Wiener-Hopf Method Applied to a Dielectric Cylinder Asymmetrically Excited By a Circular Metallic Waveguide.

Abstract

The problem of an infinite dielectric cylinder, clad by a semi-infinite metallic circular waveguide, and excited by means of an azimuthally asymmetric mode propagating down the guide towards its open end, is formulated using the Wiener-Hopf technique. The method introduced by Jones is used, and leads to a pair of coupled Wiener-Hopf equations, whose solution depends on the factorization of a second order square matrix, into two square matrices analytic in the upper and lower half-planes, respectively. That factorization appears to be beyond the capabilities of presently known methods. Instead, a perturbation solution is sought for the case of a dielectric with relative permittivity close to unity. The coupled Wiener-Hopf equations are exp and ed in Taylor series, in the propagation constant around the free space value. New sets of coupled Wiener-Hopf equations are found for the zeroth- and the first-order terms of the series expansions of the unknown functions. For each one of these sets of equations, a redefinition of the unknown functions, as linear combinations of the previous unknown functions, leads to the uncoupling of the equations. Then, the solutions for the zeroth- and the first-order terms of the unknown functions are found by solving, in each case, two independent Wiener-Hopf problems. These solutions are inverse Fourier transformed, to yield the zeroth- and the first-order terms of the components of the surface current density. The zeroth- and first-order terms of the reflection coefficients inside the metallic guide are then identified, and computations are carried out for the reflection coefficient of the dominant mode. The solutions of the zeroth- and first-order Wiener-Hopf equations are used to yield the zeroth- and first-order terms of the Fourier transforms of the components of the electric and magnetic fields for every point in space. These functions are inverse Fourier transformed, for points in the far field, by evaluating the integrals asymptotically, using the method of steepest descents. Expressions for the zeroth- and first-order terms of the far field components are found.