Multimodal Solution for a Circular Waveguide Radiating Into Multilayered Structures Using the Axially Symmetric Modes

In this article, an exact formulation is derived for the mode reflection coefficient (and in general, the full S-parameter matrix) for a circular waveguide radiating into a multilayered structure and when excited with any combination of axially symmetric modes (i.e., modes of the form TE
_0m
and TM
_0m
, where
$\mathbf {m}$
is a positive integer). This formulation solves for the fields in the waveguide, including fields resulting from higher-order evanescent modes, using Fourier analysis. This leads to an accurate calculation of the mode S-parameter matrix, which includes the reflection coefficient of each excited mode. The derived formulations were validated through comparison to those computed using full-wave 3-D electromagnetic simulations. Additional simulations demonstrated the effect of considering higher-order modes on the results. The effect of having a finite flange and sample size on the complex reflection coefficient was also shown through 3-D simulations, indicating the relative insensitivity of the TE
₀₁
probe to edge effects. Reflection coefficient measurements, using a circular waveguide probe with the TE
₀₁
mode, were performed to experimentally verify the accuracy of the formulations. Forward-iterative optimization (i.e., optimal curve fitting) techniques were then performed on the reflection coefficient measurements to demonstrate the efficacy of this method for accurately estimating the thickness and complex permittivity of thin dielectric layers.