Backward wave propagation in waveguide filled with negative permeability meta material

The waveguide filled with a lossless dispersionless material with negative permeability is analyzed. It is shown that such a waveguide supports the propagation of backward waves below the cut-off frequency. There is no propagation above the cut-off-frequency and the waveguide behaves as a low-pass filter; thus, it is a dual of the conventional waveguide. The theoretical analysis was verified by measurements of the scattering coefficients of a rectangular waveguide loaded with split ring resonators in the 10 GHz frequency band. As expected, the propagation band occurred below the cut-off frequency. It was observed that an increase of the frequency caused an increase of electrical length of the waveguide. Therefore, Foster´s theorem is satisfied for this waveguide. It was also shown that an increase of the physical length of the waveguide causes a decrease of the electrical length. This unusual behavior is a consequence of the fact that the phase of the backward wave leads along the waveguide.