Identifying Near-Perfect Tunneling in Discrete Metamaterial Loaded Waveguides

Abstract

Mu-negative and epsilon-negative loaded waveguides taken on their own are nominally cut-off. In ideal circumstances, and when paired in the correct proportions, tunneling will occur. However, due to losses and constraints imposed by finite-sized constituent elements, the ability to experimentally demonstrate tunneling may be hindered. A tunnel identification method has been developed and demonstrated to reveal tunneling behavior that is otherwise obscured. Using ABCD (voltage-current transmission) matrix formulation, the S-parameters of the mu-negative/epsilon-negative loaded waveguide junction is combined with S-parameters of an epsilon-negative loaded waveguide. The method yields symmetric scattering matrices, which allows the effect of losses to be removed to provide yet clearer identification of tunneling.

1. Introduction

The purpose of this work is to identify tunneling across an experimental microwave metamaterial heterojunction that may be obscured due to losses and constraints imposed by finite-sized constituent elements in real microwave metamaterials.

Microwave metamaterials exhibit unusual electromagnetic properties (such as negative permittivity and negative permeability) which yield a range of unusual phenomena. One such phenomenon is the tunneling of electromagnetic waves through epsilon-near-zero (ENZ) metamaterial [1,2,3,4,5], epsilon-negative (ENG) metamaterial (or empty waveguides operated at cut-off) paired with mu-negative (MNG) metamaterial [6,7,8,9,10,11,12,13,14,15], and cut-off waveguides either filled [16,17], or lined [18,19], with metamaterials. In this work, we are specifically interested in tunneling in a microwave waveguide filled by a ENG/MNG metamaterial heterojunction where the waveguide is cut-off when loaded by either the ENG or MNG metamaterial on its own [7,14].

ENG behavior can be obtained at microwave frequencies using an empty cut-off waveguide [7,14] or by loading the waveguide with an array of thin wires [20,21]. MNG behavior can be obtained at microwave frequencies by loading the waveguide with an array of split-ring resonators (SRRs) [7,14,20,22]. In both cases, lumped elements can also be used [8,9,10,13]. Practical microwave metamaterials are constructed from a discrete number of finite-sized elements. Therefore, in a practical experiment, it may not be possible to satisfy the length requirement (or attenuation condition) for perfect tunneling [6,7]. Hence, it is the purpose of this work to develop a method to identify tunneling behavior in discrete microwave ENG/MNG metamaterial junctions.

Whether using loaded waveguides or transmission lines, resistive losses impact on the ability to demonstrate tunneling [6,15]. Therefore, an additional purpose of this work is to isolate losses from experimental results to further reveal tunneling.

2. Theoretical Background

In this work we confine ourselves to time-harmonic electromagnetic waves. At a given angular frequency ω, each field component of an electromagnetic wave travelling in the positive z direction will contain the factor $\exp \left(j\omega t-\gamma z\right)$ where γ is the propagation constant and $j=\sqrt{-1}$. In general γ is complex valued and related to real valued attenuation and phase constants, α and β respectively, taking the form $\gamma =\alpha +j\beta$. For passive media, the amplitude of the field components will either decay or remain constant with z and this means α is always greater than or equal zero. For the rest of this work we will drop explicit reference to the factor $\exp \left(j\omega t-\gamma z\right)$ and describe such quantities using phasor notation. Within the waveguide, we confine ourselves to transverse-electric (TE) wave propagation and consider the lowest order mode being the TE₁₀ mode [23] since this simplifies the procedure to demonstrate tunneling.

Figure 1 shows a diagram of an idealized waveguide tunneling configuration which comprises four sections: Section 1 and Section 4 air-filled, and Section 2 and Section 3 being filled by μ-negative (MNG) and ε-negative (ENG) metamaterials respectively. For the moment, it will be assumed that the MNG and ENG metamaterials are continuous, lossless and isotropic. The waveguide cross-section is identical along the structure so the only discontinuity is a change of material along the longitudinal axis (z).

At a frequency above the TE₁₀ mode cut-off frequency of Section 1 and Section 4, Section 1 and Section 4 will support propagation. On the other hand, Section 2 and Section 3 on their own, because their permittivity and permeability are opposite signed, will be cut-off (see Appendix A). It can be shown (see Appendix B) perfect transmission from port 1 to port 2 (zero insertion loss and zero reflection coefficient) occurs when [6,7]:

$${\alpha }_{ENG}{l}_{ENG}={\alpha }_{MNG}{l}_{MNG}$$

(1)

and:

$$Z_{ENG}=-Z_{MNG}$$

(2)

where l_MNG and l_ENG are the lengths of the MNG and ENG loaded waveguides respectively, and Z_MNG and Z_ENG are the TE₁₀ mode wave-impedances of the MNG and ENG metamaterials respectively. We denote (1) and (2) the attenuation and impedance tunneling conditions respectively.

As only evanescent modes can be excited in waveguides filled with lossless isotropic MNG and ENG metamaterials, Z_MNG and Z_ENG will be purely imaginary with $\mathrm{Im}\left(Z_{MNG}\right)<0$ and $\mathrm{Im}\left(Z_{ENG}\right)>0$ (see Appendix A). Provided Z_MNG and Z_ENG are of similar magnitude, the impedance tunneling condition can be satisfied at some frequency. On the other hand, the attenuation constant is always positive valued and this requires a certain length ratio to satisfy the attenuation tunneling condition. Similar conclusions are obtained for waveguides filled with certain continuous lossless anisotropic MNG and ENG metamaterials (see Appendix C) [7].

The single ENG/MNG heterojunction contained within the waveguide of Figure 1 represents the minimum configuration for tunneling. A five section waveguide tunneling structure may be realized as shown in Figure 2 where the length of Section 3 (ENG) is 2l_ENG. The tunneling conditions of Figure 2 are the same as Figure 1, and hence equations (1) and (2) apply to Figure 2. It is the structure of Figure 2 which forms the theoretical basis for the experimental tunneling identification. This structure is symmetrical which is useful when dealing with losses.

3. Tunneling in Practical Microwave Metamaterials

Practical microwave metamaterials employing split-ring resonators (SRRs) and thin wires to obtain MNG and ENG behavior respectively, are crystalline (periodic) structures with lattice constants of the order of millimeter [7,14,16,17,20,24,25]. So whilst significantly smaller than the guide wavelength, thereby approximating the continuous medium, the dimensions of practical metamaterial elements are never-the-less finite and means that material lengths take discrete values. It is unlikely that the attenuation tunneling condition is satisfied at the same frequency as the impedance tunneling condition when the ENG and MNG lengths are restricted to discrete values.

For example, for ENG and MNG metamaterials based upon the negative-refractive-index (NRI) metamaterials described by Eccleston & Platt [24], the length of the MNG metamaterial would be integer multiples of 5 mm and the ENG metamaterial takes lengths in integer increments of 20 mm. The significant difference in lattice constants arises from the need to achieve useful permittivity values within the low gigahertz range using an entirely printed structure [24]. As will be demonstrated in Section 5, perfect tunneling in such materials is obtained at 2.73 GHz when l_MNG = 25 mm and l_ENG = 8.8 mm, which is incompatible with these lattice constraints of the materials. At higher frequencies, the lattice constant for ENG could be comparable to that of the MNG [25].

On the other hand, this problem does not arise when one of the metamaterials is continuous. For example, Baena et al. [7] use a SRR waveguide and an empty cut-off waveguide for MNG and ENG behavior respectively; l_MNG and l_ENG are discrete and continuous respectively.

4. Tunnel Identification Principle

The procedure that will be described is aimed at identifying tunneling in a MNG/ENG junction constructed entirely from discrete metamaterials. Two sets of microwave scattering parameters [23] is required: (i) ENG/MNG filled waveguide junction, and (ii) ENG filled waveguide of known length. The former structure will be denoted the device-under-test (DUT) and contains the MNG/ENG heterojunction required for tunneling.

It will be assumed that MNG behavior is obtained over a narrow bandwidth using an array of split-ring resonators (SRRs), and ENG behavior is obtained over a significantly wider bandwidth using an array of strips. The term strip is used rather than wire to acknowledge implementation in printed-circuit-board (PCB) or similar technology [24,25].

The block diagram describing tunneling identification is shown in Figure 3a. The DUT and reversed DUT blocks represent the direct measurements of the DUT, whilst the middle section, of length l_Strip2, is described by a transmission line model. The propagation constant and wave-impedance of the transmission line model are extracted from measurements of the strip loaded (ENG) waveguide. The length l_Strip2 of the middle section of Figure 3a is in general different to the length of the measured strip loaded waveguide.

The cascade of Figure 3a is equivalent to the block diagram shown in Figure 3b. Comparing Figure 3b with Figure 2, then applying (1) and (2) to Figure 3b, it can be shown for lossless media, perfect tunneling occurs when:

$$2{\alpha }_{SRR}{l}_{SRR}={\alpha }_{Strip}\left(2l_{Strip1}+l_{Strip2}\right)$$

(3)

and

$$Z_{SRR}=-Z_{Strip}$$

(4)

The ability to apply Figure 3a and its equivalent Figure 3b assumes only the dominant mode exists at the ports of each waveguide section depicted in Figure 3a, and that the strip loaded waveguide behaves as a homogenously filled waveguide. Provided that the waveguide transverse cross section is the same for all sections, the method does not introduce new discontinuities; the only discontinuity being the essential MNG/ENG (SRR/Strip) interface inherently contained in the DUT.

The block diagram of Figure 3a can be analyzed using ABCD (voltage-current transmission) parameters which are related to scattering parameters [26]. The ABCD matrix, A_a, (containing the ABCD parameters) of the circuit of Figure 3a is given by:

$${\mathbf{A}}_a={\mathbf{A}}_{DUT}{\mathbf{A}}_{Strip}(l_{Strip2}){\overline{A}}_{DUT}$$

(5)

where A_DUT and A_Strip(l_Strip2) are the ABCD matrices of the DUT and strip loaded waveguide of length l_Strip2 respectively, and ${\overline{A}}_{DUT}$ is the ABCD matrix of the DUT in the reverse direction and is related to the elements of A_DUT by [27]:

$${\overline{A}}_{DUT}=\left[\begin{array}{cc}{D}_{DUT}& B_{DUT}\\ C_{DUT}& A_{DUT}\end{array}\right]$$

(6)

where

$${\mathbf{A}}_{DUT}=\left[\begin{array}{cc}{A}_{DUT}& B_{DUT}\\ C_{DUT}& D_{DUT}\end{array}\right]$$

(7)

Applying the continuous material approximation for the strip loaded waveguide, a transmission line model for the strip loaded waveguide of length l can be used [23]:

$${\mathbf{A}}_{Strip}(l)=\left[\begin{array}{cc}\cosh \left({\gamma }_{Strip}l\right)& Z_{Strip}\sinh \left({\gamma }_{Strip}l\right)\\ \frac{\sinh \left({\gamma }_{Strip}l\right)}{Z_{Strip}}& \cosh \left({\gamma }_{Strip}l\right)\end{array}\right]$$

(8)

where γ_Strip is its complex propagation constant and Z_Strip is its wave-impedance. Both γ_Strip and Z_Strip are obtained from S-parameter measurements of a known length of strip loaded waveguide. This is achieved by relating (8) to the ABCD matrix of the measured strip loaded waveguide of known length l.

Conversion between S-parameters and ABCD parameters requires knowledge of the port reference impedance [26]. By using the same waveguide transverse cross section for both the DUT and strip loaded waveguide, the ambiguity associated with waveguide characteristic impedance [28], and port reference impedance is avoided. Hence, the port reference impedance used here is taken as directly proportional to the TE₁₀ mode wave-impedance of an empty waveguide.

The ABCD matrix A_a is calculated using (5) over a range of frequencies and l_Strip2. The overall scattering matrix of Figure 3b can be obtained from A_a. Therefore, $\left|S_{21}\right|$ can be plotted as a function of frequency and l_Strip2 from which a suitable value of l_Strip2, as well as the tunneling frequency can be identified. Negative values of l_Strip2 are permissible, but only if the total length of the ENG sections, $\left(2l_{Strip1}+l_{Strip2}\right)$, is positive and non-zero. Losses will limit the maximum transmission that can be achieved.

The complimentary method, which uses data from a SRR loaded waveguide can be used to identify tunneling in the DUT. Similarly, a method that uses data from both a SRR loaded waveguide and strip loaded waveguide can also be developed. Neither of these methods will be considered here, as losses as well as the highly resonant behavior of SRRs renders these approaches unfeasible.

5. Description and Theoretical Analysis of Experimental Metamaterials

The purpose of this section is to show that tunneling is theoretically possible in accordance to the mechanism described in Section 2, for strip loaded and SRR loaded waveguides described in [24].

5.1. Description of Metamaterials

Figure 4 gives a rendering of the DUT (MNG and ENG loaded waveguide junction) of which one layer of thickness 5 mm (or one period of 5 mm in the y direction) is shown for clarity. To fill a WR284 waveguide used in the experiment, 7 layers (or 7 periods in the y direction) are required. Also shown are 72 mm wide waveguide feeds at either end which correspond to WR284 waveguide feeds used in the experiment. Both the SRR and strip loaded regions have a length of 25 mm and a width of 65 mm.

The MNG region comprises an array of SRRs aligned transverse to the waveguide to achieve negative permeability in the x-direction and will interact with the x component of magnetic field of the TE₁₀ mode. The ENG region comprises y-directed strips to achieve negative permittivity in the y-direction thereby interacting with the TE₁₀ mode electric field. This arrangement gives anisotropic permeability and permittivity [7,24] and field analysis given in Appendix C shows that tunneling described in Section 2 is indeed possible for such metamaterial junctions. The SRRs and strips are mounted on 13 printed circuit boards (PCBs) mounted parallel to the yz plane and are spaced 5 mm in the x direction to span a width of 65 mm. On the PCBs containing the SRRs, the SRRs have 5 mm periodicity in both directions and this means that within the SRR loaded waveguide, the SRRs have three-dimensional periodicity of 5 mm.

The design approach and the structural parameters of the SRRs and strips are described in detail elsewhere [24], but the key features are: (i) broadside-coupled SRRs were used as they are compact and minimize bianisotropy [29], (ii) the SRRs were designed to resonate around 3 GHz, (iii) uses a thinned array of strips to achieve suitable values of permittivity [24]. The strip locations have a two-dimensional periodicity of 14.1 mm in the xz plane, and their locations are coincident with the PCBs.

5.2. Theoretical Predictions

Theoretical models for the SRR and strip loaded waveguides are described in earlier work [24] and are used to predict the μ_XX and ε_YY of the SRR loaded waveguide, and ε_YY of the strip loaded waveguide. These models include electric polarization of the SRRs and dielectric loading due to the PCB substrates, and use the continuous medium approximation. The other tensor permeability and permittivity diagonal elements are equal to that of free-space, and the off-diagonal elements are zero. Figure 5 shows the frequency response of μ_XX and ε_YY of the SRR loaded waveguide, and ε_YY of the strip loaded waveguide over the frequency range 2.5 GHz to 3 GHz. It is apparent that SRR loading provides negative permeability over a narrow band (2.663 GHz to 2.752 GHz) and its value is strongly dependent on frequency. On the other hand, strip loading provides negative permittivity over a much wider frequency range and is a relatively weaker function of frequency.

Using the field analysis of Appendix C, the TE₁₀ mode propagation constant and wave-impedance can be calculated from the anisotropic permittivity and permittivity (Figure 5), and are shown in Figure 6. The impedances are normalized to the TE₁₀ mode wave-impedance of an empty 65 mm wide waveguide. As the strip loaded waveguide operates in its evanescent mode over the frequency range 2.5 GHz to 3 GHz, its real part of wave-impedance and phase constant are both zero over this range and are not shown. Importantly, the imaginary parts of wave-impedance are opposite signed over the frequency range 2.663 GHz to 2.752 GHz, and hence it is useful to show the sum $\mathrm{Im}\left(Z_{Strip}\right)+\mathrm{Im}\left(Z_{SRR}\right)$ over the range 2.663 GHz to 2.752 GHz in Figure 6a.

The zero crossing of $\mathrm{Im}\left(Z_{Strip}\right)+\mathrm{Im}\left(Z_{SRR}\right)$ in Figure 6a indicates that the impedance tunneling condition (2) is satisfied at 2.728 GHz. At this frequency, Figure 6b shows that the attenuation constants are 30.1 Np/m and 85.2 Np/m for the SRR and strip loaded waveguide respectively. If the length of the SRR loaded waveguide l_MNG is 25 mm, then the strip loaded waveguide length l_ENG needs to be 8.9 mm to satisfy the attenuation tunneling condition (1). Figure 7 shows the calculated S-parameter frequency responses for the symmetrical structure of Figure 2 for the two cases (i) l_MNG = 25 mm and l_ENG = 8.9 mm, and (ii) l_MNG = 25 mm and l_ENG = 25 mm. It is apparent that for the former case, reflection-less transmission is obtained at 2.728 GHz. On the hand, the latter cases shows low coupling due to the non-satisfaction of the attenuation tunneling condition.

For the case of l_ENG equal to 8.9 mm, a very sharp peak in transmission is observed at 2.66 GHz. This frequency falls slightly outside the negative permeability band, and therefore, the SRR loaded waveguides behave as electrically long, high impedance transmission lines (see Figure 6). Due to the high electrical lengths in this vicinity, there will be a frequency (or multiple frequencies) where the length of the SRR loaded waveguides are integer multiples of a half guide-wavelength. In this situation, the SRR loaded waveguides behave as half-wave transformers effectively providing a direct coupling to the strip loaded waveguide to port 1 and 2. Due to the mismatch between the SRR loaded waveguide and empty waveguide wave-impedances, the resulting peak will be narrow. The total ENG length of Figure 2 is 17.8 mm (2l_ENG) and its attenuation is constant at 2.66 GHz is 86.3 Np/m. Hence, at 2.66 GHz its total attenuation will be 13 dB which is the level of transmission seen in Figure 7. The phenomenon is also denoted a Fabry-Perot resonance [7].

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 (l_SRR = 25 mm and l_Strip1 = 25 mm). The length of the strip loaded waveguide is 25 mm.

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 [31,32,33,34,35] 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 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, α_SRR and imaginary part of Z_SRR are both non-zero, over this range. Figure 9a also shows that the sum of Im(Z_Strip) and Im(Z_SRR), 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 $\mathrm{Im}\left(Z_{Strip}\right)+\mathrm{Im}\left(Z_{SRR}\right)$ 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 (S₂₁) 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 l_Strip2 is −40.8 mm. The l_Strip2 value of −40.8 mm means that the total length of the equivalent strip loaded section of Figure 3b, $\left(2l_{Strip1}+l_{Strip2}\right)$, is 9.2 mm. This value is comparable to the theoretical prediction of 17.8 mm given in Section 5.

Figure 11 shows the S-parameters of the cascaded structure of Figure 3a when l_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 (S₂₁) compared to the DUT on its own, as well as in improvement of input and output match. Importantly, the resulting S-parameters are symmetric (S₁₁ = S₂₂).

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

$$L=1-{\left|S_{11}\right|}^2-{\left|S_{21}\right|}^2=1-{\left|S_{22}\right|}^2-{\left|S_{12}\right|}^2$$

(9)

where S₁₁, S₂₂, S₁₂ and S₂₁ 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 $1/\sqrt{1-L}$. 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.

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%.

8. Conclusions

Due to the finite size of constituent elements and losses, the ability to experimentally demonstrate tunneling across a cascade of mu-negative and epsilon-negative loaded waveguides may be hindered. A tunnel identification method has been developed and demonstrated to reveal the tunneling behavior that is otherwise obscured. The S-parameters of a mu-negative/epsilon-negative loaded waveguide junction are combined with the S-parameters of an epsilon-negative loaded waveguide. The method yields symmetric S-matrices, which the effect of losses to be removed to provide a yet clearer demonstration of tunneling.