747–1755 (1985).
[CrossRef] [PubMed]

W. S. Jagger and P. J. Standsl, “A wide-angle gradient index optical model of the crystalline lens and eye of the octopus,” Vision Res.39, 2841–2852 (1999).
[CrossRef] [PubMed]

Other (3)

Y. Le Grand, Form and Space Vision, rev. ed., translated by M. Millodot and G. Heath (Indiana University Press, 1967).

The geometry-Invariant lens computational code. This is a computable document format (CDF) for the equations presented in Ref. [3]. Our source CDF code can be accessed via Mathematica, the computational software developed by Wolfram Research (Oct. 2011), http://optics.nuigalway.ie/people/mehdiB/CDF.html .

M. J. Kidger, Fundamental Optical Design (SPIE Press, 2002).

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Figures (7)

Fig. 1
Fig. 1

The invariant-geometry GRIN lens and its interior iso-indicial contours [3].

Fig. 2
Fig. 2

The fit of the dispersion curves at the center and the surface of the lens to the dispersion data from Palmer and Sivak for a 70 year old eye [10].

Fig. 3
Fig. 3

The fit of the dispersion curves at the center and the surface of the lens to the dispersion data from Sivak and Mandelman [11].

Fig. 4
Fig. 4

The fit of the dispersion curves at the center and the surface of the lens to the dispersion data from Sivak and Mandelman [11], and our calculated dispersion curves across the lens employing Eq. (4) for (a) p = 2.0 and (b) p = 5.0.

Fig. 5
Fig. 5

The refractive index profiles across the lens for λF = 486.1 nm, λd = 587.6 nm, and λC = 656.3 nm using Eq. (4) for (a) p = 2.0 and (b) p = 5.0 fit to the dispersion data from Sivak and Mandelman [11].

Fig. 6
Fig. 6

The quantity (nd − 1)/(ndVd) as a function of ζ (the normalized distance from the lens center) using the dispersion data from Sivak and Mandelman [11] for p = 4.0.

Fig. 7
Fig. 7

The quantity (nd − 1)/(ndVd) as a function of ζ (the normalized distance from the lens center) using the dispersion data from Palmer and Sivak [10] for p = 5.5.

Tables (9)

Tables Icon

Table 1 A typical GRIN lens geometry and the dispersive characteristics of the surrounding media at the d line. (*using Table 5 in Ref. [12].)

Tables Icon

Table 2 The chromatic coefficients of a typical GRIN lens defined in Table 1 with the dispersion data from Sivak and Mandelman [11] described by Eqs. (9) and (10).

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Table 3 The chromatic coefficients of a typical GRIN lens defined in Table 1 with the dispersion data from Sivak and Mandelman [11] on the eye lens capsule described by Eq. (21).

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Table 4 The chromatic coefficients of a typical GRIN lens defined in Table 1 with the dispersion data from Palmer and Sivak [10] described by Eqs. (7) and (8).

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Table 5 The chromatic coefficients of a typical GRIN lens defined by Table 1 using the dispersion data from Atchison and Smith (Table 5 in Ref. [12]).

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Table 6 The calculated quantities neqv and Veqv using the dispersion data from Sivak and Mandelman [11], and Palmer and Sivak [10] for a typical p = 3.0.

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Table 7 A comparison between the exact and equivalent color coefficients calculated respectively for a typical GRIN lens and its equivalent refractive index lens.

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Table 8 The axial color coefficients from the GRIN structure for three age groups (20, 40, and 60 year old) using the dispersion data provided by Palmer and Sivak [10].

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Table 9 The chromatic coefficients of a typical octopus GRIN lens using the dispersion data from Jagger and Sands [27].

Equations (27)

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n ( ζ ) = n c + ( n s n c ) ( ζ 2 ) p ;
n ( λ ) = A + B λ 2 + C λ 4 + D λ 6 +
n ( λ ) = n m + n λ 2 ( 1 λ 2 1 λ m 2 ) + n λ 4 ( 1 λ 4 1 λ m 4 ) ,
n ( ζ , λ ) = n center ( λ ) + ( n surface ( λ ) n center ( λ ) ) ( ζ 2 ) p ( λ ) ,
n center ( λ ) = n c + n c λ 2 ( 1 λ 2 1 λ m 2 ) + n c λ 4 ( 1 λ 4 1 λ m 4 ) ,
n surface ( λ ) = n s + n s λ 2 ( 1 λ 2 1 λ m 2 ) + n s λ 4 ( 1 λ 4 1 λ m 4 ) .
n center ( λ ) = 1.39879 + 6241.59 ( 1 λ 2 1 555 2 ) 2.10368 × 10 8 ( 1 λ 4 1 555 4 ) ,
n surface ( λ ) = 1.37555 7994.17 ( 1 λ 2 1 555 2 ) + 1.58549 × 10 9 ( 1 λ 4 1 555 4 ) .
n center ( λ ) = 1.40395 + 7256.06 ( 1 λ 2 1 555 2 ) 1.54846 × 10 8 ( 1 λ 4 1 555 4 ) ,
n surface ( λ ) = 1.37763 + 9260.03 ( 1 λ 2 1 555 2 ) 3.12554 × 10 8 ( 1 λ 4 1 555 4 ) .
C L = n d y ( y r + u ) ( n d 1 n d V d n d 1 n d V d ) ,
V d = n d 1 n F n c ,
δ A X = 1 n d i u i 2 C L ,
C T = n d y ( y c r + u c ) ( n d 1 n d V d n d 1 n d V d ) ,
δ T L C = 1 n d i u i C T .
V d ( ζ ) = n ( ζ , λ d ) 1 n ( ζ , λ F ) n ( ζ , λ C ) ,
δ C L = n ( z T , λ d ) y ( z ) ( y ( z ) R z + y ( z ) T ) [ n z ( z T , λ d ) ( n ( z T , λ F ) n ( z T , λ C ) ) n ( z T , λ d ) ( n z ( z T , λ F ) n z ( z T , λ C ) ) ] / n 2 ( z T , λ d ) δ z ,
C L = n aqu y 0 ( y 0 R a + u a ) ( n aqu 1 n aqu V aqu n ( 1 , λ d ) 1 n ( 1 , λ d ) V d ( 1 ) ) + + T a T p d C L + n ( 1 , λ d ) y ( T p ) ( y ( T p ) R p + u ( T p ) ) ( n ( 1 , λ d ) 1 n ( 1 , λ d ) V d ( 1 ) n vit 1 n vit V vit ) ,
δ C T = n ( z T , λ d ) y ( z ) ( y c ( z ) R z + y c ( z ) T ) [ n z ( z T , λ d ) ( n ( z T , λ F ) n ( z T , λ C ) ) n ( z T , λ d ) ( n z ( z T , λ F ) n z ( z T , λ C ) ) ] / n 2 ( z T , λ d ) δ z ,
C T = n aqu y 0 u c a ( n aqu 1 n aqu V aqu n ( 1 , λ d ) 1 n ( 1 , λ d ) V d ( 1 ) ) + + T a T p d C T + n ( 1 , λ d ) y ( T p ) ( y c ( T p ) R p + u c ( T p ) ) ( n ( 1 , λ d ) 1 n ( 1 , λ d ) V d ( 1 ) n vit 1 n vit V vit ) ,
n capsule ( λ ) = 1.37108 + 2617.71 ( 1 λ 2 1 555 2 ) + 2.5783 × 10 8 ( 1 λ 4 1 555 4 ) .
F thin = n s n aqu R a + 2 p 2 p 1 ( n c n s ) ( 1 R a + 1 R p ) + n vit n s R p .
F eqv = n eqv n aqu R a + n vit n eqv R p ,
n eqv = 2 p n c n s 2 p 1 .
n eqv ( λ ) = 2 p ( λ ) n center ( λ ) n surface ( λ ) 2 p ( λ ) 1 .
V eqv ( λ ) = V d c V d s [ n s 1 2 p ( n c 1 ) ] V dc ( n s 1 ) 2 p V d s ( n c 1 ) ,
O P L = ( T a + T p ) 2 n c p + n s 2 p + 1 ,

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