**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 [1–5], epsilon-negative (ENG) metamaterial (or empty waveguides operated at cut-off) paired with mu-negative (MNG) metamaterial [6–15], 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 [8–10,13]. 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(*jω<sup>t</sup>* − *γz*) where *γ* is the propagation constant and *j* = √−1. In general *γ* is complex valued and related to real valued attenuation and phase constants, *α* and *β* respectively, taking the form *γ* = *α* + *jβ*. 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(*jω<sup>t</sup>* − *γz*) 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: Sections 1 and 4 air-filled, and Sections 2 and 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*).

**Figure 1.** Schematic of a cascade of MNG and ENG metamaterial filled waveguide where tunneling (transmission from port 1 to port 2 or vice versa) is possible under certain conditions. (In color.).

At a frequency above the TE10 mode cut-off frequency of Sections 1 and 4, Sections 1 and 4 will support propagation. On the other hand, Sections 2 and 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]:

$$
\pi\_{ENG} l\_{ENG} = \pi\_{MNG} l\_{MNG} \tag{1}
$$

and:

$$Z\_{ENG} = -Z\_{MNG} \tag{2}$$

where *lMNG* and *lENG* are the lengths of the MNG and ENG loaded waveguides respectively, and *ZMNG* and *ZENG* 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, *ZMNG* and *ZENG* will be purely imaginary with Im(*ZMNG*) < 0 and Im(*ZENG*) > 0 (see Appendix A). Provided *ZMNG* and *ZENG* 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 2*lENG*. 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.

**Figure 2.** Schematic of a cascade of MNG, ENG and MNG metamaterial filled waveguide where tunneling (transmission from port 1 to port 2 or vice versa) is possible under certain conditions. (In color.).

#### **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 *lMNG* = 25 mm and *lENG* = 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; *lMNG* and *lENG* 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 *lStrip*2, 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 *lStrip*2 of the middle section of Figure 3a is in general different to the length of the measured strip loaded waveguide.

**Figure 3.** Block diagram of the tunneling identification method: (**a**) constituent elements, and (**b**) equivalent structure.

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:

$$2\mathfrak{a}\_{SRR}l\_{SRR} = \mathfrak{a}\_{Strip} \left( 2l\_{Strip1} + l\_{Strip2} \right) \tag{3}$$

and

$$Z\_{SRR} = -Z\_{Strip} \tag{4}$$

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***<sup>a</sup>*, (containing the ABCD parameters) of the circuit of Figure 3a is given by:

$$\mathbf{A}\_{d} = \mathbf{A}\_{DUT}\mathbf{A}\_{Strip}(l\_{Strip2})\mathbf{A}\_{DUT} \tag{5}$$

where **A***DUT* and **<sup>A</sup>***Strip*(*lStrip*2) are the ABCD matrices of the DUT and strip loaded waveguide of length *lStrip*2 respectively, and **\$***DUT* is the ABCD matrix of the DUT in the reverse direction and is related to the elements of **A***DUT* by [27]:

$$
\overline{\mathbf{A}}\_{DITT} = \begin{bmatrix}
\begin{array}{cc}
D\_{DITT} & B\_{DITT} \\
\mathbb{C}\_{DITT} & A\_{DITT}
\end{array}
\end{bmatrix} \tag{6}
$$

where

$$\mathbf{A}\_{DUT} = \begin{bmatrix} A\_{DUT} & B\_{DUT} \\ C\_{DUT} & D\_{DUT} \end{bmatrix} \tag{7}$$

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]:

$$\mathbf{A}\_{Strip}(l) = \begin{bmatrix} \cosh\left(\gamma\_{Strip}l\right) & Z\_{Strip}\sinh\left(\gamma\_{Strip}l\right) \\ \frac{\sinh\left(\gamma\_{Strip}l\right)}{Z\_{Strip}} & \cosh\left(\gamma\_{Strip}l\right) \end{bmatrix} \tag{8}$$

where *γStrip* is its complex propagation constant and *ZStrip* is its wave-impedance. Both *γStrip* and *ZStrip* 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 *lStrip*2. The overall scattering matrix of Figure 3b can be obtained from **A***<sup>a</sup>*. Therefore, |*<sup>S</sup>*21| can be plotted as a function of frequency and *lStrip*2 from which a suitable value of *lStrip*2, as well as the tunneling frequency can be identified. Negative values of *lStrip*2 are permissible, but only if the total length of the ENG sections, 2*lStrip*1 + *lStrip*2, 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 arrangemen<sup>t</sup> 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.

**Figure 4.** Rendering of one layer of the DUT containing the SRR array (MNG)/strip array (ENG) interface. (In color.).

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 *μXX* 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 *μXX* 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.

**Figure 5.** Theoretical predictions of anisotropic permittivity and permeability for the SRR and strip loaded waveguides of the type considered in this work. (In color.).

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 Im*ZStrip* + Im(*ZSRR*) over the range 2.663 GHz to 2.752 GHz in Figure 6a.

**Figure 6.** Theoretically calculated TE10 mode (**a**) normalized wave-impedance, (**b**) attenuation and phase constants for strip and SRR loaded waveguide. (In color.).

The zero crossing of Im*ZStrip* + Im(*ZSRR*) 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 *lMNG* is 25 mm, then the strip loaded waveguide length *lENG* 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) *lMNG* = 25 mm and *lENG* = 8.9 mm, and (ii) *lMNG* = 25 mm and *lENG* = 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 *lENG* 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 (2*lENG*) 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].

**Figure 7.** Theoretically predicted TE10 mode S-parameters for the structure of Figure 2 using anisotropic permittivity and permeability values of Figure 5 for the two cases of *lENG* when *lMNG* is 25 mm. (In color.).
