Abstract

We describe a new approach to the index reconstruction of three-dimensional optical systems with rotational symmetry, which is based on sampling ray paths that lie in the sagittal plane. Since the observed rays are distorted by the optical system itself, they cannot be used directly for index reconstruction. We present an iterative procedure to compute the true ray paths and then to find the index distribution. The utility of the method is verified on the model problem.

© 1998 Optical Society of America

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References

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  1. O. Pomerantzeff, M. Pankratov, G-J. Wang, P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61, 166–176 (1984).
    [CrossRef] [PubMed]
  2. W. S. Jagger, “The refractive structure and optical properties of the isolated crystalline lens of the cat,” Vision Res. 30, 723–738 (1990).
    [CrossRef] [PubMed]
  3. B. K. Pierscionek, D. Y. C. Chan, “Refractive index gradient of human lenses,” Optom. Vision Sci. 66, 822–829 (1989).
    [CrossRef]
  4. D. O. Mutti, K. Zadnik, A. J. Adams, “The equivalent refractive index of the crystalline lens in childhood,” Vision Res. 35, 1565–1573 (1995).
    [CrossRef] [PubMed]
  5. B. K. Pierscionek, “Surface refractive index of the eye lens determined with an optical fiber sensor,” J. Opt. Soc. Am. A 10, 1867–1870 (1993).
    [CrossRef]
  6. D. Y. C. Chan, J. P. Ennis, B. K. Pierscionek, G. Smith, “Determination and modelling of the 3-D gradient refractive indices in crystalline lenses,” Appl. Opt. 27, 926–931 (1988).
    [CrossRef] [PubMed]
  7. G. Beliakov, D. Y. C. Chan, “Analysis of inhomogeneous optical systems by the use of ray tracing. I. Planar systems,” Appl. Opt. 36, 5303–5309 (1997).
    [CrossRef] [PubMed]
  8. G. Beliakov, “Reconstruction of optical characteristics of waveguide lenses by the use of ray tracing,” Appl. Opt. 33, 3401–3404 (1994).
    [CrossRef] [PubMed]
  9. G. Beliakov, “Numerical evaluation of the Luneburg’s integral and ray tracing,” Appl. Opt. 35, 1011–1014 (1996).
    [CrossRef] [PubMed]
  10. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), pp. 5–6.
  11. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1992), pp. 710, 757.
  12. Encyclopedia of Mathematics (Kluwer Academic, Dordrecht, The Netherlands, 1992), Vol. 8, pp. 53, 379.
  13. C. L. Hu, L. L. Schumaker, “Bivariate spline smoothing,” in Delay Equations, Approximation, and Application, Proceedings of the International Symposium at the University of Mannheim, G. Meinardus, G. Nurnberger, eds. (Birkhauser, Basel, 1985), pp. 165–179.

1997 (1)

1996 (1)

1995 (1)

D. O. Mutti, K. Zadnik, A. J. Adams, “The equivalent refractive index of the crystalline lens in childhood,” Vision Res. 35, 1565–1573 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

1990 (1)

W. S. Jagger, “The refractive structure and optical properties of the isolated crystalline lens of the cat,” Vision Res. 30, 723–738 (1990).
[CrossRef] [PubMed]

1989 (1)

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

1988 (1)

1984 (1)

O. Pomerantzeff, M. Pankratov, G-J. Wang, P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61, 166–176 (1984).
[CrossRef] [PubMed]

Adams, A. J.

D. O. Mutti, K. Zadnik, A. J. Adams, “The equivalent refractive index of the crystalline lens in childhood,” Vision Res. 35, 1565–1573 (1995).
[CrossRef] [PubMed]

Beliakov, G.

Chan, D. Y. C.

Dufault, P.

O. Pomerantzeff, M. Pankratov, G-J. Wang, P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61, 166–176 (1984).
[CrossRef] [PubMed]

Ennis, J. P.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1992), pp. 710, 757.

Hu, C. L.

C. L. Hu, L. L. Schumaker, “Bivariate spline smoothing,” in Delay Equations, Approximation, and Application, Proceedings of the International Symposium at the University of Mannheim, G. Meinardus, G. Nurnberger, eds. (Birkhauser, Basel, 1985), pp. 165–179.

Jagger, W. S.

W. S. Jagger, “The refractive structure and optical properties of the isolated crystalline lens of the cat,” Vision Res. 30, 723–738 (1990).
[CrossRef] [PubMed]

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), pp. 5–6.

Mutti, D. O.

D. O. Mutti, K. Zadnik, A. J. Adams, “The equivalent refractive index of the crystalline lens in childhood,” Vision Res. 35, 1565–1573 (1995).
[CrossRef] [PubMed]

Pankratov, M.

O. Pomerantzeff, M. Pankratov, G-J. Wang, P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61, 166–176 (1984).
[CrossRef] [PubMed]

Pierscionek, B. K.

Pomerantzeff, O.

O. Pomerantzeff, M. Pankratov, G-J. Wang, P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61, 166–176 (1984).
[CrossRef] [PubMed]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1992), pp. 710, 757.

Schumaker, L. L.

C. L. Hu, L. L. Schumaker, “Bivariate spline smoothing,” in Delay Equations, Approximation, and Application, Proceedings of the International Symposium at the University of Mannheim, G. Meinardus, G. Nurnberger, eds. (Birkhauser, Basel, 1985), pp. 165–179.

Smith, G.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1992), pp. 710, 757.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1992), pp. 710, 757.

Wang, G-J.

O. Pomerantzeff, M. Pankratov, G-J. Wang, P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61, 166–176 (1984).
[CrossRef] [PubMed]

Zadnik, K.

D. O. Mutti, K. Zadnik, A. J. Adams, “The equivalent refractive index of the crystalline lens in childhood,” Vision Res. 35, 1565–1573 (1995).
[CrossRef] [PubMed]

Am. J. Optom. Physiol. Opt. (1)

O. Pomerantzeff, M. Pankratov, G-J. Wang, P. Dufault, “Wide-angle optical model of the eye,” Am. J. Optom. Physiol. Opt. 61, 166–176 (1984).
[CrossRef] [PubMed]

Appl. Opt. (4)

J. Opt. Soc. Am. A (1)

Optom. Vision Sci. (1)

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

Vision Res. (2)

D. O. Mutti, K. Zadnik, A. J. Adams, “The equivalent refractive index of the crystalline lens in childhood,” Vision Res. 35, 1565–1573 (1995).
[CrossRef] [PubMed]

W. S. Jagger, “The refractive structure and optical properties of the isolated crystalline lens of the cat,” Vision Res. 30, 723–738 (1990).
[CrossRef] [PubMed]

Other (4)

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), pp. 5–6.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1992), pp. 710, 757.

Encyclopedia of Mathematics (Kluwer Academic, Dordrecht, The Netherlands, 1992), Vol. 8, pp. 53, 379.

C. L. Hu, L. L. Schumaker, “Bivariate spline smoothing,” in Delay Equations, Approximation, and Application, Proceedings of the International Symposium at the University of Mannheim, G. Meinardus, G. Nurnberger, eds. (Birkhauser, Basel, 1985), pp. 165–179.

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Figures (4)

Fig. 1
Fig. 1

Model of the crystalline lens.

Fig. 2
Fig. 2

Formation of a distorted image by the crystalline lens. On the observation plane the dashed curve denotes the true ray path, and the solid curve denotes the observed distorted ray path.

Fig. 3
Fig. 3

(a) True ray paths. (b) Observed ray paths distorted by the optical system. Note the nonmonotonicity of the rays inside the lens.

Fig. 4
Fig. 4

Results of index reconstruction in the sagittal plane, compared with the model index: (a) reconstructed index, (b) model index, given by the formula for Fig. 3. The initial data consisted of the distorted image of the true rays in Fig. 3(b). As the initial approximation to the index, the constant value n(x, y) = 1 was taken. The algorithm converged in eight iterations. The maximum absolute error of reconstruction is 0.0108, and the average absolute error is 0.00256.

Equations (12)

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d d s n   d r d s = n ,
nx 1 + x 2 + y 2 + x n z - n x = 0 , ny 1 + x 2 + y 2 + y n z - n y = 0 ,
{ x ij ,   y ij :   y ij = Y x ij ,   h j ,   i = 1 ,   2 , ,   N j , j =   1 ,   2 , ,   M } ,
x z end ,   y z end = x end ,   y end ,
x z end ,   y z end = x ,   y , x z end ,   y z end = x end ,   y end
p = 1.8 , if   x 0 , 1.4 , if   x > 0 .
X ˜ = x ˜ ij ,   y ˜ ij ,   i = 1 ,   2 , ,   N j ,   j = 1 ,   2 , ,   M .
n = N [ D n - 1 X ˜ ] .
n k + 1 = N [ D n k - 1 [ X ˜ ] ] ,
W 2 m Ω = f x :   f C m - 1 Ω ,   f m L 2 Ω ,
n k + 1 = α N [ D n k - 1 [ X ˜ ] ] + 1 - α n k ,     0 < α < 1 .
x z end ,   y z end = x ˜ ij ,   y ˜ ij , x z end ,   y z end = x end ,   y end ,

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