Abstract

Employing advanced technologies in studying the crystalline lens of the eye has improved our understanding of the refractive index gradient of the lens. Reconstructing and studying such a complex structure requires models with adaptable internal geometry that can be altered to simulate geometrical and optical changes of the lens with aging. In this Letter, we introduce an optically well-defined, geometrical structure for modeling the gradient refractive index profile of the crystalline lens with the advantage of an adjustable internal structure that is not available with existing models. The refractive index profile assigned to this rotationally symmetric geometry is calculated numerically, yet it is shown that this does not limit the model. The study provides a basis for developing lens models with sophisticated external and internal structures without the need for analytical solutions to calculate refractive index profiles.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2013 (1)

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. Pierscionek, J. Mod. Opt. 60, 406 (2013).
[CrossRef]

2012 (2)

M. Bahrami and A. V. Goncharov, J. Biomed. Opt. 17, 55001 (2012).
[CrossRef]

M. Bahrami and A. V. Goncharov, Biomed. Opt. Express 3, 1684 (2012).
[CrossRef]

2011 (2)

A. de Castro, S. Barbero, S. Ortiz, and S. Marcos, Opt. Express 19, 19265 (2011).
[CrossRef]

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, PLoS ONE 6, e25140 (2011).
[CrossRef]

2010 (2)

2008 (2)

J. A. Díaz, C. Pizarro, and J. Arasa, J. Opt. Soc. Am. A 25, 250 (2008).
[CrossRef]

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, Investig. Ophthalmol. Vis. Sci. 49, 2531 (2008).
[CrossRef]

2007 (2)

2006 (1)

2005 (1)

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, Vis. Res. 45, 2352 (2005).
[CrossRef]

2004 (1)

B. K. Pierscionek, A. Belaidi, and H. H. Bruun, Eye 19, 375 (2004).
[CrossRef]

2000 (1)

H. T. Kasprzak, Ophthalmic Physiol. Optics 20, 31 (2000).
[CrossRef]

1997 (2)

1995 (1)

T. J. Ypma, SIAM Review 37, 531 (1995).
[CrossRef]

1994 (1)

1991 (1)

G. Smith, B. K. Pierscionek, and D. A. Atchison, Ophthalmic Physiol. Opt. 11, 359 (1991).
[CrossRef]

1990 (1)

B. K. Pierscionek, Clin. Exp. Optom. 73, 23 (1990).
[CrossRef]

1988 (1)

1982 (1)

1980 (1)

J. W. Blaker, J. Opt. Soc. Am. A 70, 220 (1980).
[CrossRef]

Acosta, E.

Arasa, J.

Atchison, D. A.

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, Investig. Ophthalmol. Vis. Sci. 49, 2531 (2008).
[CrossRef]

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, Vis. Res. 45, 2352 (2005).
[CrossRef]

G. Smith, B. K. Pierscionek, and D. A. Atchison, Ophthalmic Physiol. Opt. 11, 359 (1991).
[CrossRef]

Bahrami, M.

M. Bahrami and A. V. Goncharov, Biomed. Opt. Express 3, 1684 (2012).
[CrossRef]

M. Bahrami and A. V. Goncharov, J. Biomed. Opt. 17, 55001 (2012).
[CrossRef]

Barbero, S.

Belaidi, A.

B. K. Pierscionek, A. Belaidi, and H. H. Bruun, Eye 19, 375 (2004).
[CrossRef]

Bille, J. F.

Blaker, J. W.

J. W. Blaker, J. Opt. Soc. Am. A 70, 220 (1980).
[CrossRef]

Brennan, N. A.

Bruun, H. H.

B. K. Pierscionek, A. Belaidi, and H. H. Bruun, Eye 19, 375 (2004).
[CrossRef]

Chan, D. Y. C.

Dainty, C.

de Castro, A.

Díaz, J. A.

Ennis, J. P.

Gambra, E.

Garner, L.

Ghatak, A. K.

Giovanzana, S.

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. Pierscionek, J. Mod. Opt. 60, 406 (2013).
[CrossRef]

Goelz, S.

Goncharov, A. V.

González, L.

Grimm, B.

Gullstrand, A.

A. Gullstrand, Handbuch der Physiologischen Optik (Optical Society of America, 1924).

Ho, A.

R. Urs, A. Ho, F. Manns, and J. M. Parel, Vis. Res. 50, 1041 (2010).
[CrossRef]

Hoshino, M.

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, PLoS ONE 6, e25140 (2011).
[CrossRef]

Jones, C. E.

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, Vis. Res. 45, 2352 (2005).
[CrossRef]

Kasprzak, H. T.

H. T. Kasprzak, Ophthalmic Physiol. Optics 20, 31 (2000).
[CrossRef]

Kasthurirangan, S.

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, Investig. Ophthalmol. Vis. Sci. 49, 2531 (2008).
[CrossRef]

Kirby, R. D.

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. Pierscionek, J. Mod. Opt. 60, 406 (2013).
[CrossRef]

Kumar, D. V.

Liang, J.

Liou, H. L.

Manns, F.

R. Urs, A. Ho, F. Manns, and J. M. Parel, Vis. Res. 50, 1041 (2010).
[CrossRef]

Marcos, S.

Markwell, E. L.

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, Investig. Ophthalmol. Vis. Sci. 49, 2531 (2008).
[CrossRef]

Meder, R.

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, Vis. Res. 45, 2352 (2005).
[CrossRef]

Mohri, S.

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, PLoS ONE 6, e25140 (2011).
[CrossRef]

Navarro, R.

Ortiz, S.

Palos, F.

Parel, J. M.

R. Urs, A. Ho, F. Manns, and J. M. Parel, Vis. Res. 50, 1041 (2010).
[CrossRef]

Pierscionek, B.

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. Pierscionek, J. Mod. Opt. 60, 406 (2013).
[CrossRef]

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, PLoS ONE 6, e25140 (2011).
[CrossRef]

Pierscionek, B. K.

B. K. Pierscionek, A. Belaidi, and H. H. Bruun, Eye 19, 375 (2004).
[CrossRef]

B. K. Pierscionek, Exp. Eye Res. 64, 887 (1997).
[CrossRef]

G. Smith, B. K. Pierscionek, and D. A. Atchison, Ophthalmic Physiol. Opt. 11, 359 (1991).
[CrossRef]

B. K. Pierscionek, Clin. Exp. Optom. 73, 23 (1990).
[CrossRef]

D. Y. C. Chan, J. P. Ennis, B. K. Pierscionek, and G. Smith, Appl. Opt. 27, 926 (1988).
[CrossRef]

Pizarro, C.

Pope, J. M.

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, Investig. Ophthalmol. Vis. Sci. 49, 2531 (2008).
[CrossRef]

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, Vis. Res. 45, 2352 (2005).
[CrossRef]

Regini, J.

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, PLoS ONE 6, e25140 (2011).
[CrossRef]

Schachar, R. A.

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. Pierscionek, J. Mod. Opt. 60, 406 (2013).
[CrossRef]

Sharma, A.

Siedlecki, D.

Smith, G.

Talu, S.

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. Pierscionek, J. Mod. Opt. 60, 406 (2013).
[CrossRef]

Uesugi, K.

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, PLoS ONE 6, e25140 (2011).
[CrossRef]

Urs, R.

R. Urs, A. Ho, F. Manns, and J. M. Parel, Vis. Res. 50, 1041 (2010).
[CrossRef]

Vazquez, D.

Yagi, N.

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, PLoS ONE 6, e25140 (2011).
[CrossRef]

Yan, E.

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. Pierscionek, J. Mod. Opt. 60, 406 (2013).
[CrossRef]

Ypma, T. J.

T. J. Ypma, SIAM Review 37, 531 (1995).
[CrossRef]

Appl. Opt. (2)

Biomed. Opt. Express (1)

Clin. Exp. Optom. (1)

B. K. Pierscionek, Clin. Exp. Optom. 73, 23 (1990).
[CrossRef]

Exp. Eye Res. (1)

B. K. Pierscionek, Exp. Eye Res. 64, 887 (1997).
[CrossRef]

Eye (1)

B. K. Pierscionek, A. Belaidi, and H. H. Bruun, Eye 19, 375 (2004).
[CrossRef]

Investig. Ophthalmol. Vis. Sci. (1)

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, Investig. Ophthalmol. Vis. Sci. 49, 2531 (2008).
[CrossRef]

J. Biomed. Opt. (1)

M. Bahrami and A. V. Goncharov, J. Biomed. Opt. 17, 55001 (2012).
[CrossRef]

J. Mod. Opt. (1)

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. Pierscionek, J. Mod. Opt. 60, 406 (2013).
[CrossRef]

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

Ophthalmic Physiol. Opt. (1)

G. Smith, B. K. Pierscionek, and D. A. Atchison, Ophthalmic Physiol. Opt. 11, 359 (1991).
[CrossRef]

Ophthalmic Physiol. Optics (1)

H. T. Kasprzak, Ophthalmic Physiol. Optics 20, 31 (2000).
[CrossRef]

Opt. Express (2)

PLoS ONE (1)

M. Hoshino, K. Uesugi, N. Yagi, S. Mohri, J. Regini, and B. Pierscionek, PLoS ONE 6, e25140 (2011).
[CrossRef]

SIAM Review (1)

T. J. Ypma, SIAM Review 37, 531 (1995).
[CrossRef]

Vis. Res. (2)

R. Urs, A. Ho, F. Manns, and J. M. Parel, Vis. Res. 50, 1041 (2010).
[CrossRef]

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, Vis. Res. 45, 2352 (2005).
[CrossRef]

Other (1)

A. Gullstrand, Handbuch der Physiologischen Optik (Optical Society of America, 1924).

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Figures (4)

Fig. 1.
Fig. 1.

External shape of the model. Ra, Rp, Da, and Dp are always positive, and the common optical sign convention is taken into account within the equations. Ca, Cp, and Z0 are a result of the Eqs. (1)–(3) and can take positive or negative values. Y0 is the maximum height of the lens, which happens at Z0; O is the point on the optic axis that defines where Da and Dp meet.

Fig. 2.
Fig. 2.

Lens contours for X=1.0, 0.8, 0.6, and 0.4, where Ra=12, Rp=9, Da=3, Dp=2, (a) ps=1 and (b) Ps=2. The geometry of the model provides flexibility in replicating the crystalline lens with a wide range of internal contours.

Fig. 3.
Fig. 3.

Lens contours and the relevant refractive index profiles along the optical axis z; and radial axis y, where nC=1.415, nS=1.36, Ra=11, Rp=8, Da=2, Dp=1.5, (a) pz=4, py=4, (b) pz=4, and py=8. The isoindicial contours represent n equal to 1.36, 1.37, 1.38, 1.39, 1.40 and 1.41, which are equivalent to X equal to 1.0, 0.95, 0.89, 0.82, 0.72, and 0.55, respectively.

Fig. 4.
Fig. 4.

Diagrammatic representation of the ZEMAX optical design showing wide-angle ray tracing through the cornea, the aqueous, the lens model, and the vitreous, respectively, with focused light on the retina. The iris, as the stop aperture of the system, is located just before the anterior surface of the lens.

Equations (10)

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

Ca=Ra(Da+Z0)Da+Z0
Cp=Rp(DpZ0)DpZ0,
Z0=DpRpDaRaRp+Ra
{ra=RaXpsrp=RpXpsda=DaXdp=DpX,
x2+y2={2ra(da+z)(1+ca)(da+z)22rp(dpz)(1+cp)(dpz)2,
x2+y2={RaXps(DaX+z)(2(Ra+Rp)(DaX+z)RpX(Da+Dp))RpXps(DpXz)(2(Ra+Rp)(DpXz)RaX(Da+Dp)),
Xz=(2X(DaRaXDpRpX+(Ra+Rp)z))/(Da(ps+1)(DaRa(Da+2Dp)Rp)X2+2ps(DaRaDpRp)Xz+(ps1)(Ra+Rp)z2).
n=nC+(nSnC)rp,
{n(z)=nC+(nSnC)X(z)pzn(y)=nC+(nSnC)X(y)py
n(x,y,z)=nC+(nSnC)X(x,y,z)pz.

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