A Homotopy Method for Locating All the Poles of a Parallel Plate Waveguide With the PML

Abstract

In this paper, a homotopy method for locating all the poles of a parallel plate waveguide with the perfectly matched layer is proposed. As is well known, the Green's functions for an open multilayered medium can be approximated by those for a parallel plate waveguide with the perfectly matched layer, while the latter Green's functions can be theoretically expressed into series expansions of their own eigenmodes without involving any numerical integrations. The homotopy method here is applied for finding the desired mode poles. In this method, the Newton-Raphson iteration is employed to solve every node equation except the start node equation, while the dichotomy method is used to solve the start node equation. The roots of the end node equation are just the desired mode poles. With the proposed method, the Green's functions for a lossy or lossless multilayered medium can be efficiently calculated by the corresponding series expansions. Some numerical examples are provided to examine the robustness and accuracy of the proposed method.Index Terms-Berenger mode pole, evanescent mode pole, Green's function, homotopy method, multilayered medium, Newton-Raphson iteration, perfectly matched layer, surface mode pole.

Pseudo-Analytical Modeling for Electromagnetic Well-Logging Tools in Complex Geophysical Formations

“…In [11,12], the eigenvalues of a circular waveguide embedded in lossy surroundings were determined from the initial guess given by the high-frequency asymptotic analysis of the problem. The root-finding strategy employed in [57,80] starts out from the lossless eigenvalues of the structure. These eigenvalues lie on the coordinate axis and can be calculated with relatively little numerical effort [57] by using a root search along real (for propagating modes) and the imaginary axis (for evanescent modes).…”

“…These eigenvalues lie on the coordinate axis and can be calculated with relatively little numerical effort [57] by using a root search along real (for propagating modes) and the imaginary axis (for evanescent modes). In a second step, we need to gradually increase the losses in the structure and track the modes as they move from the coordinate axis into the complex plane [57,80]. The techniques used in these works proved to be numerically efficient in many cases.…”