Two numerical methods, the reflection pole method (RPM) and the wavevector density method (WDM), are introduced for determining the propagation constants of guided and leaky modes in lossless and lossy planar multilayer waveguides. These methods are based on the extraction of propagation constants from Lorentzian-type peaks of the reflection coefficient (RPM) or on the density of wavevectors of the structure (WDM). Furthermore, in the case of the RPM the propagation constants can be determined with the help of the phase variation of the denominator of the reflection coefficient in conjunction with an optimization procedure. Both methods are tested on numerically "challenging" multilayer waveguides such as a two-metal-layer waveguide, a multilayer lossy waveguide, and an ARROW waveguide. The results produced by both methods are in good agreement with other numerical techniques but are obtained without the need for solving a dispersion equation in the complex plane. In addition, an approximate but easily implementable method is proposed which verifies whether a cluster of radiation modes can be accurately represented by a single leaky mode.
Emmanuel Anemogiannis, Elias N. Glytsis, and Thomas K. Gaylord, "Determination of Guided and Leaky Modes in Lossless and Lossy Planar Multilayer Optical Waveguides: Reflection Pole Method and Wavevector Density Method," J. Lightwave Technol. 17, 929- (1999)