Abstract

We propose a wide-field schematic eye model, which provides a more realistic description of the optical system of the eye in relation to its anatomical structure. The wide-field model incorporates a gradient-index (GRIN) lens, which enables it to fulfill properties of two well-known schematic eye models, namely, Navarro’s model for off-axis aberrations and Thibos’s chromatic on-axis model (the Indiana eye). These two models are based on extensive experimental data, which makes the derived wide-field eye model also consistent with that data. A mathematical method to construct a GRIN lens with its iso-indicial contours following the optical surfaces of given asphericity is presented. The efficiency of the method is demonstrated with three variants related to different age groups. The role of the GRIN structure in relation to the lens paradox is analyzed. The wide-field model with a GRIN lens can be used as a starting design for the eye inverse problem, i.e., reconstructing the optical structure of the eye from off-axis wavefront measurements. Anatomically more accurate age-dependent optical models of the eye could ultimately help an optical designer to improve wide-field retinal imaging.

© 2007 Optical Society of America

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2006 (2)

R. Navarro, L. González, and J. L. Hernández-Matamoros, "On the prediction of optical aberrations by personalized eye models," Optom. Vision Sci. 83, 371-381 (2006).
[CrossRef]

R. Navarro, L. González, and J. L. Hernández, "Optics of the average normal cornea from general and canonical representations of its surface topography," J. Opt. Soc. Am. A 23, 219-232 (2006).
[CrossRef]

2005 (4)

M. V. Perez, C. Bao, M. T. Flores-Arias, M. A. Rama, and C. Gomez-Reino, "Description of gradient-index crystalline lens by a first-order optical system," J. Opt. A, Pure Appl. Opt. 7, 103-110 (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 (Stuttgart) 116, 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, 2352-2366 (2005).
[CrossRef] [PubMed]

J. L. Alió, P. Schimchak, H. P. Negri, and R. Montés-Micó, "Crystalline lens optical dysfunction through aging," Ophthalmology 112, 2022-2029 (2005).
[CrossRef] [PubMed]

2004 (6)

S. Aoshima, T. Nagata, and A. Minakata, "Optical characteristics of oblique incident rays in pseudophakic eyes," J. Cataract Refractive Surg. 30, 471-477 (2004).
[CrossRef]

S. Amano, Y. Amano, S. Yamagami, T. Miyai, K. Miyata, T. Samejima, and T. Oshika, "Age-related changes in corneal and ocular higher-order wavefront aberrations," Am. J. Ophthalmol. 137, 988-992 (2004).
[CrossRef] [PubMed]

J. E. Kelly, T. Mihashi, and H. C. Howland, "Compensation of corneal horizontal/vertical astigmatism, lateral coma, and spherical aberration by internal optics of the eye," J. Vision 4, 262-271 (2004).
[CrossRef]

D. A. Atchison, "Anterior corneal and internal contributions to peripheral aberrations of human eyes," J. Opt. Soc. Am. A 21, 335-359 (2004).
[CrossRef]

J. F. Koretz, S. Strenk, L. M. Strenk, and J. L. Semmlow, "Scheimpflug and high-resolution magnetic resonance imaging of the anterior segment: a comparative study," J. Opt. Soc. Am. A 21, 346-354 (2004).
[CrossRef]

D. Siedlecki, H. Kasprzak, and B. K. Pierscionek, "Schematic eye with a gradient-index lens and aspheric surfaces," Opt. Lett. 29, 1197-1199 (2004).
[CrossRef] [PubMed]

2003 (7)

J. C. He, J. Gwiazda, F. Thorn, and R. Held, "Wave-front aberrations in the anterior corneal surface and the whole eye," J. Opt. Soc. Am. A 20, 1155-1163 (2003).
[CrossRef]

A. Lleó, A. Marcos, M. Calatayud, L. Alonso, S. M. Rahhal, and J. A. Sanchis-Gimeno, "The relationship between central corneal thickness and Goldmann applanation tonometry," Clin. Exp. Optom. 86, 104-108 (2003).
[CrossRef] [PubMed]

A. K. C. Lam and J. Chan, "Corneal thickness at different reference points from Orbscan II system," Clin. Exp. Optom. 86, 230-234 (2003).
[CrossRef] [PubMed]

J. C. He, J. Gwiazda, F. Thorn, R. Held, and W. Huang, "Change in corneal shape and corneal wave-front aberrations with accommodation," J. Vision 3, 456-463 (2003).
[CrossRef]

L. Wang, E. Dai, D. D. Koch, and A. Nathoo, "Optical aberrations of the human anterior cornea," J. Cataract Refractive Surg. 29, 1514-1521 (2003).
[CrossRef]

M. Dubbelman, G. L. Van der Heijde, H. A. Weeber, and G. F. J. M. Vrensen, "Changes in the internal structure of the human crystalline lens with age and accommodation," Vision Res. 43, 2363-2375 (2003).
[CrossRef] [PubMed]

G. Smith, "The optical properties of the crystalline lens and their significance," Clin. Exp. Optom. 86, 3-18 (2003).
[CrossRef]

2002 (5)

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

M. Dubbelman, H. A. Weeber, R. G. L. van der Heijde, and H. J. Volker-Dieben, "Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography," Acta Ophthalmol. Scand. 80, 379-383 (2002).
[CrossRef] [PubMed]

B. A. Moffat, D. A. Atchison, and J. M. Pope, "Explanation of the lens paradox," Optom. Vision Sci. 79, 148-150 (2002).
[CrossRef]

P. Artal, E. Berrio, A Guirao, and P. Piers, "Contribution of the cornea and internal surfaces to the change of ocular aberrations," J. Opt. Soc. Am. A 19, 137-143 (2002).
[CrossRef]

L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, "Statistical variations of aberration structure and image quality in a normal population of healthy eyes," J. Opt. Soc. Am. A 19, 2329-2348 (2002).
[CrossRef]

2001 (7)

J. Porter, A. Guirao, I. G. Cox, and D. R. Williams, "Monochromatic aberrations of the human eye in a large population," J. Opt. Soc. Am. A 18, 1793-1803 (2001).
[CrossRef]

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, 1867-1877 (2001).
[CrossRef] [PubMed]

J. F. Koretz, C. A. Cook, and P. L. Kaufman, "Aging of the human lens: changes in lens shape at zero-diopter accommodation," J. Opt. Soc. Am. A 18, 265-272 (2001).
[CrossRef]

J. F. Koretz and C. A. Cook, "Aging of the optics of the human eye: lens refraction models and principal plane locations," Optom. Vision Sci. 78, 396-404 (2001).
[CrossRef]

M. Dubbelman, G. L. Van der Heijde, and H. A. Weeber, "The thickness of the aging human lens obtained from corrected Scheimpflug images," Optom. Vision Sci. 78, 411-416 (2001).
[CrossRef]

G. Smith and D. A. Atchison, "The gradient index and spherical aberration of the lens of the human eye," Ophthalmic Physiol. Opt. 21, 317-326 (2001).
[CrossRef] [PubMed]

G. Smith, M. J. Cox, R. Calver, and L. F. Garner, "The spherical aberration of the crystalline lens of the human eye," Vision Res. 41, 235-243 (2001).
[CrossRef] [PubMed]

2000 (5)

1999 (4)

I. Escudero-Sanz and R. Navarro, "Off-axis aberrations of a wide-angle schematic eye model," J. Opt. Soc. Am. A 16, 1881-1891 (1999).
[CrossRef]

R. I. Calver, M. J. Cox, and D. B. Elliott, "Effect of aging on the monochromatic aberrations of the human eye," J. Opt. Soc. Am. A 16, 2069-2078 (1999).
[CrossRef]

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, 41-48 (1999).
[CrossRef]

W. S. Jagger and P. J. Stands, "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 (7)

1997 (6)

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

G. Smith and D. A. Atchison, "Equivalent power of the crystalline lens of the human eye: comparison of methods of calculation," J. Opt. Soc. Am. A 14, 2537-2546 (1997).
[CrossRef]

J. Liang and D. R. Williams, "Aberrations and retinal image quality of the normal human eye," J. Opt. Soc. Am. A 14, 2873-2883 (1997).
[CrossRef]

R. Navarro and M. A. Losada, "Aberrations and relative efficiency of ray pencils in the living human eye," Optom. Vision Sci. 74, 540-547 (1997).
[CrossRef]

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

L. N. Thibos, M. Ye, X. Zhang, and A. Bradley, "Spherical aberration of the reduced schematic eye with elliptical refracting surface," Optom. Vision Sci. 74, 548-556 (1997).
[CrossRef]

1996 (3)

R. L. Woods, A. Bradley, and D. A. Atchison, "Monocular diplopia caused by ocular aberrations and hyperopic defocus," Vision Res. 36, 3597-3606 (1996).
[CrossRef] [PubMed]

H.-L. Liou and N. A. Brennan, "The prediction of spherical aberration with schematic eyes," Ophthalmic Physiol. Opt. 16, 348-354 (1996).
[CrossRef] [PubMed]

W. S. Jagger and P. J. Stands, "A wide-angle gradient index optical model of the crystalline lens and eye of the rainbow trout," Vision Res. 36, 2623-2639 (1996).
[CrossRef] [PubMed]

1995 (5)

I. H. Al-Ahdali and M. A. El-Messiery, "Examination of the effect of the fibrous structure of a lens on the optical characteristics of the human eye: a computer-simulated model," Appl. Opt. 25, 5738-5745 (1995).
[CrossRef]

D. A. Atchison and G. Smith, "Continuous gradient-index and shell models of the human lens," Vision Res. 35, 2529-2538 (1995).
[CrossRef]

B. K. Pierscionek, "Variations in refractive index and absorbance of 679 nm light with age and cataract formation in human lenses," Exp. Eye Res. 60, 407-414 (1995).
[CrossRef] [PubMed]

R. P. Hemenger, L. F. Garner, and C. S. Ooi, "Changes with age of the refractive index gradient of the human ocular lens," Invest. Ophthalmol. Visual Sci. 36, 703-707 (1995).

C. S. Ooi and T. Grosvenor, "Mechanisms of emmetropization in the aging eye," Optom. Vision Sci. 72, 60-66 (1995).
[CrossRef]

1994 (3)

J. F. Koretz, C. A. Cook, and J. R. Kuszak, "The zones of discontinuity in the human lens: development and distribution with age," Vision Res. 34, 2955-2962 (1994).
[CrossRef] [PubMed]

C. Edmund, "Posterior corneal curvature and its influence on corneal dioptric power," Acta Ophthalmol. 72, 715-720 (1994).

J. Liang, B. Grimm, S. Goez, and J. F. Bille, "Objective measurements of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor," J. Opt. Soc. Am. A 11, 1949-1957 (1994).
[CrossRef]

1993 (2)

M. Ye, X. X. Zhang, L. N. Thibos, and A. Bradley, "A new single-surface model eye that accurately predicts chromatic and spherical aberrations of the human eye," Invest. Ophthalmol. Visual Sci. 34, 777 (1993).

B. K. Pierscionek, "Surface refractive index of the eye lens determined with an optic fiber sensor," J. Opt. Soc. Am. A 10, 1867-1871 (1993).
[CrossRef]

1992 (2)

1991 (2)

1990 (1)

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

1989 (2)

M. Sheridan and W. A. Douthwaite, "Meridional variations of corneal shape," Ophthalmic Physiol. Opt. 9, 235-238 (1989).
[CrossRef] [PubMed]

B. K. Pierscionek and D. Y. C. Chan, "Refractive index gradient of human lenses," Optom. Vision Sci. 66, 822-829 (1989).
[CrossRef]

1988 (2)

B. K. Pierscionek, D. Y. C. Chan, J. P. Ennis, G. Smith, and R. C. Augusteyn, "Nondestructive method of constructing three-dimensional gradient index models for crystalline lenses: 1 theory and experiment," Am. J. Optom. Physiol. Opt. 65, 481-491 (1988).
[PubMed]

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

1987 (1)

T. Grosvenor, "Reduction in axial length with age: an emmetropizing mechanism for the adult eye?" Am. J. Optom. Physiol. Opt. 64, 657-663 (1987).
[PubMed]

1986 (3)

J. F. Koretz and G. H. Handelman, "The lens paradox and image formation in accommodating human eyes," Top. Aging Res. Eur. 6, 57-64 (1986).

H. Saunders, "A longitudinal study of the age dependence of human ocular refraction. 1. Age-dependent changes in the equivalent sphere," Ophthalmic Physiol. Opt. 6, 343-344 (1986).
[CrossRef]

M. Guillon, P. M. Lydon, and C. Wilson, "Corneal topography: a clinical model," Ophthalmic Physiol. Opt. 6, 47-56 (1986).
[CrossRef] [PubMed]

1985 (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, 166-176 (1984).
[PubMed]

1983 (1)

1981 (2)

P. P. Fagerholm, B. T. Philipson, and B. Lindstrom, "Normal human lens, the distribution of protein," Exp. Eye Res. 33, 615-620 (1981).
[CrossRef] [PubMed]

H. Saunders, "Age-dependence of human refractive errors," Ophthalmic Physiol. Opt. 1, 159-174 (1981).
[CrossRef]

1980 (1)

1979 (2)

C. S. Yu, D. Kao, and C. T. Chang, "Measurements of the length of the visual axis by ultrasonography in 1789 eyes," Chin. J. Ophthal. 15, 45-47 (1979).

M. Millodot and J. Sivak, "Contribution of the cornea and lens to the spherical aberration of the eye," Vision Res. 19, 685-687 (1979).
[CrossRef] [PubMed]

1977 (1)

M. J. Howcroft and J. A. Parker, "Aspheric curvatures for the human lens," Vision Res. 17, 1217-1223 (1977).
[CrossRef] [PubMed]

1974 (1)

N. Brown, "The change in lens curvature with age," Exp. Eye Res. 19, 175-183 (1974).
[CrossRef]

1973 (1)

N. Brown, "The change in shape and internal form of the lens of the eye on accommodation," Exp. Eye Res. 15, 441-459 (1973).
[CrossRef] [PubMed]

1971 (1)

1970 (1)

1969 (1)

S. Nakao, T. Ono, R. Nagata, and K. Iwata, "Model of refractive indices in the human crystalline lens," Jpn. J. Clin. Ophthalmol. 23, 903-906 (1969).

1963 (1)

T. Jenkins, "Aberrations of the eye and their effects on vision: part 1," Br. J. Physiol. Opt. 20, 59-91 (1963).
[PubMed]

1956 (1)

1949 (1)

1948 (1)

S. Stenström, "Investigation of the variation and the correlation of the optical elements of human eye," Am. J. Optom. Arch. Am. Acad. Optom. 25, 340-350 (1948).

Al-Ahdali, I. H.

I. H. Al-Ahdali and M. A. El-Messiery, "Examination of the effect of the fibrous structure of a lens on the optical characteristics of the human eye: a computer-simulated model," Appl. Opt. 25, 5738-5745 (1995).
[CrossRef]

Alió, J. L.

J. L. Alió, P. Schimchak, H. P. Negri, and R. Montés-Micó, "Crystalline lens optical dysfunction through aging," Ophthalmology 112, 2022-2029 (2005).
[CrossRef] [PubMed]

Alonso, L.

A. Lleó, A. Marcos, M. Calatayud, L. Alonso, S. M. Rahhal, and J. A. Sanchis-Gimeno, "The relationship between central corneal thickness and Goldmann applanation tonometry," Clin. Exp. Optom. 86, 104-108 (2003).
[CrossRef] [PubMed]

Amano, S.

S. Amano, Y. Amano, S. Yamagami, T. Miyai, K. Miyata, T. Samejima, and T. Oshika, "Age-related changes in corneal and ocular higher-order wavefront aberrations," Am. J. Ophthalmol. 137, 988-992 (2004).
[CrossRef] [PubMed]

Amano, Y.

S. Amano, Y. Amano, S. Yamagami, T. Miyai, K. Miyata, T. Samejima, and T. Oshika, "Age-related changes in corneal and ocular higher-order wavefront aberrations," Am. J. Ophthalmol. 137, 988-992 (2004).
[CrossRef] [PubMed]

Aoshima, S.

S. Aoshima, T. Nagata, and A. Minakata, "Optical characteristics of oblique incident rays in pseudophakic eyes," J. Cataract Refractive Surg. 30, 471-477 (2004).
[CrossRef]

Artal, P.

Atchison, D. A.

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]

D. A. Atchison, "Anterior corneal and internal contributions to peripheral aberrations of human eyes," J. Opt. Soc. Am. A 21, 335-359 (2004).
[CrossRef]

B. A. Moffat, D. A. Atchison, and J. M. Pope, "Explanation of the lens paradox," Optom. Vision Sci. 79, 148-150 (2002).
[CrossRef]

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

G. Smith and D. A. Atchison, "The gradient index and spherical aberration of the lens of the human eye," Ophthalmic Physiol. Opt. 21, 317-326 (2001).
[CrossRef] [PubMed]

D. A. Atchison and D. H. Scott, "Monochromatic aberrations of human eyes in the horizontal visual field," J. Opt. Soc. Am. A 19, 2180-2184 (1998).
[CrossRef]

G. Smith and D. A. Atchison, "Equivalent power of the crystalline lens of the human eye: comparison of methods of calculation," J. Opt. Soc. Am. A 14, 2537-2546 (1997).
[CrossRef]

R. L. Woods, A. Bradley, and D. A. Atchison, "Monocular diplopia caused by ocular aberrations and hyperopic defocus," Vision Res. 36, 3597-3606 (1996).
[CrossRef] [PubMed]

D. A. Atchison and G. Smith, "Continuous gradient-index and shell models of the human lens," Vision Res. 35, 2529-2538 (1995).
[CrossRef]

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]

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

Augusteyn, R. C.

B. K. Pierscionek, D. Y. C. Chan, J. P. Ennis, G. Smith, and R. C. Augusteyn, "Nondestructive method of constructing three-dimensional gradient index models for crystalline lenses: 1 theory and experiment," Am. J. Optom. Physiol. Opt. 65, 481-491 (1988).
[PubMed]

Bao, C.

M. V. Perez, C. Bao, M. T. Flores-Arias, M. A. Rama, and C. Gomez-Reino, "Description of gradient-index crystalline lens by a first-order optical system," J. Opt. A, Pure Appl. Opt. 7, 103-110 (2005).
[CrossRef]

Berrio, E.

Bescos, J.

Bille, J. F.

Blaker, J. W.

Bradley, A.

L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, "Statistical variations of aberration structure and image quality in a normal population of healthy eyes," J. Opt. Soc. Am. A 19, 2329-2348 (2002).
[CrossRef]

T. Salmon, L. N. Thibos, and A. Bradley, "Comparison of the eye's wave-front aberration measured psychophysically and with the Shack-Hartmann wave-front sensor," J. Opt. Soc. Am. A 15, 2457-2465 (1998).
[CrossRef]

L. N. Thibos, M. Ye, X. Zhang, and A. Bradley, "Spherical aberration of the reduced schematic eye with elliptical refracting surface," Optom. Vision Sci. 74, 548-556 (1997).
[CrossRef]

R. L. Woods, A. Bradley, and D. A. Atchison, "Monocular diplopia caused by ocular aberrations and hyperopic defocus," Vision Res. 36, 3597-3606 (1996).
[CrossRef] [PubMed]

M. Ye, X. X. Zhang, L. N. Thibos, and A. Bradley, "A new single-surface model eye that accurately predicts chromatic and spherical aberrations of the human eye," Invest. Ophthalmol. Visual Sci. 34, 777 (1993).

L. N. Thibos, M. Ye, X. X. Zhang, and A. Bradley, "The chromatic eye: a new model of ocular chromatic aberration," Appl. Opt. 31, 3594-3600 (1992).
[CrossRef] [PubMed]

Brennan, N. A.

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

H.-L. Liou and N. A. Brennan, "The prediction of spherical aberration with schematic eyes," Ophthalmic Physiol. Opt. 16, 348-354 (1996).
[CrossRef] [PubMed]

Brown, N.

N. Brown, "The change in lens curvature with age," Exp. Eye Res. 19, 175-183 (1974).
[CrossRef]

N. Brown, "The change in shape and internal form of the lens of the eye on accommodation," Exp. Eye Res. 15, 441-459 (1973).
[CrossRef] [PubMed]

Calatayud, M.

A. Lleó, A. Marcos, M. Calatayud, L. Alonso, S. M. Rahhal, and J. A. Sanchis-Gimeno, "The relationship between central corneal thickness and Goldmann applanation tonometry," Clin. Exp. Optom. 86, 104-108 (2003).
[CrossRef] [PubMed]

Calver, R.

G. Smith, M. J. Cox, R. Calver, and L. F. Garner, "The spherical aberration of the crystalline lens of the human eye," Vision Res. 41, 235-243 (2001).
[CrossRef] [PubMed]

Calver, R. I.

Campbell, M. C. W.

A. Glasser and M. C. W. Campbell, "Presbyopia and the optical changes in the human crystalline lens with age," Vision Res. 38, 209-229 (1998).
[CrossRef] [PubMed]

Chan, D. Y. C.

B. K. Pierscionek and D. Y. C. Chan, "Refractive index gradient of human lenses," Optom. Vision Sci. 66, 822-829 (1989).
[CrossRef]

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

B. K. Pierscionek, D. Y. C. Chan, J. P. Ennis, G. Smith, and R. C. Augusteyn, "Nondestructive method of constructing three-dimensional gradient index models for crystalline lenses: 1 theory and experiment," Am. J. Optom. Physiol. Opt. 65, 481-491 (1988).
[PubMed]

Chan, J.

A. K. C. Lam and J. Chan, "Corneal thickness at different reference points from Orbscan II system," Clin. Exp. Optom. 86, 230-234 (2003).
[CrossRef] [PubMed]

Chang, C. T.

C. S. Yu, D. Kao, and C. T. Chang, "Measurements of the length of the visual axis by ultrasonography in 1789 eyes," Chin. J. Ophthal. 15, 45-47 (1979).

Cheng, X.

Cook, C. A.

J. F. Koretz and C. A. Cook, "Aging of the optics of the human eye: lens refraction models and principal plane locations," Optom. Vision Sci. 78, 396-404 (2001).
[CrossRef]

J. F. Koretz, C. A. Cook, and P. L. Kaufman, "Aging of the human lens: changes in lens shape at zero-diopter accommodation," J. Opt. Soc. Am. A 18, 265-272 (2001).
[CrossRef]

J. F. Koretz, C. A. Cook, and J. R. Kuszak, "The zones of discontinuity in the human lens: development and distribution with age," Vision Res. 34, 2955-2962 (1994).
[CrossRef] [PubMed]

C. A. Cook and J. F. Koretz, "Acquisition of the curves of the human crystalline lens from slit lamp images: an application of the Hough transform," Appl. Opt. 30, 2088-2099 (1991).
[CrossRef] [PubMed]

Cox, I. G.

Cox, M. J.

G. Smith, M. J. Cox, R. Calver, and L. F. Garner, "The spherical aberration of the crystalline lens of the human eye," Vision Res. 41, 235-243 (2001).
[CrossRef] [PubMed]

R. I. Calver, M. J. Cox, and D. B. Elliott, "Effect of aging on the monochromatic aberrations of the human eye," J. Opt. Soc. Am. A 16, 2069-2078 (1999).
[CrossRef]

Dai, E.

L. Wang, E. Dai, D. D. Koch, and A. Nathoo, "Optical aberrations of the human anterior cornea," J. Cataract Refractive Surg. 29, 1514-1521 (2003).
[CrossRef]

Dainty, C.

A. V. Goncharov and C. Dainty are currently preparing a manuscript to be called "Aberrations of chromatic wide-field schematic eye model with a GRIN lens." (alexander.goncharov@nuigalway.ie)

Dorronsoro, C.

Douthwaite, W. A.

A. K. C. Lam and W. A. Douthwaite, "The aging effect on the central posterior corneal radius," Ophthalmic Physiol. Opt. 20, 63-69 (2000).
[CrossRef]

M. Sheridan and W. A. Douthwaite, "Meridional variations of corneal shape," Ophthalmic Physiol. Opt. 9, 235-238 (1989).
[CrossRef] [PubMed]

Dubbelman, M.

M. Dubbelman, G. L. Van der Heijde, H. A. Weeber, and G. F. J. M. Vrensen, "Changes in the internal structure of the human crystalline lens with age and accommodation," Vision Res. 43, 2363-2375 (2003).
[CrossRef] [PubMed]

M. Dubbelman, H. A. Weeber, R. G. L. van der Heijde, and H. J. Volker-Dieben, "Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography," Acta Ophthalmol. Scand. 80, 379-383 (2002).
[CrossRef] [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, 1867-1877 (2001).
[CrossRef] [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. Vision Sci. 78, 411-416 (2001).
[CrossRef]

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, 166-176 (1984).
[PubMed]

Edmund, C.

C. Edmund, "Posterior corneal curvature and its influence on corneal dioptric power," Acta Ophthalmol. 72, 715-720 (1994).

El Hage, S. G.

Y. Le Grand and S. G. El Hage, Physiological Optics (Springer-Verlag, 1980).

Elliott, D. B.

El-Messiery, M. A.

I. H. Al-Ahdali and M. A. El-Messiery, "Examination of the effect of the fibrous structure of a lens on the optical characteristics of the human eye: a computer-simulated model," Appl. Opt. 25, 5738-5745 (1995).
[CrossRef]

Ennis, J. P.

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

B. K. Pierscionek, D. Y. C. Chan, J. P. Ennis, G. Smith, and R. C. Augusteyn, "Nondestructive method of constructing three-dimensional gradient index models for crystalline lenses: 1 theory and experiment," Am. J. Optom. Physiol. Opt. 65, 481-491 (1988).
[PubMed]

Escudero-Sanz, I.

Fagerholm, P. P.

P. P. Fagerholm, B. T. Philipson, and B. Lindstrom, "Normal human lens, the distribution of protein," Exp. Eye Res. 33, 615-620 (1981).
[CrossRef] [PubMed]

Flores-Arias, M. T.

M. V. Perez, C. Bao, M. T. Flores-Arias, M. A. Rama, and C. Gomez-Reino, "Description of gradient-index crystalline lens by a first-order optical system," J. Opt. A, Pure Appl. Opt. 7, 103-110 (2005).
[CrossRef]

Garner, L. F.

G. Smith, M. J. Cox, R. Calver, and L. F. Garner, "The spherical aberration of the crystalline lens of the human eye," Vision Res. 41, 235-243 (2001).
[CrossRef] [PubMed]

L. F. Garner, C. S. Ooi, and G. Smith, "Refractive index of the crystalline lens in young and aged eyes," Clin. Exp. Optom. 81, 145-150 (1998).
[CrossRef]

R. P. Hemenger, L. F. Garner, and C. S. Ooi, "Changes with age of the refractive index gradient of the human ocular lens," Invest. Ophthalmol. Visual Sci. 36, 703-707 (1995).

Glasser, A.

A. Glasser and M. C. W. Campbell, "Presbyopia and the optical changes in the human crystalline lens with age," Vision Res. 38, 209-229 (1998).
[CrossRef] [PubMed]

Goez, S.

Gomez-Reino, C.

M. V. Perez, C. Bao, M. T. Flores-Arias, M. A. Rama, and C. Gomez-Reino, "Description of gradient-index crystalline lens by a first-order optical system," J. Opt. A, Pure Appl. Opt. 7, 103-110 (2005).
[CrossRef]

Goncharov, A. V.

A. V. Goncharov and C. Dainty are currently preparing a manuscript to be called "Aberrations of chromatic wide-field schematic eye model with a GRIN lens." (alexander.goncharov@nuigalway.ie)

González, L.

R. Navarro, L. González, and J. L. Hernández, "Optics of the average normal cornea from general and canonical representations of its surface topography," J. Opt. Soc. Am. A 23, 219-232 (2006).
[CrossRef]

R. Navarro, L. González, and J. L. Hernández-Matamoros, "On the prediction of optical aberrations by personalized eye models," Optom. Vision Sci. 83, 371-381 (2006).
[CrossRef]

Grimm, B.

Grosvenor, T.

C. S. Ooi and T. Grosvenor, "Mechanisms of emmetropization in the aging eye," Optom. Vision Sci. 72, 60-66 (1995).
[CrossRef]

T. Grosvenor, "Reduction in axial length with age: an emmetropizing mechanism for the adult eye?" Am. J. Optom. Physiol. Opt. 64, 657-663 (1987).
[PubMed]

T. Grosvenor, "Changes in spherical refraction during the adult years," in Refractive Anomalies. Research and Clinical Applications, T.Grosvenor and M.C.Flom, eds. (Butterworth-Heinemann, 1991), pp. 131-145.

Guillon, M.

M. Guillon, P. M. Lydon, and C. Wilson, "Corneal topography: a clinical model," Ophthalmic Physiol. Opt. 6, 47-56 (1986).
[CrossRef] [PubMed]

Guirao, A.

Gullstrand, A.

A. Gullstrand, "Appendix II," in Handbuch der Physiologischen Optik, 3rd ed. 1909, J.P.Southall trans., ed. (Optical Society of America, 1924), Vol. 1, pp. 351-352.

Gwiazda, J.

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Figures (9)

Fig. 1
Fig. 1

Longitudinal spherical aberration of the eye predicted by different models: 0, linear model based on Millodot and Sivak’s data [48]; 1, Liou and Brennan’s eye model with a GRIN lens [10]; 2, Indiana eye model with a single ellipsoidal refracting surface [50] ( k = 0.43 ) ; 3, Navarro’s wide-angle eye model [16]; 20U, 30U, and 40U models with an unbalanced GRIN lens; 20B, 30B, and 40B models with a balanced GRIN lens; 20S, 30S, and 40S models with a simplified GRIN lens.

Fig. 2
Fig. 2

Optimization of the eye model by reverse ray tracing. For clarity, optical systems are shown on both sides of the reference plane (vertical dashed line), where rays start to traverse backward.

Fig. 3
Fig. 3

Off-axis wavefront aberrations versus field angle ω for optimized models of the eye with an unbalanced GRIN lens. Navarro’s eye model is shown for comparison ( λ = 0.589 μ m ) .

Fig. 4
Fig. 4

Refractive index profiles in the peak plane (a) and sagittal plane (b) for the 20U, 30U, 40U, 20B, 30B, 40B, 20S, 30S, and 40S models.

Fig. 5
Fig. 5

Iso-indicial contours following the shape of the crystalline lens. The refractive index values are in increments of 0.002, starting from the anterior surface value n 0 and reaching the central contour at 1.411, 1.407, and 1.403 for the 20U, 30U, and 40U models, respectively.

Fig. 6
Fig. 6

Off-axis wavefront aberrations versus field angle ω for optimized 20B, 30B, and 40B models of the eye with a balanced GRIN lens ( λ = 0.589 μ m ) .

Fig. 7
Fig. 7

Iso-indicial contours following the anterior surface of the crystalline lens. The refractive index values are in increments of 0.004, starting from the surface value n 0 = 1.376 and reaching the central contour at 1.412 for all models ( n max = 1.416 ) .

Fig. 8
Fig. 8

Off-axis wavefront aberrations versus field angle ω for optimized 20S, 30S, and 40S models of the eye with a simplified GRIN lens ( λ = 0.589 μ m ) .

Fig. 9
Fig. 9

Iso-indicial contours within the crystalline lens. The refractive index values are in increments of 0.004, starting from the surface value n 0 = 1.362 and reaching the central contour at 1.398 for all models ( n max = 1.402 ) .

Tables (6)

Tables Icon

Table 1 Summary of Experimental Data for the Ocular and Corneal Spherical Aberrations of the Human Eye and Different Eye Models a

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Table 2 Effect of Aging on the Anatomical Structure of the Human Eye

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Table 3 Optical Parameters of the Wide-Field Eye Models with Unbalanced (U), Balanced (B), and Simplified (S) GRIN Lenses a

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Table 4 Refractive Index Coefficients for the Wide-Field Eye Models with Unbalanced (U) and Balanced (B) GRIN Lenses

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Table 5 Refractive Index Coefficients for the Wide-Field Eye Models with a Simplified (S) GRIN Lens

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Table 6 Equivalent Optical Power of the GRIN Lens in Media with Refractive Index n = n 0 and n = 1.336 (Natural Conditions)

Equations (34)

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

n a ( z , r ) = n 0 + n 1 r 2 + n 2 r 4 + n 3 z + n 4 z 2 + n 5 z 3 + n 6 z 4 ,
n p ( z , r ) = n max + n 1 r 2 + n 2 r 4 + n 3 , 2 z + n 4 , 2 z 2 + n 5 , 2 z 3 + n 6 , 2 z 4 ,
r 2 = 2 r a z ( 1 + k a ) z 2 .
r 2 = 2 r p t ( 1 + k p ) t 2 ,
z = t + d z m .
Δ n = ( n max n 0 ) ,
m = z m 2 ( 1 + k a ) 2 r a z m ,
n 1 = 2 Δ n m , n 2 = Δ n m 2 , n 3 = 4 Δ n r a m ,
n 4 = 2 Δ n [ 3 r a 2 ( r a z m ( 1 + k a ) ) 2 ] m 2 ,
n 5 = 4 Δ n ( 1 + k a ) r a m 2 , n 6 = Δ n ( 1 + k a ) 2 m 2 ,
n 3 , 2 = 0 ,
n 5 , 2 = 4 Δ n ( 1 + k p ) [ r p + ( d z m ) ( 1 + k p ) ] m 2 ,
n 6 , 2 = Δ n ( 1 + k p ) 2 m 2 ,
n 4 , 2 = ( 1 + k p ) n 1 4 n 2 r p 2 3 ( d z m ) n 5 , 2 6 ( d z m ) 2 n 6 , 2 .
( k p k a ) z m 2 2 [ d ( 1 + k p ) + r p r a ] z m + d [ d ( 1 + k p ) + 2 r p ] = 0 .
z m = d + r p ( 1 + k p ) .
n 1 = Δ n z m d 2 ( d 2 z m ) ( d z m ) m * ,
n 3 = 2 Δ n z m r a d 2 ( d 2 z m ) ( d z m ) m * ,
n 4 = Δ n d [ d 3 r a 3 d ( 3 r a + r p ) z m 2 + 4 ( 2 r a + r p ) z m 3 ] m * ,
n 5 = 2 Δ n [ d 3 r a d 2 ( 3 r a + r p ) z m + 2 ( r a + r p ) z m 3 ] m * ,
n 6 = Δ n [ d 2 r a 2 d ( 2 r a + r p ) z m + 3 ( r a + r p ) z m 2 ] m * ,
k a = 1 ( n 4 + 4 n 2 r a 2 ) m * [ Δ n z m d 2 ( d z m ) ( d 2 z m ) ] ,
k b = 1 + [ n 4 + d n 5 4 r p ( n 2 r p n 1 d ) ] m * [ Δ n z m d 2 ( d z m ) ( d 2 z m ) ] .
h ( z ) = h 0 ( 1 + g z 2 ) ,
F = 6 n 0 n 1 d ( 3 n 0 2 n 1 d 2 ) .
W 4 , 0 = S I ( 8 h 0 4 ) .
W 4 , 0 = W a + W b 1 + W b 2 + W p ,
W a = 500 ( n 1 + n 3 4 r a ) r a ,
W b 1 = 25 F 4 ( 6300 n 0 2 + 588 n 0 n 1 d 2 + n 1 d 3 ( 1260 n 3 + 1080 n 4 d + 945 n 5 d 2 + 840 n 6 d 3 184 n 1 d ) ) 1512 n 0 4 n 1 d ,
W b 2 = 25 F 4 n 2 ( 3780 n 0 3 n 1 d 2 3402 n 0 2 n 1 2 d 4 + 1236 n 0 n 1 3 d 6 187 n 1 4 d 8 2835 n 0 4 ) 1134 n 0 4 n 1 4 d 3 ,
W p = 500 [ n 1 + ( n 3 + 2 n 4 d + 3 n 5 d 2 + 4 n 6 d 3 ) 4 r p ] r p .
Δ F = n ( f + LSA ) n f ,
W 4 , 0 = 6 5 Z 4 0 ,
r c p = 0.87 r c a 0.24 .

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