**6. Experiment**

#### *6.1. Description of Metamaterials*

The metamaterials described in Section 5.1 were fabricated using a multilayer PCB fabrication process on Rogers 4000 series substrates of which individual PCBs were mounted longitudinally in waveguide test fixtures. The PCB thickness was 0.5 mm. The PCBs were 25 mm × 36 mm and each type required to construct the MNG (SRR) and ENG (strip) regions is shown in Figure 8a. The rings of the SRRs use the middle metal layers but are visible through the translucent substrate, whilst the strips are placed on the top metal layer; the bottom metal layer is unutilized. The DUT is 50 mm long (*lSRR* = 25 mm and *lStrip*1 = 25 mm). The length of the strip loaded waveguide is 25 mm.

**Figure 8.** Photo of: (**a**) each PCB type used in the experiment, (**b**) the waveguide PCB mount, and (**c**) measurement equipment shown measuring the *through* configuration for TL deembedding. (In color.).

#### *6.2. Waveguide PCB Mount*

Figure 8b shows a photo of the waveguide PCB mount to hold the PCBs associated with the DUT. The one for the strip loaded waveguide is similar but shorter in length. When fully assembled, the waveguide mounts contain 13 longitudinal slots, with width equal to the PCB thickness (0.5 mm), and depth 1 mm, to hold the PCBs with 5 mm spacing across a width of 65 mm. The waveguide PCB mount mates with WR284 (72.14 mm × 34.04 mm) waveguide, and has transverse dimensions of 65 mm × 34 mm.

For manufacturability and to aid inserting PCBs, the waveguide mount is constructed from laminations. Transverse bolts along with the laminated nature of the test fixture, creates a vice to secure the PCBs in the slots. Due to manufacturing tolerances, it is impossible to machine laminations to exactly the same longitudinal length. To ensure continuity of waveguide longitudinal currents (in top and bottom walls) 0.5 mm deep recesses are milled at each end of the mount to create very low impedance RF chokes with the WR284 flanges.

#### *6.3. Measurement Procedure*

Measurements were performed over the band 2 GHz to 4 GHz using a Keysight N9918A RF vector network analyzer (Figure 8c) calibrated to SMA coaxial connector reference planes using the short-open-load-thru (SOLT) procedure. To eliminate effects coax-to-waveguide transitions, through-line (TL) deembedding [30] (which is related to TRL deembedding) was conducted offline to obtain the *S*-parameters for the various loaded waveguide PCB mounts. These measurements are available as Supplementary Materials. The reference planes were then extended by 0.5 mm to remove the effect of the RF chokes at the ends of the waveguide PCB mounts. The *S*-parameters were finally renormalized from the wave-impedance of an empty 72 mm wide (WR284) waveguide to that of an empty 65 mm wide (waveguide mount) waveguide. As the TE10 mode cut-off frequency of the 65 mm wide waveguide is 2.31 GHz, only measurements above this frequency are useful.

#### *6.4. Strip and SRR Loaded Waveguide*

Figure 9 shows the attenuation and phase constants, and wave-impedance extracted from the strip loaded waveguide measurements over the range 2.5 GHz to 3.5 GHz. The wave-impedance is normalized to the wave-impedance of an empty 65 mm waveguide. For completeness, the attenuation and phase constants, and wave-impedance for a SRR loaded waveguide is also shown for qualitative purposes. The challenges of retrieval ambiguities [31–35] is not evident for the strip loaded waveguide, whereas it could be a factor for the SRR loaded waveguide at the lower end of the MNG passband.

**Figure 9.** Extracted TE10 mode (**a**) normalized wave-impedance, (**b**) attenuation and phase constants for strip and SRR loaded waveguide. (In color.).

Figure 9 shows that the strip loaded waveguide behaves like an ideal evanescent-mode attenuator with an essentially zero phase constant (*β*) and pure imaginary characteristic impedance over the entire frequency range of interest. The SRR loaded waveguide has almost pure-real wave-impedance and near-zero attenuation constant (*α*), when operating below about 2.6 GHz and above about 2.95 GHz, and it therefore operates in a propagating mode in these frequency bands. On the other hand, between about 2.65 GHz and 2.95 GHz, it behaves as a lossy evanescent-mode attenuator with complex valued propagation constant and wave-impedance. This lossy behavior is attributed to losses in the SRRs which are operating near resonance [24].

Although it is difficult to make use of the SRR loaded waveguide data directly, it never-the-less can be qualitatively interpreted as evidence of MNG behavior over the range 2.65 GHz to 2.95 GHz. For instance, *αSRR* and imaginary part of *ZSRR* are both non-zero, over this range. Figure 9a also shows that the sum of Im( *ZStrip*) and Im( *ZSRR*), crosses zero at several frequencies over the range 2.76 GHz to 2.86 GHz; meaning that the impedance tunneling condition (4) will be satisfied in the imaginary part within this range. In an ideal situation, the impedance matching condition (4) is satisfied at one frequency only, so the multiple zeros of Im *ZStrip* + Im(*ZSRR*) in Figure 9a is due to SRR resonant frequency variation across the PCB panels [36].

#### *6.5. Tunnel Identification*

The tunnel identification method described in Section 4 was applied to the DUT measurements. Figure 10 shows the transmission (*S*21) as a function of frequency (*f*) and middle strip loaded waveguide section length (*l*Strip2). Transmission is maximum at 2.8 GHz, and peaks when *lStrip*2 is −40.8 mm. The *lStrip*2 value of −40.8 mm means that the total length of the equivalent strip loaded section of Figure 3b, <sup>2</sup>*lStrip*1 + *lStrip*2 , is 9.2 mm. This value is comparable to the theoretical prediction of 17.8 mm given in Section 5.

**Figure 10.** Transmission (*S*21) magnitude versus frequency and length of the middle strip loaded waveguide (*l*Strip2). Peak transmission occurs at 2.8 GHz when the length of the middle strip loaded waveguide (*l*Strip2) is −40.8 mm. (In color.).

Figure 11 shows the *S*-parameters of the cascaded structure of Figure 3a when *<sup>l</sup>*Strip2 is equal to −40.8 mm. For reference, the *S*-parameters of the DUT are shown. At 2.8 GHz, the tunneling effect is clearly revealed with an 8.6 dB boost in transmission (*S*21) compared to the DUT on its own, as well as in improvement of input and output match. Importantly, the resulting S-parameters are symmetric (*S*11 = *S*22).

**Figure 11.** S-parameter magnitude frequency responses after tunnel identification and after loss is removed: (**a**) *S*21, and (**b**) *S*11 (and *S*22). The DUT response is included for comparison. (In color.).

For a symmetrical reciprocal two-port, the power loss proportion, *L*, can be calculated by power conservation:

$$L = 1 - \left|\mathbf{S}\_{11}\right|^2 - \left|\mathbf{S}\_{21}\right|^2 = 1 - \left|\mathbf{S}\_{22}\right|^2 - \left|\mathbf{S}\_{12}\right|^2\tag{9}$$

where *S*11, *S*22, *S*12 and *S*21 are the *S*-parameters of the two-port. Figure 12 shows the power loss of the DUT after tunnel identification. Significant loss occurs within the MNG band and this explains the insertion loss of 8.1 dB at 2.8 GHz after the tunnel identification process of Section 3. The effects of losses can be removed from the symmetric two-port by scaling the *S*-parameters by 1/√1 − *L*. With the effects of losses are removed (pink dashed trace in Figure 11), the resulting tunnel residual insertion loss, due to mismatch, is 2.2 dB.

**Figure 12.** Power loss frequency response for the deembedded DUT.

#### **7. Other Observations**

After the tunnel identification method has been applied, a significant transmission boost occurs at 2.45 GHz and 3.3 GHz (blue trace in Figure 11). Both these frequencies fall well outside the MNG regime of the SRR loaded waveguide and therefore cannot be attributed to tunneling. At these frequencies, the SRR loaded waveguide behaves as a transmission line that transform the empty waveguide characteristic impedance to a complex value, and at the same time, the strip loaded waveguide essentially behaves as a lossless evanescent mode attenuator. The strip-loaded waveguide can be treated as a two-port element terminated by complex valued impedances. From microwave amplifier design principles [37], this situation may result in a boost in transmission because of the non-trivial dependence on terminating impedance.

When loss is removed, a sharp peak is also revealed at 2.636 GHz (pink trace in Figure 11a). Figure 9 indicates that this frequency is slightly outside the MNG band of frequencies and therefore, this peak is attributed to Fabry-Perot resonances [7] as discussed in the theoretical investigation in Section 5. This resonance is obscured after tunnel identification (blue trace in Figure 11) as the loss at this frequency (Figure 12), is nearly 100%.
