**Appendix A**

In this appendix, the dispersion relations for a lossless isotropic ENG or MNG loaded waveguide are obtained. The field analysis of an air-filled waveguide can be extended to a dielectric filled waveguide [23], with permeability *μ* and permittivity *ε*, and the resulting propagation constant (*γ*) and wave-impedance (*Z*) are:

$$
\gamma^2 = k\_x^2 - \omega^2 \mu \varepsilon \tag{A1}
$$

and

$$Z = \frac{j\omega\mu}{\gamma} \tag{A2}$$

where *kx* is the mode cut-off wavenumber, which for TE*m*<sup>0</sup> modes in a waveguide of width *a* are:

$$k\_x = \frac{m\pi}{a} \tag{A3}$$

where *m* is a non-zero integer.

For a given mode, a transmission line analogy can be used to represent a section of waveguide [23]. For the case of a uniform cross-section, the propagation constant and characteristic impedance of the transmission line use *γ* and wave-impedance *Z* respectively.

For a lossless MNG or ENG, either *μ* or *ε* will be negative valued and the other positive, and (A1) shows that *γ* only takes pure real values and means that only evanescent modes are possible. Equation (A2) indicates that *Z* will be pure imaginary and capacitive for MNG, and inductive for ENG.
