Adjoint high-order vectorial finite elements for nonsymmetric transversally anisotropic waveguides

In this paper, a vector finite-element method (FEM) for the investigation of microwave and optical waveguides with arbitrary cross section is presented. In particular, the FEM solves the vector wave equation in the frequency domain for waveguides, containing nonsymmetric transversally anisotropic (passive and active) materials. Therefore, a bilinear form has been derived that is solved for the original and adjoint wave equation. Biorthogonality, relations of original and adjoint solutions for simpler materials, the role of nonphysical solutions for both problems, and their proper approximations are discussed in detail. Special focus has been set on shifting the propagation constants of the nonphysical modes for nonzero frequencies to infinity. Therefore, an excellent rate of convergence for iterative solvers is observed. The occurrence of nonphysical solutions is shown. Different nth-order basis functions are systematically derived, which are identical to the well-known tangential continuous finite elements. Both algebraic eigenvalue problems are solved via a generalized Lanczos algorithm, which solves both eigenvalue problems efficiently. The capability of the FEM is illustrated by critical examples.