Bidirectionality of Reciprocal, Lossy or Lossless, Uniform or Periodic Waveguides

Using the scattering-matrix equations for two antennas placed in the fields of a waveguide, it is proven that all reciprocal, lossy or lossless, uniform or periodic waveguides are bidirectional. Since Maxwell's equations imply directly that propagation constants on a lossless reciprocal waveguide come in pairs (beta,-beta*), "complex waves" on a lossless reciprocal, uniform or periodic waveguide come in quadruplets with propagation constants (beta,-beta, beta*,-beta*).