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Applied Optics

Applied Optics


  • Editor: James C. Wyant
  • Vol. 46, Iss. 36 — Dec. 20, 2007
  • pp: 8656–8667

Analytical extraction of leaky modes in circular slab waveguides with arbitrary refractive index profile

P. Sarrafi, N. Zareian, and K. Mehrany  »View Author Affiliations

Applied Optics, Vol. 46, Issue 36, pp. 8656-8667 (2007)

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Circular slab waveguides are conformally transformed into straight inhomogeneous waveguides, whereupon electromagnetic fields in the core are expanded in terms of Legendre polynomial basis functions. Thereafter, different analytical expression of electromagnetic fields in the cladding region, viz. Wentzel–Kramers–Brillouin solution, modified Airy function expansion, and the exact field solution for circular waveguides, i.e., Hankel function of complex order, are each matched to the polynomial expansion of the transverse electric field within the guide. This field matching process renders different boundary conditions to be satisfied by the set of orthogonal Legendre polynomial basis functions. In this fashion, the governing wave equation is converted into an algebraic and easy to solve eigenvalue problem, which is associated with a matrix whose elements are analytically given. Various numerical examples are presented and the accuracy of each of the abovementioned different boundary conditions is assessed. Furthermore, the computational efficiency and the convergence rate of the proposed method with increasing number of basis functions are briefly discussed.

© 2007 Optical Society of America

OCIS Codes
(130.2790) Integrated optics : Guided waves
(260.2110) Physical optics : Electromagnetic optics

ToC Category:
Integrated Optics

Original Manuscript: September 10, 2007
Revised Manuscript: November 4, 2007
Manuscript Accepted: November 15, 2007
Published: December 19, 2007

P. Sarrafi, N. Zareian, and K. Mehrany, "Analytical extraction of leaky modes in circular slab waveguides with arbitrary refractive index profile," Appl. Opt. 46, 8656-8667 (2007)

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  1. L. Lewin, D. C. Chang, and E. F. Kuester, Electromagnetic Waves and Curved Structures, Vol. 2 of IEE Electromagnetic Waves Series (Peter Peregrinus, Ltd., 1977), p. 205.
  2. F. Sporleder and H. G. Unger, Waveguides Tapers Transitions and Couplers (IEE Press, Peter Peregrinus Ltd., 1979).
  3. S. Kim and A. Gopinath, "Vector analysis of optical dielectric waveguide bends using finite-difference method," J. Lightwave Technol. 14, 2085-2092 (1996). [CrossRef]
  4. K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Ctyroky, "Analytic approach to dielectric optical bent slab waveguides," Opt. Quantum Electron. 37, 37-61 (2005). [CrossRef]
  5. T. Yamamoto and M. Koshiba, "Numerical analysis of curvature loss in optical waveguides by finite-element method," J. Lightwave Technol. 11, 1579-1583 (1993). [CrossRef]
  6. W. J. Song, G. H. Song, B. H. Ahn, and M. Kang, "Scalar BPM analyses of TE and TM polarized fields in bent waveguides," IEEE Trans. Antennas Propag. 51, 1185-1198 (2003). [CrossRef]
  7. R. Pregla, "The method of lines for the analysis of dielectric waveguide bends," J. Lightwave Technol. 14, 634-639 (1996). [CrossRef]
  8. D. Marcuse, "Bending loss of the asymmetric slab waveguide," Bell Syst. Tech. J. 50, 2551-2563 (1971).
  9. E. A. J. Marcatili, "Bends in optical dielectric guides," Bell Syst. Tech. J. 48, 2103-2132 (1969).
  10. R. Jedidi and R. Pierre, "Efficient analytical and numerical methods for the computation of bent loss in planar waveguides," J. Lightwave Technol. 23, 2278-2284 (2005). [CrossRef]
  11. W. Kim and C. Kim, "Radiation losses of bent planar waveguides," Fiber Integr. Opt. 21, 219-232 (2002). [CrossRef]
  12. M. Heiblum and J. H. Harris, "Analysis of curved optical wave-guides by conformal transformation," IEEE J. Quantum Electron. QE-11, 75-85 (1975). [CrossRef]
  13. D. Marcuse, "Curvature loss formula for optical fibers," J. Opt. Soc. Am. 66, 216-220 (1976). [CrossRef]
  14. A. Melloni, F. Carniel, R. Costa, and M. Martinelli, "Determination of bend mode characteristics in dielectric waveguides," J. Lightwave Technol. 19, 571-577 (2001). [CrossRef]
  15. K. Thyagarajan, M. R. Shenoy, and A. K. Ghatak, "Accurate numerical method for the calculation of bending loss in optical waveguides using a matrix approach," Opt. Lett. 12, 296-298 (1987). [PubMed]
  16. W. Berglund and A. Gopinath, "WKB analysis of bend losses in optical waveguides," J. Lightwave Technol. 18, 1161-1166 (2000). [CrossRef]
  17. C. Kim, Y. Kim, and W. Kim, "Leaky modes of circular slab waveguides: modified Airy functions," IEEE J. Sel. Top. Quantum Electron. 8, 1239-1245 (2002). [CrossRef]
  18. S. G. Mikhlin and K. L. Smolitskii, Approximate Method for Solution of Differential and Integral Equations (Elsevier, 1976), pp. 250-252.
  19. C. A. J. Fletcher, Computational Galerkin methods (Springer-Verlag, 1984), p. 309.
  20. J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, 2001).
  21. A. Weisshaar, "Impedance boundary method of moments for accurate and efficient analysis of planar graded-index optical waveguides," J. Lightwave Technol. 12, 1943-1951 (1994). [CrossRef]
  22. K. Mehrany and B. Rashidian, "Polynomial expansion of electromagnetic eigenmodes in layered structures," J. Opt. Soc. Am. B 20, 2434-2441 (2003). [CrossRef]
  23. M. Chamanzar, K. Mehrany, and B. Rashidian, "Legendre polynomial expansion for analysis of linear one-dimensional inhomogeneous optical structures and photonic crystals," J. Opt. Soc. Am. B 23, 969-976 (2006). [CrossRef]
  24. A. K. Ghatak, R. L. Gallawa, and I. C. Goyal, "Modified Airy function and WKB solutions to the wave equation," National Institute of Standards and Technology, Monograph 176 (United States Department of Commerce, 1991).
  25. R. E. Langer, "On the asymptotic solutions of ordinary differential equations, with an application to the Bessel functions of large order," Trans. Am. Math. Soc. 33, 23-64 (1931). [CrossRef]
  26. I. C. Goyal, R. Jindal, and A. K. Ghatak, "Planar optical waveguides with arbitrary index profile: an accurate method of analysis," J. Lightwave Technol. 15, 2179-2182 (1997). [CrossRef]
  27. D. Rownald, "Nonperturbative calculation of bend loss for a pulse in a bent planar waveguide," IEE Proc.: Optoelectron. 144, 91-96 (1997). [CrossRef]
  28. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Ninth printing (Dover, 1970), p. 376.
  29. T. Tamir and R. C. Alferness, Guided-Wave Optoelectronics 2nd ed. (Springer-Verlag, 1990). [CrossRef]

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