Exact Analytical Equations for Eigenmodes in Varying Refractive-Index Profile’s Waveguides Terminated by Perfectly Matched Layers

In this paper, the exact analytical equations for both TM and TE cases are presented, when a varying refractive-index profile's waveguide is terminated by the perfectly matched layers. In the derivation, the differential transfer matrix method and the matrix similarity transformation are mainly used. Moreover, we deduce the asymptotic solutions for Berenger modes in TM case and they can well act as the initial values of Newton's iteration. Numerical results illustrate that leaky and Berenger modes, from the exact analytical equations, have higher precision than those by the approximate equations.