Analysis of HE and EH modes for metallic rectangular waveguides

It is well known that a perfectly conducting rectangular waveguide supports the TE and TM electromagnetic modes. In a real metal case of finite conductivity, the non-zero skin depth requires more involved boundary conditions on the surfaces of the metallic waveguides so that supported modes, the so-called HE and EH modes, are deformations of the TE and TM modes. By extending Marcatili's method to the metal case, we derive explicitly the HE and EH modes and show that they reduce to the TE and TM modes in the limiting ease of a perfect conductor.