Abstract

We present an optimization method to retrieve the gradient index (GRIN) distribution of the in-vitro crystalline lens from optical path difference data extracted from OCT images. Three-dimensional OCT images of the crystalline lens are obtained in two orientations (with the anterior surface up and posterior surface up), allowing to obtain the lens geometry. The GRIN reconstruction method is based on a genetic algorithm that searches for the parameters of a 4-variable GRIN model that best fits the distorted posterior surface of the lens. Computer simulations showed that, for noise of 5 μm in the surface elevations, the GRIN is recovered with an accuracy of 0.003 and 0.010 in the refractive indices of the nucleus and surface of the lens, respectively. The method was applied to retrieve three-dimensionally the GRIN of a porcine crystalline lens in vitro. We found a refractive index ranging from 1.362 in the surface to 1.443 in the nucleus of the lens, an axial exponential decay of the GRIN profile of 2.62 and a meridional exponential decay ranging from 3.56 to 5.18. The effect of GRIN on the aberrations of the lens also studied. The estimated spherical aberration of the measured porcine lens was 2.87 μm assuming a homogenous equivalent refractive index, and the presence of GRIN shifted the spherical aberration toward negative values (−0.97 μm), for a 6-mm pupil.

© 2010 OSA

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2010 (2)

2009 (2)

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(6), 2531–2540 (2008).
[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).
[PubMed]

2007 (3)

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

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).

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).

2006 (2)

D. Vázquez, 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).

A. M. Rosen, D. B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J. M. Parel, and R. C. Augusteyn, “In vitro dimensions and curvatures of human lenses,” Vision Res. 46(6-7), 1002–1009 (2006).

2005 (1)

2004 (5)

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).

C. E. Jones and J. M. Pope, “Measuring optical properties of an eye lens using magnetic resonance imaging,” Magn. Reson. Imaging 22(2), 211–220 (2004).
[PubMed]

A. Roorda and A. Glasser, “Wave aberrations of the isolated crystalline lens,” J. Vis. 4(4), 250–261 (2004).
[PubMed]

C. E. Jones and J. M. Pope, “Measuring optical properties of an eye lens using magnetic resonance imaging,” Magn. Reson. Imaging 22(2), 211–220 (2004).
[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).

2003 (1)

R. J. Zawadzki, C. Leisser, R. Leitgeb, M. Pircher, and A. F. Fercher, “Three-dimensional ophthalmic optical coherence tomography with a refraction correction algorithm,” Proc. SPIE 5140, 8 (2003).

2002 (2)

P. Artal, E. Berrio, A. Guirao, and P. Piers, “Contribution of the cornea and internal surfaces to the change of ocular aberrations with age,” J. Opt. Soc. Am. A 19(1), 137–143 (2002).

S. Barbero, S. Marcos, and J. Merayo-Lloves, “Corneal and total optical aberrations in a unilateral aphakic patient,” J. Cataract Refract. Surg. 28(9), 1594–1600 (2002).
[PubMed]

2001 (3)

M. Dubbelman and G. L. Van der Heijde, “The shape of the aging human lens: curvature, equivalent refractive index and the lens paradox,” Vision Res. 41(14), 1867–1877 (2001).
[PubMed]

M. Dubbelman, G. L. van der Heijde, and H. A. Weeber, “The thickness of the aging human lens obtained from corrected Scheimpflug images,” Optom. Vis. Sci. 78(6), 411–416 (2001).
[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).
[PubMed]

1998 (2)

G. Beliakov and D. Y. Chan, “Analysis of Inhomogeneous Optical Systems by the Use of Ray Tracing. II. Three-dimensional Systems with Symmetry,” Appl. Opt. 37(22), 5106–5111 (1998).

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(1), 21–29 (1998).
[PubMed]

1997 (2)

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).

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).
[PubMed]

1995 (1)

1992 (1)

W. S. Jagger, “The optics of the spherical fish lens,” Vision Res. 32(7), 1271–1284 (1992).
[PubMed]

1990 (2)

B. K. Pierscionek, “Presbyopia - effect of refractive index,” Clin. Exp. Optom. 73(1), 23–30 (1990).

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).

1989 (1)

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

1988 (1)

1984 (1)

M. C. Campbell, “Measurement of refractive index in an intact crystalline lens,” Vision Res. 24(5), 409–415 (1984).
[PubMed]

1983 (1)

J. G. Sivak and R. O. Kreuzer, “Spherical aberration of the crystalline lens,” Vision Res. 23(1), 59–70 (1983).
[PubMed]

1982 (1)

1980 (1)

1973 (1)

1972 (1)

O. Pomerantzeff, H. Fish, J. Govignon, and C. L. Schepens, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 19, 387–388 (1972).

1965 (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Acosta, E.

Al-Ahdali, I. H.

Artal, P.

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).
[PubMed]

Augusteyn, R. C.

A. M. Rosen, D. B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J. M. Parel, and R. C. Augusteyn, “In vitro dimensions and curvatures of human lenses,” Vision Res. 46(6-7), 1002–1009 (2006).

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).
[PubMed]

Barbero, S.

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).

S. Barbero, S. Marcos, and J. Merayo-Lloves, “Corneal and total optical aberrations in a unilateral aphakic patient,” J. Cataract Refract. Surg. 28(9), 1594–1600 (2002).
[PubMed]

Beliakov, G.

Berny, F.

Berrio, E.

Blaker, J. W.

Borja, D.

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

A. M. Rosen, D. B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J. M. Parel, and R. C. Augusteyn, “In vitro dimensions and curvatures of human lenses,” Vision Res. 46(6-7), 1002–1009 (2006).

Brennan, N. A.

Campbell, M. C.

M. C. Campbell, “Measurement of refractive index in an intact crystalline lens,” Vision Res. 24(5), 409–415 (1984).
[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.

Denham, D. B.

A. M. Rosen, D. B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J. M. Parel, and R. C. Augusteyn, “In vitro dimensions and curvatures of human lenses,” Vision Res. 46(6-7), 1002–1009 (2006).

Dubbelman, M.

M. Dubbelman and G. L. Van der Heijde, “The shape of the aging human lens: curvature, equivalent refractive index and the lens paradox,” Vision Res. 41(14), 1867–1877 (2001).
[PubMed]

M. Dubbelman, G. L. van der Heijde, and H. A. Weeber, “The thickness of the aging human lens obtained from corrected Scheimpflug images,” Optom. Vis. Sci. 78(6), 411–416 (2001).
[PubMed]

el-Hage, S. G.

El-Messiery, M. A.

Ennis, J. P.

Fercher, A. F.

R. J. Zawadzki, C. Leisser, R. Leitgeb, M. Pircher, and A. F. Fercher, “Three-dimensional ophthalmic optical coherence tomography with a refraction correction algorithm,” Proc. SPIE 5140, 8 (2003).

Fernandez, V.

A. M. Rosen, D. B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J. M. Parel, and R. C. Augusteyn, “In vitro dimensions and curvatures of human lenses,” Vision Res. 46(6-7), 1002–1009 (2006).

Fish, H.

O. Pomerantzeff, H. Fish, J. Govignon, and C. L. Schepens, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 19, 387–388 (1972).

Forbes, G. W.

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).
[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).
[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).

A. Roorda and A. Glasser, “Wave aberrations of the isolated crystalline lens,” J. Vis. 4(4), 250–261 (2004).
[PubMed]

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 eye,” Am. J. Optom. Physiol. Opt. 19, 387–388 (1972).

Grulkowski, I.

Guirao, A.

Gupta, P.

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

Ho, A.

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

A. M. Rosen, D. B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J. M. Parel, and R. C. Augusteyn, “In vitro dimensions and curvatures of human lenses,” Vision Res. 46(6-7), 1002–1009 (2006).

Jagger, W. S.

W. S. Jagger, “The optics of the spherical fish lens,” Vision Res. 32(7), 1271–1284 (1992).
[PubMed]

Jones, C. E.

C. E. Jones and J. M. Pope, “Measuring optical properties of an eye lens using magnetic resonance imaging,” Magn. Reson. Imaging 22(2), 211–220 (2004).
[PubMed]

C. E. Jones and J. M. Pope, “Measuring optical properties of an eye lens using magnetic resonance imaging,” Magn. Reson. Imaging 22(2), 211–220 (2004).
[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).
[PubMed]

Kowalczyk, A.

Kreuzer, R. O.

J. G. Sivak and R. O. Kreuzer, “Spherical aberration of the crystalline lens,” Vision Res. 23(1), 59–70 (1983).
[PubMed]

Kumar, D. V.

Leisser, C.

R. J. Zawadzki, C. Leisser, R. Leitgeb, M. Pircher, and A. F. Fercher, “Three-dimensional ophthalmic optical coherence tomography with a refraction correction algorithm,” Proc. SPIE 5140, 8 (2003).

Leitgeb, R.

R. J. Zawadzki, C. Leisser, R. Leitgeb, M. Pircher, and A. F. Fercher, “Three-dimensional ophthalmic optical coherence tomography with a refraction correction algorithm,” Proc. SPIE 5140, 8 (2003).

Liou, H. L.

Manns, F.

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

A. M. Rosen, D. B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J. M. Parel, and R. C. Augusteyn, “In vitro dimensions and curvatures of human lenses,” Vision Res. 46(6-7), 1002–1009 (2006).

Marcos, S.

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).
[PubMed]

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

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).

S. Barbero, S. Marcos, and J. Merayo-Lloves, “Corneal and total optical aberrations in a unilateral aphakic patient,” J. Cataract Refract. Surg. 28(9), 1594–1600 (2002).
[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).
[PubMed]

Mead, R.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Merayo-Lloves, J.

S. Barbero, S. Marcos, and J. Merayo-Lloves, “Corneal and total optical aberrations in a unilateral aphakic patient,” J. Cataract Refract. Surg. 28(9), 1594–1600 (2002).
[PubMed]

Navarro, R.

Nelder, J. A.

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Ortiz, S.

Palos, F.

Parel, J.

F. Manns, A. Ho, D. Borja, and J. Parel, “Comparison of Uniform and Gradient Paraxial Models of the Crystalline Lens,” Invest. Ophthalmol. Vis. Sci. 51, E-Abstract 789 (2010).
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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).
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A. M. Rosen, D. B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J. M. Parel, and R. C. Augusteyn, “In vitro dimensions and curvatures of human lenses,” Vision Res. 46(6-7), 1002–1009 (2006).

Pascual, D.

Patel, H.

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

Piers, P.

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(1), 21–29 (1998).
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Pircher, M.

R. J. Zawadzki, C. Leisser, R. Leitgeb, M. Pircher, and A. F. Fercher, “Three-dimensional ophthalmic optical coherence tomography with a refraction correction algorithm,” Proc. SPIE 5140, 8 (2003).

Pomerantzeff, O.

O. Pomerantzeff, H. Fish, J. Govignon, and C. L. Schepens, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 19, 387–388 (1972).

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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).
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C. E. Jones and J. M. Pope, “Measuring optical properties of an eye lens using magnetic resonance imaging,” Magn. Reson. Imaging 22(2), 211–220 (2004).
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C. E. Jones and J. M. Pope, “Measuring optical properties of an eye lens using magnetic resonance imaging,” Magn. Reson. Imaging 22(2), 211–220 (2004).
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Y. Verma, K. Rao, M. Suresh, H. Patel, and P. Gupta, “Measurement of gradient refractive index profile of crystalline lens of fish eye in vivo using optical coherence tomography,” Appl. Phys. B 87(4), 607–610 (2007).

Remon, L.

Roorda, A.

A. Roorda and A. Glasser, “Wave aberrations of the isolated crystalline lens,” J. Vis. 4(4), 250–261 (2004).
[PubMed]

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A. M. Rosen, D. B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J. M. Parel, and R. C. Augusteyn, “In vitro dimensions and curvatures of human lenses,” Vision Res. 46(6-7), 1002–1009 (2006).

Schepens, C. L.

O. Pomerantzeff, H. Fish, J. Govignon, and C. L. Schepens, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 19, 387–388 (1972).

Sharma, A.

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J. G. Sivak and R. O. Kreuzer, “Spherical aberration of the crystalline lens,” Vision Res. 23(1), 59–70 (1983).
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D. Vázquez, 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).

E. Acosta, D. Vázquez, 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).

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).
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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(1), 21–29 (1998).
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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).
[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).
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Suresh, M.

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

Szkulmowski, M.

Szlag, D.

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).
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van der Heijde, G. L.

M. Dubbelman, G. L. van der Heijde, and H. A. Weeber, “The thickness of the aging human lens obtained from corrected Scheimpflug images,” Optom. Vis. Sci. 78(6), 411–416 (2001).
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M. Dubbelman and G. L. Van der Heijde, “The shape of the aging human lens: curvature, equivalent refractive index and the lens paradox,” Vision Res. 41(14), 1867–1877 (2001).
[PubMed]

Vázquez, D.

Verma, Y.

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

Weeber, H. A.

M. Dubbelman, G. L. van der Heijde, and H. A. Weeber, “The thickness of the aging human lens obtained from corrected Scheimpflug images,” Optom. Vis. Sci. 78(6), 411–416 (2001).
[PubMed]

Wojtkowski, M.

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).
[PubMed]

Zawadzki, R. J.

R. J. Zawadzki, C. Leisser, R. Leitgeb, M. Pircher, and A. F. Fercher, “Three-dimensional ophthalmic optical coherence tomography with a refraction correction algorithm,” Proc. SPIE 5140, 8 (2003).

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

O. Pomerantzeff, H. Fish, J. Govignon, and C. L. Schepens, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 19, 387–388 (1972).

Appl. Opt. (5)

Appl. Phys. B (1)

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

Clin. Exp. Optom. (1)

B. K. Pierscionek, “Presbyopia - effect of refractive index,” Clin. Exp. Optom. 73(1), 23–30 (1990).

Comput. J. (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

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).
[PubMed]

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. Parel, “Comparison of Uniform and Gradient Paraxial Models of the Crystalline Lens,” Invest. Ophthalmol. Vis. Sci. 51, E-Abstract 789 (2010).
[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).

J. Cataract Refract. Surg. (1)

S. Barbero, S. Marcos, and J. Merayo-Lloves, “Corneal and total optical aberrations in a unilateral aphakic patient,” J. Cataract Refract. Surg. 28(9), 1594–1600 (2002).
[PubMed]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (7)

J. Vis. (1)

A. Roorda and A. Glasser, “Wave aberrations of the isolated crystalline lens,” J. Vis. 4(4), 250–261 (2004).
[PubMed]

Magn. Reson. Imaging (2)

C. E. Jones and J. M. Pope, “Measuring optical properties of an eye lens using magnetic resonance imaging,” Magn. Reson. Imaging 22(2), 211–220 (2004).
[PubMed]

C. E. Jones and J. M. Pope, “Measuring optical properties of an eye lens using magnetic resonance imaging,” Magn. Reson. Imaging 22(2), 211–220 (2004).
[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(1), 21–29 (1998).
[PubMed]

Opt. Express (2)

Optom. Vis. Sci. (3)

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

M. Dubbelman, G. L. van der Heijde, and H. A. Weeber, “The thickness of the aging human lens obtained from corrected Scheimpflug images,” Optom. Vis. Sci. 78(6), 411–416 (2001).
[PubMed]

Proc. SPIE (1)

R. J. Zawadzki, C. Leisser, R. Leitgeb, M. Pircher, and A. F. Fercher, “Three-dimensional ophthalmic optical coherence tomography with a refraction correction algorithm,” Proc. SPIE 5140, 8 (2003).

Vision Res. (7)

J. G. Sivak and R. O. Kreuzer, “Spherical aberration of the crystalline lens,” Vision Res. 23(1), 59–70 (1983).
[PubMed]

M. C. Campbell, “Measurement of refractive index in an intact crystalline lens,” Vision Res. 24(5), 409–415 (1984).
[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).
[PubMed]

W. S. Jagger, “The optics of the spherical fish lens,” Vision Res. 32(7), 1271–1284 (1992).
[PubMed]

M. Dubbelman and G. L. Van der Heijde, “The shape of the aging human lens: curvature, equivalent refractive index and the lens paradox,” Vision Res. 41(14), 1867–1877 (2001).
[PubMed]

A. M. Rosen, D. B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J. M. Parel, and R. C. Augusteyn, “In vitro dimensions and curvatures of human lenses,” Vision Res. 46(6-7), 1002–1009 (2006).

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).
[PubMed]

Other (3)

D. Siedlecki, A. de Castro, S. Ortiz, D. Borja, F. Manns, and S. Marcos, “Estimation of contribution of gradient index structure to the amount of posterior surface optical distortion for in-vitro human crystalline lenses imaged by Optical Coherence Tomography,” presented at the Fifth European Meeting on Visual and Physiological Optics (2010).

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

O. N. Stavroudis, The optics of rays, wavefronts and caustics (New York Academic Press, 1972), pp. 81–95.

Supplementary Material (2)

» Media 1: AVI (923 KB)     
» Media 2: AVI (1747 KB)     

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of a genetic algorithm iteration. In this study, 85% of the population is created by crossover and mutation. The 5% best solutions are directly copied to the next solution (elite) and the rest of the population is randomly selected from previous generations.

Fig. 2
Fig. 2

Crystalline lens GRIN tested in the simulations for a young human lens (A), a middle age lens (B) and an old human lens (C). The index of refraction ranged from 1.410 in the nucleus to 1.378 in the surface. The black line represents the undistorted cuvette, the red line represents the distorted posterior surface, and the blue line the distorted cuvette, for a 6-mm pupil. These curves are the input data to the optimization algorithm.

Fig. 3
Fig. 3

(Media 1) Three dimensional image of the in-vitro posterior-up crystalline lens measured with OCT.

Fig. 4
Fig. 4

Three dimensional OCT data from images of the crystalline lens placed with the anterior surface up (left) and the posterior up (right). The blue and red points correspond to the segmented anterior and posterior surfaces of the lens, respectively. The green points correspond to the segmented cuvette surface imaged through the crystalline lens and the black points to the segmented cuvette surface without the crystalline lens. All data are fan distortion corrected, and the surfaces are also distorted due to the presence of preservation media.

Fig. 5
Fig. 5

Deviation of the parameters resulting from optimization with different amounts of error in the detection of distorted surfaces.

Fig. 6
Fig. 6

Radius of curvature of the anterior (red) and posterior (green) lens surface, in mm. and meridional exponential decay parameter p2 (black) as a function of meridional angle.

Fig. 7
Fig. 7

(Media 2) Change of the gradient index of the lens with meridian angle.

Fig. 8
Fig. 8

Ray tracing in one meridian for the crystalline lens with the reconstructed gradient index (A) and with the equivalent refractive index (B). Main differences between them are an increase in astigmatism of the lens (not visible here) and a shift of spherical aberration toward negative values.

Equations (2)

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n ( ρ , θ ) = n N Δ n ( ρ ρ s ) p ( θ ) ,
MF ( G R I N _ c o e f f s ) = RMS ( OPD calc ( Surf2 , G R I N _ c o e f f s ) , OPD exp ( Surf2 ) ) + RMS ( OPD calc ( Surf3 , G R I N _ c o e f f s ) , OPD exp ( Surf3 ) ) ,

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