Robust Calculation of the Modes in Parabolic Cylinder Metallic Waveguides by Means of a Root-Finding Method for Bivariate Functions

This paper addresses the robust mode computation of the metallic hollow cylindrical waveguide with parabolic contour. Although this waveguide can be solved by separation of variables, it has not been fully characterized in a systematic way in the past. This is not only due to its challenging manufacture, which nowadays can be addressed by modern techniques, but also because of its more complex resolution, involving root finding in a pair of coupled functions with two variables. In order to solve this system, this paper proposes the use of a recently published algorithm for bivariate problems, which is applied for the first time to waveguide mode computation. The method, properly combined with the even and odd Taylor functions, allows obtaining the modes in a systematic and robust way, avoiding, in comparison with previous works, graphical means and the use of starting points from which to iterate. All the modal solutions for symmetrical and asymmetrical cases are solved at once over a wide domain of search with proven high accuracy (relative difference with respect to results from a finite-element method in the order of 10
^-5
for more than 1600 modes).