Abstract

We describe a novel approach to refractive-index reconstruction in two-dimensional systems with no special symmetry, based on observation of traces of rays that travel through the optical system. The mathematical model of ray-tracing analysis is presented in detail, and both the analytical and numerical solutions are given. Methods of data processing in the presence of experimental errors are developed and applied to model problems.

© 1997 Optical Society of America

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References

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  1. P. L. Chu, “Nondestructive measurement of index profile of an optical-fiber preform,” Electron. Lett. 13, 736–738 (1977).
    [Crossref]
  2. P. L. Chu, “Nondestructive refractive-index profile measurement of elliptical optical fibre or preform,” Electron. Lett. 15, 357–358 (1979).
    [Crossref]
  3. P. L. Chu, T. Whitbread, “Nondestructive determination of refractive index profile of an optical fiber: fast Fourier transform method,” Appl. Opt. 18, 1117–1122 (1979).
    [Crossref] [PubMed]
  4. J. A. Ferrari, E. Frins, A. Rondoni, G. Montaldo, “Retrieval algorithm for refractive-index profile of fibers from transverse interferograms,” Opt. Commun. 117, 25–30 (1995).
    [Crossref]
  5. J. Sochacki, M. Sochacka, C. Gomez-Reino, “Reconstruction of the axisymmetric refractive-index profiles from transverse interferograms in the presence of the immersion-air separating window,” J. Opt. Soc. Am. A 7, 211–215 (1990).
    [Crossref]
  6. G. Beliakov, “Study of the mathematical model of the refractive-index reconstruction of a smooth thin film waveguide by ray tracing,” Ph.D. dissertation (People’s Friendship University of Russia, Moscow, 1992) (in Russian).
  7. G. Beliakov, “Reconstruction of optical characteristics of waveguide lenses by the use of ray tracing,” Appl. Opt. 33, 3401–3404 (1994).
    [Crossref] [PubMed]
  8. D. Y. C. Chan, J. P. Ennis, B. K. Pierscionek, G. Smith, “Determination and modeling of the 3-D gradient refractive indices in crystalline lenses,” Appl. Opt. 27, 926–931 (1988).
    [Crossref] [PubMed]
  9. I. H. Lira, “Reconstruction of an axisymmetric refractive index distribution with non-negligible refraction,” Meas. Sci. Technol. 5, 226–232 (1994).
    [Crossref]
  10. C. M. Vest, “Tomography for properties of materials that bend rays: a tutorial,” Appl. Opt. 24, 4089–4094 (1985).
    [Crossref] [PubMed]
  11. O. Sasaki, T. Kobayashi, “Beam-deflection optical tomography of the refractive-index distribution based on the Rytov approximation,” Appl. Opt. 32, 746–751 (1993).
    [Crossref] [PubMed]
  12. 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]
  13. M. Born, E. Wolf, Principles of Optics, (Pergamon, Oxford, UK, 1986), p. 114.
  14. Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, (Springer-Verlag, Heidelberg, 1990), p. 41.
  15. G. Beliakov, “Numerical evaluation of the Luneburg’s integral and ray tracing,” Appl. Opt. 25, 1011–1014 (1996).
    [Crossref]
  16. A. L. Mikaelian, “Self-focusing media with variable index of refraction,” Prog. Opt. 17, 279–345 (1980).
    [Crossref]
  17. 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]
  18. P. Craven, G. Wahba, “Smoothing noisy data with spline functions,” Numer. Math. 31, 377–403 (1979).
    [Crossref]
  19. P. Dierckx, Curve and Surface Fitting with Splines (Oxford U. Press, New York, 1993), p. 82.

1996 (1)

G. Beliakov, “Numerical evaluation of the Luneburg’s integral and ray tracing,” Appl. Opt. 25, 1011–1014 (1996).
[Crossref]

1995 (1)

J. A. Ferrari, E. Frins, A. Rondoni, G. Montaldo, “Retrieval algorithm for refractive-index profile of fibers from transverse interferograms,” Opt. Commun. 117, 25–30 (1995).
[Crossref]

1994 (2)

G. Beliakov, “Reconstruction of optical characteristics of waveguide lenses by the use of ray tracing,” Appl. Opt. 33, 3401–3404 (1994).
[Crossref] [PubMed]

I. H. Lira, “Reconstruction of an axisymmetric refractive index distribution with non-negligible refraction,” Meas. Sci. Technol. 5, 226–232 (1994).
[Crossref]

1993 (2)

1990 (1)

1988 (1)

1985 (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]

1980 (1)

A. L. Mikaelian, “Self-focusing media with variable index of refraction,” Prog. Opt. 17, 279–345 (1980).
[Crossref]

1979 (3)

P. Craven, G. Wahba, “Smoothing noisy data with spline functions,” Numer. Math. 31, 377–403 (1979).
[Crossref]

P. L. Chu, “Nondestructive refractive-index profile measurement of elliptical optical fibre or preform,” Electron. Lett. 15, 357–358 (1979).
[Crossref]

P. L. Chu, T. Whitbread, “Nondestructive determination of refractive index profile of an optical fiber: fast Fourier transform method,” Appl. Opt. 18, 1117–1122 (1979).
[Crossref] [PubMed]

1977 (1)

P. L. Chu, “Nondestructive measurement of index profile of an optical-fiber preform,” Electron. Lett. 13, 736–738 (1977).
[Crossref]

Beliakov, G.

G. Beliakov, “Numerical evaluation of the Luneburg’s integral and ray tracing,” Appl. Opt. 25, 1011–1014 (1996).
[Crossref]

G. Beliakov, “Reconstruction of optical characteristics of waveguide lenses by the use of ray tracing,” Appl. Opt. 33, 3401–3404 (1994).
[Crossref] [PubMed]

G. Beliakov, “Study of the mathematical model of the refractive-index reconstruction of a smooth thin film waveguide by ray tracing,” Ph.D. dissertation (People’s Friendship University of Russia, Moscow, 1992) (in Russian).

Born, M.

M. Born, E. Wolf, Principles of Optics, (Pergamon, Oxford, UK, 1986), p. 114.

Chan, D. Y. C.

Chu, P. L.

P. L. Chu, “Nondestructive refractive-index profile measurement of elliptical optical fibre or preform,” Electron. Lett. 15, 357–358 (1979).
[Crossref]

P. L. Chu, T. Whitbread, “Nondestructive determination of refractive index profile of an optical fiber: fast Fourier transform method,” Appl. Opt. 18, 1117–1122 (1979).
[Crossref] [PubMed]

P. L. Chu, “Nondestructive measurement of index profile of an optical-fiber preform,” Electron. Lett. 13, 736–738 (1977).
[Crossref]

Craven, P.

P. Craven, G. Wahba, “Smoothing noisy data with spline functions,” Numer. Math. 31, 377–403 (1979).
[Crossref]

Dierckx, P.

P. Dierckx, Curve and Surface Fitting with Splines (Oxford U. Press, New York, 1993), p. 82.

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.

Ferrari, J. A.

J. A. Ferrari, E. Frins, A. Rondoni, G. Montaldo, “Retrieval algorithm for refractive-index profile of fibers from transverse interferograms,” Opt. Commun. 117, 25–30 (1995).
[Crossref]

Frins, E.

J. A. Ferrari, E. Frins, A. Rondoni, G. Montaldo, “Retrieval algorithm for refractive-index profile of fibers from transverse interferograms,” Opt. Commun. 117, 25–30 (1995).
[Crossref]

Gomez-Reino, C.

Kobayashi, T.

Kravtsov, Yu. A.

Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, (Springer-Verlag, Heidelberg, 1990), p. 41.

Lira, I. H.

I. H. Lira, “Reconstruction of an axisymmetric refractive index distribution with non-negligible refraction,” Meas. Sci. Technol. 5, 226–232 (1994).
[Crossref]

Mikaelian, A. L.

A. L. Mikaelian, “Self-focusing media with variable index of refraction,” Prog. Opt. 17, 279–345 (1980).
[Crossref]

Montaldo, G.

J. A. Ferrari, E. Frins, A. Rondoni, G. Montaldo, “Retrieval algorithm for refractive-index profile of fibers from transverse interferograms,” Opt. Commun. 117, 25–30 (1995).
[Crossref]

Orlov, Yu. I.

Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, (Springer-Verlag, Heidelberg, 1990), p. 41.

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]

Rondoni, A.

J. A. Ferrari, E. Frins, A. Rondoni, G. Montaldo, “Retrieval algorithm for refractive-index profile of fibers from transverse interferograms,” Opt. Commun. 117, 25–30 (1995).
[Crossref]

Sasaki, O.

Smith, G.

Sochacka, M.

Sochacki, J.

Vest, C. M.

Wahba, G.

P. Craven, G. Wahba, “Smoothing noisy data with spline functions,” Numer. Math. 31, 377–403 (1979).
[Crossref]

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]

Whitbread, T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, (Pergamon, Oxford, UK, 1986), p. 114.

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. (6)

Electron. Lett. (2)

P. L. Chu, “Nondestructive measurement of index profile of an optical-fiber preform,” Electron. Lett. 13, 736–738 (1977).
[Crossref]

P. L. Chu, “Nondestructive refractive-index profile measurement of elliptical optical fibre or preform,” Electron. Lett. 15, 357–358 (1979).
[Crossref]

J. Opt. Soc. Am. A (2)

Meas. Sci. Technol. (1)

I. H. Lira, “Reconstruction of an axisymmetric refractive index distribution with non-negligible refraction,” Meas. Sci. Technol. 5, 226–232 (1994).
[Crossref]

Numer. Math. (1)

P. Craven, G. Wahba, “Smoothing noisy data with spline functions,” Numer. Math. 31, 377–403 (1979).
[Crossref]

Opt. Commun. (1)

J. A. Ferrari, E. Frins, A. Rondoni, G. Montaldo, “Retrieval algorithm for refractive-index profile of fibers from transverse interferograms,” Opt. Commun. 117, 25–30 (1995).
[Crossref]

Prog. Opt. (1)

A. L. Mikaelian, “Self-focusing media with variable index of refraction,” Prog. Opt. 17, 279–345 (1980).
[Crossref]

Other (4)

P. Dierckx, Curve and Surface Fitting with Splines (Oxford U. Press, New York, 1993), p. 82.

M. Born, E. Wolf, Principles of Optics, (Pergamon, Oxford, UK, 1986), p. 114.

Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, (Springer-Verlag, Heidelberg, 1990), p. 41.

G. Beliakov, “Study of the mathematical model of the refractive-index reconstruction of a smooth thin film waveguide by ray tracing,” Ph.D. dissertation (People’s Friendship University of Russia, Moscow, 1992) (in Russian).

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Figures (6)

Fig. 1
Fig. 1

Ray paths through an optical system.

Fig. 2
Fig. 2

Refractive index reconstructed from precise data.

Fig. 3
Fig. 3

Model of sampling rays: precise values plus Gaussian noise (zero mean, standard deviation σ = 0.001).

Fig. 4
Fig. 4

Results of the refractive-index reconstruction using noisy data. (a) Contour plot of the reconstructed index, (b) section y = 0, (c) section x = 0. Reconstructed index profile is plotted in solid curves. True refractive index, given by n(x, y) = 1.42{1 - 0.2[(x - 0.1)2 + (0.7y)2]}1/2, is plotted in dashed curves. Standard deviation of the noise, σ = 0.001; maximum error of reconstruction, 0.0028; average error, 0.00080. Confidence intervals (p = 0.95) are also shown.

Fig. 5
Fig. 5

Sampling rays with Gaussian noise (zero mean, standard deviation σ = 0.0003) used for reconstruction of the refractive index in Fig. 2.

Fig. 6
Fig. 6

Results of refractive index reconstruction (a) section y = 0, (b) section x = 0. Reconstructed index is shown by the solid curve, and the true index is shown by the dashed curve. Standard deviation of the noise, σ = 0.0003; maximum error of reconstruction, 0.0031; average error, 0.0011. The confidence intervals (p = 0.95) are shown.

Equations (26)

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Sr2=nr2,
ddsndrds=n,
-y wx+wy-y1+y2=0,
yx=Yx, h, h-hh+,
Cx, ywx+wy=Fx, y,
Cx, y=-Yxx, y|h=hx,y,  Fx, y=Yxxx, y1+Yx2x, y|h=hx,y.
dxdy=Cx, y,dwdy=Fx, y.
dydx=-gxgxfyfy,
 gxgxdx=- fyfydy+C1,
wx, y=lnnx, y= FXy, C1, ydyC1=Sx,y+C2.
nx, y=exp{ FXy, C1, ydy|C1=Sx,y+ΦSx, y},
ϕSrSr21/2=nr.
Sx, y=coshycosx=C1.
d2hx, ydx2=0,
Fx, y=-tanhy.
nx, y=1coshyΦcoshycosx,
ny=n0coshy,
xij, yij: yij=Yxij, hj, i=1, 2, , Nj,  j=1,2, , M,
wXy, y=y0yFXy, ydy+w0,
nx, y=1.421-0.2x-0.12+0.7y21/2+0.01/cosh200x2+y22.
S2m-2x=C,
p abSmx2dx+i=1NSxi-yi2,
ds2=dx2+dy2,  dds=ddxdxds=ddx1+dydx2-1/2,  d2ds2=d2dx21+dydx2-1.
dndxdxds2=nx,dndxdxdsdyds+n d2ydx21+dydx2-1=ny.
dndx=1+y2nx.
nxy+ny1+y2-1=ny,

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