A Variational Method to Solve Waveguide Problems by Employing Dynamic Programming Technique

Abstract

A new variational method for the modal analysis of 2-D waveguide structures is proposed in this paper. Maxwell's equations describing the modal properties of scalar and semivectorial guided waves in the 2-D waveguides are expressed as variational/minimization problems, which are then discretized in terms of a finite-difference scheme on a finite rectangular computational window. By applying a dynamic programming technique to solve such variational problems, a 1-D equation representing the relation between the modal fields on any pair of adjacent columns in the computational window can be derived. By using such 1-D equation in a stepwise fashion from one boundary column toward the other boundary column, a system of linear equations with the unknown column modal fields can be derived and then solved to give both the accurate modal indexes and the discrete modal fields. In the examples of one weakly guiding rib-type dielectric waveguide and another strongly guiding silicon-on-insulator waveguide, computational results show that a small size of the coefficient matrix for such a system of linear equations is adequate to cause a relative error of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> -10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-6</sup> in the evaluation of the modal indexes reachable in an efficient manner. The results of the convergence tests show that the proposed method is at least an order of magnitude faster than the conventional finite-difference beam propagation method because of the transformation of a 2-D problem into a 1-D problem. Moreover, the proposed method is applied to investigate the modal properties of the conductor-gap-silicon plasmonic waveguide. The feature of the hybrid guided-mode profile is also observable from the modal field calculated by the proposed method.