Abstract

The refractive index distribution of the geometry-invariant gradient refractive index lens (GIGL) model is derived as a function of Cartesian coordinates. The adjustable external geometry of the GIGL model aims to mimic the shape of the human and animal crystalline lens. The refractive index distribution is based on an adjustable power-law profile, which provides additional flexibility of the model. An analytical method for layer-by-layer finite ray tracing through the GIGL model is developed and used to calculate aberrations of the GIGL model. The result of the finite ray tracing aberrations of the GIGL model are compared to those obtained with paraxial ray tracing. The derived analytical expression for the refractive index distribution can be employed in the reconstruction processes of the eye using the conventional ray tracing methods. The layer-by-layer finite ray tracing approach would be an asset in ray tracing through a modified GIGL model, where the refractive index distribution cannot be described analytically. Using the layer-by-layer finite ray-tracing method, the potential of the GIGL model in representing continuous as well as shell-like layered structures is illustrated and the results for both cases are presented and analysed.

© 2014 Optical Society of America

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    [Crossref] [PubMed]
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2014 (1)

2012 (4)

M. Bahrami and A. V. Goncharov, “Geometry-invariant GRIN lens: iso-dispersive contours,” Biomed. Opt. Express 3, 1684–1700 (2012).
[Crossref] [PubMed]

J. A. Díaz, J. Fernández-Dorado, and F. Sorroche, “Role of the human lens gradient-index profile in the compensation of third-order ocular aberrations,” J. Biomed. Opt. 17, 075003 (2012).
[Crossref] [PubMed]

B. K. Pierscionek and J. W. Regini, “The gradient index lens of the eye: an opto-biological synchrony,” Prog. Retin. Eye Res. 31, 332–349 (2012).
[Crossref] [PubMed]

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

2011 (1)

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, “Optical properties of in situ eye lenses measured with X-ray talbot interferometry: a novel measure of growth processes,” PLoS ONE 6, e25140 (2011).
[Crossref] [PubMed]

2010 (2)

R. Urs, A. Ho, F. Manns, and JM Parel, “Age-dependent Fourier model of the shape of the isolated ex vivo human crystalline lens,” Vision Res. 50, 1041–1047 (2010).
[Crossref] [PubMed]

A. de Castro, S. Ortiz, E. Gambra, D. Siedlecki, and S. Marcos, “Three-dimensional reconstruction of the crystalline lens gradient index distribution from OCT imaging,” Opt. Express 18, 21905–21917 (2010).
[Crossref] [PubMed]

2009 (2)

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Vis. Sci. 50, 281–289 (2009).
[Crossref]

G. Smith, D. Atchison, D. Iskander, J. Jones, and J. Pope, “Mathematical models for describing the shape of the in vitro unstretched human crystalline lens,” Vision Res. 49, 2442–2452 (2009).
[Crossref] [PubMed]

2008 (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]

J. A. Díaz, “ABCD matrix of the human lens gradient-index profile: applicability of the calculation methods,” Appl. Opt. 47, 195–205 (2008).
[Crossref] [PubMed]

2007 (2)

2006 (1)

2005 (1)

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]

2000 (1)

H. T. Kasprzak, “New approximation for the whole profile of the human crystalline lens,” Ophthalmic Physiol. Opt. 20, 31–43 (2000).
[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 (3)

A. Tardieu and M. Delaye, “Eye Lens proteins and transparency: from light transmission theory to solution X-ray structural analysis,” Annu. Rev. Biophys. Biomol. Struct. 17, 47–70 (1998).
[Crossref]

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. Beliakov and D. Y. C. Chan, “Analysis of inhomogeneous optical systems by the use of ray tracing. II. three-dimensional systems with symmetry,” Appl. Opt. 37, 5106–5111 (1998).
[Crossref]

1997 (2)

1994 (1)

1992 (1)

1989 (1)

B. K. Pierscionek and D. Y C. Chan, “Refractive index gradient of human lenses,” Optom. Vis. Sci. 66, 822–829 (1989).
[Crossref] [PubMed]

1981 (1)

P. P. Fagerholm, B. T. Philipson, and B. Lindström, “Normal human lens, the distribution of protein,” Exp. Eye Res. 33, 615–620 (1981).
[Crossref] [PubMed]

1968 (1)

S. Nakao, S. Fujimoto, R. Nagata, and K. Iwata, “Model of refractive-index distribution in the rabbit crystalline lens,” J. Opt. Soc. Am. A 58, 1125–1130 (1968).
[Crossref]

1964 (1)

Acosta, E.

Atchison, D.

G. Smith, D. Atchison, D. Iskander, J. Jones, and J. Pope, “Mathematical models for describing the shape of the in vitro unstretched human crystalline lens,” Vision Res. 49, 2442–2452 (2009).
[Crossref] [PubMed]

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]

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. A 9, 2111–2117 (1992).
[Crossref] [PubMed]

Bahrami, M.

C. J. Sheil, M. Bahrami, and A. V. Goncharov, “An analytical method for predicting the geometrical and optical properties of the human lens under accommodation,” Biomed. Opt. Express 5, 1649–1663 (2014).
[Crossref] [PubMed]

M. Bahrami and A. V. Goncharov, “Geometry-invariant GRIN lens: iso-dispersive contours,” Biomed. Opt. Express 3, 1684–1700 (2012).
[Crossref] [PubMed]

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

M. Bahrami and A. V. Goncharov, “The geometry-Invariant lens computational code,” This is a computable document format (CDF) for the equations presented in [20], http://optics.nuigalway.ie/people/mehdiB/CDF.html , (Oct.2011).

Beliakov, G.

Bille, J. F.

Chan, D. Y C.

B. K. Pierscionek and D. Y C. Chan, “Refractive index gradient of human lenses,” Optom. Vis. Sci. 66, 822–829 (1989).
[Crossref] [PubMed]

Chan, D. Y. C.

Creath, K.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, Volume 11, R. R. Shannon and J. C. Wyant, eds. (Academic Press, 1992), pp. 1–53.

Dainty, C.

de Castro, A.

Delaye, M.

A. Tardieu and M. Delaye, “Eye Lens proteins and transparency: from light transmission theory to solution X-ray structural analysis,” Annu. Rev. Biophys. Biomol. Struct. 17, 47–70 (1998).
[Crossref]

Díaz, J. A.

J. A. Díaz, J. Fernández-Dorado, and F. Sorroche, “Role of the human lens gradient-index profile in the compensation of third-order ocular aberrations,” J. Biomed. Opt. 17, 075003 (2012).
[Crossref] [PubMed]

J. A. Díaz, “ABCD matrix of the human lens gradient-index profile: applicability of the calculation methods,” Appl. Opt. 47, 195–205 (2008).
[Crossref] [PubMed]

Dubbelman, M.

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Vis. Sci. 50, 281–289 (2009).
[Crossref]

Fagerholm, P. P.

P. P. Fagerholm, B. T. Philipson, and B. Lindström, “Normal human lens, the distribution of protein,” Exp. Eye Res. 33, 615–620 (1981).
[Crossref] [PubMed]

Fernández-Dorado, J.

J. A. Díaz, J. Fernández-Dorado, and F. Sorroche, “Role of the human lens gradient-index profile in the compensation of third-order ocular aberrations,” J. Biomed. Opt. 17, 075003 (2012).
[Crossref] [PubMed]

Fujimoto, S.

S. Nakao, S. Fujimoto, R. Nagata, and K. Iwata, “Model of refractive-index distribution in the rabbit crystalline lens,” J. Opt. Soc. Am. A 58, 1125–1130 (1968).
[Crossref]

Gambra, E.

Garner, L.

Goelz, S.

Goncharov, A. V.

C. J. Sheil, M. Bahrami, and A. V. Goncharov, “An analytical method for predicting the geometrical and optical properties of the human lens under accommodation,” Biomed. Opt. Express 5, 1649–1663 (2014).
[Crossref] [PubMed]

M. Bahrami and A. V. Goncharov, “Geometry-invariant GRIN lens: iso-dispersive contours,” Biomed. Opt. Express 3, 1684–1700 (2012).
[Crossref] [PubMed]

M. Bahrami and A. V. Goncharov, “Geometry-invariant gradient refractive index 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. A 24, 2157–2174 (2007).
[Crossref]

M. Bahrami and A. V. Goncharov, “The geometry-Invariant lens computational code,” This is a computable document format (CDF) for the equations presented in [20], http://optics.nuigalway.ie/people/mehdiB/CDF.html , (Oct.2011).

González, L.

Grimm, B.

Heethaar, R. M.

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Vis. Sci. 50, 281–289 (2009).
[Crossref]

Hermans, E. A.

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Vis. Sci. 50, 281–289 (2009).
[Crossref]

Ho, A.

R. Urs, A. Ho, F. Manns, and JM Parel, “Age-dependent Fourier model of the shape of the isolated ex vivo human crystalline lens,” Vision Res. 50, 1041–1047 (2010).
[Crossref] [PubMed]

Hoshino, M.

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, “Optical properties of in situ eye lenses measured with X-ray talbot interferometry: a novel measure of growth processes,” PLoS ONE 6, e25140 (2011).
[Crossref] [PubMed]

Iskander, D.

G. Smith, D. Atchison, D. Iskander, J. Jones, and J. Pope, “Mathematical models for describing the shape of the in vitro unstretched human crystalline lens,” Vision Res. 49, 2442–2452 (2009).
[Crossref] [PubMed]

Iwata, K.

S. Nakao, S. Fujimoto, R. Nagata, and K. Iwata, “Model of refractive-index distribution in the rabbit crystalline lens,” J. Opt. Soc. Am. A 58, 1125–1130 (1968).
[Crossref]

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]

Jones, J.

G. Smith, D. Atchison, D. Iskander, J. Jones, and J. Pope, “Mathematical models for describing the shape of the in vitro unstretched human crystalline lens,” Vision Res. 49, 2442–2452 (2009).
[Crossref] [PubMed]

Kasprzak, H. T.

H. T. Kasprzak, “New approximation for the whole profile of the human crystalline lens,” Ophthalmic Physiol. Opt. 20, 31–43 (2000).
[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]

Kuijer, J. P.

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Vis. Sci. 50, 281–289 (2009).
[Crossref]

Liang, J.

Lindström, B.

P. P. Fagerholm, B. T. Philipson, and B. Lindström, “Normal human lens, the distribution of protein,” Exp. Eye Res. 33, 615–620 (1981).
[Crossref] [PubMed]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (Brown University, 1944).

Manns, F.

R. Urs, A. Ho, F. Manns, and JM Parel, “Age-dependent Fourier model of the shape of the isolated ex vivo human crystalline lens,” Vision Res. 50, 1041–1047 (2010).
[Crossref] [PubMed]

Marcos, S.

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]

Mohri, S.

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, “Optical properties of in situ eye lenses measured with X-ray talbot interferometry: a novel measure of growth processes,” PLoS ONE 6, e25140 (2011).
[Crossref] [PubMed]

Nagata, R.

S. Nakao, S. Fujimoto, R. Nagata, and K. Iwata, “Model of refractive-index distribution in the rabbit crystalline lens,” J. Opt. Soc. Am. A 58, 1125–1130 (1968).
[Crossref]

Nakao, S.

S. Nakao, S. Fujimoto, R. Nagata, and K. Iwata, “Model of refractive-index distribution in the rabbit crystalline lens,” J. Opt. Soc. Am. A 58, 1125–1130 (1968).
[Crossref]

Navarro, R.

Ortiz, S.

Palos, F.

Parel, JM

R. Urs, A. Ho, F. Manns, and JM Parel, “Age-dependent Fourier model of the shape of the isolated ex vivo human crystalline lens,” Vision Res. 50, 1041–1047 (2010).
[Crossref] [PubMed]

Philipson, B. T.

P. P. Fagerholm, B. T. Philipson, and B. Lindström, “Normal human lens, the distribution of protein,” Exp. Eye Res. 33, 615–620 (1981).
[Crossref] [PubMed]

Pierscionek, B.

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, “Optical properties of in situ eye lenses measured with X-ray talbot interferometry: a novel measure of growth processes,” PLoS ONE 6, e25140 (2011).
[Crossref] [PubMed]

Pierscionek, B. K.

B. K. Pierscionek and J. W. Regini, “The gradient index lens of the eye: an opto-biological synchrony,” Prog. Retin. Eye Res. 31, 332–349 (2012).
[Crossref] [PubMed]

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]

B. K. Pierscionek, “Refractive index contours in the human lens,” Exp. Eye Res. 64, 887–893 (1997).
[Crossref] [PubMed]

G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modeling the power of the aging human eye,” J. Opt. Soc. Am. A 9, 2111–2117 (1992).
[Crossref] [PubMed]

B. K. Pierscionek and D. Y C. Chan, “Refractive index gradient of human lenses,” Optom. Vis. Sci. 66, 822–829 (1989).
[Crossref] [PubMed]

Pope, J.

G. Smith, D. Atchison, D. Iskander, J. Jones, and J. Pope, “Mathematical models for describing the shape of the in vitro unstretched human crystalline lens,” Vision Res. 49, 2442–2452 (2009).
[Crossref] [PubMed]

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]

Pouwels, P. J.

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Vis. Sci. 50, 281–289 (2009).
[Crossref]

Regini, J.

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, “Optical properties of in situ eye lenses measured with X-ray talbot interferometry: a novel measure of growth processes,” PLoS ONE 6, e25140 (2011).
[Crossref] [PubMed]

Regini, J. W.

B. K. Pierscionek and J. W. Regini, “The gradient index lens of the eye: an opto-biological synchrony,” Prog. Retin. Eye Res. 31, 332–349 (2012).
[Crossref] [PubMed]

Ruben, P. L.

Sheil, C. J.

Siedlecki, D.

Smith, G.

G. Smith, D. Atchison, D. Iskander, J. Jones, and J. Pope, “Mathematical models for describing the shape of the in vitro unstretched human crystalline lens,” Vision Res. 49, 2442–2452 (2009).
[Crossref] [PubMed]

D. Vazquez, E. Acosta, G. Smith, and L. Garner, “Tomographic method for measurement of the gradient refractive index of the crystalline lens. II. The rotationally symmetrical lens,” J. Opt. Soc. Am. A 23, 2551–2565 (2006).
[Crossref]

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. A 9, 2111–2117 (1992).
[Crossref] [PubMed]

Sorroche, F.

J. A. Díaz, J. Fernández-Dorado, and F. Sorroche, “Role of the human lens gradient-index profile in the compensation of third-order ocular aberrations,” J. Biomed. Opt. 17, 075003 (2012).
[Crossref] [PubMed]

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]

Tardieu, A.

A. Tardieu and M. Delaye, “Eye Lens proteins and transparency: from light transmission theory to solution X-ray structural analysis,” Annu. Rev. Biophys. Biomol. Struct. 17, 47–70 (1998).
[Crossref]

Uesugi, K.

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, “Optical properties of in situ eye lenses measured with X-ray talbot interferometry: a novel measure of growth processes,” PLoS ONE 6, e25140 (2011).
[Crossref] [PubMed]

Urs, R.

R. Urs, A. Ho, F. Manns, and JM Parel, “Age-dependent Fourier model of the shape of the isolated ex vivo human crystalline lens,” Vision Res. 50, 1041–1047 (2010).
[Crossref] [PubMed]

van der Heijde, R. G.

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Vis. Sci. 50, 281–289 (2009).
[Crossref]

Vazquez, D.

Wyant, J. C.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, Volume 11, R. R. Shannon and J. C. Wyant, eds. (Academic Press, 1992), pp. 1–53.

Yagi, N.

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, “Optical properties of in situ eye lenses measured with X-ray talbot interferometry: a novel measure of growth processes,” PLoS ONE 6, e25140 (2011).
[Crossref] [PubMed]

Annu. Rev. Biophys. Biomol. Struct. (1)

A. Tardieu and M. Delaye, “Eye Lens proteins and transparency: from light transmission theory to solution X-ray structural analysis,” Annu. Rev. Biophys. Biomol. Struct. 17, 47–70 (1998).
[Crossref]

Appl. Opt. (3)

Biomed. Opt. Express (2)

Exp. Eye Res. (2)

P. P. Fagerholm, B. T. Philipson, and B. Lindström, “Normal human lens, the distribution of protein,” Exp. Eye Res. 33, 615–620 (1981).
[Crossref] [PubMed]

B. K. Pierscionek, “Refractive index contours in the human lens,” Exp. Eye Res. 64, 887–893 (1997).
[Crossref] [PubMed]

Invest. Ophthalmol. Vis. Sci. (2)

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Vis. Sci. 50, 281–289 (2009).
[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]

J. Biomed. Opt. (2)

J. A. Díaz, J. Fernández-Dorado, and F. Sorroche, “Role of the human lens gradient-index profile in the compensation of third-order ocular aberrations,” J. Biomed. Opt. 17, 075003 (2012).
[Crossref] [PubMed]

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Ophthalmic Physiol. Opt. (2)

H. T. Kasprzak, “New approximation for the whole profile of the human crystalline lens,” Ophthalmic Physiol. Opt. 20, 31–43 (2000).
[Crossref] [PubMed]

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]

Opt. Express (1)

Optom. Vis. Sci. (1)

B. K. Pierscionek and D. Y C. Chan, “Refractive index gradient of human lenses,” Optom. Vis. Sci. 66, 822–829 (1989).
[Crossref] [PubMed]

PLoS ONE (1)

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, “Optical properties of in situ eye lenses measured with X-ray talbot interferometry: a novel measure of growth processes,” PLoS ONE 6, e25140 (2011).
[Crossref] [PubMed]

Prog. Retin. Eye Res. (1)

B. K. Pierscionek and J. W. Regini, “The gradient index lens of the eye: an opto-biological synchrony,” Prog. Retin. Eye Res. 31, 332–349 (2012).
[Crossref] [PubMed]

Vision Res. (4)

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]

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. Atchison, D. Iskander, J. Jones, and J. Pope, “Mathematical models for describing the shape of the in vitro unstretched human crystalline lens,” Vision Res. 49, 2442–2452 (2009).
[Crossref] [PubMed]

R. Urs, A. Ho, F. Manns, and JM Parel, “Age-dependent Fourier model of the shape of the isolated ex vivo human crystalline lens,” Vision Res. 50, 1041–1047 (2010).
[Crossref] [PubMed]

Other (3)

M. Bahrami and A. V. Goncharov, “The geometry-Invariant lens computational code,” This is a computable document format (CDF) for the equations presented in [20], http://optics.nuigalway.ie/people/mehdiB/CDF.html , (Oct.2011).

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, Volume 11, R. R. Shannon and J. C. Wyant, eds. (Academic Press, 1992), pp. 1–53.

R. K. Luneburg, Mathematical Theory of Optics (Brown University, 1944).

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Figures (5)

Fig. 1
Fig. 1 The geometry-invariant GRIN lens model geometry and its iso-indicial contours described in Eqs. (1). The Blue lines indicate the anterior part of the lens and the red lines specify the posterior part. The dashed lines depict the equatorial interface of the lens. In a 3-D presentation the equatorial interface is in the shape of a cone.
Fig. 2
Fig. 2 A typical geometry-invariant GRIN lens with its iso-indicial contours (a) and the corresponding normalized refractive index profile along radial direction from the center of the lens (b).
Fig. 3
Fig. 3 Schematic refraction of a ray passing through iso-indicial layers of the GRIN lens.
Fig. 4
Fig. 4 Numerical ray tracing for an aligned lens (a), and a tilted (8 degrees) and decentered (2 mm) lens (b). The tilt and decenteration are extremely exaggerated to make the effect more visible in the demonstration.
Fig. 5
Fig. 5 The impact of the number of shells on effective focal length.

Tables (1)

Tables Icon

Table 1 The difference (in percentage) between the aberrations calculated from paraxial Seidel coefficients and Zernike coefficients for spherical aberration, ΔW131, coma, ΔW131, and astigmatism, ΔW222, see Eq. (29).

Equations (34)

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ρ 2 = { 2 r a ( t a + z ) ( 1 + k a ) ( t a + z ) 2 + b a ( t a + z ) 3 t a z < z c 2 r p ( t p z ) ( 1 + k p ) ( t p z ) 2 + b p ( t p z ) 3 z c z t p ,
n ( ζ ) = n c + ( n s n c ) ( ζ 2 ) p ;
ρ 2 = { 2 R a ζ ( T a ζ + z ) ( 1 + k a ) ( T a ζ + z ) 2 + ( B a / ζ ) ( T a ζ + z ) 3 T a ζ z < Z c ζ 2 R p ζ ( T p ζ z ) ( 1 + k p ) ( T p ζ z ) 2 + ( B p / ζ ) ( T p ζ z ) 3 Z c ζ z T p ζ .
ρ 2 2 R a ζ ( T a ζ + z ) + ( 1 + k a ) ( T a ζ + z ) 2 ( B a / ζ ) ( T a ζ + z ) 3 = 0 .
α a 0 z 3 + ( α a 1 z 2 + ρ 2 ) ζ + α a 2 z ζ 2 + α a 3 ζ 3 = 0
α a 0 = B a , α a 1 = 1 + k a 3 B a T a , α a 2 = 2 R a + 2 ( 1 + k a ) T a 3 B a T a 2 , and α a 3 = 2 R a T a + ( 1 + k a ) T a 2 B a T a 3 .
α p 0 z 3 + ( α p 1 z 2 + ρ 2 ) ζ + α p 2 z ζ 2 + α p 3 ζ 3 = 0 ,
α p 0 = B p , α p 1 = 1 + k p 3 B p T p , α p 2 = 2 R p 2 ( 1 + k p ) T p + 3 B p T p 2 , and α p 3 = 2 R p T p + ( 1 + k p ) T p 2 B p T p 3 .
α 0 + α 1 ζ + α 2 ζ 2 + α 3 ζ 3 = 0 ,
ζ = χ α 2 3 α 3 ,
β 0 + β 1 χ + χ 3 = 0 ,
β 0 = 2 α 2 3 9 α 3 α 2 α 1 + 27 α 3 2 α 0 27 α 3 3 , and β 1 = 3 α 3 α 1 α 2 2 3 α 3 2 .
4 cos 3 ( θ ) 3 cos ( θ ) 3 β 0 2 β 1 3 β 1 = 0 .
4 cos 3 ( θ ) 3 cos ( θ ) cos ( 3 θ ) = 0 ,
cos ( 3 θ ) = 3 β 0 2 β 1 3 β 1 ,
χ j = 2 β 1 3 cos ( 1 3 arccos ( 3 β 0 2 β 1 3 β 1 ) j 2 π 3 ) for j = 0 , 1 , 2 .
ζ = 4 z 2 β a 1 12 α a 3 ρ 2 9 α a 3 2 cos ( 1 2 arccos ( z 3 β a 0 9 α a 2 α a 3 z ρ 2 6 α a 3 ( 3 α a 3 ρ 2 z 2 β a 1 ) 3 α a 3 2 z 2 β a 1 3 α a 3 ρ 2 ) ) α a 2 3 α a 3 z
ζ = 4 z 2 β p 1 12 α p 3 ρ 2 9 α p 3 2 cos ( 1 3 arccos ( z 3 β p 0 9 α p 2 α p 3 z ρ 2 6 α p 3 ( 3 α p 3 ρ 2 z 2 β p 1 ) 3 α p 3 2 z 2 β p 1 3 α p 3 p 2 ) ) α p 2 3 α p 3 z
β a 1 = α a 2 2 3 α a 1 α a 3 , β a 0 = 2 α a 2 3 9 α a 1 α a 2 α a 3 + 27 α a 0 α a 3 2 , β p 1 = α p 2 2 3 α p 1 α p 3 , and
ζ ( x , y , z ) = x 2 + y 2 + z 2 R 0 x 2 + y 2 + z 2 R 2 .
n ( x , y , z ) = n c + ( n s n c ) ( ζ 2 ( x , y , z ) ) p .
{ x 1 = x 0 + L Δ , y 1 = y 0 + M Δ , z 1 = z 0 + N Δ .
x 1 2 + y 1 2 = 2 r a ( t a + z 1 ) ( 1 + k a ) ( t a + z 1 ) 2 + b a ( t a + z 1 ) 3 .
δ 0 + δ 1 Δ + δ 2 Δ 2 + δ 3 Δ 3 = 0 ,
δ 0 = x 0 2 + y 0 2 ( t a + z 0 ) { 2 r + ( t a + z 0 ) [ 1 k a + b a ( t a + z 0 ) ] } , δ 1 = 2 ( L x 0 + M y 0 ) + N { 2 r a ( t a + z 0 ) [ 2 2 k a + 3 b a ( t a + z 0 ) ] } , δ 2 = L 2 + M 2 + N 2 [ 1 + k a 3 b a ( t a + z 0 ) ] , and δ 3 = b a N 3 .
N = F x i + F y j + F z k ( F x ) 2 + ( F y ) 2 + ( F z ) 2 .
x 2 + y 2 2 r a ( t a + z ) + ( 1 + k a ) ( t a + z ) 2 b a ( t a + z ) 3 = 0 .
N = 2 x i + 2 y j + [ 2 r a + 2 ( 1 + k a ) ( t a + z ) 3 b a ( t a + z ) 2 ] k 4 ( x 2 + y 2 ) + { 2 r a + ( t a + z ) [ 2 k a + 3 b a ( t a + z ) ] } 2 .
n 0 ( r × N ) = n 1 ( r × N ) .
n ( x , y , z ) = n c + n s n c R 2 [ x 2 + y 2 + ( z R ) 2 ] ,
W ( ρ , θ ) = W 040 ρ 4 + W 131 H ¯ ρ 3 cos θ + W 222 H ¯ 2 ρ 2 cos 2 θ + ,
W 040 = S I 8 ; W 131 = S II 2 ; W 222 = S III 2 ,
W 040 = 6 5 Z 11 ; W 131 = 3 8 Z 7 ; W 222 = 2 6 Z 6 ,
Δ W 040 = S I 8 6 5 Z 11 S I 8 × 100 % ,

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