**Appendix C**

In this appendix, the fields are formulated for TE*m*<sup>0</sup> modes in a rectangular waveguide filled with anisotropic dielectrics. The outcome of this analysis is the attenuation constant and wave-impedance for the special case of anisotropic ENG and MNG dielectric fill.

Consider an anisotropic dielectric material whose permeability and permittivity tensors in rectangular coordinates are respectively:

$$
\mu = \begin{bmatrix}
\mu\_{xx} & 0 & 0 \\
0 & \mu\_{yy} & 0 \\
0 & 0 & \mu\_{zz}
\end{bmatrix} \tag{A6}
$$

$$
\boldsymbol{\varepsilon} = \begin{bmatrix}
\varepsilon\_{xx} & 0 & 0 \\
0 & \varepsilon\_{yy} & 0 \\
0 & 0 & \varepsilon\_{zz}
\end{bmatrix} \tag{A7}
$$

where *μxx*, *μyy* and *μzz* are the *x*, *y* and *z* directed permeability respectively, and *εxx*, *<sup>ε</sup>yy* and *εzz* are the *x*, *y* and *z* directed permittivity respectively. It is therefore assumed that the material is non-bianisotropic, non-gyrotropic and non-chiral which give zero off-diagonal elements. This assumption is reasonable for the types of metamaterials considered in this work comprising either broad-side-coupled SRRs, or thin strips.

*Electronics* **2019**, *8*, 84

Consider a rectangular waveguide directed in the *z* direction and whose perfect-electric-conducting walls are located at *x* = 0, *x* = *a*, *y* = 0 and *y* = *b*. All field component phasors of a wave propagating in the z direction will contain the common factor exp(- *γz*) where *γ* is the complex valued propagation constant. If we restrict the waves to be TE (transverse electric) waves, and further restrict the wave to containing no *y* dependence (i.e., TEm0 modes), substitution of (A6) and (A7) into Maxwell's curl equations in phasor form, yields:

$$
\gamma E\_y = -j\omega\mu\_{xx}H\_x \tag{A8a}
$$

$$\frac{\partial E\_y}{\partial x} = -j\omega\mu\_{zz}H\_z \tag{A8b}$$

$$-\gamma H\_{\rm x} - \frac{\Im H\_z}{\partial \mathbf{x}} = j\omega \varepsilon\_{yy} E\_y \tag{A8C}$$

Equations (A8a)–(A8c) shows that only the tensor components *μxx*, *<sup>ε</sup>yy* and *μzz* play a role in TE*m*<sup>0</sup> mode behavior. Based upon the waveguide boundary conditions, the solution of the non-zero field phasors, that also satisfy (A8a) and (A8b) must take the form:

$$H\_{\mathbf{x}} = \frac{-\gamma A}{j\omega\mu\_{\text{xx}}} \sin k\_{\mathbf{x}} \mathbf{x} \tag{A9a}$$

$$E\_{\mathcal{Y}} = A \sin k\_x \mathfrak{x} \tag{A9b}$$

$$H\_z = \frac{-k\_x A}{j\omega\mu\_{zz}} \cos k\_x x \tag{A9c}$$

where the cut-off wavenumber *kx* is given by (A3), *A* is an amplitude (which could be complex valued), and *m* is a non-zero integer. Substituting (A9) into (A8c) yields the dispersion relation:

$$
\gamma^2 = \frac{\mu\_{xx}}{\mu\_{zz}} k\_x^{\;2} - \omega^2 \mu\_{xx} \varepsilon\_{yy} \tag{A10}
$$

For an evanescent mode, *γ* will be pure real and equal to the attenuation constant *α* which is positive valued. For a TE wave travelling (or decaying) in the *z* direction, the wave-impedance is <sup>−</sup>*Ey*/*Hx*. Hence from (A9a) and (A9b):

$$Z\_{TE} = \frac{\mathbf{j}\omega\mu\_{xx}}{\mathfrak{a}}\tag{A11}$$

Thus from (A11), *ZTE* will be pure imaginary whose is sign is that of *μxx*.

There are two types of anisotropic dielectric that are of interest here. They are MNG with negative permeability confined to the *x* direction, and ENG with negative permittivity confined to the *y* direction.

#### *C.1. Anisotropic MNG*

For the first case, *μxx* is negative valued, *μzz* equals *μ*0 the permeability of a vacuum, and *<sup>ε</sup>yy* is positive valued. The permittivity *<sup>ε</sup>yy* is greater than the permittivity of a free-space (*ε*0) due to the inherent electric polarizability of the constituent elements comprising the metamaterial, and the presence of insulating substrates to support such elements. Hence, from (A10):

$$
\alpha = \sqrt{\frac{-\mu\_{xx}}{\mu\_0} \left(\omega^2 \mu\_0 \varepsilon\_{yy} - k\_x^{-2}\right)}\tag{A12}
$$

and is real (i.e., evanescent mode) only if *<sup>ω</sup>*<sup>2</sup>*μ*0*εyy* > *kx* 2 which is denoted the "anti-cut-off" property [7]. Clearly from (A11), *ZTE* will be a capacitive reactance.

### *C.2. Anisotropic ENG*

For the second case, *μxx* = *μzz* = *μ*0, and *<sup>ε</sup>yy* is negative valued. Hence, from (A10): 

$$
\mathfrak{a} = \sqrt{k\_x^2 - \omega^2 \mu\_0 \varepsilon\_{yy}} \tag{A13}
$$

and is real for all frequencies provided *<sup>ε</sup>yy* is negative valued. Clearly from (A11), *ZTE* will be an inductive reactance.

The foregoing indicates that pairing of the above anisotropic ENG and MNG filled waveguides can yield the required impedance tunneling condition. Namely, apart from the anti-cut-off property, the conclusions are similar to the isotropic ENG and MNG cases.

In practice, for SRR realizations of anisotropic MNG, and strip realization of ENG, *μxx* and *<sup>ε</sup>yy* are frequency dependent and are only negative over a restricted frequency range; particularly for the SRR loaded waveguide. Therefore, the above theoretical analysis needs to be interpreted in that context.
