An Efficient Algorithm for Locating TE and TM Poles for a Class of Multiscale Inhomogeneous Media Problems

The Mittag-Leffler (ML) expansion method has been minimally studied in the electromagnetic theory community. Therefore, in this article, the ML expansion functions are effectively utilized to locate initial poles for practical dielectric materials backed by a ground plane. The initial poles from the ML expansion and other existing methods are, then, checked for convergence, and found that the proposed method is highly efficient for electrically thick substrates (ETSs). To understand why, a fundamental and intrinsic convergence (in terms of relative or absolute errors) analysis of ML expansion and Taylor series expansion functions has been carried out and it was found out that the ML expansion converges significantly faster. In addition, Padè approximant has been successfully applied at no computational expense to improve the accuracy of initial pole locations. Moreover, proper and improper surface wave (SW) modes have been clearly distinguished from the poles obtained using the ML expansion method. Finally, the spectral region between leaky and SW poles for ETSs is identified using the initial pole locations obtained from using the ML expansion function on the transverse electric characteristic mode equation.