Are There the Pure TE and TM Modes in the Closed Waveguide Filled With a Homogeneous, Anisotropic and Lossless Medium?

Using the Abelian group theory and the classic electromagnetic waveguide theory, we propose a sufficient condition for the existence of the pure TE and TM modes in the closed waveguide filled with a homogeneous, anisotropic, and lossless medium. This sufficient condition is the relation between the permittivity and permeability tensors. Four eigenvalue problems that can be used to simulate the pure TE and TM modes in the anisotropic waveguide are derived from source-free Maxwell's equations. Two of them are eigenvalue problems of the elliptic differential operator, and the other two are eigenvalue problems of the curl-curl operator. Moreover, we solve these four eigenvalue problems by the finite element method. Finally, we carry out some numerical experiments to verify our theoretical results.