Direct computation of parabolic waveguide modes via a bivariate root-finding algorithm

The cylindrical parabolic waveguide is one of the waveguides whose modal spectra can be characterized quasi-analytically. The associated non-linear system of equations has already been accurately solved in the past, combining homotopy with shooting methods. However, a novel algorithm for bivariate problems has been found to be a good alternative for the root-finding process, allowing to solve all roots at once in a specified subdomain, without any starting points and covering a wide range of the involved variables. Accurate root values are reached, proving the good performance of the algorithm.