Abstract

The purpose of this manuscript is to introduce a new age-dependent model of the human lens with two GRIN power distributions (axial and radial) that allow decoupling of its refractive power and axial optical path length. The aspect ratio of the lens core can be held constant under accommodation, as well as the lens volume by varying the asphericity of the lens external surfaces. The spherical aberration calculated by exact raytracing is shown to be in line with experimental data. The proposed model is compared to previous GRIN models from the literature, and it is concluded that the features of the new model will be useful for GRIN reconstruction in future experimental studies; in particular, studies of the accommodation-dependent properties of the ageing human eye. A proposed logarithmic model of the lens core enables decoupling of three fundamental optical characteristics of the lens, namely axial optical path length, optical power and third-order spherical aberration, without changing the external shape of the lens. Conversely, the near-surface GRIN structure conforms to the external shape of the lens, which is necessary for accommodation modelling.

© 2016 Optical Society of America

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References

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  1. 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, 1649–1663 (2014).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  4. S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Vis. Sci. 49, 2531–2540 (2008).
    [Crossref] [PubMed]
  5. 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, 2175–2185 (2007).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  8. A. de Castro, S. Ortiz, E. Gambra, D. Siedlecki, and S. Marcos, “Three-dimensional reconstruction of the crystalline lens gradient index distribution from OCT imaging,” Opt. Express 18, 21905–21917 (2010).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  12. B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).
  13. M. Bahrami, A. V. Goncharov, and B. K. Pierscionek, “Adjustable internal structure for reconstructing gradient index profile of crystalline lens,” Opt. Lett. 39, 1310–1313 (2014).
    [Crossref] [PubMed]
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    [Crossref]
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2015 (1)

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).

2014 (4)

2013 (1)

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. K. Pierscionek, “Evaluation of equations for describing the human crystalline lens,” J. Mod. Opt. 60, 406–413 (2013).
[Crossref]

2012 (1)

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

2010 (2)

A. de Castro, S. Ortiz, E. Gambra, D. Siedlecki, and S. Marcos, “Three-dimensional reconstruction of the crystalline lens gradient index distribution from OCT imaging,” Opt. Express 18, 21905–21917 (2010).
[Crossref] [PubMed]

F. Manns, A. Ho, D. Borja, and J.-M. Parel, “Comparison of uniform and gradient paraxial models of the crystalline lens,” Invest. Ophthalmol. Vis. Sci. 51, 789 (2010).

2008 (2)

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Vis. Sci. 49, 2531–2540 (2008).
[Crossref] [PubMed]

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

2007 (2)

2005 (1)

C. Jones, D. Atchison, R. Meder, and J. 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]

2003 (1)

G. Smith, “The optical properties of the crystalline lens and their significance,” Clin. Exp. Optom. 86, 3–18 (2003).
[Crossref] [PubMed]

2002 (1)

2001 (2)

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

1997 (2)

1992 (1)

1991 (1)

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

1990 (2)

1982 (1)

Arasa, J.

Atchison, D.

C. Jones, D. Atchison, R. Meder, and J. 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]

Atchison, D. A.

W. N. Charman and D. A. Atchison, “Age-dependence of the average and equivalent refractive indices of the crystalline lens,” Biomed. Opt. Express 5, 31–39 (2014).
[Crossref] [PubMed]

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Vis. Sci. 49, 2531–2540 (2008).
[Crossref] [PubMed]

G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modeling the power of the aging human eye,” J. Opt. Soc. Am. A 9, 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] [PubMed]

Bahrami, M.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).

M. Bahrami, A. V. Goncharov, and B. K. Pierscionek, “Adjustable internal structure for reconstructing gradient index profile of crystalline lens,” Opt. Lett. 39, 1310–1313 (2014).
[Crossref] [PubMed]

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, 1649–1663 (2014).
[Crossref] [PubMed]

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

Borja, D.

F. Manns, A. Ho, D. Borja, and J.-M. Parel, “Comparison of uniform and gradient paraxial models of the crystalline lens,” Invest. Ophthalmol. Vis. Sci. 51, 789 (2010).

Bradley, A.

Brennan, N. A.

Charman, W. N.

Cheng, X.

Creath, K.

J. C. Wyant and K. Creath, Applied Optics and Optical Engineering (Academic Press, 1992), vol. 11, chap. 1: Basic Wavefront Aberration Theory for Optical Metrology.

Dainty, C.

de Castro, A.

Díaz, J. A.

Dubbelman, M.

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

Forbes, G. W.

Gambra, E.

Ghatak, A. K.

Giovanzana, S.

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. K. Pierscionek, “Evaluation of equations for describing the human crystalline lens,” J. Mod. Opt. 60, 406–413 (2013).
[Crossref]

Goncharov, A. V.

González, L.

Ho, A.

F. Manns, A. Ho, D. Borja, and J.-M. Parel, “Comparison of uniform and gradient paraxial models of the crystalline lens,” Invest. Ophthalmol. Vis. Sci. 51, 789 (2010).

Hong, X.

Hoshino, M.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).

Jones, C.

C. Jones, D. Atchison, R. Meder, and J. 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]

Kasthurirangan, S.

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Vis. Sci. 49, 2531–2540 (2008).
[Crossref] [PubMed]

Kirby, R. D.

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. K. Pierscionek, “Evaluation of equations for describing the human crystalline lens,” J. Mod. Opt. 60, 406–413 (2013).
[Crossref]

Kumar, D. V.

Liou, H.-L.

Manns, F.

F. Manns, A. Ho, D. Borja, and J.-M. Parel, “Comparison of uniform and gradient paraxial models of the crystalline lens,” Invest. Ophthalmol. Vis. Sci. 51, 789 (2010).

Marcos, S.

Markwell, E. L.

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Vis. Sci. 49, 2531–2540 (2008).
[Crossref] [PubMed]

Meder, R.

C. Jones, D. Atchison, R. Meder, and J. 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]

Navarro, R.

Ortiz, S.

Palos, F.

Parel, J.-M.

F. Manns, A. Ho, D. Borja, and J.-M. Parel, “Comparison of uniform and gradient paraxial models of the crystalline lens,” Invest. Ophthalmol. Vis. Sci. 51, 789 (2010).

Pierscionek, B.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).

Pierscionek, B. K.

M. Bahrami, A. V. Goncharov, and B. K. Pierscionek, “Adjustable internal structure for reconstructing gradient index profile of crystalline lens,” Opt. Lett. 39, 1310–1313 (2014).
[Crossref] [PubMed]

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. K. Pierscionek, “Evaluation of equations for describing the human crystalline lens,” J. Mod. Opt. 60, 406–413 (2013).
[Crossref]

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

G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modeling the power of the aging human eye,” J. Opt. Soc. Am. A 9, 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] [PubMed]

B. K. Pierscionek, “Presbyopia—effect of refractive index,” Clin. Exp. Optom. 73, 23–30 (1990).
[Crossref]

Pizarro, C.

Pope, J.

C. Jones, D. Atchison, R. Meder, and J. 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]

Pope, J. M.

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Vis. Sci. 49, 2531–2540 (2008).
[Crossref] [PubMed]

Regini, J.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).

Schachar, R. A.

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. K. Pierscionek, “Evaluation of equations for describing the human crystalline lens,” J. Mod. Opt. 60, 406–413 (2013).
[Crossref]

Sharma, A.

Sheil, C. J.

Siedlecki, D.

Smith, G.

G. Smith, “The optical properties of the crystalline lens and their significance,” Clin. Exp. Optom. 86, 3–18 (2003).
[Crossref] [PubMed]

G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modeling the power of the aging human eye,” J. Opt. Soc. Am. A 9, 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] [PubMed]

Stone, B. D.

Talu, S.

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. K. Pierscionek, “Evaluation of equations for describing the human crystalline lens,” J. Mod. Opt. 60, 406–413 (2013).
[Crossref]

Thibos, L. N.

Uesugi, K.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).

van der Heijde, G.

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

Weeber, H.

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

Wyant, J. C.

J. C. Wyant and K. Creath, Applied Optics and Optical Engineering (Academic Press, 1992), vol. 11, chap. 1: Basic Wavefront Aberration Theory for Optical Metrology.

Yagi, N.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).

Yan, E.

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. K. Pierscionek, “Evaluation of equations for describing the human crystalline lens,” J. Mod. Opt. 60, 406–413 (2013).
[Crossref]

Appl. Opt. (1)

Biomed. Opt. Express (2)

Clin. Exp. Optom. (2)

G. Smith, “The optical properties of the crystalline lens and their significance,” Clin. Exp. Optom. 86, 3–18 (2003).
[Crossref] [PubMed]

B. K. Pierscionek, “Presbyopia—effect of refractive index,” Clin. Exp. Optom. 73, 23–30 (1990).
[Crossref]

Exp. Eye Res. (1)

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

Invest. Ophthalmol. Vis. Sci. (2)

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Vis. Sci. 49, 2531–2540 (2008).
[Crossref] [PubMed]

F. Manns, A. Ho, D. Borja, and J.-M. Parel, “Comparison of uniform and gradient paraxial models of the crystalline lens,” Invest. Ophthalmol. Vis. Sci. 51, 789 (2010).

J. Biomed. Opt. (1)

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

J. Mod. Opt. (1)

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. K. Pierscionek, “Evaluation of equations for describing the human crystalline lens,” J. Mod. Opt. 60, 406–413 (2013).
[Crossref]

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

J. Vis. (1)

R. Navarro, “Adaptive model of the aging emmetropic eye and its changes with accommodation,” J. Vis. 1413, 21 (2014).
[Crossref] [PubMed]

Oncotarget (1)

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).

Ophthalmic Physiol. Opt. (1)

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

Opt. Express (1)

Opt. Lett. (1)

Optom. Vis. Sci. (1)

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

Vision Res. (2)

M. Dubbelman and G. 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]

C. Jones, D. Atchison, R. Meder, and J. 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]

Other (2)

Alexander V. Goncharov and Conor J. Sheil, Applied Optics Group, National University of Ireland, Galway, are preparing a manuscript on raytracing through GRIN media defined as an infinite number of thin shells with a known shape.

J. C. Wyant and K. Creath, Applied Optics and Optical Engineering (Academic Press, 1992), vol. 11, chap. 1: Basic Wavefront Aberration Theory for Optical Metrology.

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Figures (9)

Fig. 1
Fig. 1 Lenticular plots showing iso-indicial contours for young (a) and old (b). Negative m shows a bunching of contours towards the equator (c), a behaviour seen in recent studies [4, 12]. The three lenses have the same external geometry and parameter P representative of a 30 year old, after [5, 21, 22]: Ra = 11.2 mm, Rp = 6.0 mm, Ka = −4.5, Kp = −1.1, lens thickness T = 3.6 mm, P = 3, nc = 1.415 and ns = 1.37.
Fig. 2
Fig. 2 Comparing the axial (a) and radial (b) refractive index profiles of the lenses of Fig. 1.
Fig. 3
Fig. 3 Lens power F vs m for different values of P, calculated using the thin-lens formula of Eq. (8) (solid lines) and exact raytracing (data points) with a ray height of 20 μm. Ra = 11.2 mm, Rp = 6.0 mm, Ka = −4.5, Kp = −1.1, lens thickness T = 3.6 mm, P = 3, nc = 1.415 and ns = 1.37.
Fig. 4
Fig. 4 Plots of LSA for different values of m. Ra = 11.2 mm, Rp = 6.0 mm, Ka = −4.5, Kp = −1.1, lens thickness T = 3.6 mm, P = 3, nc = 1.415 and ns = 1.37. Z 4 0 is calculated for 4 mm pupil diameter.
Fig. 5
Fig. 5 SA (4 mm) vs m for different values of P. Ra = 11.2 mm, Rp = 6.0 mm, Ka = −4.5, Kp = −1.1, lens thickness T = 3.6 mm, nc = 1.415 and ns = 1.37.
Fig. 6
Fig. 6 The normalised aspect ratio (A/A0) of an internal contour using Eq. (11).
Fig. 7
Fig. 7 Plot of optical power vs m for different values of w, using the log function. Ra = 11.2 mm, Rp = 6.0 mm, Ka = −4.5, Kp = −1.1, lens thickness T = 3.6 mm, P = 3, nc = 1.415 and ns = 1.37. Note that m = 0 gives the GIGL model.
Fig. 8
Fig. 8 SA (4 mm) vs m for different values of w using the log function; P = 3. Ra = 11.2 mm, Rp = 6.0 mm, Ka = −4.5, Kp = −1.1, lens thickness T = 3.6 mm, nc = 1.415 and ns = 1.37.
Fig. 9
Fig. 9 Plots of LSA for different values of m and w using the log function. Ra = 11.2 mm, Rp = 6.0 mm, Ka = −4.5, Kp = −1.1, lens thickness T = 3.6 mm, P = 3, nc = 1.415 and ns = 1.37. Z 4 0 is calculated for 4 mm pupil diameter.

Tables (1)

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Table 1 A compilation from the literature of previous models of the human lens, including the AVOCADO model.

Equations (25)

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n ( ζ ) = n c + ( n s n c ) ( ζ 2 ) P ,
ρ 2 = x 2 + y 2 = 2 R a ( T a + z ) ( 1 + K a ) ( T a + z ) 2 + B a ( T a + z ) 3 ,
ρ 2 = 2 r a ( t a + z ) ( 1 + k a ) ( t a + z ) 2 + b a ( t a + z ) 3 ,
ρ 2 = 2 ζ R a ( ζ T a + z ) ( 1 + K a ) ( ζ T a + z ) 2 + ζ 1 B a ( ζ T a + z ) 3 .
ρ 2 = 2 ζ 2 m + 1 R a ( ζ T a + z ) ζ 2 m ( 1 + K a ) ( ζ T a + z ) 2 + ζ 2 m 1 B a ( ζ T a + z ) 3 ,
n ( ρ , z ) | ρ = 0 = n c ( Δ n / T p 2 P ) z 2 P ;
ρ 2 = 2 ζ 2 m + 1 R p ( ζ T p ) ζ 2 m ( 1 + K p ) ( ζ T p ) 2 + ζ 2 m 1 B p ( ζ T p ) 3 = ζ 2 m + 2 ( 2 R p T p ( 1 + K p ) T p 2 + B p T p 3 )
ρ 0 2 = 2 R p T p ( 1 + K p ) T p 2 + B p T p 3 ;
ζ = ( ρ / ρ 0 ) 1 m + 1 .
n ( ρ , z ) | z = 0 = n c ( Δ n / ρ 0 2 P m + 1 ) ρ 2 P m + 1 ;
OPL = T a T p n ( z ) d z ,
OPL = n av Δ z ,
OPL = T a 0 n c Δ n ( z T a ) 2 P d z + 0 T p n c Δ n ( z T p ) 2 P d z .
n av n c Δ n / ( 2 P + 1 ) .
F thin = ( n s n ) ( 1 R a + 1 R p ) 4 P Δ n Δ R β R a R p ,
F eq = ( n eq n ) ( 1 R a + 1 R p + T ( n n eq ) R a R p n eq ) = F thin .
n eq = 1 β ( T Δ R ) ( n T β γ Δ R + u ) ,
Z 4 0 = W 4 , 0 6 5 .
V = 1 6 π [ 5 R p Z p 2 + 5 R a Z a 2 ( 1 + K p ) Z p 3 ( 1 + K a ) Z a 3 ] ,
v ( ζ ) = ζ 2 m + 3 V .
A = 2 ρ ( ζ ) t ( ζ ) = ζ m + 1 ζ 2 ρ 0 T = ζ m A 0 .
A = ζ m ( ζ ) A 0 .
m = ( 1 + w w ) m 0 ln ( 1 + w ζ 1 + w ) ln ( ζ ) .
A A 0 ( 1 1 + w ) m 0 ( 1 + 1 w ) .
ρ 2 = 2 ζ ϕ R a ( ζ T a + z ) ϕ ( 1 + K a ) ( ζ T a + z ) 2 + ζ 1 ϕ B a ( ζ T a + z ) 3 ,

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