OSA's Digital Library

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)

View Full Text Article

Enhanced HTML    Acrobat PDF (244 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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 [Non-linear Stochastic Systems Theory and Applications to Physics (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

Original Manuscript: July 7, 1997
Revised Manuscript: December 23, 1997
Manuscript Accepted: January 20, 1998
Published: May 1, 1998

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

Sort:  Author  |  Year  |  Journal  |  Reset  


  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, 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). [CrossRef]
  2. N. Froman, P. O. Froman, JWKB Approximation: Contributions to the Theory (North-Holland, Amsterdam, 1965).
  3. C. M. Bender, S. A. Orzsag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
  4. A. K. Ghatak, R. L. Gallawa, I. C. Goyal, Modified Airy Function and WKB Solutions to the Wave Equation, National Institute of Standards and Technology Monogr. 176 (U.S. GPO, Washington, D.C., 1991).
  5. G. Baym, Lectures on Quantum Mechanics (Benjamin, New York, 1969).
  6. I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “An approximate solution to the wave equation—revisited,” J. Electromagn. Waves Appl. 5, 623–636 (1991). [CrossRef]
  7. I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “Approximate solutions to the scalar wave equations for optical waveguides,” Appl. Opt. 29, 2985–2990 (1991). [CrossRef]
  8. A. K. Ghatak, R. L. Gallawa, I. C. Goyal, “Accurate solutions to Schrodinger’s equation using modified Airy functions,” IEEE J. Quantum Electron. 28, 400–403 (1992). [CrossRef]
  9. M. L. Calvo, V. Lakshminarayanan, “Light propagation in optical waveguides: a dynamic programming approach,” J. Opt. Soc. Am. A 14, 872–881 (1997). [CrossRef]
  10. G. Adomian, “Convergent series solution of non-linear equations,” J. Comput. Appl. Math. 11, 225–230 (1984). [CrossRef]
  11. G. Adomian, “A new approach to the heat equation—an application of the decomposition method,” J. Math. Anal. Appl. 113, 202–209 (1986). [CrossRef]
  12. G. Adomian, “An investigation of asymptotic decomposition method for non-linear equations in physics,” Appl. Math. Comput. 24, 1–17 (1987). [CrossRef]
  13. G. Adomian, “Non-linear oscillations in physical systems,” Math Comput. Sim. 29, 275–284 (1987). [CrossRef]
  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, V. Lakshminarayanan, eds. (Allied, New Delhi, 1995), pp. 47–57.
  16. R. Rach, “On the Adomian decomposition method and comparisons with Picard’s method,” J. Math. Anal. Appl. 128, 480–483 (1987). [CrossRef]
  17. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).
  18. M. Abramowitz, 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, D. Yevick, “Application of the bidirectional parabolic equations to optical waveguide facets,” J. Opt. Soc. Am. A 14, 1448–1450 (1997). [CrossRef]
  21. I. R. Bellman, G. Adomian, Partial Differential Equations: New Methods for Their Treatment and Solution (Reidel, Dordrecht, The Netherlands, 1985).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited