Abstract

The lens is a complex optical component of the human eye because of its physiological structure: the surface is aspherical and the structural entities create a gradient refractive index (GRIN). Most existent models of the lens deal with its external shape independently of the refractive index and, subsequently, through optimization processes, adjust the imaging properties. In this paper, we propose a physiologically realistic GRIN model of the lens based on a single function for the whole lens that accurately describes different accommodative states simultaneously providing the corresponding refractive index distribution and the external shape of the lens by changing a single parameter that we associate with the function of the ciliary body. This simple, but highly accurate model, is incorporated into a schematic eye constructed with reported experimental biometric data and accommodation is simulated over a range of 0 to 6 diopters to select the optimum levels of image quality. Changes with accommodation in equatorial and total axial lens thicknesses, as well as aberrations, are found to lie within reported biometric data ranges.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  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).
  2. B. K. Pierscionek and D. Y. C. Chan, “Refractive index gradient of human lenses,” Optom. Vis. Sci. 66(12), 822–829 (1989).
    [Crossref]
  3. 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]
  4. 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]
  5. S. Ortiz, P. Pérez-Merino, E. Gambra, A. de Castro, and S. Marcos, “In vivo human crystalline lens topography,” Biomed. Opt. Express 3(10), 2471–2488 (2012).
    [Crossref]
  6. J. J. Esteve-Taboada, R. Montés-Micó, and T. Ferrer-Blasco, “Schematic eye models to mimic the behavior of the accommodating human eye,” J. Cataract Refractive Surg. 44(5), 627–641 (2018).
    [Crossref]
  7. B. K. Pierscionek and J. W. Regini, “The gradient index lens of the eye: an opto-biological synchrony,” Prog. Retinal Eye Res. 31(4), 332–349 (2012).
    [Crossref]
  8. G. Smith and D. A. Atchison, The Eye and Visual Optical Instruments (Cambridge University Press, 1997).
  9. D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinemann, 2000).
  10. H. H. Emsley, Visual Optics (Butterworth, 1952).
  11. Y. Le Grand and S. G. El Hage, Physiological Optics (Springer-Verlag, 1980).
  12. L. Matthiesen, “Ueber Begriff und Answerthung des sogenannten Totalindex der Krystalllinse,” Pfluegers Arch. 36(1), 72–100 (1885).
    [Crossref]
  13. H. Helmholtz, Helmholtz’s Treatise on Physiological Optics (Dover, 1962), Vol. 1. Appendix II.
  14. D. A. Atchison and L. N. Thibos, “Optical models of the human eye,” Clin. Exp. Optom. 99(2), 99–106 (2016).
    [Crossref]
  15. Y. Wu, A. Liu, H. Lv, X. Yi, Q. Li, X. Wang, Y. Ding, and J. Tong, “Finite Schematic Eye Model with Maxwell Fish-eye Spherical lens,” in 2010 Symposium on Photonics and Optoelectronics (IEEE, 2010), pp. 1–4.
  16. R. G. Zainullin, A. B. Kravtsov, and E. P. Shaitor, “The crystalline lens as a Luneburg lens,” Biofizica 19, 913–915 (1974).
  17. T. Liu and L. N. Thibos, “Customized models of ocular aberrations across the visual field during accommodation.,” J. Vis. 19(9), 13 (2019).
    [Crossref]
  18. 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]
  19. R. A. Schachar, “Growth patterns of fresh human crystalline lenses measured by in vitro photographic biometry,” J. Anat. 206, 575–580 (2005).
    [Crossref]
  20. 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).
    [Crossref]
  21. Y. Shao, A. Tao, H. Jiang, M. Shen, J. Zhong, F. Lu, and J. Wang, “Simultaneous real-time imaging of the ocular anterior segment including the ciliary muscle during accommodation,” Biomed. Opt. Express 4(3), 466–480 (2013).
    [Crossref]
  22. M. Shen, L. Cui, M. Li, D. Zhu, M. R. Wang, and J. Wang, “Extended scan depth optical coherence tomography for evaluating ocular surface shape,” J. Biomed. Opt. 16(5), 056007 (2011).
    [Crossref]
  23. C. Du, D. Zhu, M. Shen, M. Li, M. R. Wang, and J. Wang, “Novel optical coherence tomography for imaging the entire anterior segment of the eye,” Invest. Ophthalmol. Visual Sci. 52(2), 987 (2011).
    [Crossref]
  24. L. A. Lossing, L. T. Sinnott, C. Y. Kao, K. Richdale, and M. D. Bailey, “Measuring changes in ciliary muscle thickness with accommodation in young adults,” Optom. Vis. Sci. 89(5), 719–726 (2012).
    [Crossref]
  25. 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]
  26. D. Siedlecki, A. de Castro, E. Gambra, S. Ortiz, D. Borja, S. Uhlhorn, F. Manns, S. Marcos, and J. M. Parel, “Distortion correction of OCT images of the crystalline lens: gradient index approach,” Optom. Vis. Sci. 89(5), E709–E718 (2012).
    [Crossref]
  27. A. de Castro, J. Birkenfeld, B. M. Heilman, M. Ruggeri, E. Arrieta, J. M. Parel, F. Manns, and S. Marcos, “Off-axis optical coherence tomography imaging of the crystalline lens to reconstruct the gradient refractive index using optical methods,” Biomed. Opt. Express 10(7), 3622–3634 (2019).
    [Crossref]
  28. J. Yao, J. Huang, P. Meemon, M. Ponting, and J. P. Rolland, “Simultaneous estimation of thickness and refractive index of layered gradient refractive index optics using a hybrid confocal-scan swept-source optical coherence tomography system,” Opt. Express 23(23), 30149–30164 (2015).
    [Crossref]
  29. 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]
  30. A. V. Goncharov and C. Dainty, “Wide-field schematic eye models with gradient index of the lens,” J. Opt. Soc. Am. A 24(8), 2157–2174 (2007).
    [Crossref]
  31. R. Navarro, F. Palos, and L. González, “Adaptive model of the gradient index of the human lens. I. Formulation and model of aging ex vivo lenses,” J. Opt. Soc. Am. A 24(8), 2175–2185 (2007).
    [Crossref]
  32. 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]
  33. M. Bahrami and A. V. Goncharov, “Geometry-invariant GRIN lens: analytical ray tracing,” J. Biomed. Opt. 17(5), 055001 (2012).
    [Crossref]
  34. C. J. Sheil, M. Bahrami, and A. V. Goncharov, “An analytical method for predicting the geometrical and optical properties of the human lens under accommodation,” Biomed. Opt. Express 5(5), 1649–1663 (2014).
    [Crossref]
  35. C. J. Sheil and A. V. Goncharov, “Accommodating volume-constant age-dependent optical (AVOCADO) model of the crystalline GRIN lens,” Biomed. Opt. Express 7(5), 1985–1999 (2016).
    [Crossref]
  36. J. E. Gómez-Correa, S. E. Balderas-Mata, B. K. Pierscionek, and S. Chávez-Cerda, “Composite modified Luneburg model of human eye lens,” Opt. Lett. 40(17), 3990–3993 (2015).
    [Crossref]
  37. J. E. Gómez-Correa, V. Coello, A. Garza-Rivera, N. P. Puente, and S. Chávez-Cerda, “Three-dimensional ray tracing in spherical and elliptical generalized Luneburg lenses for application in the human eye lens,” Appl. Opt. 55(8), 2002–2010 (2016).
    [Crossref]
  38. L. Moser, “Über das Auge,” Dove’s Rep. Phys. 5, 337–349 (1844).
  39. H. Helmholtz, Helmholtz’s Treatise on Physiological Optics (Dover, 1962), Vol. 1.
  40. J. W. Blaker, “Toward an adaptive model of the human eye,” J. Opt. Soc. Am. 70(2), 220–223 (1980).
    [Crossref]
  41. R. Navarro, J. Santamaría, and J. Bescos, “Accommodation-dependent model of the human eye with aspherics,” J. Opt. Soc. Am. A 2(8), 1273–1281 (1985).
    [Crossref]
  42. Y. Huang and D. T. Moore, “Human eye modeling using a single equation of gradient index crystalline lens for relaxed and accommodated states,” Proc. SPIE 6342, 634201 (2006).
    [Crossref]
  43. H. T. Kasprzak, “New approximation for the whole profile of the human crystalline lens,” Oph. Phys. Optics 20(1), 31–43 (2000).
    [Crossref]
  44. 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]
  45. A. Popiolek-Masajada and H. Kasprzak, “Model of the optical system of the human eye during accommodation,” Oph. Phys. Optics 22(3), 201–208 (2002).
    [Crossref]
  46. A. Popiolek-Masajada and H. T. Kasprzak, “A new schematic eye model incorporating accommodation,” Optom. Vis. Sci. 76(10), 720–727 (1999).
    [Crossref]
  47. A. Khan, J. M. Pope, P. K. Verkicharla, M. Suheimat, and D. A. Atchison, “Change in human lens dimensions, lens refractive index distribution and ciliary body ring diameter with accommodation,” Biomed. Opt. Express 9(3), 1272–1282 (2018).
    [Crossref]
  48. T. Yamane, Statistics An Introductory Analysis (Harper & Row, 1967).
  49. F. A. Haight, Handbook of the Poisson Distribution (John Wiley, 1967).
  50. W. H. Carter, “Spot size and divergence for Hermite Gaussian beams of any order,” Appl. Opt. 19(7), 1027–1029 (1980).
    [Crossref]
  51. G. Smith, P. Bedggood, R. Ashman, M. Daaboul, and A. Metha, “Exploring ocular aberrations with a schematic human eye model,” Optom. Vis. Sci. 85(5), 330–340 (2008).
    [Crossref]
  52. E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Visual Sci. 50(1), 281–289 (2009).
    [Crossref]
  53. R. Navarro and N. López-Gil, “Impact of internal curvature gradient on the power and accommodation of the crystalline lens,” Optica 4(3), 334–340 (2017).
    [Crossref]
  54. D. Siedlecki, H. Kasprzak, and B. K. Pierscionek, “Schematic eye with a gradient-index lens and aspheric surfaces,” Opt. Lett. 29(11), 1197–1199 (2004).
    [Crossref]
  55. M. Guillon, D. P. Lydon, and C. Wilson, “Corneal topography: a clinical model,” Oph. Phys. Optics 6(1), 47–56 (1986).
    [Crossref]
  56. M. T. Sheehan, A. V. Goncharov, V. M. O’Dwyer, V. Toal, and C. Dainty, “Population study of the variation in monochromatic aberrations of the normal human eye over the central visual field,” Opt. Express 15(12), 7367–7380 (2007).
    [Crossref]
  57. H. Cheng, J. K. Barnett, A. S. Vilupuru, J. D. Marsack, S. Kasthurirangan, R. A. Applegate, and A. Roorda, “A population study on changes in wave aberrations with accomodation,” J. Vis. 4(8), 272–280 (2004).
    [Crossref]
  58. E. Tepichín-Rodríguez, A. S. Cruz Felix, E. López-Olazagasti, and S. Balderas-Mata, “Emmetropic eyes: objective performance and clinical reference,” Proc. SPIE 8785, 878551 (2013).
    [Crossref]
  59. P. Artal, ed., Handbook of Visual Optics: Fundamentals and Eye Optics (CRC Press, 2017).
  60. A. S. Vilupuru and A. Glasser, “Optical and biometric relationships of the isolated pig crystalline lens,” Oph. Phys. Optics 21(4), 296–311 (2001).
    [Crossref]
  61. F. Manns, V. Fernandez, S. Zipper, S. Sandadi, M. Hamaoui, A. Ho, and J. M. Parel, “Radius of curvature and asphericity of the anterior and posterior surface of human cadaver crystalline lenses,” Exp. Eye Res. 78(1), 39–51 (2004).
    [Crossref]
  62. A. Perez-Escudero, C. Dorronsoro, and S. Marcos, “Correlation between radius and asphericity in surfaces fitted by conics,” J. Opt. Soc. Am. A 27(7), 1541–1548 (2010).
    [Crossref]
  63. E. Martinez-Enriquez, M. Sun, M. Velasco-Ocana, J. Birkenfeld, P. Pérez-Merino, and S. Marcos, “Optical coherence tomography based estimates of crystalline lens volume, equatorial diameter, and plane position,” Invest. Ophthalmol. Visual Sci. 57(9), OCT600 (2016).
    [Crossref]
  64. R. Navarro, L. González, and J. L. Hernández-Matamoros, “On the prediction of optical aberrations by personalized eye models,” Optom. Vis. Sci. 83(6), 371–381 (2006).
    [Crossref]
  65. A. de Castro, J. Birkenfeld, B. Maceo, F. Manns, E. Arrieta, J. M. Parel, and S. Marcos, “Influence of shape and gradient refractive index in the accommodative changes of spherical aberration in nonhuman primate crystalline lenses,” Invest. Ophthalmol. Visual Sci. 54(9), 6197–6207 (2013).
    [Crossref]
  66. R. L. Phillips and L. C. Andrews, “Spot size and divergence for Laguerre Gaussian beams of any order,” Appl. Opt. 22(5), 643–644 (1983).
    [Crossref]
  67. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, 1972).

2019 (2)

2018 (2)

J. J. Esteve-Taboada, R. Montés-Micó, and T. Ferrer-Blasco, “Schematic eye models to mimic the behavior of the accommodating human eye,” J. Cataract Refractive Surg. 44(5), 627–641 (2018).
[Crossref]

A. Khan, J. M. Pope, P. K. Verkicharla, M. Suheimat, and D. A. Atchison, “Change in human lens dimensions, lens refractive index distribution and ciliary body ring diameter with accommodation,” Biomed. Opt. Express 9(3), 1272–1282 (2018).
[Crossref]

2017 (1)

2016 (4)

E. Martinez-Enriquez, M. Sun, M. Velasco-Ocana, J. Birkenfeld, P. Pérez-Merino, and S. Marcos, “Optical coherence tomography based estimates of crystalline lens volume, equatorial diameter, and plane position,” Invest. Ophthalmol. Visual Sci. 57(9), OCT600 (2016).
[Crossref]

D. A. Atchison and L. N. Thibos, “Optical models of the human eye,” Clin. Exp. Optom. 99(2), 99–106 (2016).
[Crossref]

C. J. Sheil and A. V. Goncharov, “Accommodating volume-constant age-dependent optical (AVOCADO) model of the crystalline GRIN lens,” Biomed. Opt. Express 7(5), 1985–1999 (2016).
[Crossref]

J. E. Gómez-Correa, V. Coello, A. Garza-Rivera, N. P. Puente, and S. Chávez-Cerda, “Three-dimensional ray tracing in spherical and elliptical generalized Luneburg lenses for application in the human eye lens,” Appl. Opt. 55(8), 2002–2010 (2016).
[Crossref]

2015 (2)

2014 (1)

2013 (3)

Y. Shao, A. Tao, H. Jiang, M. Shen, J. Zhong, F. Lu, and J. Wang, “Simultaneous real-time imaging of the ocular anterior segment including the ciliary muscle during accommodation,” Biomed. Opt. Express 4(3), 466–480 (2013).
[Crossref]

E. Tepichín-Rodríguez, A. S. Cruz Felix, E. López-Olazagasti, and S. Balderas-Mata, “Emmetropic eyes: objective performance and clinical reference,” Proc. SPIE 8785, 878551 (2013).
[Crossref]

A. de Castro, J. Birkenfeld, B. Maceo, F. Manns, E. Arrieta, J. M. Parel, and S. Marcos, “Influence of shape and gradient refractive index in the accommodative changes of spherical aberration in nonhuman primate crystalline lenses,” Invest. Ophthalmol. Visual Sci. 54(9), 6197–6207 (2013).
[Crossref]

2012 (5)

L. A. Lossing, L. T. Sinnott, C. Y. Kao, K. Richdale, and M. D. Bailey, “Measuring changes in ciliary muscle thickness with accommodation in young adults,” Optom. Vis. Sci. 89(5), 719–726 (2012).
[Crossref]

S. Ortiz, P. Pérez-Merino, E. Gambra, A. de Castro, and S. Marcos, “In vivo human crystalline lens topography,” Biomed. Opt. Express 3(10), 2471–2488 (2012).
[Crossref]

B. K. Pierscionek and J. W. Regini, “The gradient index lens of the eye: an opto-biological synchrony,” Prog. Retinal Eye Res. 31(4), 332–349 (2012).
[Crossref]

M. Bahrami and A. V. Goncharov, “Geometry-invariant GRIN lens: analytical ray tracing,” J. Biomed. Opt. 17(5), 055001 (2012).
[Crossref]

D. Siedlecki, A. de Castro, E. Gambra, S. Ortiz, D. Borja, S. Uhlhorn, F. Manns, S. Marcos, and J. M. Parel, “Distortion correction of OCT images of the crystalline lens: gradient index approach,” Optom. Vis. Sci. 89(5), E709–E718 (2012).
[Crossref]

2011 (2)

M. Shen, L. Cui, M. Li, D. Zhu, M. R. Wang, and J. Wang, “Extended scan depth optical coherence tomography for evaluating ocular surface shape,” J. Biomed. Opt. 16(5), 056007 (2011).
[Crossref]

C. Du, D. Zhu, M. Shen, M. Li, M. R. Wang, and J. Wang, “Novel optical coherence tomography for imaging the entire anterior segment of the eye,” Invest. Ophthalmol. Visual Sci. 52(2), 987 (2011).
[Crossref]

2010 (2)

2009 (1)

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Visual Sci. 50(1), 281–289 (2009).
[Crossref]

2008 (3)

G. Smith, P. Bedggood, R. Ashman, M. Daaboul, and A. Metha, “Exploring ocular aberrations with a schematic human eye model,” Optom. Vis. Sci. 85(5), 330–340 (2008).
[Crossref]

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]

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]

2007 (4)

2006 (3)

R. Navarro, L. González, and J. L. Hernández-Matamoros, “On the prediction of optical aberrations by personalized eye models,” Optom. Vis. Sci. 83(6), 371–381 (2006).
[Crossref]

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).
[Crossref]

Y. Huang and D. T. Moore, “Human eye modeling using a single equation of gradient index crystalline lens for relaxed and accommodated states,” Proc. SPIE 6342, 634201 (2006).
[Crossref]

2005 (2)

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]

R. A. Schachar, “Growth patterns of fresh human crystalline lenses measured by in vitro photographic biometry,” J. Anat. 206, 575–580 (2005).
[Crossref]

2004 (3)

F. Manns, V. Fernandez, S. Zipper, S. Sandadi, M. Hamaoui, A. Ho, and J. M. Parel, “Radius of curvature and asphericity of the anterior and posterior surface of human cadaver crystalline lenses,” Exp. Eye Res. 78(1), 39–51 (2004).
[Crossref]

H. Cheng, J. K. Barnett, A. S. Vilupuru, J. D. Marsack, S. Kasthurirangan, R. A. Applegate, and A. Roorda, “A population study on changes in wave aberrations with accomodation,” J. Vis. 4(8), 272–280 (2004).
[Crossref]

D. Siedlecki, H. Kasprzak, and B. K. Pierscionek, “Schematic eye with a gradient-index lens and aspheric surfaces,” Opt. Lett. 29(11), 1197–1199 (2004).
[Crossref]

2002 (2)

A. Popiolek-Masajada and H. Kasprzak, “Model of the optical system of the human eye during accommodation,” Oph. Phys. Optics 22(3), 201–208 (2002).
[Crossref]

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]

2001 (1)

A. S. Vilupuru and A. Glasser, “Optical and biometric relationships of the isolated pig crystalline lens,” Oph. Phys. Optics 21(4), 296–311 (2001).
[Crossref]

2000 (1)

H. T. Kasprzak, “New approximation for the whole profile of the human crystalline lens,” Oph. Phys. Optics 20(1), 31–43 (2000).
[Crossref]

1999 (1)

A. Popiolek-Masajada and H. T. Kasprzak, “A new schematic eye model incorporating accommodation,” Optom. Vis. Sci. 76(10), 720–727 (1999).
[Crossref]

1997 (1)

1989 (1)

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

1986 (1)

M. Guillon, D. P. Lydon, and C. Wilson, “Corneal topography: a clinical model,” Oph. Phys. Optics 6(1), 47–56 (1986).
[Crossref]

1985 (1)

1983 (1)

1980 (2)

1974 (1)

R. G. Zainullin, A. B. Kravtsov, and E. P. Shaitor, “The crystalline lens as a Luneburg lens,” Biofizica 19, 913–915 (1974).

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).

1885 (1)

L. Matthiesen, “Ueber Begriff und Answerthung des sogenannten Totalindex der Krystalllinse,” Pfluegers Arch. 36(1), 72–100 (1885).
[Crossref]

1844 (1)

L. Moser, “Über das Auge,” Dove’s Rep. Phys. 5, 337–349 (1844).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, 1972).

Andrews, L. C.

Applegate, R. A.

H. Cheng, J. K. Barnett, A. S. Vilupuru, J. D. Marsack, S. Kasthurirangan, R. A. Applegate, and A. Roorda, “A population study on changes in wave aberrations with accomodation,” J. Vis. 4(8), 272–280 (2004).
[Crossref]

Arasa, J.

Arrieta, E.

A. de Castro, J. Birkenfeld, B. M. Heilman, M. Ruggeri, E. Arrieta, J. M. Parel, F. Manns, and S. Marcos, “Off-axis optical coherence tomography imaging of the crystalline lens to reconstruct the gradient refractive index using optical methods,” Biomed. Opt. Express 10(7), 3622–3634 (2019).
[Crossref]

A. de Castro, J. Birkenfeld, B. Maceo, F. Manns, E. Arrieta, J. M. Parel, and S. Marcos, “Influence of shape and gradient refractive index in the accommodative changes of spherical aberration in nonhuman primate crystalline lenses,” Invest. Ophthalmol. Visual Sci. 54(9), 6197–6207 (2013).
[Crossref]

Ashman, R.

G. Smith, P. Bedggood, R. Ashman, M. Daaboul, and A. Metha, “Exploring ocular aberrations with a schematic human eye model,” Optom. Vis. Sci. 85(5), 330–340 (2008).
[Crossref]

Atchison, D. A.

A. Khan, J. M. Pope, P. K. Verkicharla, M. Suheimat, and D. A. Atchison, “Change in human lens dimensions, lens refractive index distribution and ciliary body ring diameter with accommodation,” Biomed. Opt. Express 9(3), 1272–1282 (2018).
[Crossref]

D. A. Atchison and L. N. Thibos, “Optical models of the human eye,” Clin. Exp. Optom. 99(2), 99–106 (2016).
[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]

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]

G. Smith and D. A. Atchison, The Eye and Visual Optical Instruments (Cambridge University Press, 1997).

D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinemann, 2000).

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).
[Crossref]

Bahrami, M.

Bailey, M. D.

L. A. Lossing, L. T. Sinnott, C. Y. Kao, K. Richdale, and M. D. Bailey, “Measuring changes in ciliary muscle thickness with accommodation in young adults,” Optom. Vis. Sci. 89(5), 719–726 (2012).
[Crossref]

Balderas-Mata, S.

E. Tepichín-Rodríguez, A. S. Cruz Felix, E. López-Olazagasti, and S. Balderas-Mata, “Emmetropic eyes: objective performance and clinical reference,” Proc. SPIE 8785, 878551 (2013).
[Crossref]

Balderas-Mata, S. E.

Barnett, J. K.

H. Cheng, J. K. Barnett, A. S. Vilupuru, J. D. Marsack, S. Kasthurirangan, R. A. Applegate, and A. Roorda, “A population study on changes in wave aberrations with accomodation,” J. Vis. 4(8), 272–280 (2004).
[Crossref]

Bedggood, P.

G. Smith, P. Bedggood, R. Ashman, M. Daaboul, and A. Metha, “Exploring ocular aberrations with a schematic human eye model,” Optom. Vis. Sci. 85(5), 330–340 (2008).
[Crossref]

Bescos, J.

Birkenfeld, J.

A. de Castro, J. Birkenfeld, B. M. Heilman, M. Ruggeri, E. Arrieta, J. M. Parel, F. Manns, and S. Marcos, “Off-axis optical coherence tomography imaging of the crystalline lens to reconstruct the gradient refractive index using optical methods,” Biomed. Opt. Express 10(7), 3622–3634 (2019).
[Crossref]

E. Martinez-Enriquez, M. Sun, M. Velasco-Ocana, J. Birkenfeld, P. Pérez-Merino, and S. Marcos, “Optical coherence tomography based estimates of crystalline lens volume, equatorial diameter, and plane position,” Invest. Ophthalmol. Visual Sci. 57(9), OCT600 (2016).
[Crossref]

A. de Castro, J. Birkenfeld, B. Maceo, F. Manns, E. Arrieta, J. M. Parel, and S. Marcos, “Influence of shape and gradient refractive index in the accommodative changes of spherical aberration in nonhuman primate crystalline lenses,” Invest. Ophthalmol. Visual Sci. 54(9), 6197–6207 (2013).
[Crossref]

Blaker, J. W.

Borja, D.

D. Siedlecki, A. de Castro, E. Gambra, S. Ortiz, D. Borja, S. Uhlhorn, F. Manns, S. Marcos, and J. M. Parel, “Distortion correction of OCT images of the crystalline lens: gradient index approach,” Optom. Vis. Sci. 89(5), E709–E718 (2012).
[Crossref]

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]

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).
[Crossref]

Brennan, N. A.

Carter, W. H.

Chan, D. Y. C.

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

Chávez-Cerda, S.

Cheng, H.

H. Cheng, J. K. Barnett, A. S. Vilupuru, J. D. Marsack, S. Kasthurirangan, R. A. Applegate, and A. Roorda, “A population study on changes in wave aberrations with accomodation,” J. Vis. 4(8), 272–280 (2004).
[Crossref]

Coello, V.

Cruz Felix, A. S.

E. Tepichín-Rodríguez, A. S. Cruz Felix, E. López-Olazagasti, and S. Balderas-Mata, “Emmetropic eyes: objective performance and clinical reference,” Proc. SPIE 8785, 878551 (2013).
[Crossref]

Cui, L.

M. Shen, L. Cui, M. Li, D. Zhu, M. R. Wang, and J. Wang, “Extended scan depth optical coherence tomography for evaluating ocular surface shape,” J. Biomed. Opt. 16(5), 056007 (2011).
[Crossref]

Daaboul, M.

G. Smith, P. Bedggood, R. Ashman, M. Daaboul, and A. Metha, “Exploring ocular aberrations with a schematic human eye model,” Optom. Vis. Sci. 85(5), 330–340 (2008).
[Crossref]

Dainty, C.

de Castro, A.

A. de Castro, J. Birkenfeld, B. M. Heilman, M. Ruggeri, E. Arrieta, J. M. Parel, F. Manns, and S. Marcos, “Off-axis optical coherence tomography imaging of the crystalline lens to reconstruct the gradient refractive index using optical methods,” Biomed. Opt. Express 10(7), 3622–3634 (2019).
[Crossref]

A. de Castro, J. Birkenfeld, B. Maceo, F. Manns, E. Arrieta, J. M. Parel, and S. Marcos, “Influence of shape and gradient refractive index in the accommodative changes of spherical aberration in nonhuman primate crystalline lenses,” Invest. Ophthalmol. Visual Sci. 54(9), 6197–6207 (2013).
[Crossref]

D. Siedlecki, A. de Castro, E. Gambra, S. Ortiz, D. Borja, S. Uhlhorn, F. Manns, S. Marcos, and J. M. Parel, “Distortion correction of OCT images of the crystalline lens: gradient index approach,” Optom. Vis. Sci. 89(5), E709–E718 (2012).
[Crossref]

S. Ortiz, P. Pérez-Merino, E. Gambra, A. de Castro, and S. Marcos, “In vivo human crystalline lens topography,” Biomed. Opt. Express 3(10), 2471–2488 (2012).
[Crossref]

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).
[Crossref]

Díaz, J. A.

Ding, Y.

Y. Wu, A. Liu, H. Lv, X. Yi, Q. Li, X. Wang, Y. Ding, and J. Tong, “Finite Schematic Eye Model with Maxwell Fish-eye Spherical lens,” in 2010 Symposium on Photonics and Optoelectronics (IEEE, 2010), pp. 1–4.

Dorronsoro, C.

Du, C.

C. Du, D. Zhu, M. Shen, M. Li, M. R. Wang, and J. Wang, “Novel optical coherence tomography for imaging the entire anterior segment of the eye,” Invest. Ophthalmol. Visual Sci. 52(2), 987 (2011).
[Crossref]

Dubbelman, M.

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Visual Sci. 50(1), 281–289 (2009).
[Crossref]

El Hage, S. G.

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

Emsley, H. H.

H. H. Emsley, Visual Optics (Butterworth, 1952).

Esteve-Taboada, J. J.

J. J. Esteve-Taboada, R. Montés-Micó, and T. Ferrer-Blasco, “Schematic eye models to mimic the behavior of the accommodating human eye,” J. Cataract Refractive Surg. 44(5), 627–641 (2018).
[Crossref]

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).
[Crossref]

F. Manns, V. Fernandez, S. Zipper, S. Sandadi, M. Hamaoui, A. Ho, and J. M. Parel, “Radius of curvature and asphericity of the anterior and posterior surface of human cadaver crystalline lenses,” Exp. Eye Res. 78(1), 39–51 (2004).
[Crossref]

Ferrer-Blasco, T.

J. J. Esteve-Taboada, R. Montés-Micó, and T. Ferrer-Blasco, “Schematic eye models to mimic the behavior of the accommodating human eye,” J. Cataract Refractive Surg. 44(5), 627–641 (2018).
[Crossref]

Gambra, E.

D. Siedlecki, A. de Castro, E. Gambra, S. Ortiz, D. Borja, S. Uhlhorn, F. Manns, S. Marcos, and J. M. Parel, “Distortion correction of OCT images of the crystalline lens: gradient index approach,” Optom. Vis. Sci. 89(5), E709–E718 (2012).
[Crossref]

S. Ortiz, P. Pérez-Merino, E. Gambra, A. de Castro, and S. Marcos, “In vivo human crystalline lens topography,” Biomed. Opt. Express 3(10), 2471–2488 (2012).
[Crossref]

Garza-Rivera, A.

Glasser, A.

A. S. Vilupuru and A. Glasser, “Optical and biometric relationships of the isolated pig crystalline lens,” Oph. Phys. Optics 21(4), 296–311 (2001).
[Crossref]

Gómez-Correa, J. E.

Goncharov, A. V.

González, L.

R. Navarro, F. Palos, and L. González, “Adaptive model of the gradient index of the human lens. I. Formulation and model of aging ex vivo lenses,” J. Opt. Soc. Am. A 24(8), 2175–2185 (2007).
[Crossref]

R. Navarro, L. González, and J. L. Hernández-Matamoros, “On the prediction of optical aberrations by personalized eye models,” Optom. Vis. Sci. 83(6), 371–381 (2006).
[Crossref]

González, L. M.

Grulkowski, I.

Guillon, M.

M. Guillon, D. P. Lydon, and C. Wilson, “Corneal topography: a clinical model,” Oph. Phys. Optics 6(1), 47–56 (1986).
[Crossref]

Haight, F. A.

F. A. Haight, Handbook of the Poisson Distribution (John Wiley, 1967).

Hamaoui, M.

F. Manns, V. Fernandez, S. Zipper, S. Sandadi, M. Hamaoui, A. Ho, and J. M. Parel, “Radius of curvature and asphericity of the anterior and posterior surface of human cadaver crystalline lenses,” Exp. Eye Res. 78(1), 39–51 (2004).
[Crossref]

Heethaar, R. M.

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Visual Sci. 50(1), 281–289 (2009).
[Crossref]

Heilman, B. M.

Helmholtz, H.

H. Helmholtz, Helmholtz’s Treatise on Physiological Optics (Dover, 1962), Vol. 1.

H. Helmholtz, Helmholtz’s Treatise on Physiological Optics (Dover, 1962), Vol. 1. Appendix II.

Hermans, E. A.

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Visual Sci. 50(1), 281–289 (2009).
[Crossref]

Hernández-Matamoros, J. L.

R. Navarro, L. González, and J. L. Hernández-Matamoros, “On the prediction of optical aberrations by personalized eye models,” Optom. Vis. Sci. 83(6), 371–381 (2006).
[Crossref]

Ho, A.

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).
[Crossref]

F. Manns, V. Fernandez, S. Zipper, S. Sandadi, M. Hamaoui, A. Ho, and J. M. Parel, “Radius of curvature and asphericity of the anterior and posterior surface of human cadaver crystalline lenses,” Exp. Eye Res. 78(1), 39–51 (2004).
[Crossref]

Huang, J.

Huang, Y.

Y. Huang and D. T. Moore, “Human eye modeling using a single equation of gradient index crystalline lens for relaxed and accommodated states,” Proc. SPIE 6342, 634201 (2006).
[Crossref]

Iwata, K.

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).

Jiang, H.

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]

Kao, C. Y.

L. A. Lossing, L. T. Sinnott, C. Y. Kao, K. Richdale, and M. D. Bailey, “Measuring changes in ciliary muscle thickness with accommodation in young adults,” Optom. Vis. Sci. 89(5), 719–726 (2012).
[Crossref]

Kasprzak, H.

D. Siedlecki, H. Kasprzak, and B. K. Pierscionek, “Schematic eye with a gradient-index lens and aspheric surfaces,” Opt. Lett. 29(11), 1197–1199 (2004).
[Crossref]

A. Popiolek-Masajada and H. Kasprzak, “Model of the optical system of the human eye during accommodation,” Oph. Phys. Optics 22(3), 201–208 (2002).
[Crossref]

Kasprzak, H. T.

H. T. Kasprzak, “New approximation for the whole profile of the human crystalline lens,” Oph. Phys. Optics 20(1), 31–43 (2000).
[Crossref]

A. Popiolek-Masajada and H. T. Kasprzak, “A new schematic eye model incorporating accommodation,” Optom. Vis. Sci. 76(10), 720–727 (1999).
[Crossref]

Kasthurirangan, S.

H. Cheng, J. K. Barnett, A. S. Vilupuru, J. D. Marsack, S. Kasthurirangan, R. A. Applegate, and A. Roorda, “A population study on changes in wave aberrations with accomodation,” J. Vis. 4(8), 272–280 (2004).
[Crossref]

Khan, A.

Kravtsov, A. B.

R. G. Zainullin, A. B. Kravtsov, and E. P. Shaitor, “The crystalline lens as a Luneburg lens,” Biofizica 19, 913–915 (1974).

Kuijer, J. P.

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Visual Sci. 50(1), 281–289 (2009).
[Crossref]

Le Grand, Y.

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

Li, M.

C. Du, D. Zhu, M. Shen, M. Li, M. R. Wang, and J. Wang, “Novel optical coherence tomography for imaging the entire anterior segment of the eye,” Invest. Ophthalmol. Visual Sci. 52(2), 987 (2011).
[Crossref]

M. Shen, L. Cui, M. Li, D. Zhu, M. R. Wang, and J. Wang, “Extended scan depth optical coherence tomography for evaluating ocular surface shape,” J. Biomed. Opt. 16(5), 056007 (2011).
[Crossref]

Li, Q.

Y. Wu, A. Liu, H. Lv, X. Yi, Q. Li, X. Wang, Y. Ding, and J. Tong, “Finite Schematic Eye Model with Maxwell Fish-eye Spherical lens,” in 2010 Symposium on Photonics and Optoelectronics (IEEE, 2010), pp. 1–4.

Liou, H. L.

Liu, A.

Y. Wu, A. Liu, H. Lv, X. Yi, Q. Li, X. Wang, Y. Ding, and J. Tong, “Finite Schematic Eye Model with Maxwell Fish-eye Spherical lens,” in 2010 Symposium on Photonics and Optoelectronics (IEEE, 2010), pp. 1–4.

Liu, T.

T. Liu and L. N. Thibos, “Customized models of ocular aberrations across the visual field during accommodation.,” J. Vis. 19(9), 13 (2019).
[Crossref]

López-Gil, N.

López-Olazagasti, E.

E. Tepichín-Rodríguez, A. S. Cruz Felix, E. López-Olazagasti, and S. Balderas-Mata, “Emmetropic eyes: objective performance and clinical reference,” Proc. SPIE 8785, 878551 (2013).
[Crossref]

Lossing, L. A.

L. A. Lossing, L. T. Sinnott, C. Y. Kao, K. Richdale, and M. D. Bailey, “Measuring changes in ciliary muscle thickness with accommodation in young adults,” Optom. Vis. Sci. 89(5), 719–726 (2012).
[Crossref]

Lu, F.

Lv, H.

Y. Wu, A. Liu, H. Lv, X. Yi, Q. Li, X. Wang, Y. Ding, and J. Tong, “Finite Schematic Eye Model with Maxwell Fish-eye Spherical lens,” in 2010 Symposium on Photonics and Optoelectronics (IEEE, 2010), pp. 1–4.

Lydon, D. P.

M. Guillon, D. P. Lydon, and C. Wilson, “Corneal topography: a clinical model,” Oph. Phys. Optics 6(1), 47–56 (1986).
[Crossref]

Maceo, B.

A. de Castro, J. Birkenfeld, B. Maceo, F. Manns, E. Arrieta, J. M. Parel, and S. Marcos, “Influence of shape and gradient refractive index in the accommodative changes of spherical aberration in nonhuman primate crystalline lenses,” Invest. Ophthalmol. Visual Sci. 54(9), 6197–6207 (2013).
[Crossref]

Manns, F.

A. de Castro, J. Birkenfeld, B. M. Heilman, M. Ruggeri, E. Arrieta, J. M. Parel, F. Manns, and S. Marcos, “Off-axis optical coherence tomography imaging of the crystalline lens to reconstruct the gradient refractive index using optical methods,” Biomed. Opt. Express 10(7), 3622–3634 (2019).
[Crossref]

A. de Castro, J. Birkenfeld, B. Maceo, F. Manns, E. Arrieta, J. M. Parel, and S. Marcos, “Influence of shape and gradient refractive index in the accommodative changes of spherical aberration in nonhuman primate crystalline lenses,” Invest. Ophthalmol. Visual Sci. 54(9), 6197–6207 (2013).
[Crossref]

D. Siedlecki, A. de Castro, E. Gambra, S. Ortiz, D. Borja, S. Uhlhorn, F. Manns, S. Marcos, and J. M. Parel, “Distortion correction of OCT images of the crystalline lens: gradient index approach,” Optom. Vis. Sci. 89(5), E709–E718 (2012).
[Crossref]

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]

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).
[Crossref]

F. Manns, V. Fernandez, S. Zipper, S. Sandadi, M. Hamaoui, A. Ho, and J. M. Parel, “Radius of curvature and asphericity of the anterior and posterior surface of human cadaver crystalline lenses,” Exp. Eye Res. 78(1), 39–51 (2004).
[Crossref]

Marcos, S.

A. de Castro, J. Birkenfeld, B. M. Heilman, M. Ruggeri, E. Arrieta, J. M. Parel, F. Manns, and S. Marcos, “Off-axis optical coherence tomography imaging of the crystalline lens to reconstruct the gradient refractive index using optical methods,” Biomed. Opt. Express 10(7), 3622–3634 (2019).
[Crossref]

E. Martinez-Enriquez, M. Sun, M. Velasco-Ocana, J. Birkenfeld, P. Pérez-Merino, and S. Marcos, “Optical coherence tomography based estimates of crystalline lens volume, equatorial diameter, and plane position,” Invest. Ophthalmol. Visual Sci. 57(9), OCT600 (2016).
[Crossref]

A. de Castro, J. Birkenfeld, B. Maceo, F. Manns, E. Arrieta, J. M. Parel, and S. Marcos, “Influence of shape and gradient refractive index in the accommodative changes of spherical aberration in nonhuman primate crystalline lenses,” Invest. Ophthalmol. Visual Sci. 54(9), 6197–6207 (2013).
[Crossref]

D. Siedlecki, A. de Castro, E. Gambra, S. Ortiz, D. Borja, S. Uhlhorn, F. Manns, S. Marcos, and J. M. Parel, “Distortion correction of OCT images of the crystalline lens: gradient index approach,” Optom. Vis. Sci. 89(5), E709–E718 (2012).
[Crossref]

S. Ortiz, P. Pérez-Merino, E. Gambra, A. de Castro, and S. Marcos, “In vivo human crystalline lens topography,” Biomed. Opt. Express 3(10), 2471–2488 (2012).
[Crossref]

A. Perez-Escudero, C. Dorronsoro, and S. Marcos, “Correlation between radius and asphericity in surfaces fitted by conics,” J. Opt. Soc. Am. A 27(7), 1541–1548 (2010).
[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]

Marsack, J. D.

H. Cheng, J. K. Barnett, A. S. Vilupuru, J. D. Marsack, S. Kasthurirangan, R. A. Applegate, and A. Roorda, “A population study on changes in wave aberrations with accomodation,” J. Vis. 4(8), 272–280 (2004).
[Crossref]

Martinez-Enriquez, E.

E. Martinez-Enriquez, M. Sun, M. Velasco-Ocana, J. Birkenfeld, P. Pérez-Merino, and S. Marcos, “Optical coherence tomography based estimates of crystalline lens volume, equatorial diameter, and plane position,” Invest. Ophthalmol. Visual Sci. 57(9), OCT600 (2016).
[Crossref]

Matthiesen, L.

L. Matthiesen, “Ueber Begriff und Answerthung des sogenannten Totalindex der Krystalllinse,” Pfluegers Arch. 36(1), 72–100 (1885).
[Crossref]

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]

Meemon, P.

Metha, A.

G. Smith, P. Bedggood, R. Ashman, M. Daaboul, and A. Metha, “Exploring ocular aberrations with a schematic human eye model,” Optom. Vis. Sci. 85(5), 330–340 (2008).
[Crossref]

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]

Montés-Micó, R.

J. J. Esteve-Taboada, R. Montés-Micó, and T. Ferrer-Blasco, “Schematic eye models to mimic the behavior of the accommodating human eye,” J. Cataract Refractive Surg. 44(5), 627–641 (2018).
[Crossref]

Moore, D. T.

Y. Huang and D. T. Moore, “Human eye modeling using a single equation of gradient index crystalline lens for relaxed and accommodated states,” Proc. SPIE 6342, 634201 (2006).
[Crossref]

Moser, L.

L. Moser, “Über das Auge,” Dove’s Rep. Phys. 5, 337–349 (1844).

Nagata, R.

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).

Nakao, S.

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).

Navarro, R.

O’Dwyer, V. M.

Ono, T.

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).

Ortiz, S.

Palos, F.

Parel, J. M.

A. de Castro, J. Birkenfeld, B. M. Heilman, M. Ruggeri, E. Arrieta, J. M. Parel, F. Manns, and S. Marcos, “Off-axis optical coherence tomography imaging of the crystalline lens to reconstruct the gradient refractive index using optical methods,” Biomed. Opt. Express 10(7), 3622–3634 (2019).
[Crossref]

A. de Castro, J. Birkenfeld, B. Maceo, F. Manns, E. Arrieta, J. M. Parel, and S. Marcos, “Influence of shape and gradient refractive index in the accommodative changes of spherical aberration in nonhuman primate crystalline lenses,” Invest. Ophthalmol. Visual Sci. 54(9), 6197–6207 (2013).
[Crossref]

D. Siedlecki, A. de Castro, E. Gambra, S. Ortiz, D. Borja, S. Uhlhorn, F. Manns, S. Marcos, and J. M. Parel, “Distortion correction of OCT images of the crystalline lens: gradient index approach,” Optom. Vis. Sci. 89(5), E709–E718 (2012).
[Crossref]

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]

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).
[Crossref]

F. Manns, V. Fernandez, S. Zipper, S. Sandadi, M. Hamaoui, A. Ho, and J. M. Parel, “Radius of curvature and asphericity of the anterior and posterior surface of human cadaver crystalline lenses,” Exp. Eye Res. 78(1), 39–51 (2004).
[Crossref]

Pascual, D.

Perez-Escudero, A.

Pérez-Merino, P.

E. Martinez-Enriquez, M. Sun, M. Velasco-Ocana, J. Birkenfeld, P. Pérez-Merino, and S. Marcos, “Optical coherence tomography based estimates of crystalline lens volume, equatorial diameter, and plane position,” Invest. Ophthalmol. Visual Sci. 57(9), OCT600 (2016).
[Crossref]

S. Ortiz, P. Pérez-Merino, E. Gambra, A. de Castro, and S. Marcos, “In vivo human crystalline lens topography,” Biomed. Opt. Express 3(10), 2471–2488 (2012).
[Crossref]

Phillips, R. L.

Pierscionek, B. K.

J. E. Gómez-Correa, S. E. Balderas-Mata, B. K. Pierscionek, and S. Chávez-Cerda, “Composite modified Luneburg model of human eye lens,” Opt. Lett. 40(17), 3990–3993 (2015).
[Crossref]

B. K. Pierscionek and J. W. Regini, “The gradient index lens of the eye: an opto-biological synchrony,” Prog. Retinal Eye Res. 31(4), 332–349 (2012).
[Crossref]

D. Siedlecki, H. Kasprzak, and B. K. Pierscionek, “Schematic eye with a gradient-index lens and aspheric surfaces,” Opt. Lett. 29(11), 1197–1199 (2004).
[Crossref]

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

Pizarro, C.

Ponting, M.

Pope, J. M.

A. Khan, J. M. Pope, P. K. Verkicharla, M. Suheimat, and D. A. Atchison, “Change in human lens dimensions, lens refractive index distribution and ciliary body ring diameter with accommodation,” Biomed. Opt. Express 9(3), 1272–1282 (2018).
[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]

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]

Popiolek-Masajada, A.

A. Popiolek-Masajada and H. Kasprzak, “Model of the optical system of the human eye during accommodation,” Oph. Phys. Optics 22(3), 201–208 (2002).
[Crossref]

A. Popiolek-Masajada and H. T. Kasprzak, “A new schematic eye model incorporating accommodation,” Optom. Vis. Sci. 76(10), 720–727 (1999).
[Crossref]

Pouwels, P. J.

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Visual Sci. 50(1), 281–289 (2009).
[Crossref]

Puente, N. P.

Regini, J. W.

B. K. Pierscionek and J. W. Regini, “The gradient index lens of the eye: an opto-biological synchrony,” Prog. Retinal Eye Res. 31(4), 332–349 (2012).
[Crossref]

Remon, L.

Richdale, K.

L. A. Lossing, L. T. Sinnott, C. Y. Kao, K. Richdale, and M. D. Bailey, “Measuring changes in ciliary muscle thickness with accommodation in young adults,” Optom. Vis. Sci. 89(5), 719–726 (2012).
[Crossref]

Rolland, J. P.

Roorda, A.

H. Cheng, J. K. Barnett, A. S. Vilupuru, J. D. Marsack, S. Kasthurirangan, R. A. Applegate, and A. Roorda, “A population study on changes in wave aberrations with accomodation,” J. Vis. 4(8), 272–280 (2004).
[Crossref]

Rosen, A. M.

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).
[Crossref]

Ruggeri, M.

Sandadi, S.

F. Manns, V. Fernandez, S. Zipper, S. Sandadi, M. Hamaoui, A. Ho, and J. M. Parel, “Radius of curvature and asphericity of the anterior and posterior surface of human cadaver crystalline lenses,” Exp. Eye Res. 78(1), 39–51 (2004).
[Crossref]

Santamaría, J.

Schachar, R. A.

R. A. Schachar, “Growth patterns of fresh human crystalline lenses measured by in vitro photographic biometry,” J. Anat. 206, 575–580 (2005).
[Crossref]

Shaitor, E. P.

R. G. Zainullin, A. B. Kravtsov, and E. P. Shaitor, “The crystalline lens as a Luneburg lens,” Biofizica 19, 913–915 (1974).

Shao, Y.

Sheehan, M. T.

Sheil, C. J.

Shen, M.

Y. Shao, A. Tao, H. Jiang, M. Shen, J. Zhong, F. Lu, and J. Wang, “Simultaneous real-time imaging of the ocular anterior segment including the ciliary muscle during accommodation,” Biomed. Opt. Express 4(3), 466–480 (2013).
[Crossref]

M. Shen, L. Cui, M. Li, D. Zhu, M. R. Wang, and J. Wang, “Extended scan depth optical coherence tomography for evaluating ocular surface shape,” J. Biomed. Opt. 16(5), 056007 (2011).
[Crossref]

C. Du, D. Zhu, M. Shen, M. Li, M. R. Wang, and J. Wang, “Novel optical coherence tomography for imaging the entire anterior segment of the eye,” Invest. Ophthalmol. Visual Sci. 52(2), 987 (2011).
[Crossref]

Siedlecki, D.

Sinnott, L. T.

L. A. Lossing, L. T. Sinnott, C. Y. Kao, K. Richdale, and M. D. Bailey, “Measuring changes in ciliary muscle thickness with accommodation in young adults,” Optom. Vis. Sci. 89(5), 719–726 (2012).
[Crossref]

Smith, G.

G. Smith, P. Bedggood, R. Ashman, M. Daaboul, and A. Metha, “Exploring ocular aberrations with a schematic human eye model,” Optom. Vis. Sci. 85(5), 330–340 (2008).
[Crossref]

G. Smith and D. A. Atchison, The Eye and Visual Optical Instruments (Cambridge University Press, 1997).

D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinemann, 2000).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, 1972).

Suheimat, M.

Sun, M.

E. Martinez-Enriquez, M. Sun, M. Velasco-Ocana, J. Birkenfeld, P. Pérez-Merino, and S. Marcos, “Optical coherence tomography based estimates of crystalline lens volume, equatorial diameter, and plane position,” Invest. Ophthalmol. Visual Sci. 57(9), OCT600 (2016).
[Crossref]

Tao, A.

Tepichín-Rodríguez, E.

E. Tepichín-Rodríguez, A. S. Cruz Felix, E. López-Olazagasti, and S. Balderas-Mata, “Emmetropic eyes: objective performance and clinical reference,” Proc. SPIE 8785, 878551 (2013).
[Crossref]

Thibos, L. N.

T. Liu and L. N. Thibos, “Customized models of ocular aberrations across the visual field during accommodation.,” J. Vis. 19(9), 13 (2019).
[Crossref]

D. A. Atchison and L. N. Thibos, “Optical models of the human eye,” Clin. Exp. Optom. 99(2), 99–106 (2016).
[Crossref]

Toal, V.

Tong, J.

Y. Wu, A. Liu, H. Lv, X. Yi, Q. Li, X. Wang, Y. Ding, and J. Tong, “Finite Schematic Eye Model with Maxwell Fish-eye Spherical lens,” in 2010 Symposium on Photonics and Optoelectronics (IEEE, 2010), pp. 1–4.

Uhlhorn, S.

D. Siedlecki, A. de Castro, E. Gambra, S. Ortiz, D. Borja, S. Uhlhorn, F. Manns, S. Marcos, and J. M. Parel, “Distortion correction of OCT images of the crystalline lens: gradient index approach,” Optom. Vis. Sci. 89(5), E709–E718 (2012).
[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]

van der Heijde, R. G.

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Visual Sci. 50(1), 281–289 (2009).
[Crossref]

Velasco-Ocana, M.

E. Martinez-Enriquez, M. Sun, M. Velasco-Ocana, J. Birkenfeld, P. Pérez-Merino, and S. Marcos, “Optical coherence tomography based estimates of crystalline lens volume, equatorial diameter, and plane position,” Invest. Ophthalmol. Visual Sci. 57(9), OCT600 (2016).
[Crossref]

Verkicharla, P. K.

Vilupuru, A. S.

H. Cheng, J. K. Barnett, A. S. Vilupuru, J. D. Marsack, S. Kasthurirangan, R. A. Applegate, and A. Roorda, “A population study on changes in wave aberrations with accomodation,” J. Vis. 4(8), 272–280 (2004).
[Crossref]

A. S. Vilupuru and A. Glasser, “Optical and biometric relationships of the isolated pig crystalline lens,” Oph. Phys. Optics 21(4), 296–311 (2001).
[Crossref]

Wang, J.

Y. Shao, A. Tao, H. Jiang, M. Shen, J. Zhong, F. Lu, and J. Wang, “Simultaneous real-time imaging of the ocular anterior segment including the ciliary muscle during accommodation,” Biomed. Opt. Express 4(3), 466–480 (2013).
[Crossref]

C. Du, D. Zhu, M. Shen, M. Li, M. R. Wang, and J. Wang, “Novel optical coherence tomography for imaging the entire anterior segment of the eye,” Invest. Ophthalmol. Visual Sci. 52(2), 987 (2011).
[Crossref]

M. Shen, L. Cui, M. Li, D. Zhu, M. R. Wang, and J. Wang, “Extended scan depth optical coherence tomography for evaluating ocular surface shape,” J. Biomed. Opt. 16(5), 056007 (2011).
[Crossref]

Wang, M. R.

M. Shen, L. Cui, M. Li, D. Zhu, M. R. Wang, and J. Wang, “Extended scan depth optical coherence tomography for evaluating ocular surface shape,” J. Biomed. Opt. 16(5), 056007 (2011).
[Crossref]

C. Du, D. Zhu, M. Shen, M. Li, M. R. Wang, and J. Wang, “Novel optical coherence tomography for imaging the entire anterior segment of the eye,” Invest. Ophthalmol. Visual Sci. 52(2), 987 (2011).
[Crossref]

Wang, X.

Y. Wu, A. Liu, H. Lv, X. Yi, Q. Li, X. Wang, Y. Ding, and J. Tong, “Finite Schematic Eye Model with Maxwell Fish-eye Spherical lens,” in 2010 Symposium on Photonics and Optoelectronics (IEEE, 2010), pp. 1–4.

Wilson, C.

M. Guillon, D. P. Lydon, and C. Wilson, “Corneal topography: a clinical model,” Oph. Phys. Optics 6(1), 47–56 (1986).
[Crossref]

Wojtkowski, M.

Wu, Y.

Y. Wu, A. Liu, H. Lv, X. Yi, Q. Li, X. Wang, Y. Ding, and J. Tong, “Finite Schematic Eye Model with Maxwell Fish-eye Spherical lens,” in 2010 Symposium on Photonics and Optoelectronics (IEEE, 2010), pp. 1–4.

Yamane, T.

T. Yamane, Statistics An Introductory Analysis (Harper & Row, 1967).

Yao, J.

Yi, X.

Y. Wu, A. Liu, H. Lv, X. Yi, Q. Li, X. Wang, Y. Ding, and J. Tong, “Finite Schematic Eye Model with Maxwell Fish-eye Spherical lens,” in 2010 Symposium on Photonics and Optoelectronics (IEEE, 2010), pp. 1–4.

Zainullin, R. G.

R. G. Zainullin, A. B. Kravtsov, and E. P. Shaitor, “The crystalline lens as a Luneburg lens,” Biofizica 19, 913–915 (1974).

Zhong, J.

Zhu, D.

C. Du, D. Zhu, M. Shen, M. Li, M. R. Wang, and J. Wang, “Novel optical coherence tomography for imaging the entire anterior segment of the eye,” Invest. Ophthalmol. Visual Sci. 52(2), 987 (2011).
[Crossref]

M. Shen, L. Cui, M. Li, D. Zhu, M. R. Wang, and J. Wang, “Extended scan depth optical coherence tomography for evaluating ocular surface shape,” J. Biomed. Opt. 16(5), 056007 (2011).
[Crossref]

Zipper, S.

F. Manns, V. Fernandez, S. Zipper, S. Sandadi, M. Hamaoui, A. Ho, and J. M. Parel, “Radius of curvature and asphericity of the anterior and posterior surface of human cadaver crystalline lenses,” Exp. Eye Res. 78(1), 39–51 (2004).
[Crossref]

Appl. Opt. (3)

Biofizica (1)

R. G. Zainullin, A. B. Kravtsov, and E. P. Shaitor, “The crystalline lens as a Luneburg lens,” Biofizica 19, 913–915 (1974).

Biomed. Opt. Express (6)

Clin. Exp. Optom. (1)

D. A. Atchison and L. N. Thibos, “Optical models of the human eye,” Clin. Exp. Optom. 99(2), 99–106 (2016).
[Crossref]

Dove’s Rep. Phys. (1)

L. Moser, “Über das Auge,” Dove’s Rep. Phys. 5, 337–349 (1844).

Exp. Eye Res. (1)

F. Manns, V. Fernandez, S. Zipper, S. Sandadi, M. Hamaoui, A. Ho, and J. M. Parel, “Radius of curvature and asphericity of the anterior and posterior surface of human cadaver crystalline lenses,” Exp. Eye Res. 78(1), 39–51 (2004).
[Crossref]

Invest. Ophthalmol. Visual Sci. (4)

E. Martinez-Enriquez, M. Sun, M. Velasco-Ocana, J. Birkenfeld, P. Pérez-Merino, and S. Marcos, “Optical coherence tomography based estimates of crystalline lens volume, equatorial diameter, and plane position,” Invest. Ophthalmol. Visual Sci. 57(9), OCT600 (2016).
[Crossref]

A. de Castro, J. Birkenfeld, B. Maceo, F. Manns, E. Arrieta, J. M. Parel, and S. Marcos, “Influence of shape and gradient refractive index in the accommodative changes of spherical aberration in nonhuman primate crystalline lenses,” Invest. Ophthalmol. Visual Sci. 54(9), 6197–6207 (2013).
[Crossref]

E. A. Hermans, P. J. Pouwels, M. Dubbelman, J. P. Kuijer, R. G. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: an MRI and Scheimpflug study,” Invest. Ophthalmol. Visual Sci. 50(1), 281–289 (2009).
[Crossref]

C. Du, D. Zhu, M. Shen, M. Li, M. R. Wang, and J. Wang, “Novel optical coherence tomography for imaging the entire anterior segment of the eye,” Invest. Ophthalmol. Visual Sci. 52(2), 987 (2011).
[Crossref]

J. Anat. (1)

R. A. Schachar, “Growth patterns of fresh human crystalline lenses measured by in vitro photographic biometry,” J. Anat. 206, 575–580 (2005).
[Crossref]

J. Biomed. Opt. (2)

M. Shen, L. Cui, M. Li, D. Zhu, M. R. Wang, and J. Wang, “Extended scan depth optical coherence tomography for evaluating ocular surface shape,” J. Biomed. Opt. 16(5), 056007 (2011).
[Crossref]

M. Bahrami and A. V. Goncharov, “Geometry-invariant GRIN lens: analytical ray tracing,” J. Biomed. Opt. 17(5), 055001 (2012).
[Crossref]

J. Cataract Refractive Surg. (1)

J. J. Esteve-Taboada, R. Montés-Micó, and T. Ferrer-Blasco, “Schematic eye models to mimic the behavior of the accommodating human eye,” J. Cataract Refractive Surg. 44(5), 627–641 (2018).
[Crossref]

J. Opt. Soc. Am. (1)

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

J. Vis. (2)

T. Liu and L. N. Thibos, “Customized models of ocular aberrations across the visual field during accommodation.,” J. Vis. 19(9), 13 (2019).
[Crossref]

H. Cheng, J. K. Barnett, A. S. Vilupuru, J. D. Marsack, S. Kasthurirangan, R. A. Applegate, and A. Roorda, “A population study on changes in wave aberrations with accomodation,” J. Vis. 4(8), 272–280 (2004).
[Crossref]

Jpn. J. Clin. Ophthalmol. (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).

Oph. Phys. Optics (4)

M. Guillon, D. P. Lydon, and C. Wilson, “Corneal topography: a clinical model,” Oph. Phys. Optics 6(1), 47–56 (1986).
[Crossref]

A. Popiolek-Masajada and H. Kasprzak, “Model of the optical system of the human eye during accommodation,” Oph. Phys. Optics 22(3), 201–208 (2002).
[Crossref]

H. T. Kasprzak, “New approximation for the whole profile of the human crystalline lens,” Oph. Phys. Optics 20(1), 31–43 (2000).
[Crossref]

A. S. Vilupuru and A. Glasser, “Optical and biometric relationships of the isolated pig crystalline lens,” Oph. Phys. Optics 21(4), 296–311 (2001).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Optica (1)

Optom. Vis. Sci. (6)

G. Smith, P. Bedggood, R. Ashman, M. Daaboul, and A. Metha, “Exploring ocular aberrations with a schematic human eye model,” Optom. Vis. Sci. 85(5), 330–340 (2008).
[Crossref]

A. Popiolek-Masajada and H. T. Kasprzak, “A new schematic eye model incorporating accommodation,” Optom. Vis. Sci. 76(10), 720–727 (1999).
[Crossref]

R. Navarro, L. González, and J. L. Hernández-Matamoros, “On the prediction of optical aberrations by personalized eye models,” Optom. Vis. Sci. 83(6), 371–381 (2006).
[Crossref]

D. Siedlecki, A. de Castro, E. Gambra, S. Ortiz, D. Borja, S. Uhlhorn, F. Manns, S. Marcos, and J. M. Parel, “Distortion correction of OCT images of the crystalline lens: gradient index approach,” Optom. Vis. Sci. 89(5), E709–E718 (2012).
[Crossref]

L. A. Lossing, L. T. Sinnott, C. Y. Kao, K. Richdale, and M. D. Bailey, “Measuring changes in ciliary muscle thickness with accommodation in young adults,” Optom. Vis. Sci. 89(5), 719–726 (2012).
[Crossref]

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

Pfluegers Arch. (1)

L. Matthiesen, “Ueber Begriff und Answerthung des sogenannten Totalindex der Krystalllinse,” Pfluegers Arch. 36(1), 72–100 (1885).
[Crossref]

Proc. SPIE (2)

Y. Huang and D. T. Moore, “Human eye modeling using a single equation of gradient index crystalline lens for relaxed and accommodated states,” Proc. SPIE 6342, 634201 (2006).
[Crossref]

E. Tepichín-Rodríguez, A. S. Cruz Felix, E. López-Olazagasti, and S. Balderas-Mata, “Emmetropic eyes: objective performance and clinical reference,” Proc. SPIE 8785, 878551 (2013).
[Crossref]

Prog. Retinal Eye Res. (1)

B. K. Pierscionek and J. W. Regini, “The gradient index lens of the eye: an opto-biological synchrony,” Prog. Retinal Eye Res. 31(4), 332–349 (2012).
[Crossref]

Vision Res. (4)

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]

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]

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]

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).
[Crossref]

Other (11)

H. Helmholtz, Helmholtz’s Treatise on Physiological Optics (Dover, 1962), Vol. 1. Appendix II.

Y. Wu, A. Liu, H. Lv, X. Yi, Q. Li, X. Wang, Y. Ding, and J. Tong, “Finite Schematic Eye Model with Maxwell Fish-eye Spherical lens,” in 2010 Symposium on Photonics and Optoelectronics (IEEE, 2010), pp. 1–4.

G. Smith and D. A. Atchison, The Eye and Visual Optical Instruments (Cambridge University Press, 1997).

D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinemann, 2000).

H. H. Emsley, Visual Optics (Butterworth, 1952).

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

H. Helmholtz, Helmholtz’s Treatise on Physiological Optics (Dover, 1962), Vol. 1.

P. Artal, ed., Handbook of Visual Optics: Fundamentals and Eye Optics (CRC Press, 2017).

T. Yamane, Statistics An Introductory Analysis (Harper & Row, 1967).

F. A. Haight, Handbook of the Poisson Distribution (John Wiley, 1967).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, 1972).

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Figures (9)

Fig. 1.
Fig. 1. The Poisson distribution $P(z;m)$ Eq. (1) (a) for $m=1,2, 3, 4$ , along the real axis and (b) its extension onto the $z-y$ plane for $m=2$ . (c) The Gaussian function $\mbox {exp}{(-z^2/a_z^2-r^2/a_r^2)}$ and (d) the Poisson-Gauss function Eq. (2) for $m=2$ , $b=1$ , and $a_z=a_r=0.9$ .
Fig. 2.
Fig. 2. (a) Intersection of a plane with the Poisson-Gauss function Eq. (2) in the whole half plane $z>0$ for $m=2$ , $a_z=0.9$ , $a_r=0.9$ and $b=1$ . (b) The domain resulting from the projection of the level curve by the intersection shown in a).
Fig. 3.
Fig. 3. The intersection shown in Fig. 2(a) for $m=2$ , $b=1$ , $a_z=0.9$ , $a_r=0.9$ . The level curves between $l$ and $h$ are presented. The iso-indicial contours are shown as the projections of the level curves of the PG function onto the $y-z$ plane, see inset.
Fig. 4.
Fig. 4. (a) 3D view of the external surface of the Poisson-Gauss lens. (b) The process of lens accommodation is modelled by increasing the value of the Poisson parameter $m$ through $m=2.8$ (light blue, color online only), $m=3.5$ (purple), $m=4.4$ (green), $m=5.7$ (blue), $m=7.4$ (orange), $m=10$ (brown), $m=14.2$ (red) and the fixed values $b=0.67$ and $a_z=3.15$ , corresponding to an accommodated dioptrical power of 0 D, 1.01 D, 2.02 D, 3.07 D, 4.03 D, 5.01 D, 6.02 D, respectively (see Table 3). The dimensions of the lens are in mm. During the accommodation process the position of the equator is constant. The volume of the lens (106.5 mm $^3$ ) is unchanged by means of Eq. (15). (c) The physiological parameters of the lens.
Fig. 5.
Fig. 5. Ray tracing through the schematic eye. The outermost ray height in both cases is 2 mm for $z=-1$ (at the beginning of the Figs. (a) and (c)). The dimensions of the plots are in mm. The values of the parameters of the PG GRIN lens are the given in the second and the fourth rows of Table 1 for figures (a) and (c), respectively. Figures (b) and (d) show the detail of the rays through the lens. [It should be noted that given that the distance between the object and the eye is 270 mm and the axial length of the eye is under 25 mm, rays diverging from the object appear to be parallel at incidence on the cornea].
Fig. 6.
Fig. 6. Axial radius of curvature of anterior (red) and posterior (blue) external surfaces of the lens as a function of m (the corresponding accommodative amplitude to each value of m is given in Table 3).
Fig. 7.
Fig. 7. Refractive index profile of the lens in the (dashed blue) unaccommodated and (red) the 5.01 D accommodative states, measured from the center at the equatorial axis (see point of intersection in Figure 4(c)) to the periphery as function of (a) the anterior axial distance (measured from 0 to the anterior apex), (b) the posterior axial distance (measured from 0 to the posterior apex), and (c) the equatorial distance (measured from 0 to the equatorial apex).
Fig. 8.
Fig. 8. Longitudinal spherical aberration (dashed blue) and its polynomial fit of order 8th, with only even order terms (red), for the (a) unaccommodated and (b) 4.03 D accommodative states.
Fig. 9.
Fig. 9. $h$ , Eq. (5), as function of $m$ .

Tables (3)

Tables Icon

Table 1. Poisson-Gauss and physiological parameters with b = 0.67 and a z = 3.15 . The parameters are written in mm, with exception of m which is dimensionless.

Tables Icon

Table 2. Biometric data: the radius of curvature of the external R a c and internal R p c surfaces of the cornea and their respective conic constants Q a and Q p , the thickness of the cornea d c , the distance between cornea and lens d c c , in mm. The refractive indices of the cornea n c o r and of the aqueous humor n a . The central and peripheral refractive indices of the lens, n c and n s , resp., are taken from Ref. [54].

Tables Icon

Table 3. PG GRIN parameters. The parameters are written in mm, with exception of m which is dimensionless and A that is the accommodation written in Diopters. As in Table 1, b = 0.67 and a z = 3.15 . The accommodation process is shown in Fig. 4(b)

Equations (37)

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

P ( z ; m ) = z m m ! e z ,
P G ( r , z ; m , b , a z , a r ) = ( b z ) m exp ( b z z 2 a z 2 r 2 a r 2 ) ,
y e = 0 , z e = 1 4 ( a z 2 b + a z 4 b 2 + 8 a z 2 m ) .
l = P G ( y e , z e ) .
h = P G ( 0 , μ z + σ z ) .
z 2 a z 2 + y 2 a r 2 + b z m ln ( b z ) = ln ( 1 / c ) ,
h c l ,
D 0 = { ( y , z ) R 2 | y 2 a r 2 + z 2 a z 2 + b z m ln ( b z ) ln ( 1 / h ) } ,
D = { ( x , y , z ) R 3 | r 2 a r 2 + z 2 a z 2 + b z m ln ( b z ) ln ( 1 / h ) } .
y ± ( z ; h ) = ± a r ln [ ( b z ) m h ] b z z 2 a z 2 .
R e = y + ( z e , h ) = a r ln [ ( b z e ) m h ] b z e z e 2 a z 2 .
z p = μ z + σ z z e .
n ( r , z ) = ( n c n s ) ( b z ) m e b z z 2 / a z 2 r 2 / a r 2 h l h + n s ,
V = π z e z a z e + z p y 2 ( z ) d z = a r 2 π ( z ln ( 1 / h ) m z b z 2 2 z 3 3 a z 2 + m z ln ( b z ) ) | μ z + σ z d μ z + σ z const. ,
a r 2 = V π L 0 .
R ( z ) = a r 2 2 | b + m z 2 z a z 2 | .
y 2 = 2 R c z ( 1 + Q ) z 2 ,
y ( z ) 2 = y ( z 0 ) 2 + a r 2 ( b + m z 0 2 z 0 a z 2 ) ( z z 0 ) a r 2 ( 1 a z 2 + m 2 z 0 2 ) ( z z 0 ) 2 +
Q l = a r 2 ( 1 a z 2 + m 2 z 0 2 ) 1.
σ z 2 = 2 0 ( z μ z ) 2 f ( y , z ) d y d z 0 f ( y , z ) d y d z ,
μ z = 0 z f ( y , z ) d z d y 0 f ( y , z ) d z d y .
σ y 2 = 2 0 ( y μ y ) 2 f ( y , z ) d y d z 0 f ( y , z ) d y d z , μ y = 0 y f ( y , z ) d y d z 0 f ( y , z ) d y d z .
P G ( y , z ) = ( b z ) m exp ( b z z 2 / a z 2 y 2 / a r 2 ) ,
μ y = 0 , while σ y 2 = a y 2 .
μ z = p 1 / 2 ( m 2 + 1 ) a z b p 3 / 2 ( m + 3 2 ) 1 a z p 1 / 2 ( m + 1 2 ) b p 3 / 2 ( m 2 + 1 ) ,
σ z 2 = a z 2 Δ [ a z b m 2 ( a z 2 b + 4 μ z ) p 3 / 2 ( m + 1 2 ) b ( a z 2 ( m + 1 ) + 2 μ z 2 ) p 3 / 2 ( m 2 + 1 ) + a z 2 ( 2 ( m + 1 ) + ( a z b + 2 μ z a z ) 2 ) p 1 / 2 ( m + 1 2 ) ( a z 2 b + 4 μ z ) p 1 / 2 ( m 2 + 1 ) ] ,
Δ = a z p 1 / 2 ( m + 1 2 ) a z 2 b p 3 / 2 ( m 2 + 1 ) ,
p q ( a ) = Γ ( a ) 1 F 1 ( a , q , 1 4 a z 2 b 2 ) .
R = ( 1 + ( d z d r ) 2 ) 3 2 | d 2 z d r 2 | .
y ( z ) = a r ln ( 1 / h ) + m ln ( b z ) b z z 2 / a z 2 ,
R = 1 | d 2 z d y 2 | .
d 2 z d y 2 = ( d 2 y d z 2 ) 2 ( d y d z ) 3 .
R = | a r 2 ( a z 2 ( m b z ) 2 z 2 ) 3 Ω | ,
Ω = 2 a z 4 z [ a z 2 ( m ( 2 ln ( 1 / h ) + m 4 b z ) + b 2 z 2 ) + ( 4 ln ( 1 / h ) 6 m ) z 2 + 2 m ( a z 2 m + 2 z 2 ) ln ( z ) ] .
a r ln ( 1 / h ) + m ln ( b z ) b z z 2 / a z 2 = 0 ,
m ln ( b z ) = ln ( 1 / h ) + b z + z 2 / a z 2 .
R ( z ) = a r 2 2 | b + m z 2 z a z 2 | .

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