A Necessary and Sufficient Condition for Having Independent TE and TM Modes in an Anisotropic Waveguide

In a metallic waveguide filled with a lossless anisotropic medium, do independent transverse electric (TE) and transverse magnetic (TM) modes exist? If so, under what conditions? This is a fundamental problem in the electromagnetic waveguide theory, but so far no definitive answers have been published in existing references. This paper proposes a sufficient and necessary condition for this theoretic electromagnetic waveguide problem based on both classic waveguide theory in electromagnetics and basic knowledge in mathematics. Moreover, we prove that for the independent TE modes, the propagation constants obtained from both the longitudinal scalar magnetic field simulation and the transverse vector electric field simulation are the same, and for the independent TM modes, the propagation constants obtained from both the longitudinal scalar electric field simulation and the transverse vector magnetic field simulation are the same. Finally, we validate the correctness of this sufficient and necessary condition by several numerical experiments based on the finite-element method. Therefore, this is a new theoretical result in the electromagnetic waveguide theory, and is expected to be valuable for the design of waveguides filled with anisotropic media.