Stability analysis of Cartesian, cylindrical and spherical perfectly matched layers

Abstract

The analytic continuation of Maxwell's equations to complex space is a powerful tool to achieve the reflectionless absorption of electromagnetic waves in coordinate systems. Some limitations of this approach are presented particularly, that the analytic continuation should preserve the connection between the violation of causality and the dynamical instability of the resultant time-domain scheme. The Cartesian perfectly matched layer (PML) does not violate causality with the frequency dependence. The complex-space formulation of concave PML in cylindrical and spherical coordinates preserves the analyticity of the solutions on the upper-half plane. The convex PML violates causality resulting on an unstable finite-difference time-domain (FDTD) method.