Abstract

Optical models of the human cornea and tear film typically employ a single homogeneous cornea with an average refractive index. I propose to use a more realistic multilayer model based on morphological data from the literature. The mathematical methodology to derive the refractive power equation of this model is presented. Special attention is given to the axial gradient index of the refraction structure of the stroma layer because of its optical implications. The importance of considering this multilayer model is quantified in a specific example (orthokeratology) with the help of the derived power equation.

© 2006 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [PubMed]
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    [PubMed]
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    [PubMed]
  10. M. Dubbelman, H. A. Weeber, R. G. L. van der Heijde, and H. J. Volker-Dieben, 'Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography,' Acta Ophthalmol. Scand. 80, 379-383 (2002).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  27. H. Helmholtz, Helmholtz's Treatise on Physiological Optics (Optical Society of America, 1924).
  28. G. Smith, 'The optical properties of the crystalline lens and their significance,' Clin. Exp. Optom. 86, 3-18 (2003).
    [Crossref] [PubMed]
  29. M. Kline, Mathematical Thought from Ancient to Modern Times (Oxford U. Press, 1972).

2005 (2)

M. V. Pérez, C. Bao, M. T. Flores-Arias, M. A. Rama, and C. Gómez-Reino, 'Description of gradient-index crystalline lens by a first-order optical system,' J. Opt. A, Pure Appl. Opt. 1, 103-110 (2005).
[Crossref]

D. A. Atchison and G. Smith, 'Chromatic dispersions of the ocular media of human eyes,' J. Opt. Soc. Am. A 22, 29-37 (2005).
[Crossref]

2004 (1)

X. Cheng, A. Bradley, and L. N. Thibos, 'Predicting subjective judgment of best focus with objective image quality metrics,' J. Math. Imaging Vision 4, 310-321 (2004).

2003 (2)

A. Alharbi and H. A. Swarbrick, 'The effects of overnight orthokeratology lens wear on corneal thickness,' Invest. Ophthalmol. Visual Sci. 44, 2518-2523 (2003).
[Crossref]

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

2002 (1)

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

1998 (1)

S. Patel, D. Z. Reinstein, R. H. Silverman, and D. J. Coleman, 'The shape of Bowman's layer in the human cornea,' J. Refract. Surg. 14, 636-640 (1998).
[PubMed]

1997 (1)

1996 (1)

1995 (2)

S. Patel, J. Marshall, and F. W. Fitzke, 'Refractive-index of the human corneal epithelium and stroma,' J. Refract. Surg. 11, 100-105 (1995).
[PubMed]

J. P. Craig, P. A. Simmons, S. Patel, and A. Tomlinson, 'Refractive-index and osmolality of human tears,' Optom. Vision Sci. 72, 718-724 (1995).
[Crossref]

1990 (1)

1985 (1)

1973 (1)

1971 (1)

1970 (1)

1963 (1)

1957 (1)

D. M. Maurice, 'The structure and transparency of the cornea,' J. Physiol. 136, 263-286 (1957).
[PubMed]

Alharbi, A.

A. Alharbi and H. A. Swarbrick, 'The effects of overnight orthokeratology lens wear on corneal thickness,' Invest. Ophthalmol. Visual Sci. 44, 2518-2523 (2003).
[Crossref]

Atchison, D. A.

Bao, C.

M. V. Pérez, C. Bao, M. T. Flores-Arias, M. A. Rama, and C. Gómez-Reino, 'Description of gradient-index crystalline lens by a first-order optical system,' J. Opt. A, Pure Appl. Opt. 1, 103-110 (2005).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1980).
[PubMed]

Bradley, A.

X. Cheng, A. Bradley, and L. N. Thibos, 'Predicting subjective judgment of best focus with objective image quality metrics,' J. Math. Imaging Vision 4, 310-321 (2004).

Brennan, N. A.

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

Cheng, X.

X. Cheng, A. Bradley, and L. N. Thibos, 'Predicting subjective judgment of best focus with objective image quality metrics,' J. Math. Imaging Vision 4, 310-321 (2004).

Coleman, D. J.

S. Patel, D. Z. Reinstein, R. H. Silverman, and D. J. Coleman, 'The shape of Bowman's layer in the human cornea,' J. Refract. Surg. 14, 636-640 (1998).
[PubMed]

Craig, J. P.

J. P. Craig, P. A. Simmons, S. Patel, and A. Tomlinson, 'Refractive-index and osmolality of human tears,' Optom. Vision Sci. 72, 718-724 (1995).
[Crossref]

Dijksterhuis, F. J.

F. J. Dijksterhuis, Lenses and Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century, Vol. 9 of Archimedes Series (Springer, 2004).

Dubbelman, M.

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

Fitzke, F. W.

S. Patel, J. Marshall, and F. W. Fitzke, 'Refractive-index of the human corneal epithelium and stroma,' J. Refract. Surg. 11, 100-105 (1995).
[PubMed]

Flores-Arias, M. T.

M. V. Pérez, C. Bao, M. T. Flores-Arias, M. A. Rama, and C. Gómez-Reino, 'Description of gradient-index crystalline lens by a first-order optical system,' J. Opt. A, Pure Appl. Opt. 1, 103-110 (2005).
[Crossref]

Gómez-Reino, C.

M. V. Pérez, C. Bao, M. T. Flores-Arias, M. A. Rama, and C. Gómez-Reino, 'Description of gradient-index crystalline lens by a first-order optical system,' J. Opt. A, Pure Appl. Opt. 1, 103-110 (2005).
[Crossref]

Helmholtz, H.

H. Helmholtz, Helmholtz's Treatise on Physiological Optics (Optical Society of America, 1924).

Keating, M. P.

M. P. Keating, Geometric, Physical, and Visual Optics (Butterworth-Heinemann, 2002).

Kline, M.

M. Kline, Mathematical Thought from Ancient to Modern Times (Oxford U. Press, 1972).

Korb, D. R.

D. R. Korb, The Tear Film: Structure, Function, and Clinical Examination (Butterworth-Heinemann, Oxford, 2002).

Kumar, G.

Liou, H. L.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

Marshall, J.

S. Patel, J. Marshall, and F. W. Fitzke, 'Refractive-index of the human corneal epithelium and stroma,' J. Refract. Surg. 11, 100-105 (1995).
[PubMed]

Maurice, D. M.

D. M. Maurice, 'The structure and transparency of the cornea,' J. Physiol. 136, 263-286 (1957).
[PubMed]

Miyamoto, K.

Moore, D. T.

Navarro, R.

Oyster, C. W.

C. W. Oyster, The Human Eye: Structure and Function (Sinauer, 1999).

Patel, S.

S. Patel, D. Z. Reinstein, R. H. Silverman, and D. J. Coleman, 'The shape of Bowman's layer in the human cornea,' J. Refract. Surg. 14, 636-640 (1998).
[PubMed]

J. P. Craig, P. A. Simmons, S. Patel, and A. Tomlinson, 'Refractive-index and osmolality of human tears,' Optom. Vision Sci. 72, 718-724 (1995).
[Crossref]

S. Patel, J. Marshall, and F. W. Fitzke, 'Refractive-index of the human corneal epithelium and stroma,' J. Refract. Surg. 11, 100-105 (1995).
[PubMed]

Pérez, M. V.

M. V. Pérez, C. Bao, M. T. Flores-Arias, M. A. Rama, and C. Gómez-Reino, 'Description of gradient-index crystalline lens by a first-order optical system,' J. Opt. A, Pure Appl. Opt. 1, 103-110 (2005).
[Crossref]

Rama, M. A.

M. V. Pérez, C. Bao, M. T. Flores-Arias, M. A. Rama, and C. Gómez-Reino, 'Description of gradient-index crystalline lens by a first-order optical system,' J. Opt. A, Pure Appl. Opt. 1, 103-110 (2005).
[Crossref]

Reinstein, D. Z.

S. Patel, D. Z. Reinstein, R. H. Silverman, and D. J. Coleman, 'The shape of Bowman's layer in the human cornea,' J. Refract. Surg. 14, 636-640 (1998).
[PubMed]

Sands, P. J.

Santamaria, J.

Schmitt, J. M.

Silverman, R. H.

S. Patel, D. Z. Reinstein, R. H. Silverman, and D. J. Coleman, 'The shape of Bowman's layer in the human cornea,' J. Refract. Surg. 14, 636-640 (1998).
[PubMed]

Simmons, P. A.

J. P. Craig, P. A. Simmons, S. Patel, and A. Tomlinson, 'Refractive-index and osmolality of human tears,' Optom. Vision Sci. 72, 718-724 (1995).
[Crossref]

Smith, G.

D. A. Atchison and G. Smith, 'Chromatic dispersions of the ocular media of human eyes,' J. Opt. Soc. Am. A 22, 29-37 (2005).
[Crossref]

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

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

Smolin, G.

G. Smolin and R. A. Thoft, The Cornea: Scientific Foundations and Clinical Practice (Little, Brown,1994).

Swarbrick, H. A.

A. Alharbi and H. A. Swarbrick, 'The effects of overnight orthokeratology lens wear on corneal thickness,' Invest. Ophthalmol. Visual Sci. 44, 2518-2523 (2003).
[Crossref]

Thibos, L. N.

X. Cheng, A. Bradley, and L. N. Thibos, 'Predicting subjective judgment of best focus with objective image quality metrics,' J. Math. Imaging Vision 4, 310-321 (2004).

Thoft, R. A.

G. Smolin and R. A. Thoft, The Cornea: Scientific Foundations and Clinical Practice (Little, Brown,1994).

Tomlinson, A.

J. P. Craig, P. A. Simmons, S. Patel, and A. Tomlinson, 'Refractive-index and osmolality of human tears,' Optom. Vision Sci. 72, 718-724 (1995).
[Crossref]

van der Heijde, R. G. L.

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

Volker-Dieben, H. J.

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

Wang, D. Y.

Weeber, H. A.

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1980).
[PubMed]

Acta Ophthalmol. Scand. (1)

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

Appl. Opt. (2)

Clin. Exp. Optom. (1)

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

Invest. Ophthalmol. Visual Sci. (1)

A. Alharbi and H. A. Swarbrick, 'The effects of overnight orthokeratology lens wear on corneal thickness,' Invest. Ophthalmol. Visual Sci. 44, 2518-2523 (2003).
[Crossref]

J. Math. Imaging Vision (1)

X. Cheng, A. Bradley, and L. N. Thibos, 'Predicting subjective judgment of best focus with objective image quality metrics,' J. Math. Imaging Vision 4, 310-321 (2004).

J. Opt. A, Pure Appl. Opt. (1)

M. V. Pérez, C. Bao, M. T. Flores-Arias, M. A. Rama, and C. Gómez-Reino, 'Description of gradient-index crystalline lens by a first-order optical system,' J. Opt. A, Pure Appl. Opt. 1, 103-110 (2005).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

J. Physiol. (1)

D. M. Maurice, 'The structure and transparency of the cornea,' J. Physiol. 136, 263-286 (1957).
[PubMed]

J. Refract. Surg. (2)

S. Patel, D. Z. Reinstein, R. H. Silverman, and D. J. Coleman, 'The shape of Bowman's layer in the human cornea,' J. Refract. Surg. 14, 636-640 (1998).
[PubMed]

S. Patel, J. Marshall, and F. W. Fitzke, 'Refractive-index of the human corneal epithelium and stroma,' J. Refract. Surg. 11, 100-105 (1995).
[PubMed]

Opt. Lett. (1)

Optom. Vision Sci. (1)

J. P. Craig, P. A. Simmons, S. Patel, and A. Tomlinson, 'Refractive-index and osmolality of human tears,' Optom. Vision Sci. 72, 718-724 (1995).
[Crossref]

Other (11)

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

D. R. Korb, The Tear Film: Structure, Function, and Clinical Examination (Butterworth-Heinemann, Oxford, 2002).

C. W. Oyster, The Human Eye: Structure and Function (Sinauer, 1999).

G. Smolin and R. A. Thoft, The Cornea: Scientific Foundations and Clinical Practice (Little, Brown,1994).

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

F. J. Dijksterhuis, Lenses and Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century, Vol. 9 of Archimedes Series (Springer, 2004).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1980).
[PubMed]

M. P. Keating, Geometric, Physical, and Visual Optics (Butterworth-Heinemann, 2002).

H. Helmholtz, Helmholtz's Treatise on Physiological Optics (Optical Society of America, 1924).

M. Kline, Mathematical Thought from Ancient to Modern Times (Oxford U. Press, 1972).

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Figures (5)

Fig. 1
Fig. 1

Refractive index along the optical axis coordinate (in millimeters) in the cornea model. The y axis of the graphic is broken from 1.02 to 1.32 for better visualization.

Fig. 2
Fig. 2

Cardinal points for the tear–cornea model. Distances from the left to the right are positive sign. F and F , focal points; H and H , principal points; V and V , anterior and posterior vertex of the tear–cornea, respectively; C, location of the physical pupil; C , location of the entrance pupil. If V C is set to be 3.5 mm , VF = 23.5659 mm , V F = 30.9858 mm , HF = 23.5243 mm , H F = 31.4285 mm , VH = 0.0416 mm , V H = 0.4427 mm .

Fig. 3
Fig. 3

Spherical aberration coefficient W 40 versus pupil radius (in millimeters) in the gradient multilayer model (solid curve), Liou’s model (short-dashed curve) and Navarro’s model (long-dashed curve).

Fig. 4
Fig. 4

Plane of best focus (in millimeters) with respect to the exit pupil versus on-axis object position with respect to tear anterior vertex. The solid curve shows the gradient and the dashed curve the homogeneous stroma model.

Fig. 5
Fig. 5

Spherical aberration coefficient W 40 versus on-axis object position with respect to the cornea. The solid curve shows the gradient and the dashed curve the homogeneous stroma model.

Tables (4)

Tables Icon

Table 1 Parameters of the Cornea and Tear-Film Multilayer Model

Tables Icon

Table 2 Paraxial Power Contributions (D) of the Terms in Eq. (5) in the Different Models

Tables Icon

Table 3 Wave Primary Spherical Aberration ( W 40 ) Contributions ( mm 3 )

Tables Icon

Table 4 Parameters of the Cornea Multilayer Model before and after Orthokeratology Treatment Considering Only Corneal Tissue Redistribution

Equations (23)

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f = f parax 2 σ 1 ρ 2 3 v .
[ Y 1 U 1 ] = [ A B C D ] [ Y 0 U 0 ] ,
T = [ 1 t 0 1 ] , R = [ 1 0 P / n n / n ] .
[ A B C D ] = [ 1 0 P 4 / n 4 n 32 / n 4 ] [ 1 t 3 n 31 ln ( n 32 / n 31 ) ( n 31 n 31 ) 0 n 31 / n 32 ] [ 1 0 P 3 / n 31 n 2 / n 31 ] [ 1 t 2 0 1 ] [ 1 0 P 2 / n 2 n 1 / n 2 ] [ 1 t 1 0 1 ] [ 1 0 P 1 / n 1 1 / n 1 ] ,
P = i = 1 4 P i i = 1 3 T i + T g ,
T 1 = t 1 ( P 2 + P 3 + P 4 ) n 1 , T 2 = t 2 ( P 1 + P 2 ) ( P 3 + P 4 ) n 2 ;
T 3 = t 3 ( P 1 + P 2 + P 3 ) n 3
T g = t 1 t 2 P 1 P 2 ( P 3 + P 4 ) n 1 n 2 + t 2 t 3 P 3 P 4 ( P 1 + P 2 ) n 2 n 3 + t 1 t 3 P 1 P 4 ( P 2 + P 3 ) n 1 n 3 t 1 t 2 t 3 P 1 P 2 P 3 P 4 n 1 n 2 n 3 ,
T 3 = t 31 ( P 1 + P 2 + P 3 ) log ( n 32 / n 31 ) n 31 ( n 32 n 31 )
T g = log ( n 32 / n 31 ) ( n 32 n 31 ) ( t 1 t 2 P 1 P 2 ( P 3 + P 4 ) n 1 n 2 + t 2 t 3 P 3 P 4 ( P 1 + P 2 ) n 2 n 3 + t 1 t 3 P 1 P 4 ( P 2 + P 3 ) n 1 n 3 t 1 t 2 t 3 P 1 P 2 P 3 P 4 n 1 n 2 n 3 ) .
W 40 = 1 8 i h i [ Q i R i 3 ( n i n i 1 ) + ( n i 1 ( 1 R i 1 s i ) ) 2 ( 1 n i s i 1 n i 1 s i ) ] .
σ 1 = 4 W 40 ( s i / n i ) .
σ 1 = Δ n h i ( h i / s i ) .
δ N ( r ) d s = 0 ,
d s = ( d Y 2 + d z 2 ) 1 / 2 = d z ( d Y 2 d z 2 + 1 ) 1 / 2 ,
δ N ( Y , z ) ( 1 + Y 2 ) 1 / 2 d z = δ L ( Y , Y , z ) d z = 0 ,
L ( Y , Y , z ) Y d d z [ L ( Y , Y , z ) Y ] = 0 .
d d z { N ( z ) [ ( 1 + Y 2 ) 1 / 2 ] Y } = d d z { N ( z ) Y ( 1 + Y 2 ) 1 / 2 } = 0 .
d d z ( N ( z ) Y ) = 0 .
Y ( z ) = N ( 0 ) Y ( 0 ) N ( z ) .
0 t Y ( z ) d z = Y ( z ) Y ( 0 ) = N ( 0 ) Y ( 0 ) 0 t d z N ( z ) ,
Y ( z ) = Y ( 0 ) + N a Y ( 0 ) t ln ( N ( z ) / N a ) ( N p N a ) .
A = 1 , B = t N a ln ( N ( z ) / N a ) ( N p N a ) , C = 0 , D = N ( 0 ) N ( z ) .

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