Determination of the Optimal Value of the Radius of a Circular Cylindrical Post in a Rectangular Waveguide for Measurement of the Dielectric Permittivity

The goal of this paper is to determine the optimal value of the radius of a circular cylindrical post in a rectangular waveguide, that is, such a value of the radius of the post at which measurement uncertainty takes minimum value. To that end, we employ the well-known Monte Carlo method. Probability distributions for input quantities of the Monte Carlo model are assigned according to the principle of maximum entropy. 1. INTRODUCTION This paper is inspired by a recently published paper (1), in which a method for determination of the optimal radius of a dielectric post in a cylindrical metallic resonator for measurement of the dielectric permittivity has been proposed. This task requires a lot of computation time, since its based upon the Monte Carlo method that, in turn, requires a very large number of trials in order for estimation of the measurement uncertainty to be reliable and each of these trials requires solution of the inverse scattering problem, that in our case is being solved by iteratively minimizing the distance between simulated and measurement data. In order to accomplish this task in a reasonable amount of time, we need a very fast approach for solving the direct scattering problem, since the computation time required to solve the inverse scattering problem is proportional to that required by an algorithm for solving the corresponding direct scattering problem. A simpler way to evaluate measurement uncertainty is to follow GUM framework. Unfortunately, validity of the GUM framework is restricted only to models that lend themselves to an adequate linear approximation. The problem of scattering of the dominant mode in a rectangular waveguide by a cylindrical post was flrst treated in Notes on lectures by Julian Schwinger (2) by using the variational method. Numerical results obtained by application of this method also may be found in (3). Unfortunately, it has been found that results obtained by the variational method are highly inaccurate for posts of large electric radius, particularly near resonance frequencies. An attempt to improve the accuracy of the variational method was made in (4) by using the so-called second order approximation, that is, by using one more term in both series expansions instead of using just one term. Also many other approaches have been proposed by various authors over the last several decades (5{9). A flrst approach based on division of geometry of a problem into subregions of simple regular geometry has been proposed by Nielsen (10). Since the flelds in regular regions are expressed in terms of a series of solutions of the homogenous Helmholtz equation, it allows one to solve the boundary problem on the interfaces between layers of the post analytically that, in turn, considerably reduce computational efiorts as well as provides su-ciently high resolution of the fleld distribution. In order to flnd unknown expansion coe-cients, inflnite series are truncated and then by applying the point matching procedure the system of linear equations is set up and solved for unknown expansion coe-cients. Unfortunately, this approach converges to the correct result only for post of small electric radius (the product of the radius of the post and wavenumber). It was found that this limitation can be overcome by replacing the rectangular interaction region with the circular one, where the center of the circular interaction region (the region where point-matching procedure applies) coincides with the axis of the post and its radius is equal to one-half the width of the broader wall of the waveguide (11). It was also found that applying numerical integration on the surface of the interaction region instead of the point matching procedure yields faster convergence (12). 2. DISCUSSION