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Journal of the Optical Society of America B

Journal of the Optical Society of America B

| OPTICAL PHYSICS

  • Vol. 15, Iss. 4 — Apr. 1, 1998
  • pp: 1335–1345

Simulation of Hamiltonian light-beam propagation in nonlinear media

Menashe Sonnenschein and Dan Censor  »View Author Affiliations


JOSA B, Vol. 15, Issue 4, pp. 1335-1345 (1998)
http://dx.doi.org/10.1364/JOSAB.15.001335


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Abstract

The simulation of nonlinear wave propagation in the ray regime, i.e., in the limit of geometrical optics, is discussed. The medium involved is nonlinear, which means that the field amplitudes affect the constitutive parameters (e.g., dielectric constant) involved in the propagation formalism. Conventionally, linear ray propagation is computed by the use of Hamilton’s ray equations whose terms are derived from the appropriate dispersion equation. The formalism used to solve such a set of equations is the Runge–Kutta algorithm in one of its variants. In the present case of nonlinear propagation, a proper dispersion equation must first be established from which the rays can be computed. Linear ray tracing with Hamilton’s ray theory allows for the computation of ray trajectories and wave fronts. The convergence or divergence of rays suggests heuristic methods for computing the variation of amplitudes. Here, terms appearing in the Hamiltonian ray equations involve field amplitudes, which themselves are determined by the convergence (or divergence) of the rays. This dictates the simultaneous computation of a beam comprising many rays, so it is necessary to modify the original Runge–Kutta scheme by building into it some iteration mechanism such that the process converges to the values that take into account the amplitude effect. This research attempts to modify the existing propagation formalism and apply the new algorithm to simple problems of nonlinear ray propagation. The results display self-focusing effects characteristic of nonlinear optics problems. The influence of weak losses on the beam propagation and its self focusing is also discussed. Some displayed results obtained by simulating the modified formalism seem to be physically plausible and are in excellent agreement with experimental results reported in the literature.

© 1998 Optical Society of America

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(080.2710) Geometric optics : Inhomogeneous optical media
(160.4330) Materials : Nonlinear optical materials
(260.5950) Physical optics : Self-focusing

Citation
Menashe Sonnenschein and Dan Censor, "Simulation of Hamiltonian light-beam propagation in nonlinear media," J. Opt. Soc. Am. B 15, 1335-1345 (1998)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-15-4-1335


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References

  1. D. Censor, “Ray tracing in weakly nonlinear moving medium,” J. Plasma Phys. 16, 415–426 (1976).
  2. D. Censor, “Ray theoretic analysis of spatial and temporal self-focusing in general weakly nonlinear medium,” Phys. Rev. A 16, 1673–1677 (1977).
  3. D. Censor, “Ray propagation and self-focusing in nonlinear absorbing medium,” Phys. Rev. A 18, 2614–2617 (1978).
  4. D. Censor, “Scattering by weakly nonlinear objects,” SIAM J. Appl. Math. 43, 1400–1417 (1983).
  5. D. Censor, “Waveguide and cavity oscillations in the presence of nonlinear medium,” IEEE Trans. Microwave Theory Tech. 33, 296–301 (1985).
  6. I. Gurwich and D. Censor, “Steady state electromagnetic wave propagation in weakly nonlinear medium,” IEEE Trans. Magn. 30, 3192–3195 (1994).
  7. I. Gurwich and D. Censor, “Existence problems in steady state theory for electromagnetic waves in weakly nonlinear medium,” J. Electromagn. Waves Appl. 9, 1115–1139 (1995).
  8. I. Gurwich and D. Censor, “On the propagation of multi-band spectrum electromagnetic waves in weakly nonlinear medium,” J. Electromagn. Waves Appl. 10, 889–907 (1996).
  9. M. A. Hasan and P. L. E. Uslenghi, “Electromagnetic scattering from nonlinear anisotropic cylinders. I. Fundamental frequency,” IEEE Trans. Antennas Propag. 38, 523–533 (1990).
  10. M. A. Hasan and P. L. E. Uslenghi, “Higher-order harmonics in electromagnetic scattering from a nonlinear anisotropic cylinder,” Electromagnetics 11, 377–391 (1991).
  11. D. Censor, I. Gurwich, and M. Sonnenschein, “Volterra’s functionals series and wave propagation in weakly nonlinear medium: the problematics of first-principles physical modeling,” in Volterra Equations and Applications, Proceedings of The Volterra Centennial Symposium, C. Corduneanu and I. W. Sandberg, eds. (Marcel Dekker, New York, 1998).
  12. D. Censor and Y. Ben-Shimol, “Wave propagation in weakly nonlinear bi-anisotropic and bi-isotropic medium,” J. Electromagn. Waves Appl. 11, 1763–1779 (1997).
  13. D. Censor, “Application-oriented ray theory,” Int. J. Electr. Eng. Educ. 15, 215–223 (1978).
  14. J. Molcho and D. Censor, “A simple derivation and a classroom example for Hamiltonian ray propagation,” Am. J. Phys. 54, 351–353 (1986).
  15. B. Meier and A. Penzkofer, “Determination of nonlinear refractive indices by external self-focusing,” Appl. Phys. B 49, 513–519 (1989).
  16. J. Reintjes and R. L. Carman, “Direct observation of the orientational Kerr effect in the self-focusing of picosecond pulses,” Phys. Rev. Lett. 28, 1697–1700 (1972).

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