Abstract

We have developed a ray-tracing simulation procedure for optically isotropic gradient refractive-index media. The procedure can take discrete points of arbitrary distribution for the definition of refractive-index distributions and lens surfaces. It is useful for simulating ray trajectories in real lens systems. The procedure is applied to a ray-tracing simulation of the Luneburg lens and a radial gradient optical fiber. The simulation results are compared with the analytical solutions, and it is shown that they are in precise agreement.

© 2011 Optical Society of America

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References

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  1. D. W. Sweeney and C. M. Vest, “Reconstruction of three-dimensional refractive index fields from multidirectional interferometric data,” Appl. Opt. 12, 2649–2554 (1973).
    [CrossRef] [PubMed]
  2. M. H. Maleki, A. J. Devaney, and A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
    [CrossRef]
  3. H. Suhara, “Interferometric measurement of the refractive-index distribution in plastic lenses by use of computed tomography,” Appl. Opt. 41, 5317–5325 (2002).
    [CrossRef] [PubMed]
  4. W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Investigation of the refractive index distribution in precision compression glass molding by use of 3D tomography,” Meas. Sci. Technol. 20, 055109 (2009).
    [CrossRef]
  5. W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Refractive index and dispersion variation in precision optical glass molding by computed tomography,” Appl. Opt. 48, 3588–3595 (2009).
    [CrossRef] [PubMed]
  6. S. Cha and C. M. Vest, “Interferometry and reconstruction of strongly refracting asymmetric-refractive-index fields,” Opt. Lett. 4, 311–313 (1979).
    [CrossRef] [PubMed]
  7. S. Cha and C. M. Vest, “Tomographic reconstruction of strongly refracting fields and its application to interferometric measurement of boundary layers,” Appl. Opt. 20, 2787–2794(1981).
    [CrossRef] [PubMed]
  8. E. W. Marchand, Gradient Index Optics (Academic, 1978).
  9. M. Born and E. Wolf, Principles of Optics I, 7th ed. (Cambridge Univ. Press, 1999).
  10. A. Sharma, D. V. Kumar, and A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. 21, 984–987 (1982).
    [CrossRef] [PubMed]
  11. A. Sharma, “Computing optical path length in gradient index media: a fast and accurate method,” Appl. Opt. 24, 4367–4370(1985).
    [CrossRef] [PubMed]
  12. A. Sharma and A. K. Ghatak, “Ray tracing in gradient-index lenses: computation of ray-surface intersection,” Appl. Opt. 25, 3409–3412 (1986).
    [CrossRef] [PubMed]
  13. H. Nishi, H. Ichikawa, M. Toyama, and I. Kitano, “Gradient-index objective lens for the compact disk system,” Appl. Opt. 25, 3340–3344 (1986).
    [CrossRef] [PubMed]
  14. D. Siedlecki, H. Kasprzak, and B. K. Pierscionek, “Schematic eye with a gradient-index lens and aspheric surfaces,” Opt. Lett. 29, 1197–1199 (2004).
    [CrossRef] [PubMed]
  15. Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
    [CrossRef] [PubMed]
  16. C. C. Handapangoda and M. Premaratne, “An approximate numerical technique for characterizing optical pulse propagation in inhomogeneous biological tissue,” J. Biomed. Biotechnol. 2008, 784354 (2008).
    [CrossRef] [PubMed]
  17. J. Arrue, F. Jiménez, G. Aldabaldetreku, G. Durana, J. Zubia, M. Lomer, and J. Mateo, “Analysis of the use of tapered graded-index polymer optical fibers for refractive-index sensors,” Opt. Express 16, 16616–16631 (2008).
    [CrossRef] [PubMed]
  18. I. Ihrke, G. Ziegler, A. Tevs, C. Theobalt, M. Magnor, and H.-P. Seidel, “Eikonal rendering: efficient light transport in refractive objects,” ACM Trans. Graph. 26(3), 59 (2007).
    [CrossRef]
  19. T. Sakamoto, “Ray trace algorithms for GRIN media,” Appl. Opt. 26, 2943–2946 (1987).
    [CrossRef] [PubMed]
  20. B. D. Stone and G. W. Forbes, “Optimal interpolants for Runge–Kutta ray tracing in inhomogeneous media,” J. Opt. Soc. Am. A 7, 248–254 (1990).
    [CrossRef]
  21. J. Puchalski, “Numerical determination of ray tracing: a new method,” Appl. Opt. 31, 6789–6799 (1992).
    [CrossRef] [PubMed]
  22. W. C. Carpenter and P. A. T. Gill, “Automated solutions of time-dependent problems,” in The Mathematics of Finite Elements and Applications II, J.Whiteman, ed. (Academic, 1973), pp. 495–506.
  23. B. Richerzhagen, “Finite element ray tracing: a new method for ray tracing in gradient-index media,” Appl. Opt. 35, 6186–6189 (1996).
    [CrossRef] [PubMed]
  24. S. Deng, X. Li, Z. Cen, and S. Jian, “Simulation of the inhomogeneous medium with a self-adapting grid,” Appl. Opt. 46, 3102–3106 (2007).
    [CrossRef] [PubMed]
  25. D. A. Atchison and G. Smith, “Continuous gradient index and shell models of human lens,” Vis. Res. 35, 2529–2538(1995).
    [CrossRef] [PubMed]
  26. S. Morita, Y. Nishidate, T. Nagata, Y. Yamagata, and C. Teodosiu, “Ray-tracing simulation method using piecewise quadratic interpolant for aspheric optical systems,” Appl. Opt. 49, 3442–3451 (2010).
    [CrossRef] [PubMed]
  27. T. Nagata, “Simple local interpolation of surfaces using normal vectors,” Comput. Aided Geom. Des. 22, 327–347 (2005).
    [CrossRef]
  28. R. Courant, K. Friedrichs, and H. Lewy, “On the partial difference equations of mathematical physics,” IBM J. Res. Dev. 11, 215–234 (1967).
    [CrossRef]
  29. T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin methods,” Int. J. Numer. Meth. Eng. 37, 229–256 (1994).
    [CrossRef]
  30. S. N. Atluri and T. Zhu, “A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics,” Comput. Mech. 22, 117–127 (1998).
    [CrossRef]
  31. X. Jin, G. Li, and N. R. Aluru, “On the equivalence between least-squares and kernel approximations in meshless methods,” Comput. Model. Eng. Sci. 2, 447–462 (2001).
    [CrossRef]
  32. F. T. Wong and W. Kanok-Nukulchai, “Kriging-based finite element method: element-by-element Kriging interpolation,” Civil Eng. Dimens. 11, 15–22 (2009).
  33. E. Fehlberg, “Low-order classical Runge–Kutta formulas with step size control and their application to some heat transfer problems,” Technical report 315 (NASA, 1969).

2010 (1)

2009 (3)

F. T. Wong and W. Kanok-Nukulchai, “Kriging-based finite element method: element-by-element Kriging interpolation,” Civil Eng. Dimens. 11, 15–22 (2009).

W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Investigation of the refractive index distribution in precision compression glass molding by use of 3D tomography,” Meas. Sci. Technol. 20, 055109 (2009).
[CrossRef]

W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Refractive index and dispersion variation in precision optical glass molding by computed tomography,” Appl. Opt. 48, 3588–3595 (2009).
[CrossRef] [PubMed]

2008 (2)

C. C. Handapangoda and M. Premaratne, “An approximate numerical technique for characterizing optical pulse propagation in inhomogeneous biological tissue,” J. Biomed. Biotechnol. 2008, 784354 (2008).
[CrossRef] [PubMed]

J. Arrue, F. Jiménez, G. Aldabaldetreku, G. Durana, J. Zubia, M. Lomer, and J. Mateo, “Analysis of the use of tapered graded-index polymer optical fibers for refractive-index sensors,” Opt. Express 16, 16616–16631 (2008).
[CrossRef] [PubMed]

2007 (2)

I. Ihrke, G. Ziegler, A. Tevs, C. Theobalt, M. Magnor, and H.-P. Seidel, “Eikonal rendering: efficient light transport in refractive objects,” ACM Trans. Graph. 26(3), 59 (2007).
[CrossRef]

S. Deng, X. Li, Z. Cen, and S. Jian, “Simulation of the inhomogeneous medium with a self-adapting grid,” Appl. Opt. 46, 3102–3106 (2007).
[CrossRef] [PubMed]

2005 (1)

T. Nagata, “Simple local interpolation of surfaces using normal vectors,” Comput. Aided Geom. Des. 22, 327–347 (2005).
[CrossRef]

2004 (1)

2002 (1)

2001 (2)

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

X. Jin, G. Li, and N. R. Aluru, “On the equivalence between least-squares and kernel approximations in meshless methods,” Comput. Model. Eng. Sci. 2, 447–462 (2001).
[CrossRef]

1998 (1)

S. N. Atluri and T. Zhu, “A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics,” Comput. Mech. 22, 117–127 (1998).
[CrossRef]

1996 (1)

1995 (1)

D. A. Atchison and G. Smith, “Continuous gradient index and shell models of human lens,” Vis. Res. 35, 2529–2538(1995).
[CrossRef] [PubMed]

1994 (1)

T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin methods,” Int. J. Numer. Meth. Eng. 37, 229–256 (1994).
[CrossRef]

1992 (2)

1990 (1)

1987 (1)

1986 (2)

1985 (1)

1982 (1)

1981 (1)

1979 (1)

1973 (1)

1967 (1)

R. Courant, K. Friedrichs, and H. Lewy, “On the partial difference equations of mathematical physics,” IBM J. Res. Dev. 11, 215–234 (1967).
[CrossRef]

Agard, D. A.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Aldabaldetreku, G.

Aluru, N. R.

X. Jin, G. Li, and N. R. Aluru, “On the equivalence between least-squares and kernel approximations in meshless methods,” Comput. Model. Eng. Sci. 2, 447–462 (2001).
[CrossRef]

Arrue, J.

Atchison, D. A.

D. A. Atchison and G. Smith, “Continuous gradient index and shell models of human lens,” Vis. Res. 35, 2529–2538(1995).
[CrossRef] [PubMed]

Atluri, S. N.

S. N. Atluri and T. Zhu, “A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics,” Comput. Mech. 22, 117–127 (1998).
[CrossRef]

Belytschko, T.

T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin methods,” Int. J. Numer. Meth. Eng. 37, 229–256 (1994).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics I, 7th ed. (Cambridge Univ. Press, 1999).

Carpenter, W. C.

W. C. Carpenter and P. A. T. Gill, “Automated solutions of time-dependent problems,” in The Mathematics of Finite Elements and Applications II, J.Whiteman, ed. (Academic, 1973), pp. 495–506.

Cen, Z.

Cha, S.

Chen, Y.

W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Refractive index and dispersion variation in precision optical glass molding by computed tomography,” Appl. Opt. 48, 3588–3595 (2009).
[CrossRef] [PubMed]

W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Investigation of the refractive index distribution in precision compression glass molding by use of 3D tomography,” Meas. Sci. Technol. 20, 055109 (2009).
[CrossRef]

Courant, R.

R. Courant, K. Friedrichs, and H. Lewy, “On the partial difference equations of mathematical physics,” IBM J. Res. Dev. 11, 215–234 (1967).
[CrossRef]

Deng, S.

Devaney, A. J.

Durana, G.

Fehlberg, E.

E. Fehlberg, “Low-order classical Runge–Kutta formulas with step size control and their application to some heat transfer problems,” Technical report 315 (NASA, 1969).

Forbes, G. W.

Friedrichs, K.

R. Courant, K. Friedrichs, and H. Lewy, “On the partial difference equations of mathematical physics,” IBM J. Res. Dev. 11, 215–234 (1967).
[CrossRef]

Ghatak, A. K.

Gill, P. A. T.

W. C. Carpenter and P. A. T. Gill, “Automated solutions of time-dependent problems,” in The Mathematics of Finite Elements and Applications II, J.Whiteman, ed. (Academic, 1973), pp. 495–506.

Gu, L.

T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin methods,” Int. J. Numer. Meth. Eng. 37, 229–256 (1994).
[CrossRef]

Gustafsson, M. G. L.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Handapangoda, C. C.

C. C. Handapangoda and M. Premaratne, “An approximate numerical technique for characterizing optical pulse propagation in inhomogeneous biological tissue,” J. Biomed. Biotechnol. 2008, 784354 (2008).
[CrossRef] [PubMed]

Hanser, B.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Ichikawa, H.

Ihrke, I.

I. Ihrke, G. Ziegler, A. Tevs, C. Theobalt, M. Magnor, and H.-P. Seidel, “Eikonal rendering: efficient light transport in refractive objects,” ACM Trans. Graph. 26(3), 59 (2007).
[CrossRef]

Jian, S.

Jiménez, F.

Jin, X.

X. Jin, G. Li, and N. R. Aluru, “On the equivalence between least-squares and kernel approximations in meshless methods,” Comput. Model. Eng. Sci. 2, 447–462 (2001).
[CrossRef]

Kam, Z.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Kanok-Nukulchai, W.

F. T. Wong and W. Kanok-Nukulchai, “Kriging-based finite element method: element-by-element Kriging interpolation,” Civil Eng. Dimens. 11, 15–22 (2009).

Kasprzak, H.

Kitano, I.

Kumar, D. V.

Lewy, H.

R. Courant, K. Friedrichs, and H. Lewy, “On the partial difference equations of mathematical physics,” IBM J. Res. Dev. 11, 215–234 (1967).
[CrossRef]

Li, G.

X. Jin, G. Li, and N. R. Aluru, “On the equivalence between least-squares and kernel approximations in meshless methods,” Comput. Model. Eng. Sci. 2, 447–462 (2001).
[CrossRef]

Li, X.

Lomer, M.

Lu, Y. Y.

T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin methods,” Int. J. Numer. Meth. Eng. 37, 229–256 (1994).
[CrossRef]

Magnor, M.

I. Ihrke, G. Ziegler, A. Tevs, C. Theobalt, M. Magnor, and H.-P. Seidel, “Eikonal rendering: efficient light transport in refractive objects,” ACM Trans. Graph. 26(3), 59 (2007).
[CrossRef]

Maleki, M. H.

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, 1978).

Mateo, J.

Morita, S.

Nagata, T.

Nishi, H.

Nishidate, Y.

Pierscionek, B. K.

Premaratne, M.

C. C. Handapangoda and M. Premaratne, “An approximate numerical technique for characterizing optical pulse propagation in inhomogeneous biological tissue,” J. Biomed. Biotechnol. 2008, 784354 (2008).
[CrossRef] [PubMed]

Puchalski, J.

Richerzhagen, B.

Sakamoto, T.

Schatzberg, A.

Sedat, J. W.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Seidel, H.-P.

I. Ihrke, G. Ziegler, A. Tevs, C. Theobalt, M. Magnor, and H.-P. Seidel, “Eikonal rendering: efficient light transport in refractive objects,” ACM Trans. Graph. 26(3), 59 (2007).
[CrossRef]

Sharma, A.

Shen, L.

W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Investigation of the refractive index distribution in precision compression glass molding by use of 3D tomography,” Meas. Sci. Technol. 20, 055109 (2009).
[CrossRef]

W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Refractive index and dispersion variation in precision optical glass molding by computed tomography,” Appl. Opt. 48, 3588–3595 (2009).
[CrossRef] [PubMed]

Siedlecki, D.

Smith, G.

D. A. Atchison and G. Smith, “Continuous gradient index and shell models of human lens,” Vis. Res. 35, 2529–2538(1995).
[CrossRef] [PubMed]

Stone, B. D.

Suhara, H.

Sweeney, D. W.

Teodosiu, C.

Tevs, A.

I. Ihrke, G. Ziegler, A. Tevs, C. Theobalt, M. Magnor, and H.-P. Seidel, “Eikonal rendering: efficient light transport in refractive objects,” ACM Trans. Graph. 26(3), 59 (2007).
[CrossRef]

Theobalt, C.

I. Ihrke, G. Ziegler, A. Tevs, C. Theobalt, M. Magnor, and H.-P. Seidel, “Eikonal rendering: efficient light transport in refractive objects,” ACM Trans. Graph. 26(3), 59 (2007).
[CrossRef]

Toyama, M.

Vest, C. M.

Wolf, E.

M. Born and E. Wolf, Principles of Optics I, 7th ed. (Cambridge Univ. Press, 1999).

Wong, F. T.

F. T. Wong and W. Kanok-Nukulchai, “Kriging-based finite element method: element-by-element Kriging interpolation,” Civil Eng. Dimens. 11, 15–22 (2009).

Yamagata, Y.

Yi, A. Y.

W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Refractive index and dispersion variation in precision optical glass molding by computed tomography,” Appl. Opt. 48, 3588–3595 (2009).
[CrossRef] [PubMed]

W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Investigation of the refractive index distribution in precision compression glass molding by use of 3D tomography,” Meas. Sci. Technol. 20, 055109 (2009).
[CrossRef]

Zhao, W.

W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Refractive index and dispersion variation in precision optical glass molding by computed tomography,” Appl. Opt. 48, 3588–3595 (2009).
[CrossRef] [PubMed]

W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Investigation of the refractive index distribution in precision compression glass molding by use of 3D tomography,” Meas. Sci. Technol. 20, 055109 (2009).
[CrossRef]

Zhu, T.

S. N. Atluri and T. Zhu, “A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics,” Comput. Mech. 22, 117–127 (1998).
[CrossRef]

Ziegler, G.

I. Ihrke, G. Ziegler, A. Tevs, C. Theobalt, M. Magnor, and H.-P. Seidel, “Eikonal rendering: efficient light transport in refractive objects,” ACM Trans. Graph. 26(3), 59 (2007).
[CrossRef]

Zubia, J.

ACM Trans. Graph. (1)

I. Ihrke, G. Ziegler, A. Tevs, C. Theobalt, M. Magnor, and H.-P. Seidel, “Eikonal rendering: efficient light transport in refractive objects,” ACM Trans. Graph. 26(3), 59 (2007).
[CrossRef]

Appl. Opt. (13)

T. Sakamoto, “Ray trace algorithms for GRIN media,” Appl. Opt. 26, 2943–2946 (1987).
[CrossRef] [PubMed]

A. Sharma, D. V. Kumar, and A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. 21, 984–987 (1982).
[CrossRef] [PubMed]

A. Sharma, “Computing optical path length in gradient index media: a fast and accurate method,” Appl. Opt. 24, 4367–4370(1985).
[CrossRef] [PubMed]

A. Sharma and A. K. Ghatak, “Ray tracing in gradient-index lenses: computation of ray-surface intersection,” Appl. Opt. 25, 3409–3412 (1986).
[CrossRef] [PubMed]

H. Nishi, H. Ichikawa, M. Toyama, and I. Kitano, “Gradient-index objective lens for the compact disk system,” Appl. Opt. 25, 3340–3344 (1986).
[CrossRef] [PubMed]

D. W. Sweeney and C. M. Vest, “Reconstruction of three-dimensional refractive index fields from multidirectional interferometric data,” Appl. Opt. 12, 2649–2554 (1973).
[CrossRef] [PubMed]

H. Suhara, “Interferometric measurement of the refractive-index distribution in plastic lenses by use of computed tomography,” Appl. Opt. 41, 5317–5325 (2002).
[CrossRef] [PubMed]

W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Refractive index and dispersion variation in precision optical glass molding by computed tomography,” Appl. Opt. 48, 3588–3595 (2009).
[CrossRef] [PubMed]

S. Cha and C. M. Vest, “Tomographic reconstruction of strongly refracting fields and its application to interferometric measurement of boundary layers,” Appl. Opt. 20, 2787–2794(1981).
[CrossRef] [PubMed]

J. Puchalski, “Numerical determination of ray tracing: a new method,” Appl. Opt. 31, 6789–6799 (1992).
[CrossRef] [PubMed]

B. Richerzhagen, “Finite element ray tracing: a new method for ray tracing in gradient-index media,” Appl. Opt. 35, 6186–6189 (1996).
[CrossRef] [PubMed]

S. Deng, X. Li, Z. Cen, and S. Jian, “Simulation of the inhomogeneous medium with a self-adapting grid,” Appl. Opt. 46, 3102–3106 (2007).
[CrossRef] [PubMed]

S. Morita, Y. Nishidate, T. Nagata, Y. Yamagata, and C. Teodosiu, “Ray-tracing simulation method using piecewise quadratic interpolant for aspheric optical systems,” Appl. Opt. 49, 3442–3451 (2010).
[CrossRef] [PubMed]

Civil Eng. Dimens. (1)

F. T. Wong and W. Kanok-Nukulchai, “Kriging-based finite element method: element-by-element Kriging interpolation,” Civil Eng. Dimens. 11, 15–22 (2009).

Comput. Aided Geom. Des. (1)

T. Nagata, “Simple local interpolation of surfaces using normal vectors,” Comput. Aided Geom. Des. 22, 327–347 (2005).
[CrossRef]

Comput. Mech. (1)

S. N. Atluri and T. Zhu, “A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics,” Comput. Mech. 22, 117–127 (1998).
[CrossRef]

Comput. Model. Eng. Sci. (1)

X. Jin, G. Li, and N. R. Aluru, “On the equivalence between least-squares and kernel approximations in meshless methods,” Comput. Model. Eng. Sci. 2, 447–462 (2001).
[CrossRef]

IBM J. Res. Dev. (1)

R. Courant, K. Friedrichs, and H. Lewy, “On the partial difference equations of mathematical physics,” IBM J. Res. Dev. 11, 215–234 (1967).
[CrossRef]

Int. J. Numer. Meth. Eng. (1)

T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin methods,” Int. J. Numer. Meth. Eng. 37, 229–256 (1994).
[CrossRef]

J. Biomed. Biotechnol. (1)

C. C. Handapangoda and M. Premaratne, “An approximate numerical technique for characterizing optical pulse propagation in inhomogeneous biological tissue,” J. Biomed. Biotechnol. 2008, 784354 (2008).
[CrossRef] [PubMed]

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

Meas. Sci. Technol. (1)

W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Investigation of the refractive index distribution in precision compression glass molding by use of 3D tomography,” Meas. Sci. Technol. 20, 055109 (2009).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Proc. Natl. Acad. Sci. USA (1)

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Vis. Res. (1)

D. A. Atchison and G. Smith, “Continuous gradient index and shell models of human lens,” Vis. Res. 35, 2529–2538(1995).
[CrossRef] [PubMed]

Other (4)

W. C. Carpenter and P. A. T. Gill, “Automated solutions of time-dependent problems,” in The Mathematics of Finite Elements and Applications II, J.Whiteman, ed. (Academic, 1973), pp. 495–506.

E. Fehlberg, “Low-order classical Runge–Kutta formulas with step size control and their application to some heat transfer problems,” Technical report 315 (NASA, 1969).

E. W. Marchand, Gradient Index Optics (Academic, 1978).

M. Born and E. Wolf, Principles of Optics I, 7th ed. (Cambridge Univ. Press, 1999).

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Figures (9)

Fig. 1
Fig. 1

Sampling point acquisitions for reconstructions of the refractive-index distribution at position x. In the grid-based acquisition (a), points of label 1 are used but of label 2 are not. On the other hand, in the distance-based acquisition (b), sampling points are more reasonably selected.

Fig. 2
Fig. 2

For local sampling point acquisition, the space is partitioned into cells and all points are registered into one of the cells according to its position. In this case, calculation of distances between the center (red point) with points belonging to regions 0–8 are enough for MLS sampling point acquisition.

Fig. 3
Fig. 3

Nagata patch interpolant of an edge and a surface. The curvature parameter vector c is uniquely determined for each edge. Accordingly, the surface interpolant is orthogonal to the normal vectors at the vertices.

Fig. 4
Fig. 4

Ray trajectory near the exit of GRIN media. When a ray intersects with the exit surface, an intermediate point of the line connecting the start and end points is considered as the contact position. It can cause significant loss of precision because the curvature of the ray trajectory may not be negligible.

Fig. 5
Fig. 5

Schematic of the iterative procedure to find a better contact position.

Fig. 6
Fig. 6

The Luneburg lens is a spherical isoindicial lens that converges every parallel ray to a point on the lens surface. The optical axis coincides with the z coordinate axis.

Fig. 7
Fig. 7

Spot radius and the incident ray positions on the x y plane at the focal point.

Fig. 8
Fig. 8

Radial isoindicial optical fiber with refraction at the lens surface. The incident ray grazes the top and the bottom surfaces of the fiber, and it is refracted at the exit surface.

Fig. 9
Fig. 9

Error of the ray-tracing simulation in the optical fiber and comparison of the error with and without treatment near the GRIN media exit.

Tables (2)

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Table 1 Coefficients a i j , b i , b ^ i , and c i for the RKF45

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Table 2 Algorithm 1: Iterative Procedure

Equations (34)

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d d s [ n ( r ) d r d s ] = n ( r ) ,
{ d r d t = T d T d t = n ( r ) n ( r ) ,
n ˜ ( x ) = p T ( x ) a ( x ) ,
[ M ( x ) ] a ( x ) = [ N ( x ) ] n ,
[ M ( x ) ] = [ P T ] [ W ( x ) ] [ P ] ,
[ N ( x ) ] = [ P T ] [ W ( x ) ] .
P ( x ) = [ p T ( s 1 ) p T ( s n ) ] ,
W ( x ) = diag { w 1 ( x ) , , w i ( x ) , , w n ( x ) } ,
n ˜ x = p T x a .
p T = { 1 , x , y , z , x y , y z , z x , x y z , x 2 , y 2 , z 2 , x 2 y , x 2 z , y 2 x , y 2 z , z 2 x , z 2 y , x 3 , y 3 , z 3 } ,
w i ( x ) = 1 6 l i 2 + 8 l i 3 3 l i 4 ,
d x d t = f ( t , x ) ,
x t + Δ t = x t + Δ t i = 1 6 b i k i + O ( Δ t 5 ) ,
x ^ t + Δ t = x t + Δ t i = 1 6 b ^ i k i + O ( Δ t 6 ) ,
k i = f ( t + c i Δ t , x t + j = 1 i 1 a i j k j ) ,
δ = max j | x ^ t + Δ t ( j ) x t + Δ t ( j ) | / ε ( j ) = max j | i = 1 6 ( b ^ i b i ) k i ( j ) | / ε ( j ) ,
Δ t n + 1 = S Δ t n δ 1 5 ,
x ( η , ζ ) = c 00 + c 10 η + c 01 ζ + c 11 η ζ + c 20 η 2 + c 02 ζ 2 ,
c 00 = x 00 , c 10 = x 10 x 00 c 1 , c 01 = x 11 x 10 + c 1 c 3 , c 11 = c 3 c 1 c 2 , c 20 = c 1 , c 02 = c 2 ,
c ( d , n 0 , n 1 ) = { [ n 0 , n 1 ] 1 c 2 [ 1 c c 1 ] { n 0 · d n 1 · d } ( c ± 1 ) 0 ( c = ± 1 ) ,
x ( λ ) = x a + λ d ,
e 3 = d / | d | , e 1 = e 3 × c 02 | e 3 × c 02 | , e 2 = e 3 × e 1 .
0 = α 00 + α 10 η + α 01 ζ + α 11 η ζ + α 20 η 2 ,
0 = β 00 + β 10 η + β 01 ζ + β 11 η ζ + β 20 η 2 + β 02 ζ 2 ,
λ d = γ 00 + γ 10 η + γ 01 ζ + γ 11 η ζ + γ 20 η 2 + γ 02 ζ 2 ,
α i j = { e 1 · ( c i j x a ) ( i = j = 0 ) e 1 · c i j ( otherwise ) ,
β i j = { e 2 · ( c i j x a ) ( i = j = 0 ) e 2 · c i j ( otherwise ) ,
γ i j = { e 3 · ( c i j x a ) ( i = j = 0 ) e 3 · c i j ( otherwise ) .
n ( r ) = 2 ( r R ) 2 ,
n ( r x y ) = 2.5 ( r x y R ) 2 ,
y in ( z ) = y 0 cos z n 0 ,
v = 1 n c { 0 , q c , n 0 } ,
v = 1 n c { 0 , n c q c , n 0 2 + ( 1 n c 2 ) q c 2 } .
y out ( z ) = n c q c n 0 2 + ( 1 n c 2 ) q c 2 ( z z c ) + y in ( z c ) .

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