Abstract

We present an iterative method for simulating beam propagation in nonlinear media using Hamiltonian ray tracing. The Wigner distribution function of the input beam is computed at the entrance plane and is used as the initial condition for solving the Hamiltonian equations. Examples are given for the study of periodic self-focusing, spatial solitons, and Gaussian–Schell model in Kerr-effect media. Simulation results show good agreement with the split-step beam propagation method. The main advantage of ray tracing, even in the nonlinear case, is that ray diagrams are intuitive and easy to interpret in terms of traditional optical engineering terms, such as aberrations, ray-intercept plots, etc.

© 2010 Optical Society of America

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  1. G. Agrawal, Nonlinear Fiber Optics, 3rd ed.(Academic, 2001).
  2. H. J. W. M. Hoekstra, Opt. Quantum Electron. 29, 157 (1997).
    [CrossRef]
  3. M. Soljačić, C. Luo, J. Joannopoulos, and S. Fan, Opt. Lett. 28, 637 (2003).
    [CrossRef] [PubMed]
  4. J. Bravo-Abad, S. Fan, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, J. Lightwave Technol. 25, 2539 (2007).
    [CrossRef]
  5. M. Tsang, D. Psaltis, and F. G. Omenetto, Opt. Lett. 28, 1873 (2003).
    [CrossRef] [PubMed]
  6. O. V. Sinkin, R. Holzlöhner, J. Zweck, and C. R. Menyuk, J. Lightwave Technol. 21, 61 (2003).
    [CrossRef]
  7. V. V. Shkunov and D. Z. Anderson, Phys. Rev. Lett. 81, 2683 (1998).
    [CrossRef]
  8. M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
    [CrossRef]
  9. D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997).
    [CrossRef]
  10. D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 35601 (2001).
    [CrossRef]
  11. D. Dragoman, Appl. Opt. 35, 4142 (1996).
    [CrossRef] [PubMed]
  12. B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
    [CrossRef]
  13. K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).
  14. A. Walther, J. Opt. Soc. Am. 58, 1256 (1968).
    [CrossRef]
  15. E. Wolf, J. Opt. Soc. Am. 68, 6 (1978).
    [CrossRef]
  16. M. J. Bastiaans, Opt. Acta 26, 1265 (1979).
    [CrossRef]
  17. M. Alonso, J. Opt. Soc. Am. A 18, 902 (2001).
    [CrossRef]
  18. M. J. Bastiaans, J. Opt. Soc. Am. A 3, 1227 (1986).
    [CrossRef]
  19. C. Sun, D. V. Dylov, and J. W. Fleischer, Opt. Lett. 34, 3003 (2009).
    [CrossRef] [PubMed]
  20. H. Gao, S. Takahashi, L. Tian, and G. Barbastathis, in Optical MEMS and Nanophotonics 2010, Paper Th1-2, pp. 179–180.
  21. Y. Jiao, S. Fan, and D. A. B. Miller, Phys. Rev. E 70, 036612 (2004).
    [CrossRef]
  22. P. S. J. Russel and T. A. Birks, J. Lightwave Technol. 17, 1982 (1999).
    [CrossRef]
  23. R. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).
  24. R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
    [CrossRef]
  25. P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
    [CrossRef]
  26. Y. Kivshar and G. Agrawal, Optical Solitons (Academic, 2003).

2009

2007

2004

Y. Jiao, S. Fan, and D. A. B. Miller, Phys. Rev. E 70, 036612 (2004).
[CrossRef]

2003

2002

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

2001

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 35601 (2001).
[CrossRef]

M. Alonso, J. Opt. Soc. Am. A 18, 902 (2001).
[CrossRef]

1999

1998

V. V. Shkunov and D. Z. Anderson, Phys. Rev. Lett. 81, 2683 (1998).
[CrossRef]

1997

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

H. J. W. M. Hoekstra, Opt. Quantum Electron. 29, 157 (1997).
[CrossRef]

1996

1986

1979

M. J. Bastiaans, Opt. Acta 26, 1265 (1979).
[CrossRef]

1978

1968

1965

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

1964

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, 3rd ed.(Academic, 2001).

Y. Kivshar and G. Agrawal, Optical Solitons (Academic, 2003).

Alonso, M.

Anderson, D.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Anderson, D. Z.

V. V. Shkunov and D. Z. Anderson, Phys. Rev. Lett. 81, 2683 (1998).
[CrossRef]

Barbastathis, G.

H. Gao, S. Takahashi, L. Tian, and G. Barbastathis, in Optical MEMS and Nanophotonics 2010, Paper Th1-2, pp. 179–180.

Bastiaans, M. J.

Birks, T. A.

Boyd, R.

R. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

Bravo-Abad, J.

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Christodoulides, D. N.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 35601 (2001).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

Coskun, T. H.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 35601 (2001).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

Dragoman, D.

Dylov, D. V.

Eugenieva, E. D.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 35601 (2001).
[CrossRef]

Fan, S.

Fedele, R.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Fleischer, J. W.

Gao, H.

H. Gao, S. Takahashi, L. Tian, and G. Barbastathis, in Optical MEMS and Nanophotonics 2010, Paper Th1-2, pp. 179–180.

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Hall, B.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Hoekstra, H. J. W. M.

H. J. W. M. Hoekstra, Opt. Quantum Electron. 29, 157 (1997).
[CrossRef]

Holzlöhner, R.

Jiao, Y.

Y. Jiao, S. Fan, and D. A. B. Miller, Phys. Rev. E 70, 036612 (2004).
[CrossRef]

Joannopoulos, J.

Joannopoulos, J. D.

Johnson, S. G.

Kelley, P. L.

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

Kivshar, Y.

Y. Kivshar and G. Agrawal, Optical Solitons (Academic, 2003).

Lisak, M.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Luo, C.

Menyuk, C. R.

Miller, D. A. B.

Y. Jiao, S. Fan, and D. A. B. Miller, Phys. Rev. E 70, 036612 (2004).
[CrossRef]

Mitchell, M.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 35601 (2001).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

Omenetto, F. G.

Psaltis, D.

Russel, P. S. J.

Segev, M.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 35601 (2001).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

Semenov, V. E.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Shkunov, V. V.

V. V. Shkunov and D. Z. Anderson, Phys. Rev. Lett. 81, 2683 (1998).
[CrossRef]

Sinkin, O. V.

Soljacic, M.

Sun, C.

Takahashi, S.

H. Gao, S. Takahashi, L. Tian, and G. Barbastathis, in Optical MEMS and Nanophotonics 2010, Paper Th1-2, pp. 179–180.

Tian, L.

H. Gao, S. Takahashi, L. Tian, and G. Barbastathis, in Optical MEMS and Nanophotonics 2010, Paper Th1-2, pp. 179–180.

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Tsang, M.

Walther, A.

Wolf, E.

Wolf, K. B.

K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).

Zweck, J.

Appl. Opt.

J. Lightwave Technol.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

M. J. Bastiaans, Opt. Acta 26, 1265 (1979).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

H. J. W. M. Hoekstra, Opt. Quantum Electron. 29, 157 (1997).
[CrossRef]

Phys. Rev. E

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 35601 (2001).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Y. Jiao, S. Fan, and D. A. B. Miller, Phys. Rev. E 70, 036612 (2004).
[CrossRef]

Phys. Rev. Lett.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

V. V. Shkunov and D. Z. Anderson, Phys. Rev. Lett. 81, 2683 (1998).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

Other

K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).

H. Gao, S. Takahashi, L. Tian, and G. Barbastathis, in Optical MEMS and Nanophotonics 2010, Paper Th1-2, pp. 179–180.

Y. Kivshar and G. Agrawal, Optical Solitons (Academic, 2003).

R. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

G. Agrawal, Nonlinear Fiber Optics, 3rd ed.(Academic, 2001).

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Figures (5)

Fig. 1
Fig. 1

Block diagram of the iterative method for nonlinear beam propagation.

Fig. 2
Fig. 2

(a) WDF of the Gaussian beam at the input plane in 1D and (b) the Gaussian beam generated from rays with initial condition specified by the WDF. Here x is position and p is momentum. Numbers are scaled according to the waist w 0 . Wavelength is λ = w 0 / 20 . Nine points on the WDF plane [white dots in (a)] correspond to nine rays [arrows in (b)]. Three different ray directions are shown for each of the three positions (A–C). The lengths of arrows are proportional to the generalized radiance of the WDF.

Fig. 3
Fig. 3

Periodic self-focusing of Gaussian beam in Kerr medium produced by (a) the iterative method and (b) the BPM. Periodic focusing of two Gaussian beams in a Kerr medium produced by (c) the iterative method and (d) the BPM. The white curves are sampled from the set of all 10,100 rays used in the simulation. The color shading indicates intensity, computed as projection from the WDF. (see text).

Fig. 4
Fig. 4

(a) Spatial soliton in a Kerr medium produced by the proposed iterative method and a subset of all 10,100 rays used in the simulation. (b) Comparison of intensity distribution along transverse direction between the results from the iterative method and the analytical solution. (c) Intensity difference in percentage of spatial soliton along the transverse direction between the iterative method and the analytical solution.

Fig. 5
Fig. 5

(a) Gaussian–Schell model beam propagation in a weak Kerr-effect medium. (b) FWHM at different Δ n for a fixed propagation length.

Equations (1)

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d q d σ = H p = p | p | , d p d σ = H q = n q ,

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