**Appendix B**

In this appendix, the tunneling conditions derivation are derived. This derivation is based upon voltage-current chain or ABCD parameters [23] noting this problem can that alternatively formulated using wave-transmission parameters [7].

We consider the cascade of a pair of waveguide sections with identical transverse cross sections and therefore the only discontinuity is due to the different materials that fill each of these sections. These two sections could correspond to Sections 2 and 3 of Figure 1 which contain MNG and ENG fills respectively. As both sections are operating in their evanescent modes, they can be described by a pure real propagation constant (being the attenuation constant) and a pure imaginary wave-impedance *α*1 and *jX*1 respectively for Section 1, *α*2 and *jX*2 respectively for Section 2. The values of *α*1 and *α*2 are assumed to take positive values for passive media, whereas the *X*1 and *X*2 can take positive and negative values. For a given mode, a section of waveguide can be modelled as a transmission line described by its propagation constant and characteristic impedance. Using an ABCD parameter formulation [23], the ABCD matrix of the cascade of the two waveguide sections of lengths *l*1 and *l*2 is:

$$ABCD\_{\text{case2def}} = \begin{bmatrix} \cosh(a\_1l\_1) & jX\_1\sinh(a\_1l\_1) \\ \frac{1}{jX\_1}\sinh(a\_1l\_1) & \cosh(a\_1l\_1) \end{bmatrix} \begin{bmatrix} \cosh(a\_2l\_2) & jX\_2\sinh(a\_2l\_2) \\ \frac{1}{jX\_2}\sinh(a\_2l\_2) & \cosh(a\_2l\_2) \end{bmatrix} \tag{A4}$$

Expanding:

$$\begin{aligned} ABCD\_{\text{causal}} &= \begin{bmatrix} \cosh(a\_1l\_1)\cosh(a\_2l\_2) + \frac{X\_1}{X\_2}\sinh(a\_1l\_1)\sinh(a\_2l\_2) \\ \frac{1}{jX\_2}\sinh(a\_2l\_2)\cosh(a\_1l\_1) + \frac{1}{jX\_1}\sinh(a\_1l\_1)\cosh(a\_2l\_2) \end{bmatrix} \\ &\quad jX\_2\sinh(a\_2l\_2)\cosh(a\_1l\_1) + jX\_1\sinh(a\_1l\_1)\cosh(a\_2l\_2) \\ &\quad \cosh(a\_1l\_1)\cosh(a\_2l\_2) + \frac{X\_2}{X\_1}\sinh(a\_1l\_1)\sinh(a\_2l\_2) \end{bmatrix} \end{aligned} \tag{A5}$$

An identity ABCD corresponds to perfect transmission with zero reflection. Equation (A5) is equal to the identity matrix when *<sup>α</sup>*1*l*1 = *<sup>α</sup>*2*l*2 and *X*1 = − *X*2 which are denoted the attenuation and impedance tunneling conditions respectively. The latter condition requires *X*1 and *X*2 to be opposite signed and represents resonance across the boundary between Sections 1 and 2 [6,8].
