Abstract

A simple method of constructing 3-D gradient refractive-index profiles in crystalline lenses is proposed. The input data are derived from 2-D refraction measurements of rays in the equatorial plane of the lens. In this paper, the isoindicial contours within the lens are modeled as a family of concentric ellipses; however, other physically more appropriate models may also be constructed. This method is illustrated by using it to model the 3-D refractive-index profile of a bovine lens.

© 1988 Optical Society of America

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References

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  1. P. L. Chu, “Nondestructive Measurements of Index Profile of an Optical-Fibre Preform,” Electron. Lett. 12, 136 (1977).
  2. M. C. W. Campbell, “Measurement of Refractive Index in an Intact Crystalline Lens,” Vision Res. 24, 409 (1984).
    [Crossref] [PubMed]
  3. K. F. Barrell, C. Pask, “Nondestructive Index Profile Measurement of Non-circular Optical Fibre Preforms,” Opt. Commun. 27, 230 (1978).
    [Crossref]
  4. P. L. Chu, “Nondestructive Refractive-Index Profile Measurement of Elliptical Optical Fibre or Preform,” Electron. Lett. 15, 357 (1979).
    [Crossref]
  5. M. J. Howcroft, J. A. Parker, “Aspheric Curvatures for the Human Lens,” Vision Res. 17, 1217 (1977).
    [Crossref] [PubMed]
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).
  7. C. Pask, “The Theory of Non-destructive Lens Index Distribution Measurement,” in Modelling the Eye with Graded Index Optics, A. Hughes, Ed. (Cambridge U. P., London, 198X) (in press); see also Ref. 2.
  8. P. L. Chu, T. Whitbread, “Nondestructive Determination of Refractive Index Profile of an Optical Fiber: Fast Fourier Transform Method,” Appl. Opt. 18, 1117 (1979).
    [Crossref] [PubMed]
  9. P-L. Francois, I. Sasaki, M. J. Adams, “Practical Three-Dimensional Profiling of Optical Fiber Preforms,” IEEE J. Quantum Electron. 18, 524 (1982).
    [Crossref]
  10. B. K. Pierscionek, D. Y. C. Chan, J. P. Ennis, G. Smith, R. C. Augusteyn, “A Nondestructive Method of Constructing Three-dimensional Gradient Index Models for Crystalline Lenses: I. Theory and Experiment,” Am. J. Optom. Physiol. Optics (in press).

1984 (1)

M. C. W. Campbell, “Measurement of Refractive Index in an Intact Crystalline Lens,” Vision Res. 24, 409 (1984).
[Crossref] [PubMed]

1982 (1)

P-L. Francois, I. Sasaki, M. J. Adams, “Practical Three-Dimensional Profiling of Optical Fiber Preforms,” IEEE J. Quantum Electron. 18, 524 (1982).
[Crossref]

1979 (2)

P. L. Chu, T. Whitbread, “Nondestructive Determination of Refractive Index Profile of an Optical Fiber: Fast Fourier Transform Method,” Appl. Opt. 18, 1117 (1979).
[Crossref] [PubMed]

P. L. Chu, “Nondestructive Refractive-Index Profile Measurement of Elliptical Optical Fibre or Preform,” Electron. Lett. 15, 357 (1979).
[Crossref]

1978 (1)

K. F. Barrell, C. Pask, “Nondestructive Index Profile Measurement of Non-circular Optical Fibre Preforms,” Opt. Commun. 27, 230 (1978).
[Crossref]

1977 (2)

M. J. Howcroft, J. A. Parker, “Aspheric Curvatures for the Human Lens,” Vision Res. 17, 1217 (1977).
[Crossref] [PubMed]

P. L. Chu, “Nondestructive Measurements of Index Profile of an Optical-Fibre Preform,” Electron. Lett. 12, 136 (1977).

Adams, M. J.

P-L. Francois, I. Sasaki, M. J. Adams, “Practical Three-Dimensional Profiling of Optical Fiber Preforms,” IEEE J. Quantum Electron. 18, 524 (1982).
[Crossref]

Augusteyn, R. C.

B. K. Pierscionek, D. Y. C. Chan, J. P. Ennis, G. Smith, R. C. Augusteyn, “A Nondestructive Method of Constructing Three-dimensional Gradient Index Models for Crystalline Lenses: I. Theory and Experiment,” Am. J. Optom. Physiol. Optics (in press).

Barrell, K. F.

K. F. Barrell, C. Pask, “Nondestructive Index Profile Measurement of Non-circular Optical Fibre Preforms,” Opt. Commun. 27, 230 (1978).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

Campbell, M. C. W.

M. C. W. Campbell, “Measurement of Refractive Index in an Intact Crystalline Lens,” Vision Res. 24, 409 (1984).
[Crossref] [PubMed]

Chan, D. Y. C.

B. K. Pierscionek, D. Y. C. Chan, J. P. Ennis, G. Smith, R. C. Augusteyn, “A Nondestructive Method of Constructing Three-dimensional Gradient Index Models for Crystalline Lenses: I. Theory and Experiment,” Am. J. Optom. Physiol. Optics (in press).

Chu, P. L.

P. L. Chu, T. Whitbread, “Nondestructive Determination of Refractive Index Profile of an Optical Fiber: Fast Fourier Transform Method,” Appl. Opt. 18, 1117 (1979).
[Crossref] [PubMed]

P. L. Chu, “Nondestructive Refractive-Index Profile Measurement of Elliptical Optical Fibre or Preform,” Electron. Lett. 15, 357 (1979).
[Crossref]

P. L. Chu, “Nondestructive Measurements of Index Profile of an Optical-Fibre Preform,” Electron. Lett. 12, 136 (1977).

Ennis, J. P.

B. K. Pierscionek, D. Y. C. Chan, J. P. Ennis, G. Smith, R. C. Augusteyn, “A Nondestructive Method of Constructing Three-dimensional Gradient Index Models for Crystalline Lenses: I. Theory and Experiment,” Am. J. Optom. Physiol. Optics (in press).

Francois, P-L.

P-L. Francois, I. Sasaki, M. J. Adams, “Practical Three-Dimensional Profiling of Optical Fiber Preforms,” IEEE J. Quantum Electron. 18, 524 (1982).
[Crossref]

Howcroft, M. J.

M. J. Howcroft, J. A. Parker, “Aspheric Curvatures for the Human Lens,” Vision Res. 17, 1217 (1977).
[Crossref] [PubMed]

Parker, J. A.

M. J. Howcroft, J. A. Parker, “Aspheric Curvatures for the Human Lens,” Vision Res. 17, 1217 (1977).
[Crossref] [PubMed]

Pask, C.

K. F. Barrell, C. Pask, “Nondestructive Index Profile Measurement of Non-circular Optical Fibre Preforms,” Opt. Commun. 27, 230 (1978).
[Crossref]

C. Pask, “The Theory of Non-destructive Lens Index Distribution Measurement,” in Modelling the Eye with Graded Index Optics, A. Hughes, Ed. (Cambridge U. P., London, 198X) (in press); see also Ref. 2.

Pierscionek, B. K.

B. K. Pierscionek, D. Y. C. Chan, J. P. Ennis, G. Smith, R. C. Augusteyn, “A Nondestructive Method of Constructing Three-dimensional Gradient Index Models for Crystalline Lenses: I. Theory and Experiment,” Am. J. Optom. Physiol. Optics (in press).

Sasaki, I.

P-L. Francois, I. Sasaki, M. J. Adams, “Practical Three-Dimensional Profiling of Optical Fiber Preforms,” IEEE J. Quantum Electron. 18, 524 (1982).
[Crossref]

Smith, G.

B. K. Pierscionek, D. Y. C. Chan, J. P. Ennis, G. Smith, R. C. Augusteyn, “A Nondestructive Method of Constructing Three-dimensional Gradient Index Models for Crystalline Lenses: I. Theory and Experiment,” Am. J. Optom. Physiol. Optics (in press).

Whitbread, T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

Appl. Opt. (1)

Electron. Lett. (2)

P. L. Chu, “Nondestructive Measurements of Index Profile of an Optical-Fibre Preform,” Electron. Lett. 12, 136 (1977).

P. L. Chu, “Nondestructive Refractive-Index Profile Measurement of Elliptical Optical Fibre or Preform,” Electron. Lett. 15, 357 (1979).
[Crossref]

IEEE J. Quantum Electron. (1)

P-L. Francois, I. Sasaki, M. J. Adams, “Practical Three-Dimensional Profiling of Optical Fiber Preforms,” IEEE J. Quantum Electron. 18, 524 (1982).
[Crossref]

Opt. Commun. (1)

K. F. Barrell, C. Pask, “Nondestructive Index Profile Measurement of Non-circular Optical Fibre Preforms,” Opt. Commun. 27, 230 (1978).
[Crossref]

Vision Res. (2)

M. C. W. Campbell, “Measurement of Refractive Index in an Intact Crystalline Lens,” Vision Res. 24, 409 (1984).
[Crossref] [PubMed]

M. J. Howcroft, J. A. Parker, “Aspheric Curvatures for the Human Lens,” Vision Res. 17, 1217 (1977).
[Crossref] [PubMed]

Other (3)

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

C. Pask, “The Theory of Non-destructive Lens Index Distribution Measurement,” in Modelling the Eye with Graded Index Optics, A. Hughes, Ed. (Cambridge U. P., London, 198X) (in press); see also Ref. 2.

B. K. Pierscionek, D. Y. C. Chan, J. P. Ennis, G. Smith, R. C. Augusteyn, “A Nondestructive Method of Constructing Three-dimensional Gradient Index Models for Crystalline Lenses: I. Theory and Experiment,” Am. J. Optom. Physiol. Optics (in press).

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of rays in the equatorial (xy) and sagittal (yz) planes: y, Y are the beam displacements; ψ Ψ are the emergent angles formed by the beams; ns is the refractive index of the surrounding medium; ρ is the equatorial; σ is the semiminor axis for the posterior curvature in the illustrated case.

Fig. 2
Fig. 2

Error in the computed refractive-index profile Δn(r) defined in Eq. (16) for data sets A and B given in Table I: open symbols, ten data points, = 0.1; solid symbols, twenty data points, = 0.05.

Fig. 3
Fig. 3

Equatorial refractive-index profile for data set A and the isoindicial contours of the concentric ellipse model. From the centre to the boundary the refractive-index values on the contours are 1.41, 1.40, 1.38, 1.36, and 1.35 (the lens boundary).

Fig. 4
Fig. 4

Deflection angle Ψ (in degrees) as a function of the position (Y/ρ) of incident meridional rays parallel to the z axis: solid symbols, data set A; crosses, data set B.

Fig. 5
Fig. 5

(a) Experimental measurements of the deflection angle ψ (in radians) as a function of the beam position y for rays in the equatorial plane of a bovine lens. (b) Refractive-index profile n(r) in the equatorial plane deduced from the data in (a). (c) Comparison of the deflection angle angle Ψ (in radians) as a function of beam position Y for rays in the sagittal plane. Points correspond to experimental data, and predicted values based on concentric elliptical isoindicial contours and the results of (b) are given by the continuous line.

Tables (1)

Tables Icon

Table I Input Data Sets for the Parabolic Index Profile [Eq. (14)] and the Extrapolated Refractive Index at the Lens Boundary using Eq. (13) using Ten and Twenty Data Points Corresponding to = 0.1 and 0.05, Respectively.

Equations (17)

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ψ ( y ) = 2 cos 1 ( y / ρ ) + 2 y n s r m ρ d r r [ r 2 n 2 ( r ) y 2 n s 2 ] 1 / 2 .
r m n m ( r m ) = y n s .
n s ξ r n ( r ) ,
n ( r ) n ( ρ ) exp [ g ( ξ ) ] ,
ψ ( y ) = 2 { sin 1 ( y / ρ ) sin 1 ( y / n ¯ ρ ) } 2 y y n ¯ ρ d ξ g ( ξ ) ( ξ 2 y 2 ) 1 / 2 ,
n ¯ n ( ρ ) / n s
g ( ξ ) = 1 π ξ ρ d y ψ ( y ) ( y 2 ξ 2 ) 1 / 2 .
r = ξ exp [ g ( ξ ) ] ;
n ( r ) = n ( ρ ) exp [ g ( ξ ) ] ;
y ¯ y / n ¯ ,
ψ ¯ ψ ( y ) 2 { sin 1 ( y / ρ ) sin 1 ( y / n ¯ ρ ) } .
ψ ( [ 1 ] ρ ) = 2 2 { [ 1 ρ g ( ρ ) ] ( + δ ) } + O ( 3 / 2 , δ 3 / 2 ) .
n ¯ = 1 + ( { ψ ( [ 1 ] ρ ) + 2 2 } / ψ ( ρ ) ) λ 2 1 .
n ( r ) = n 0 [ 1 2 Δ ( r / ρ ) 2 ] ,
ψ ( y ) = π / 2 2 cos 1 ( y / ρ ) + sin 1 { 1 2 ( 1 2 Δ ) ( y / n ¯ ρ ) 2 [ 1 8 Δ ( 1 2 Δ ) ( y / n ¯ ρ ) 2 ] 1 / 2 } .
Δ n ( r ) n computed ( r ) n true ( r ) .
α 2 ( r / ρ ) 2 + ( z / σ ) 2 , r ( x 2 + y 2 ) ,

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