  Whether there exist independent transverse electric (TE) and transverse
magnetic (TM) modes in a metallic waveguide filled with an anisotropic medium
is a fundamental question in electromagnetics waveguide theory, but so far no
definitive answers have been published. This paper establishes a necessary and
sufficient condition for having independent TE and TM modes in a waveguide
filled with a homogeneous lossless anisotropic medium based on both waveguide
theory in electromagnetics and basic knowledge in mathematics. Moreover, for
the independent TE modes, we prove the propagation constants obtained from both
the longitudinal scalar magnetic field stimulation and the transverse vector
electric field stimulation are the same; for the independent TM modes, the
propagation constants obtained from both the longitudinal scalar electric field
stimulation and the transverse vector magnetic field stimulation are the same.
This necessary and sufficient condition is a new theoretical result in
electromagnetic waveguide theory, and is expected to be valuable for the design
of waveguides filled with anisotropic media.
