Guided and Leaky Modes of Planar Waveguides: Computation via High Order Finite Elements and Iterative Methods

Guided and leaky modes of planar dielectric waveguides are eigensolutions of a singular Sturm-Liouville problem. This paper describes how this problem can be transformed into a quartic eigenvalue problem, which in turn can be converted into a generalized eigenvalue problem. Thus standard iterative methods, such as Arnoldi methods, can be used to compute the spectrum. We show how the shifts in the Arnoldi methods must be selected to obtain convergence to the dominant modes. In addition, by using high-order finite elements, the resulting solutions can be made extremely accurate. Numerical examples demonstrate the speed and accuracy as well as the stability of the method.