Figure 2. Number of TEM modes in the waveguide with PEC and PMC boundaries.

Schematic of a waveguide having an arbitrary cross-section and satisfying the ɛ(z)=μ(−z) symmetry (a) with two PECs and two PMCs. When TEM modes are considered, Maxwell equations reduce to a Poisson's equation ∇²ϕ=0 with four boundary conditions. The two PMCs require the vertical component of electric field to be zero (E_⊥=0), while the two PECs require the whole boundary has an equal potential ϕ₁ or ϕ₂. By applying the uniqueness theorem, one finds that this configuration has one spin-filtered TEM mode in each propagating direction. (b) A waveguide with three PECs and three PMCs. This configuration has two TEM modes in each direction. Spin-filtered feature is not guaranteed in this case.