Abstract
Guided modes of a planar dielectric waveguide which encounter a nondiagonal permittivity tensor are neither TE nor TM, but hybrid. They are described by a pair of coupled second-order differential equations for the transversal electric and magnetic field components. We construct a real-valued function which plays the role of the transversal electric or magnetic field in the uncoupled Sturm-Liouville differential equation for TE or TM modes. The number of zeroes, or nodes, of this function labels the modes. The nodes increase with the prospective propagation constant. This fact is proven by constructing suitable self-adjoint operators and referring to the minimax principle. The nodal properties allow to formulate an efficient bisection algorithm for effective indices and field distributions of guided hybrid modes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Yamamoto, Y. Koyamada, T. Makimoto: J. Appl. Phys.43, 5090–5097 (1972)
W.K. Burns, J. Warner: J. Opt. Soc. Am.64, 441–446 (1974)
D. Marcuse: IEEE J. QE-14, 736–741 (1978)
D. Marcuse, I.P. Kaminow: IEEE J. QE-15, 92–101 (1979)
J. Čtyroký, M. Čada: IEEE J. QE-17, 1064–1070 (1981)
E.A. Kolosovskiî, D.V. Petrov, I.B. Yakovkin: Sov. J. Quantum Electron.13, 1179–1182 (1983)
M. Koshiba, H. Kumagami, M. Suzuki: J. Lightwave Tech. LT-3, 773–778 (1985)
T. Kato:Perturbation Theory for Linear Operators (Springer, New York 1966) see Chap. VI, Theorem 2.1
D.W. Robinson:The Thermodynamic Pressure in Quantum Statistical Mechanics (Springer, Berlin, Heidelberg 1971) see Proposition 1.2.14
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Menzler, H.P., Hertel, P. & Pape, H. Bisection algorithm to calculate hybrid modes of birefringent planar graded index waveguides. Appl. Phys. B 44, 205–209 (1987). https://doi.org/10.1007/BF00692123
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00692123