Abstract

The accuracy of the reconstruction of the Gradient Refractive Index (GRIN) of the crystalline lens from optimization methods was evaluated. Different input data, including direction cosines of deflected rays, ray impacts obtained from ray tracing and optical path differences from Optical Coherence Tomography (OCT) were studied. Three different GRIN models were analyzed. The experimental errors of the different experimental input data were estimated from comparisons of measurements and simulations using artificial lenses of known geometries. The experimental errors in the surfaces shape measurement and the influence of the number of rays were also considered. OCT-based input data produced the most accurate GRIN reconstructions. We found that optimization methods (combining global and local search algorithms) allow GRIN reconstructions with acceptable accuracies for moderate noise level.

© 2011 OSA

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2011 (1)

A. de Castro, D. Siedlecki, D. Borja, S. Uhlhorn, J. M. Parel, F. Manns, and S. Marcos, “Age-dependent variation of the gradient index profile in human crystalline lenses,” J. Mod. Opt. (2011).
[CrossRef]

2010 (4)

2009 (2)

2008 (4)

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(6), 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(27), 2732–2738 (2008).
[CrossRef] [PubMed]

J. A. Díaz, C. Pizarro, and J. Arasa, “Single dispersive gradient-index profile for the aging human lens,” J. Opt. Soc. Am. A 25(1), 250–261 (2008).
[CrossRef] [PubMed]

A. V. Goncharov, M. Nowakowski, M. T. Sheehan, and C. Dainty, “Reconstruction of the optical system of the human eye with reverse ray-tracing,” Opt. Express 16(3), 1692–1703 (2008).
[CrossRef] [PubMed]

2007 (3)

2006 (1)

2005 (5)

M. A. Rama, M. V. Pérez, C. Bao, M. T. Flores-Arias, and C. Gómez-Reino, “Gradient-index crystalline lens model: A new method for determining the paraxial properties by the axial and field rays,” Opt. Commun. 249(4-6), 595–609 (2005).
[CrossRef]

Y.-J. Liu, Z.-Q. Wang, L.-P. Song, and G.-G. Mu, “An anatomically accurate eye model with a shell-structure lens,” Optik Internat. J. Light Electron. Opt. 116(6), 241–246 (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(18), 2352–2366 (2005).
[CrossRef] [PubMed]

D. A. Atchison and G. Smith, “Chromatic dispersions of the ocular media of human eyes,” J. Opt. Soc. Am. A 22(1), 29–37 (2005).
[CrossRef] [PubMed]

E. Acosta, D. Vazquez, L. Garner, and G. Smith, “Tomographic method for measurement of the gradient refractive index of the crystalline lens. I. The spherical fish lens,” J. Opt. Soc. Am. A 22(3), 424–433 (2005).
[CrossRef] [PubMed]

2004 (2)

S. Barbero, A. Glasser, C. Clark, and S. Marcos, “Accuracy and possibilities for evaluating the lens gradient-index using a ray tracing tomography global optimization strategy,” Invest. Ophthalmol. Vis. Sci. 45, 1723 (2004).

S. Ortiz, S. Barbero, and S. Marcos, “Computer simulations of optical coherence tomography A-scans: what can we learn about refractive index distribution?” Invest. Ophthalmol. Vis. Sci. 45, 2781 (2004).

2002 (1)

B. A. Moffat, D. A. Atchison, and J. M. Pope, “Age-related changes in refractive index distribution and power of the human lens as measured by magnetic resonance micro-imaging in vitro,” Vision Res. 42(13), 1683–1693 (2002).
[CrossRef] [PubMed]

2001 (2)

E. Moreno-Barriuso, S. Marcos, R. Navarro, and S. A. Burns, “Comparing laser ray tracing, the spatially resolved refractometer, and the Hartmann-Shack sensor to measure the ocular wave aberration,” Optom. Vis. Sci. 78(3), 152–156 (2001).
[CrossRef] [PubMed]

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vision Res. 41(8), 973–979 (2001).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

A. Popiolek Masajada, “Numerical study of the influence of the shell structure of the crystalline lens on the refractive properties of the human eye,” Ophthalmic Physiol. Opt. 19(1), 41–49 (1999).
[CrossRef] [PubMed]

1998 (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9(1), 112–147 (1998).
[CrossRef]

1997 (4)

G. Beliakov and D. Y. Chan, “Analysis of inhomogeneous optical systems by the use of ray tracing. I. Planar systems,” Appl. Opt. 36(22), 5303–5309 (1997).
[CrossRef] [PubMed]

H. L. Liou and N. A. Brennan, “Anatomically accurate, finite model eye for optical modeling,” J. Opt. Soc. Am. A 14(8), 1684–1695 (1997).
[CrossRef] [PubMed]

L. F. Garner and G. Smith, “Changes in equivalent and gradient refractive index of the crystalline lens with accommodation,” Optom. Vis. Sci. 74(2), 114–119 (1997).
[CrossRef] [PubMed]

B. K. Pierscionek, “Refractive index contours in the human lens,” Exp. Eye Res. 64(6), 887–893 (1997).
[CrossRef] [PubMed]

1995 (2)

1992 (1)

1991 (1)

G. Smith, B. K. Pierscionek, and D. A. Atchison, “The optical modelling of the human lens,” Ophthalmic Physiol. Opt. 11(4), 359–369 (1991).
[CrossRef] [PubMed]

1990 (1)

1989 (1)

B. K. Pierscionek and D. Y. Chan, “Refractive index gradient of human lenses,” Optom. Vis. Sci. 66(12), 822–829 (1989).
[CrossRef] [PubMed]

1988 (1)

1984 (1)

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

1982 (1)

1981 (1)

M. C. Campbell and A. Hughes, “An analytic, gradient index schematic lens and eye for the rat which predicts aberrations for finite pupils,” Vision Res. 21(7), 1129–1148 (1981).
[CrossRef] [PubMed]

1980 (1)

1971 (1)

O. Pomerantzeff, H. Fish, J. Govignon, and C. L. Schepens, “Wide angle optical model of the human eye,” Ann. Ophthalmol. 3(8), 815–819 (1971).
[PubMed]

1965 (1)

J. A. Nelder and R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7, 308–313 (1965).

Acosta, E.

Adams, A. J.

D. O. Mutti, K. Zadnik, and A. J. Adams, “The equivalent refractive index of the crystalline lens in childhood,” Vision Res. 35(11), 1565–1573 (1995).
[CrossRef] [PubMed]

Al-Ahdali, I. H.

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(6), 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(18), 2352–2366 (2005).
[CrossRef] [PubMed]

D. A. Atchison and G. Smith, “Chromatic dispersions of the ocular media of human eyes,” J. Opt. Soc. Am. A 22(1), 29–37 (2005).
[CrossRef] [PubMed]

B. A. Moffat, D. A. Atchison, and J. M. Pope, “Age-related changes in refractive index distribution and power of the human lens as measured by magnetic resonance micro-imaging in vitro,” Vision Res. 42(13), 1683–1693 (2002).
[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(12), 2111–2117 (1992).
[CrossRef] [PubMed]

G. Smith, B. K. Pierscionek, and D. A. Atchison, “The optical modelling of the human lens,” Ophthalmic Physiol. Opt. 11(4), 359–369 (1991).
[CrossRef] [PubMed]

Augusteyn, R. C.

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vision Res. 41(8), 973–979 (2001).
[CrossRef] [PubMed]

Bao, C.

M. A. Rama, M. V. Pérez, C. Bao, M. T. Flores-Arias, and C. Gómez-Reino, “Gradient-index crystalline lens model: A new method for determining the paraxial properties by the axial and field rays,” Opt. Commun. 249(4-6), 595–609 (2005).
[CrossRef]

Barbero, S.

S. Ortiz, S. Barbero, and S. Marcos, “Computer simulations of optical coherence tomography A-scans: what can we learn about refractive index distribution?” Invest. Ophthalmol. Vis. Sci. 45, 2781 (2004).

S. Barbero, A. Glasser, C. Clark, and S. Marcos, “Accuracy and possibilities for evaluating the lens gradient-index using a ray tracing tomography global optimization strategy,” Invest. Ophthalmol. Vis. Sci. 45, 1723 (2004).

Beliakov, G.

Blaker, J. W.

Borja, D.

A. de Castro, D. Siedlecki, D. Borja, S. Uhlhorn, J. M. Parel, F. Manns, and S. Marcos, “Age-dependent variation of the gradient index profile in human crystalline lenses,” J. Mod. Opt. (2011).
[CrossRef]

F. Manns, A. Ho, D. Borja, and J. M. Parel, “Comparison of Uniform and Gradient Paraxial Models of the Crystalline Lens,” Invest. Ophthalmol. Vis. Sci. 51, 789 (2010).

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(27), 2732–2738 (2008).
[CrossRef] [PubMed]

Brennan, N. A.

Burns, S. A.

E. Moreno-Barriuso, S. Marcos, R. Navarro, and S. A. Burns, “Comparing laser ray tracing, the spatially resolved refractometer, and the Hartmann-Shack sensor to measure the ocular wave aberration,” Optom. Vis. Sci. 78(3), 152–156 (2001).
[CrossRef] [PubMed]

Campbell, C. E.

Campbell, M. C.

M. C. Campbell and A. Hughes, “An analytic, gradient index schematic lens and eye for the rat which predicts aberrations for finite pupils,” Vision Res. 21(7), 1129–1148 (1981).
[CrossRef] [PubMed]

Chan, D. Y.

Clark, C.

S. Barbero, A. Glasser, C. Clark, and S. Marcos, “Accuracy and possibilities for evaluating the lens gradient-index using a ray tracing tomography global optimization strategy,” Invest. Ophthalmol. Vis. Sci. 45, 1723 (2004).

Dainty, C.

de Castro, A.

A. de Castro, D. Siedlecki, D. Borja, S. Uhlhorn, J. M. Parel, F. Manns, and S. Marcos, “Age-dependent variation of the gradient index profile in human crystalline lenses,” J. Mod. Opt. (2011).
[CrossRef]

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(21), 21905–21917 (2010).
[CrossRef] [PubMed]

Díaz, J. A.

Dufault, P.

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

El-Messiery, M. A.

Ennis, J. P.

Fish, H.

O. Pomerantzeff, H. Fish, J. Govignon, and C. L. Schepens, “Wide angle optical model of the human eye,” Ann. Ophthalmol. 3(8), 815–819 (1971).
[PubMed]

Flores-Arias, M. T.

M. A. Rama, M. V. Pérez, C. Bao, M. T. Flores-Arias, and C. Gómez-Reino, “Gradient-index crystalline lens model: A new method for determining the paraxial properties by the axial and field rays,” Opt. Commun. 249(4-6), 595–609 (2005).
[CrossRef]

Forbes, G. W.

Gambra, E.

Garner, L.

Garner, L. F.

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vision Res. 41(8), 973–979 (2001).
[CrossRef] [PubMed]

L. F. Garner and G. Smith, “Changes in equivalent and gradient refractive index of the crystalline lens with accommodation,” Optom. Vis. Sci. 74(2), 114–119 (1997).
[CrossRef] [PubMed]

Ghatak, A. K.

Glasser, A.

S. Barbero, A. Glasser, C. Clark, and S. Marcos, “Accuracy and possibilities for evaluating the lens gradient-index using a ray tracing tomography global optimization strategy,” Invest. Ophthalmol. Vis. Sci. 45, 1723 (2004).

Gómez-Reino, C.

M. A. Rama, M. V. Pérez, C. Bao, M. T. Flores-Arias, and C. Gómez-Reino, “Gradient-index crystalline lens model: A new method for determining the paraxial properties by the axial and field rays,” Opt. Commun. 249(4-6), 595–609 (2005).
[CrossRef]

Goncharov, A. V.

González, L. M.

Gora, M.

Gorczynska, I.

Govignon, J.

O. Pomerantzeff, H. Fish, J. Govignon, and C. L. Schepens, “Wide angle optical model of the human eye,” Ann. Ophthalmol. 3(8), 815–819 (1971).
[PubMed]

Grulkowski, I.

Gupta, P. K.

Y. Verma, K. D. Rao, M. K. Suresh, H. S. Patel, and P. K. Gupta, “Measurement of gradient refractive index profile of crystalline lens of fisheye in vivo using optical coherence tomography,” Appl. Phys. B 87(4), 607–610 (2007).
[CrossRef]

Ho, A.

F. Manns, A. Ho, D. Borja, and J. M. Parel, “Comparison of Uniform and Gradient Paraxial Models of the Crystalline Lens,” Invest. Ophthalmol. Vis. Sci. 51, 789 (2010).

Hughes, A.

M. C. Campbell and A. Hughes, “An analytic, gradient index schematic lens and eye for the rat which predicts aberrations for finite pupils,” Vision Res. 21(7), 1129–1148 (1981).
[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(18), 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(6), 2531–2540 (2008).
[CrossRef] [PubMed]

Kowalczyk, A.

Kumar, D. V.

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9(1), 112–147 (1998).
[CrossRef]

Liou, H. L.

Liu, Y.-J.

Y.-J. Liu, Z.-Q. Wang, L.-P. Song, and G.-G. Mu, “An anatomically accurate eye model with a shell-structure lens,” Optik Internat. J. Light Electron. Opt. 116(6), 241–246 (2005).
[CrossRef]

Manns, F.

A. de Castro, D. Siedlecki, D. Borja, S. Uhlhorn, J. M. Parel, F. Manns, and S. Marcos, “Age-dependent variation of the gradient index profile in human crystalline lenses,” J. Mod. Opt. (2011).
[CrossRef]

F. Manns, A. Ho, D. Borja, and J. M. Parel, “Comparison of Uniform and Gradient Paraxial Models of the Crystalline Lens,” Invest. Ophthalmol. Vis. Sci. 51, 789 (2010).

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(27), 2732–2738 (2008).
[CrossRef] [PubMed]

Marcos, S.

A. de Castro, D. Siedlecki, D. Borja, S. Uhlhorn, J. M. Parel, F. Manns, and S. Marcos, “Age-dependent variation of the gradient index profile in human crystalline lenses,” J. Mod. Opt. (2011).
[CrossRef]

S. Ortiz, D. Siedlecki, I. Grulkowski, L. Remon, D. Pascual, M. Wojtkowski, and S. Marcos, “Optical distortion correction in optical coherence tomography for quantitative ocular anterior segment by three-dimensional imaging,” Opt. Express 18(3), 2782–2796 (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(21), 21905–21917 (2010).
[CrossRef] [PubMed]

S. Ortiz, D. Siedlecki, L. Remon, and S. Marcos, “Optical coherence tomography for quantitative surface topography,” Appl. Opt. 48(35), 6708–6715 (2009).
[CrossRef] [PubMed]

I. Grulkowski, M. Gora, M. Szkulmowski, I. Gorczynska, D. Szlag, S. Marcos, A. Kowalczyk, and M. Wojtkowski, “Anterior segment imaging with Spectral OCT system using a high-speed CMOS camera,” Opt. Express 17(6), 4842–4858 (2009).
[CrossRef] [PubMed]

S. Barbero, A. Glasser, C. Clark, and S. Marcos, “Accuracy and possibilities for evaluating the lens gradient-index using a ray tracing tomography global optimization strategy,” Invest. Ophthalmol. Vis. Sci. 45, 1723 (2004).

S. Ortiz, S. Barbero, and S. Marcos, “Computer simulations of optical coherence tomography A-scans: what can we learn about refractive index distribution?” Invest. Ophthalmol. Vis. Sci. 45, 2781 (2004).

E. Moreno-Barriuso, S. Marcos, R. Navarro, and S. A. Burns, “Comparing laser ray tracing, the spatially resolved refractometer, and the Hartmann-Shack sensor to measure the ocular wave aberration,” Optom. Vis. Sci. 78(3), 152–156 (2001).
[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(6), 2531–2540 (2008).
[CrossRef] [PubMed]

Mead, R.

J. A. Nelder and R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7, 308–313 (1965).

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(18), 2352–2366 (2005).
[CrossRef] [PubMed]

Moffat, B. A.

B. A. Moffat, D. A. Atchison, and J. M. Pope, “Age-related changes in refractive index distribution and power of the human lens as measured by magnetic resonance micro-imaging in vitro,” Vision Res. 42(13), 1683–1693 (2002).
[CrossRef] [PubMed]

Moreno-Barriuso, E.

E. Moreno-Barriuso, S. Marcos, R. Navarro, and S. A. Burns, “Comparing laser ray tracing, the spatially resolved refractometer, and the Hartmann-Shack sensor to measure the ocular wave aberration,” Optom. Vis. Sci. 78(3), 152–156 (2001).
[CrossRef] [PubMed]

E. Moreno-Barriuso and R. Navarro, “Laser Ray Tracing versus Hartmann-Shack sensor for measuring optical aberrations in the human eye,” J. Opt. Soc. Am. A 17(6), 974–985 (2000).
[CrossRef] [PubMed]

Mu, G.-G.

Y.-J. Liu, Z.-Q. Wang, L.-P. Song, and G.-G. Mu, “An anatomically accurate eye model with a shell-structure lens,” Optik Internat. J. Light Electron. Opt. 116(6), 241–246 (2005).
[CrossRef]

Mutti, D. O.

D. O. Mutti, K. Zadnik, and A. J. Adams, “The equivalent refractive index of the crystalline lens in childhood,” Vision Res. 35(11), 1565–1573 (1995).
[CrossRef] [PubMed]

Navarro, R.

Nelder, J. A.

J. A. Nelder and R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7, 308–313 (1965).

Nowakowski, M.

Ortiz, S.

Palos, F.

Pankratov, M.

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

Parel, J. M.

A. de Castro, D. Siedlecki, D. Borja, S. Uhlhorn, J. M. Parel, F. Manns, and S. Marcos, “Age-dependent variation of the gradient index profile in human crystalline lenses,” J. Mod. Opt. (2011).
[CrossRef]

F. Manns, A. Ho, D. Borja, and J. M. Parel, “Comparison of Uniform and Gradient Paraxial Models of the Crystalline Lens,” Invest. Ophthalmol. Vis. Sci. 51, 789 (2010).

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(27), 2732–2738 (2008).
[CrossRef] [PubMed]

Pascual, D.

Patel, H. S.

Y. Verma, K. D. Rao, M. K. Suresh, H. S. Patel, and P. K. Gupta, “Measurement of gradient refractive index profile of crystalline lens of fisheye in vivo using optical coherence tomography,” Appl. Phys. B 87(4), 607–610 (2007).
[CrossRef]

Pérez, M. V.

M. A. Rama, M. V. Pérez, C. Bao, M. T. Flores-Arias, and C. Gómez-Reino, “Gradient-index crystalline lens model: A new method for determining the paraxial properties by the axial and field rays,” Opt. Commun. 249(4-6), 595–609 (2005).
[CrossRef]

Pierscionek, B. K.

B. K. Pierscionek, “Refractive index contours in the human lens,” Exp. Eye Res. 64(6), 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(12), 2111–2117 (1992).
[CrossRef] [PubMed]

G. Smith, B. K. Pierscionek, and D. A. Atchison, “The optical modelling of the human lens,” Ophthalmic Physiol. Opt. 11(4), 359–369 (1991).
[CrossRef] [PubMed]

B. K. Pierscionek and D. Y. Chan, “Refractive index gradient of human lenses,” Optom. Vis. Sci. 66(12), 822–829 (1989).
[CrossRef] [PubMed]

D. Y. Chan, J. P. Ennis, B. K. Pierscionek, and G. Smith, “Determination and modeling of the 3-D gradient refractive indices in crystalline lenses,” Appl. Opt. 27(5), 926–931 (1988).
[CrossRef] [PubMed]

Pizarro, C.

Pomerantzeff, O.

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

O. Pomerantzeff, H. Fish, J. Govignon, and C. L. Schepens, “Wide angle optical model of the human eye,” Ann. Ophthalmol. 3(8), 815–819 (1971).
[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(6), 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(18), 2352–2366 (2005).
[CrossRef] [PubMed]

B. A. Moffat, D. A. Atchison, and J. M. Pope, “Age-related changes in refractive index distribution and power of the human lens as measured by magnetic resonance micro-imaging in vitro,” Vision Res. 42(13), 1683–1693 (2002).
[CrossRef] [PubMed]

Popiolek Masajada, A.

A. Popiolek Masajada, “Numerical study of the influence of the shell structure of the crystalline lens on the refractive properties of the human eye,” Ophthalmic Physiol. Opt. 19(1), 41–49 (1999).
[CrossRef] [PubMed]

Rama, M. A.

M. A. Rama, M. V. Pérez, C. Bao, M. T. Flores-Arias, and C. Gómez-Reino, “Gradient-index crystalline lens model: A new method for determining the paraxial properties by the axial and field rays,” Opt. Commun. 249(4-6), 595–609 (2005).
[CrossRef]

Rao, K. D.

Y. Verma, K. D. Rao, M. K. Suresh, H. S. Patel, and P. K. Gupta, “Measurement of gradient refractive index profile of crystalline lens of fisheye in vivo using optical coherence tomography,” Appl. Phys. B 87(4), 607–610 (2007).
[CrossRef]

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9(1), 112–147 (1998).
[CrossRef]

Remon, L.

Schepens, C. L.

O. Pomerantzeff, H. Fish, J. Govignon, and C. L. Schepens, “Wide angle optical model of the human eye,” Ann. Ophthalmol. 3(8), 815–819 (1971).
[PubMed]

Sharma, A.

Sheehan, M. T.

Siedlecki, D.

Smith, G.

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(10), 2551–2565 (2006).
[CrossRef] [PubMed]

E. Acosta, D. Vazquez, L. Garner, and G. Smith, “Tomographic method for measurement of the gradient refractive index of the crystalline lens. I. The spherical fish lens,” J. Opt. Soc. Am. A 22(3), 424–433 (2005).
[CrossRef] [PubMed]

D. A. Atchison and G. Smith, “Chromatic dispersions of the ocular media of human eyes,” J. Opt. Soc. Am. A 22(1), 29–37 (2005).
[CrossRef] [PubMed]

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vision Res. 41(8), 973–979 (2001).
[CrossRef] [PubMed]

L. F. Garner and G. Smith, “Changes in equivalent and gradient refractive index of the crystalline lens with accommodation,” Optom. Vis. Sci. 74(2), 114–119 (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(12), 2111–2117 (1992).
[CrossRef] [PubMed]

G. Smith, B. K. Pierscionek, and D. A. Atchison, “The optical modelling of the human lens,” Ophthalmic Physiol. Opt. 11(4), 359–369 (1991).
[CrossRef] [PubMed]

D. Y. Chan, J. P. Ennis, B. K. Pierscionek, and G. Smith, “Determination and modeling of the 3-D gradient refractive indices in crystalline lenses,” Appl. Opt. 27(5), 926–931 (1988).
[CrossRef] [PubMed]

Song, L.-P.

Y.-J. Liu, Z.-Q. Wang, L.-P. Song, and G.-G. Mu, “An anatomically accurate eye model with a shell-structure lens,” Optik Internat. J. Light Electron. Opt. 116(6), 241–246 (2005).
[CrossRef]

Stone, B. D.

Suresh, M. K.

Y. Verma, K. D. Rao, M. K. Suresh, H. S. Patel, and P. K. Gupta, “Measurement of gradient refractive index profile of crystalline lens of fisheye in vivo using optical coherence tomography,” Appl. Phys. B 87(4), 607–610 (2007).
[CrossRef]

Szkulmowski, M.

Szlag, D.

Uhlhorn, S.

A. de Castro, D. Siedlecki, D. Borja, S. Uhlhorn, J. M. Parel, F. Manns, and S. Marcos, “Age-dependent variation of the gradient index profile in human crystalline lenses,” J. Mod. Opt. (2011).
[CrossRef]

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(27), 2732–2738 (2008).
[CrossRef] [PubMed]

Vazquez, D.

Verma, Y.

Y. Verma, K. D. Rao, M. K. Suresh, H. S. Patel, and P. K. Gupta, “Measurement of gradient refractive index profile of crystalline lens of fisheye in vivo using optical coherence tomography,” Appl. Phys. B 87(4), 607–610 (2007).
[CrossRef]

Wang, G. J.

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

Wang, Z.-Q.

Y.-J. Liu, Z.-Q. Wang, L.-P. Song, and G.-G. Mu, “An anatomically accurate eye model with a shell-structure lens,” Optik Internat. J. Light Electron. Opt. 116(6), 241–246 (2005).
[CrossRef]

Wojtkowski, M.

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9(1), 112–147 (1998).
[CrossRef]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9(1), 112–147 (1998).
[CrossRef]

Yao, S.

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vision Res. 41(8), 973–979 (2001).
[CrossRef] [PubMed]

Zadnik, K.

D. O. Mutti, K. Zadnik, and A. J. Adams, “The equivalent refractive index of the crystalline lens in childhood,” Vision Res. 35(11), 1565–1573 (1995).
[CrossRef] [PubMed]

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

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

Ann. Ophthalmol. (1)

O. Pomerantzeff, H. Fish, J. Govignon, and C. L. Schepens, “Wide angle optical model of the human eye,” Ann. Ophthalmol. 3(8), 815–819 (1971).
[PubMed]

Appl. Opt. (5)

Appl. Phys. B (1)

Y. Verma, K. D. Rao, M. K. Suresh, H. S. Patel, and P. K. Gupta, “Measurement of gradient refractive index profile of crystalline lens of fisheye in vivo using optical coherence tomography,” Appl. Phys. B 87(4), 607–610 (2007).
[CrossRef]

Comput. J. (1)

J. A. Nelder and R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7, 308–313 (1965).

Exp. Eye Res. (1)

B. K. Pierscionek, “Refractive index contours in the human lens,” Exp. Eye Res. 64(6), 887–893 (1997).
[CrossRef] [PubMed]

Invest. Ophthalmol. Vis. Sci. (4)

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(6), 2531–2540 (2008).
[CrossRef] [PubMed]

S. Barbero, A. Glasser, C. Clark, and S. Marcos, “Accuracy and possibilities for evaluating the lens gradient-index using a ray tracing tomography global optimization strategy,” Invest. Ophthalmol. Vis. Sci. 45, 1723 (2004).

S. Ortiz, S. Barbero, and S. Marcos, “Computer simulations of optical coherence tomography A-scans: what can we learn about refractive index distribution?” Invest. Ophthalmol. Vis. Sci. 45, 2781 (2004).

F. Manns, A. Ho, D. Borja, and J. M. Parel, “Comparison of Uniform and Gradient Paraxial Models of the Crystalline Lens,” Invest. Ophthalmol. Vis. Sci. 51, 789 (2010).

J. Mod. Opt. (1)

A. de Castro, D. Siedlecki, D. Borja, S. Uhlhorn, J. M. Parel, F. Manns, and S. Marcos, “Age-dependent variation of the gradient index profile in human crystalline lenses,” J. Mod. Opt. (2011).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (11)

G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modeling the power of the aging human eye,” J. Opt. Soc. Am. A 9(12), 2111–2117 (1992).
[CrossRef] [PubMed]

D. A. Atchison and G. Smith, “Chromatic dispersions of the ocular media of human eyes,” J. Opt. Soc. Am. A 22(1), 29–37 (2005).
[CrossRef] [PubMed]

E. Acosta, D. Vazquez, L. Garner, and G. Smith, “Tomographic method for measurement of the gradient refractive index of the crystalline lens. I. The spherical fish lens,” J. Opt. Soc. Am. A 22(3), 424–433 (2005).
[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(10), 2551–2565 (2006).
[CrossRef] [PubMed]

A. V. Goncharov and C. Dainty, “Wide-field schematic eye models with gradient-index lens,” J. Opt. Soc. Am. A 24(8), 2157–2174 (2007).
[CrossRef] [PubMed]

R. Navarro, F. Palos, and L. M. González, “Adaptive model of the gradient index of the human lens. II. Optics of the accommodating aging lens,” J. Opt. Soc. Am. A 24(9), 2911–2920 (2007).
[CrossRef] [PubMed]

J. A. Díaz, C. Pizarro, and J. Arasa, “Single dispersive gradient-index profile for the aging human lens,” J. Opt. Soc. Am. A 25(1), 250–261 (2008).
[CrossRef] [PubMed]

E. Moreno-Barriuso and R. Navarro, “Laser Ray Tracing versus Hartmann-Shack sensor for measuring optical aberrations in the human eye,” J. Opt. Soc. Am. A 17(6), 974–985 (2000).
[CrossRef] [PubMed]

H. L. Liou and N. A. Brennan, “Anatomically accurate, finite model eye for optical modeling,” J. Opt. Soc. Am. A 14(8), 1684–1695 (1997).
[CrossRef] [PubMed]

B. D. Stone and G. W. Forbes, “Optimal interpolants for Runge-Kutta ray tracing in inhomogeneous media,” J. Opt. Soc. Am. A 7(2), 248–254 (1990).
[CrossRef]

C. E. Campbell, “Nested shell optical model of the lens of the human eye,” J. Opt. Soc. Am. A 27(11), 2432–2441 (2010).
[CrossRef] [PubMed]

Ophthalmic Physiol. Opt. (2)

G. Smith, B. K. Pierscionek, and D. A. Atchison, “The optical modelling of the human lens,” Ophthalmic Physiol. Opt. 11(4), 359–369 (1991).
[CrossRef] [PubMed]

A. Popiolek Masajada, “Numerical study of the influence of the shell structure of the crystalline lens on the refractive properties of the human eye,” Ophthalmic Physiol. Opt. 19(1), 41–49 (1999).
[CrossRef] [PubMed]

Opt. Commun. (1)

M. A. Rama, M. V. Pérez, C. Bao, M. T. Flores-Arias, and C. Gómez-Reino, “Gradient-index crystalline lens model: A new method for determining the paraxial properties by the axial and field rays,” Opt. Commun. 249(4-6), 595–609 (2005).
[CrossRef]

Opt. Express (4)

Optik Internat. J. Light Electron. Opt. (1)

Y.-J. Liu, Z.-Q. Wang, L.-P. Song, and G.-G. Mu, “An anatomically accurate eye model with a shell-structure lens,” Optik Internat. J. Light Electron. Opt. 116(6), 241–246 (2005).
[CrossRef]

Optom. Vis. Sci. (3)

L. F. Garner and G. Smith, “Changes in equivalent and gradient refractive index of the crystalline lens with accommodation,” Optom. Vis. Sci. 74(2), 114–119 (1997).
[CrossRef] [PubMed]

B. K. Pierscionek and D. Y. Chan, “Refractive index gradient of human lenses,” Optom. Vis. Sci. 66(12), 822–829 (1989).
[CrossRef] [PubMed]

E. Moreno-Barriuso, S. Marcos, R. Navarro, and S. A. Burns, “Comparing laser ray tracing, the spatially resolved refractometer, and the Hartmann-Shack sensor to measure the ocular wave aberration,” Optom. Vis. Sci. 78(3), 152–156 (2001).
[CrossRef] [PubMed]

SIAM J. Optim. (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9(1), 112–147 (1998).
[CrossRef]

Vision Res. (6)

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(27), 2732–2738 (2008).
[CrossRef] [PubMed]

D. O. Mutti, K. Zadnik, and A. J. Adams, “The equivalent refractive index of the crystalline lens in childhood,” Vision Res. 35(11), 1565–1573 (1995).
[CrossRef] [PubMed]

M. C. Campbell and A. Hughes, “An analytic, gradient index schematic lens and eye for the rat which predicts aberrations for finite pupils,” Vision Res. 21(7), 1129–1148 (1981).
[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(18), 2352–2366 (2005).
[CrossRef] [PubMed]

B. A. Moffat, D. A. Atchison, and J. M. Pope, “Age-related changes in refractive index distribution and power of the human lens as measured by magnetic resonance micro-imaging in vitro,” Vision Res. 42(13), 1683–1693 (2002).
[CrossRef] [PubMed]

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, “Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods,” Vision Res. 41(8), 973–979 (2001).
[CrossRef] [PubMed]

Other (6)

A. Huggert, The iso-indicial surfaces of the human crystalline lens (Acta Ophthalmologica supplementum, Stockholm, 1948).

R. Weale, “The ageing eye,” in The Lens, H. K. Lewis, ed. (London, 1963).

A. Gullstrand, Handbuch der Physiologitschen Optik, 3rd ed. 1909. J. P. Southall trans., (Optical Society of America, 1924).

O. N. Stavroudis, “Ray tracing,” in The optics of rays, wavefronts and caustics, N. Y. A. Press, ed. (1972).

D. Vasiljevic, Classical and evolutionary algorithms in the optimization of optical systems (Kluwer Academc Publishers Boston/Dordrecht/London, 2002), p. 296.

J. H. Holland, Adaptation in natural and artificial systems (The University of Michigan press, 1975).

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Figures (8)

Fig. 1
Fig. 1

(a) Experimental setup for lateral ray tracing system (b) Corresponding ray images (integrated image of five rays) (c) Single-pass ray tracing configuration to measure the spot diagram in a plane after the lens. (d). Corresponding spot images for five rays.

Fig. 2
Fig. 2

(a) Estimated error in simulated data (lateral deviations, impacts or OPD) as a function of the step size in the Sharma algorithm. (b) Computational time to ray trace and to estimate the OPD as a function of the step size. The OPD was calculated either using straight segments or a Hermite polynomial interpolation. Simulations were performed with Goncharov crystalline lens model 20S.

Fig. 3
Fig. 3

Difference between nominal and reconstructed GRIN (GRIN RMS difference) as a function of the experimental error of the input data. Data are the mean across 50 repetitions, and the error bars represent standard deviations.

Fig. 4
Fig. 4

Difference between experimental and simulated data: Mean and standard deviation for five repeated measurements. (a) Direction cosines of the outgoing rays, (b) Impacts on a plane after the lens and (c) OPD through the lens.

Fig. 5
Fig. 5

Difference between reconstructed and nominal GRIN (GRIN RMS difference) for the 5 proposed experimental configurations and three different levels (L R and H) of added experimental error in the input data (L: lower than expected errors; R: realistic errors; H: higher than expected errors). Data are the average GRIN RMS difference and the error bars the standard deviation of 50 realizations of the GRIN reconstruction algorithm. (a), (b) and (c) are the results for three different Goncharov models with increasing complexity (20U, 20B and 20S).

Fig. 6
Fig. 6

Axial (a) and Meridional (b) deviation from the nominal GRIN profile for the proposed experimental configurations, for realistic input data error levels (R) and the 4-variable Goncharov model. Data represent the mean value and the error bars the standard deviations of 50 realizations of the reconstruction algorithm.

Fig. 7
Fig. 7

GRIN RMS difference of the reconstructed and the nominal GRIN data, with realistic error level (R) in the input data without error in the surface shape (solid colors), and with a random deviation of the surface geometry of 1% (light colors). The bars represent average data and the error bars, the standard deviation of 50 realizations of the reconstruction algorithm.

Fig. 8
Fig. 8

GRIN RMS difference of the reconstruction for the 5 proposed experimental configurations and with realistic error level R, versus number of rays. Pupil radius was set to 6-mm pupil. Data represents mean value, and the error bars, the standard deviation of 50 repetitions of the reconstruction algorithm.

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