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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 15, Iss. 5 — May. 1, 1998
  • pp: 1394–1400

Approximate solutions to the scalar wave equation: the decomposition method

Vasudevan Lakshminarayanan and Srinivasa Varadharajan  »View Author Affiliations

JOSA A, Vol. 15, Issue 5, pp. 1394-1400 (1998)

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Exact solutions can be obtained for electromagnetic wave propagation in a medium with a simple uniform refractive-index distribution. For more-complex distributions, approximate or numerical methods have to be utilized. We describe an elegant approximation scheme called the decomposition method for nonlinear differential equations, which was introduced by Adomian [<i>Non-linear Stochastic Systems Theory and Applications to Physics</i> (Kluwer, Dordrecht, The Netherlands, 1989)]. The method is described and applied to waveguide problems (planar waveguides with step and parabolic refractive-index profiles), and the results are compared with those obtained by JWKB and modified Airy function methods.

© 1998 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(000.4430) General : Numerical approximation and analysis
(060.2310) Fiber optics and optical communications : Fiber optics
(350.7420) Other areas of optics : Waves

Vasudevan Lakshminarayanan and Srinivasa Varadharajan, "Approximate solutions to the scalar wave equation: the decomposition method," J. Opt. Soc. Am. A 15, 1394-1400 (1998)

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  1. For a discussion of variational method see, for example, H. Goldstein, Classical Mechanics, 2nd ed., Addison-Wesley Series in Physics (Addison-Wesley, Reading, Mass., 1980), Chap. 2. For applications to waveguides see A. Sharma and P. Bindal, “Analysis of diffused planar and channel waveguide,” IEEE J. Quantum Electron. 29, 150–153 (1993); “An accurate variational analysis of single mode diffused channel waveguides,” Opt. Quantum Electron. 24, 1359–1371 (1992).
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  11. G. Adomian, “A new approach to the heat equation—an application of the decomposition method,” J. Math. Anal. Appl. 113, 202–209 (1986).
  12. G. Adomian, “An investigation of asymptotic decomposition method for non-linear equations in physics,” Appl. Math. Comput. 24, 1–17 (1987).
  13. G. Adomian, “Non-linear oscillations in physical systems,” Math Comput. Sim. 29, 275–284 (1987).
  14. G. Adomian, Non-linear Stochastic Systems Theory and Applications to Physics (Kluwer, Dordrecht, The Netherlands, 1989).
  15. G. Adomian, “Non-linear stochastic differential equations,” in Selected Topics in Mathematical Physics, R. Sridhar, K. Srinivasa Rao, and V. Lakshminarayanan, eds. (Allied, New Delhi, 1995), pp. 47–57.
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  18. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1972).
  19. All the computations were done with the commercial software MAPLE V RELEASE 4, is a product of Waterloo Maple, Inc., Waterloo, Ontario, Canada.
  20. C. Yu and D. Yevick, “Application of the bidirectional parabolic equations to optical waveguide facets,” J. Opt. Soc. Am. A 14, 1448–1450 (1997).
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