## 2 Circuit Modeling of the BPF and the WG ZOR, and Its Geometry
### Devising the Metamaterial Resonators in the Waveguide  
The making of a thin resonator within the cross-section of the waveguide starts at the equivalent circuit modeling for the zeroth-order resonator. Unlike the previous ZOR filters or metamaterial passive devices, which were formed mainly in the longitudinal direction as the microstrip-line or CPW, the novel ZOR is proposed to be formed in the transverse directions on the WG cross-section as a novel approach. A CRLH circuit is built by considering the up, down, left and right metallic walls. In Figure 3a, the E-CRLH circuit is given as combination of shunt L, shunt C, series L and series C which go well with four metallic surfaces on the waveguide cross-section. C\({}_{1}\) as series C from _Wall_left_ to _Wall_right_, and L\({}_{2}\) as shunt L from _Wall_left_ to _Wall_right_ are slots and short-circuited with the metal walls, spread in the transverse directions. Thinking of the vertical electric field of TE\({}_{10}\)-mode, a metal plate gets in the way to capture the E-field and is divided into the upper and lower patches modeled as L\({}_{\text{R}}\)C\({}_{\text{R}}\) sub-resonators connected through a strip equivalent to L\({}_{1}\). Combining the electrical attributes of the elements, for the purpose of the metamaterial resonance at the center frequency, the circuit values are obtained as follows and they are written in Table 3.  
Figure 3: _Cont._Figure 3: Circuit model of the CRLH resonator and move into the waveguide. (**a**) E\(-\)CRLH circuit of the resonator applied to the waveguide cross\(-\)section. (**b**) S\({}_{11}\) and S\({}_{21}\) of the resonator circuit. (**c**) Physical shape of the thin resonator. (**d**) The resonator is longitudinally flat. (**e**) S\({}_{11}\) and S\({}_{21}\) of the resonator structure. Using the values, S\({}_{21}\) and S\({}_{11}\) in the plot of Figure 3b show the resonance as planned. As mentioned, the geometrical information of the slots, short-circuiting lines, metal patches and their connecting strip in Figure 3c,d is figured out by electromagnetically simulating the structure in a full-wave analysis program, on the basis of C\({}_{1}\), L\({}_{2}\), L\({}_{R}\)C\({}_{R}\) and L\({}_{1}\) given in Table 3 to get the same frequency response as Figure 3b. As a result, S\({}_{21}\) and S\({}_{11}\) of the flat WG metamaterial resonator of the physical dimensions written in Table 4 are obtained as Figure 3e agreeing with Figure 3b. Using the frequency response and electromagnetic simulation, the ZOR as the metamaterial characteristics is verifiable. In the area of the conventional passive components, they resonate at the half-wave long TX-line segment, and when becoming much shorter than the half-wavelength, they are not resonant but evanescent. The field is strongly resonant on the structure far shorter than the half-wavelength as in Figure 4a. This field is observed at the target frequency. Especially in the side-view, there are two vertical bars whose gap is roughly the half-wavelength, and the thickness of the proposed resonator is a small fraction of quarter-wavelength. The electromagnetic wave entering the leftmost side (input port) propagates in the longitudinal direction, which is not blocked by this thin structure. Not as the evanescent mode, this propagation mode is consistent with the beta equal to zero occurring at the same frequency in reference to the dispersion diagram. As the relationship between beta (propagation constant) and frequency, the non-linear curve is formed from the LH (left-handed) region of negative beta through the ZOR point to the RH (right-handed) region of positive beta. For the conventional WG bandpass filters, cavities are designed as the half-wave long waveguide sections for the resonance of interest and cascaded through irises for inter-resonator coupling. Generally, irises as H-plane or E-plane steps are not resonant but have reactance. On the contrary, the proposed resonator as the sub-wavelength structure passes the RF signal on at a specific frequency.