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Theory of Supercoupling, Squeezing Wave Energy, and Field Confinement in Narrow Channels and Tight Bends Using Epsilon-Near-Zero Metamaterials

In this work, we investigate the detailed theory of the supercoupling, anomalous tunneling effect, and field confinement originally identified in [M. Silveirinha, N. Engheta, Phys. Rev. Lett. 97, 157403, (2006)], where we demonstrated the possibility of using materials with permittivity near zero to drastically improve the transmission of electromagnetic energy through a narrow irregular channel with very subwavelength transverse cross-section. Here, we present additional physical insights, describe new applications of the tunneling effect in relevant waveguide scenarios (e.g., the"perfect"or"super"waveguide coupling), study the effect of metal losses in the metallic walls, and the possibility of using epsilon-near zero materials to confine energy in a subwavelength cavity with gigantic field enhancement. In addition, we systematically study the propagation of electromagnetic waves through narrow channels filled with anisotropic epsilon-near zero materials. It is demonstrated that these materials may have interesting potentials, and that for some particular geometries the reflectivity of the channel is independent of the specific dimensions or parameters of epsilon-near zero transition. We also describe several realistic metamaterial implementations of the studied problems, based on standard metallic waveguides, microstrip line configurations, and wire media.


I. Introduction
In recent years there has been a growing interest in the development of new technologies or approaches that potentially allow confining and guiding electromagnetic energy with mode sizes below the diffraction limit. This may have key applications in several fields such as telecommunications (e.g. realization of compact cavities or waveguides [1,2]), imaging with subwavelength resolution [3,4], devices with increased storage capacity, delivery and concentration (nanofocusing) of the optical radiation energy on the nanoscale [5,6,7], and realization of compact optical resonators [8]. Most of these proposals rely on the excitation of surface plasmon polaritons (SPP) supported by metallic structures with negative permittivity. In our recent work [9], we proposed a different paradigm to break the diffraction limit and squeeze light through channels and bends with subwavelength cross-section. We theoretically demonstrated that if a narrow channel is filled with a ε-near-zero (ENZ) material then, in the lossless limit, its reflectivity only depends on the volume of the channel, being independent of its specific geometry and of the transverse cross-section (relatively to the direction of propagation).
Moreover, in a counterintuitive way, our results establish that the transmission through the ENZ channel is improved when the transverse cross-section is made more and more narrow. These properties suggest that ENZ materials may have interesting potentials in efficiently transmitting energy through subwavelength regions and effectively providing 'supercoupling" between two ports/waveguides.
In [10], we have further shown that by loading the ENZ material with dielectric or metallic particles it is possible to tailor the magnetic permeability of the material. In this way, by suitably designing the inclusions it is possible to tune the permeability of the -3-ENZ filling material, without changing its electric properties. In particular, it may be feasible to design a matched zero-index metamaterial having both permittivity and permeability near zero, and thus improved transmission characteristics. A remarkable property of such matched zero-index materials is that the way they interact with electromagnetic waves is independent of the granularity of the composite material or of the specific lattice arrangement.
The objective of the present work is to study with more detail the theory of the supercoupling, field confinement, and tunneling effect reported in [9], giving new physical insights of this phenomenon, and describing its emergence and potential applications in new configurations and scenarios. Namely, here we investigate the group velocity of the wave as it travels through the ENZ channel, evaluate the effect of losses in the metallic walls, and demonstrate the possibility of concentrating and confining the electromagnetic fields in a very subwavelength air cavity with enormous electric field enhancement. Moreover, we propose a realistic (metamaterial) emulation of the studied propagation scenarios at microwaves using standard 'empty' metallic waveguides or alternatively using a microstrip line configuration. We report results obtained with a full wave electromagnetic simulator that demonstrate the emergence of a similar tunneling and supercoupling effect in these very realistic, and arguably simple, setups.
In addition, following the ideas of [9], we exploit the possibility of using anisotropic ENZ materials to create an analogous tunneling effect. In fact, as referred to in [9], while at infrared and optical frequencies (isotropic) materials with 0 ε ≈ may be readily available in nature (e.g. some metals [11], semiconductors [12], or polar dielectrics [13]), at microwaves these materials are not readily accessible (an exception is the electron gas, of which the ionosphere is a well-known example at radiowaves). Nevertheless, these materials may in principle be fabricated as artificial microstructured materials. However, nowadays the fabrication of isotropic ENZ materials is still relatively more challenging due to the complexity of the required isotropic microstructure of the material. To circumvent this problem, in [9] we suggested using anisotropic ENZ materials − which are comparatively simpler to synthesize − in order to obtain the same tunneling effect. In this work, we further develop these concepts, and develop a detailed theory for the propagation of electromagnetic waves through narrow channels filled with anisotropic ENZ materials. We also discuss the design of wire-medium-based implementations of these anisotropic materials at microwaves.
This paper is organized as follows. In section II we investigate the supercoupling properties of metallic channels filled with isotropic ENZ materials. The effect of metallic and dielectric losses is discussed, as well as possible applications (e.g., for waveguide coupling) and specific features of the tunneling phenomenon. It is also described how to emulate the studied propagation scenarios at microwaves using realistic 3D configurations based on the concept of artificial plasma. In section III, we study the propagation of waves through anisotropic ENZ materials. It is shown that for some configurations the scattering parameters may be independent of the specific geometry of the structure. The design of anisotropic ENZ slabs using wire media is discussed in detail,

II. Supercoupling and Squeezing Energy through ENZ Isotropic Channels
In this section, we present new physical insights related with the propagation of electromagnetic waves through metallic channels filled with ENZ isotropic materials.
Important aspects such as the group velocity and the effect of losses in the metallic walls are analyzed. The possibility of concentrating the electric field in a small air cavity with gigantic enhancement is suggested.

A. Overview and physical insights of the tunneling effect
Here, we briefly review the main results derived in [9] and describe the emergence of the tunneling effect in several propagation scenarios. In [9] we studied a generic twodimensional problem (with geometry invariant along the z-direction), and we assumed that the polarization of the fields is such that where 0 / k c ω = is the free-space wave number, 1 a and 2 a define the spacing between the metallic plates of the input and output waveguides, respectively, , r p μ is the relative permeability of the ENZ material and p A is the area of the cross-section of the ENZ channel (contained in the x-y plane). The transmission coefficient is given by The geometry of the problem is depicted in Fig. 1 for a very specific ENZ channel in which the transition is shaped as an "U", but we underline here that (1) is valid independently of the precise geometry of the ENZ transition. As discussed in [9], Eq. (1) demonstrates that in the case 1 2 a a a = ≡ , it may be possible to squeeze more and more energy through the channel, as the transverse section of the channel (relatively to the direction of propagation) is made more and more tight, i.e. as / p A a is made increasingly small. Such effect was demonstrated in [9] for the case of a waveguide with an 180ºbend. Next we further explore some alternative possibilities, namely we present results of full wave simulations computed with CST Microwave Studio TM [14], which demonstrate the emergence of the same tunneling effect in the geometry of Fig. 1 ("U-shaped" transition channel). We will assume that 1 2 a a a = ≡ since this situation favors the transmission of energy through the channel. It is clear that for such U-shaped channel the area of the cross-section is ( ) Thus, for a and L arbitrarily fixed it is evident that (in the lossless limit) we can make / p A a arbitrarily small (and consequently make the reflection coefficient approach zero), by reducing more and more the transverse section of the channel ch a , and by making the transition regions 1 L and 2 L more and more thin. Thus, even though the impedance contrast between the free-space region and the ENZ material is infinite the wave may effectively tunnel through the narrow channel with high transmissivity. Note that the previous discussion holds independently of the electrical size of the channel, i.e. of the value of 0 k L .
In practice, in a realistic physical system, such an effect is limited by finite losses in the ENZ material and/or by dielectric breakdown. In Fig. 2  simulation the effect of losses in the metallic walls was neglected and will be discussed later in the paper. The results of Fig. 2  ) the transmission can also be greatly enhanced due to the excitation of 'quasi-static' localized resonances (local plasmon resonance) characteristic of systems with objects with negative permittivity [15]. These resonances are further analyzed in Appendix A.
-10- It is also interesting to characterize the behavior of the ENZ material in a waveguide configuration with a 90º-bend. The geometry is illustrated in the upper panel of Fig. 3.
The incoming wave propagates along the negative y-direction. The distance between the parallel metallic plates is a (in both the input and output waveguides). The transition between the two waveguides is filled with two thin perpendicular (and connected) ENZ   Let in Z′ and L Z′ be the transverse wave impedances at the input and output planes ( 0 x = and 1 2 x L L L = + + , respectively), as shown in Fig. 4. Note that in the 0 ε = limit these impedances can be defined unambiguously because the wave in the air regions is a superimposition of TEM waves [9] (in particular, 0 L Z η ′ = ). It is also evident that in order that the incoming wave can tunnel through the ENZ material it is necessary that -13-  The fact that the transmission may be improved when the transverse cross-section of channel is made tighter, can also be explained using the theory and concepts proposed in our previous work [10], where we showed that by loading a ENZ material with metallic or dielectric inclusions it is possible to tailor its magnetic permeability. In fact, for TEM wave incidence and assuming 1 Another fundamental property of ENZ materials that gives a different perspective of the tunneling phenomenon is discussed next. As mentioned before, for the 2D problem under study, inside the ENZ material. This result has a very important implication: since the electric current density over the metallic plates is where ν is the unit normal vector (directed to the ENZ region), it follows that where ˆˆˆz = × t ν u is the vector tangent to the metallic surface. Thus, the amplitude of the current density is constant, and independent of the specific shape of the plates. Hence, the ENZ material forces the current to follow the path defined by the shape of the metallic plates, and thus the current injected into each PEC plate (left-hand side interface; see Fig. 1), appears undisturbed on the right-hand side interface. That is, the ENZ material,

B. Field concentration and confinement in a subwavelength cavity
As pointed out in [9], when an electromagnetic wave is squeezed through an ENZ channel the electric field inside the ENZ material may be greatly enhanced. This phenomenon is easy to understand using the principle of conservation of energy. In fact, in the lossless limit, the flux of the real part of the Poynting vector through an arbitrary transverse cross-section of the channel must be invariant. Thus, since the magnetic field is constant inside the ENZ material, this requires that the electric field inside the channel is roughly inversely proportional to the height of the channel. This property is also consistent with the discussion of section II.A, where we showed that for the U-shaped geometry the transverse wave impedances at the 0 Fig. 4). This result implies that the electric field y E must be such that . In order to illustrate this property, we computed the amplitude of the electric field inside the ENZ channel using CST Microwave Studio TM [14]. The parameters characteristic of the U-   The results of Fig. 6 demonstrate that the electric field may be significantly concentrated inside the air cavity (and also in the ENZ material). It is seen that the electric field inside the cavity is about 9 times (5 times) larger than the incoming field for / 0.001 , respectively (these values are only marginally smaller than those reported in Fig. 5 for the case where the transition is completely filled with the ENZ material). It is also worth noting that the magnetic field is nearly constant inside the two sections of the ENZ material for the case / 0.001 p ω Γ = , consistently with the theory of [9]. Inside the air cavity the magnetic field may vary appreciably, and to a good approximation is a linear function of x. Fig. 5 also shows that when the losses become moderate , the magnetic field is not anymore uniform inside the ENZ material.
In the 0 ε ≈ lossless limit it is possible to calculate the electric field inside the air cavity in closed analytical form. The approach is similar to that of [9] and so the details are omitted here. We only mention that in the 0 ε ≈ lossless limit the field inside the air cavity is a superimposition of TEM waves, and that the magnetic field inside each ENZ region is constant. Detailed calculations show that the electric field evaluated at the righthand side interface of the air cavity with the ENZ material is given by, Thus, provided the cavity is electrically small, 0 1 k d , the electric field inside the cavity is enhanced by a factor of / ch a a as compared to the amplitude of the incoming wave, -20-consistently with the results of Fig. 6. These results confirm that ENZ materials may have interesting potentials in concentrating the electric field in a small very subwavelength cavity.

C. Effect of losses in the metallic walls
In the previous examples it was assumed that the metallic walls were perfect electric conductors. However all metals have losses, and these can be significant at terahertz, infrared and higher frequencies. Thus, it is crucial to evaluate the inevitable effect of metallic losses in the proposed tunneling mechanism. In what follows, we generalize the theory introduced in our previous work [9], so that it can describe the effect of the finite conductivity of the metallic walls.
Here, we consider that the metallic walls are characterized by the bulk conductivity σ . Thus, the effective permittivity of the metal is of the form: η is the free-space impedance. As proved next, within such an approximation it is possible to calculate in closed analytical form the reflection and transmission coefficients in the 0 ε = lossless limit, and for a generic waveguide scenario.
To begin with, consider a generic waveguide configuration as in [9] where an ENZ filled channel is illuminated by the fundamental TEM waveguide mode (the geometry of Fig. 1 corresponds to the particular case in which the ENZ transition has the U-shape). As proved in [9], in the 0 where p A is the cross-sectional area of the ENZ channel, and the first line integral is over the interfaces with air ( 0 x = and 0 x′ = in Fig. 1) whereas the second line integral is over the metallic walls. Using the Leontovitch boundary condition (6)  , where ρ is the reflection coefficient for the magnetic field. Substituting these results in (7) we readily obtain that: The transmission coefficient is given by 1 T ρ = + . The above formula is the generalization of (1) to the case where the metallic walls have finite conductivity. We underline that the formula is completely general (valid for channels with nearly arbitrary geometry), and only assumes the conditions implicit in the Leontovitch boundary condition approximation and that 0 ε = in the ENZ material. Equation (8)

D. Group velocity
It is interesting to analyze the group velocity for a packet of electromagnetic waves that tunnels through an ENZ filled channel. We consider for simplicity that the dielectric permittivity ε of the ENZ material is characterized by a Drude-type dispersion model, represents the group velocity of a packet of waves that propagates in an unbounded ENZ material, and thus this formula does not necessarily apply for propagation in a finite thickness ENZ sample, where significant coupling between the two interfaces may effectively permit that a packet of waves is squeezed through the ENZ channel.
In fact, consider again the geometry of Fig. 1 where the TEM fundamental waveguide mode illuminates a U-shaped ENZ channel. In general, the transmitted wave (after it emerges from the ENZ transition) may consist of a superposition of waveguide modes.
However, as proved in [9], when 0 ε = no evanescent modes are excited in the region 0 x′ > , and thus it is a good approximation to consider that in the ENZ limit the Thus, the envelope of the incoming field is reproduced at the output plane with a delay of where 0 0 0 k ω ε μ = is the free-space wave number. We will use the above formula to calculate v g inside the ENZ channel. Again, we underline that (11) is not equivalent to the result v g c ε = , which represents the group velocity inside an unbounded medium, and thus, in general, does not characterize v g in the ENZ channel.
In order to estimate the group velocity for a U-shaped channel (Fig. 1), we computed the transmission coefficient ( ) T ω using a full wave electromagnetic simulator [14], and then we evaluated v g using (11). In the simulations, it was assumed that 1

E. Plasma simulation at microwaves
The simulations and results presented in the previous subsections consider that the problem under study is intrinsically two-dimensional (2D), and that the ENZ material is readily available in nature. In what follows, we prove that it is possible to emulate such propagation scenarios at microwaves, and reproduce the supercoupling, field concentration, and tunneling effects using a realistic three-dimensional (3D) setup that only requires using standard dielectrics and metals with an empty section!
The main idea is to simulate the 2D artificial plasma using two parallel metallic plates coplanar with the electric field and with the plane of propagation (i.e. parallel to the x-y plane). It is known [19,20,21] that such configuration may mimic the behavior of a plasmonic material with the Drude-type dispersion as ( ) the relative permittivity of the dielectric between the plates, and s is the distance between the plates. Such concepts were used in our previous work [10] to emulate a medium with ε and μ simultaneously near zero at microwaves. Here, we use similar concepts to demonstrate the supercoupling, field confinement, and tunneling effect under study here at microwaves.
In the first example, we analyze a closed 3D metallic waveguide environment. We consider that the cross-section of the waveguide (contained in the x-y plane) is as shown in Fig. 1, i.e. the metallic walls have the U-shape. In order to simulate an artificial plasma -29-two parallel-plates are inserted at the planes 0 z = and z s = . As discussed in [10], the 3D metallic waveguide is filled with two different dielectrics, so that the artificial plasma may emulate (at the design frequency) a dielectric with , 1.0 eff air ε = in the "free-space Fig. 1), and an ENZ material with  (calculated using CST Microwave Studio TM [14]).
-32-It is supposed that the microstrip conductor has an abrupt transition shaped as an "U" (Fig. 10). The transverse cross-section of the line (relative to the z-direction) has geometry similar to that of Fig. 1. As in previous simulations, we choose 1 2 0.1 ch L L a a = = = and 1.0 L a = . The corresponding S-parameters calculated using the full wave simulator [14] are depicted in Fig. 10 (red lines). It is seen that consistently with the results obtained for the geometry of Fig. 1, the wave may tunnel through the narrow channel around the plasma frequency. As described next, an artificial plasma realization of the ENZ material may also enable a similar tunneling phenomenon. As in the first example of this subsection, the simulated plasma is designed using a parallel-plate configuration. In this way, the microstrip conductor is short-circuited to the ground plane through two parallel metallic plates (normal to the z-direction) placed at the U-shaped transition. The metallic plates are located at the planes / 2 z W = − and / 2 z W = + , supposing that the conducting strip is defined by the region  Fig. 10 (blue lines). The results demonstrate that consistently with our theory the wave may completely tunnel through the U-shaped artificial plasma transition. It is worth noting that the results for the artificial plasma realization are even slightly better than those obtained for the ideal ENZ material, perhaps because the effect of losses in the simulated plasma is negligible. These results confirm that the described supercoupling tunneling effect may, in fact, be demonstrated at microwave frequencies using a realistic 3D setup, which only involves standard dielectric and conducting materials with arguably lower material losses.

III. Supercoupling and Squeezing Energy through ENZ Anisotropic Channels
As discussed in the Introduction and in [9], even though ENZ materials may readily available in nature at certain specific frequencies of the electromagnetic spectrum, in the general case we may need to synthesize them as microstructured materials (metamaterials). Since anisotropic materials may be easier to fabricate as compared to isotropic materials, in this section we investigate the potentials of anisotropic materials with permittivity near zero. In addition, we discuss the realization of these artificial materials using wire media. -35-

A. Characterization of ENZ Anisotropic Channels
In this subsection, we extend the theory developed in [9] to the case in which the ENZ material is anisotropic. We will prove that when the metallic channel is filled with an anisotropic material such that the permittivity component normal to the interface is near zero, then, in certain conditions, the scattering problem may be solved in closed analytical form. Moreover, we will show that for certain geometries the scattering parameters may be made independent of the specific geometry of the channel, in analogy with the results obtained in [9]. The geometry of the problem is depicted in panel (a) of Fig. 11. Two parallel-plate metallic waveguides are interfaced by an abrupt transition filled with an ENZ anisotropic material. We suppose that the structure is invariant to translations along the z-direction and that / 0 z ∂ ∂ = , so that the problem is effectively two-dimensional. In the region By solving the differential equation (which in general depends on the specific geometry of the problem) we can obtain ( ) z z H H x = inside the ENZ anisotropic region, and afterwards using (17) the unknown scattering parameters ρ and T . Hence, using (19) it is found that for (independent of its length) without affecting the transmission efficiency. Note that the profile ( ) u y y x = is completely arbitrary and may have very abrupt or curvy variations.
This result confirms that ENZ anisotropic materials may have interesting potentials in improving the transmission characteristic along 'bendy' channels. In order to verify the proposed results, we used CST Microwave Studio™ [14] to calculate the S-parameters of the structure for the particular geometry shown in the inset of Fig. 12 where and slabs of a material with negative permittivity (e.g. semiconductor or a noble metal). -42-

B. Waveguides with Sharp Walls
An interesting geometry that caught our attention is depicted in panel (b) of Fig. 11. It consists of an abrupt transition with thickness d and two 90º-bends. In what follows, we study the effect of filling such waveguide transition with an ENZ anisotropic material. To begin with, it is important to point out that Eq. (21) is not valid for the geometry of Fig.   11b. In fact, (21) was derived under the hypothesis that ( ) ( ) , and thus at these points the second term of (19) has a singularity. Hence, to obtain the correct limit solution one needs to proceed with special care.
To this end, instead of considering sharp walls with an abrupt transition, we will suppose that the profile of the metallic plates varies continuously in some transition region. To be specific consider the interface 0 x = in Fig. 11b, where the upper metallic plate has an abrupt bend. In order to avoid the singularity in the second term of (19), we suppose that the profile of the upper metallic plate, ( ) u y x , varies linearly from 1 a to ch a over some transition region with thickness δ , as illustrated in the inset of Fig. 11b. Notice that the thickness δ may be arbitrarily small. In what follows we will obtain the solution of (19) in the limit 0 δ + → .
In the thin transition region 0 x δ < < (filled with ENZ material), (19) can be rewritten as We are now ready to calculate the scattering parameters for this propagation problem. As in the previous section, it is clear the magnetic field in the air region is given by (17), whereas the magnetic field in the ENZ region 0 x d + − < < is given by (20). To compute channel. The latter condition is precisely the one derived in our previous work [9] for propagation across isotropic ENZ channels, while the former condition is easily satisfied for standard material values ( 1 yy ε ∼ , , 1 r p μ = ) and for long channels ( ch a a ). Thus, we conclude that for the geometry of panel b) of Fig. 11 and for typical material parameters, the transmissivity of a narrow long channel is approximately independent of the channel being filled with an isotropic ENZ material or an anisotropic ENZ material (with 0 xx ε = ). In particular, it may be possible to squeeze more and more energy through the narrow channel provided the thickness d is made increasingly small. In this regime, the ENZ channel supports a zero-order resonance, consistently with the discussion of section II.A. This result confirms that, in fact, in some propagation scenarios ENZ anisotropic materials may have potentials similar to those of ENZ isotropic materials. (The results of [9] also support such conclusions for a geometry analogous to that of Fig. 11b, but with an 180º bend. The formula for the reflection coefficient presented in [9] can be derived using arguments similar to those employed here).
In order to illustrate some of the suggested potentials, we have used CST Microwave Studio TM [14] to characterize a very narrow waveguide transition ( Fig. 11b)   As discussed in the beginning of this section, ENZ anisotropic materials may be synthesized as metamaterials. As in [9], next we explore the possibility of using wire -47-media [22]- [23] to simulate a continuous ENZ anisotropic material. To this end, we consider that an array of PEC wires is placed inside the ENZ transition. The wires stand in air and are centered at the points ( ) . The S21-parameter dispersion calculated for such metamaterial is depicted in Fig. 13 (solid blue line). It is seen that as predicted by our theory, the transmission is significantly enhanced near  Fig. 13). The transmission characteristic for a channel filled with such metamaterial is shown in Fig. 13 (dashed blue line). As seen, the peak of transmission is moved to higher frequencies, consistently to what one would expect from the previous analysis. As in the case 0.005 R a = , for 0.01 R a = the peak of transmission is also slightly downshifted with respect to the design frequency. The reason for this property may be related with the discreteness of the metamaterial design (note that we use only 8 rows of wires), and probably also to the fact that the wires corresponding to indices 0 m = and 7 m = are not able to emulate the ENZ property since one of their extremities is unconnected.
In conclusion, we have demonstrated theoretically and numerically that ENZ anisotropic materials may have common features with ENZ isotropic materials in some propagation scenarios. The condition 0 xx ε = effectively freezes the wavefronts inside the anisotropic ENZ material along the y-direction (i.e. the wavefronts are normal to the x-direction). It was discussed that for some geometries the transmission along anisotropic ENZ channels may be independent of the specific geometry of channel, and that a zero-order resonance may enable a tunneling effect similar to that reported in [9]. In addition, we have further discussed the possible realization of ENZ metamaterials using wire media, and enlightened some key features of such metamaterial implementations.

IV. Conclusions
In this work, we have further investigated the applications and properties of the supercoupling, field confinement, and tunneling phenomenon identified in [9]. It was shown that waveguide channels filled with ENZ materials support a zero-order -49-resonance, which may enable anomalous transmission and supercoupling of energy through very narrow channels or abrupt bends, independent of their specific geometry.
We demonstrated that the root of the tunneling effect is the property that the current injected in a metallic plate remains constant inside the ENZ channel. It was analytically proved that the effect of metallic losses on the waveguide walls may be of second order, and thus that in principle the dielectric loss in the ENZ material is the dominant loss mechanism. It was suggested that ENZ materials may enable electric field concentration and confinement in a subwavelength air cavity. Two different realistic 3D configurations based on the concept of simulated plasma were proposed to emulate the studied propagation scenarios at microwaves. In addition, we developed a theory to describe the propagation of electromagnetic waves through anisotropic ENZ materials. It was proved that in some very generic propagation scenarios the scattering coefficients can be calculated in closed analytical form. Our analysis demonstrates that in some circumstances the scattering coefficients may be made independent of the specific geometry of channel. Furthermore, in some cases, ENZ anisotropic channels may also enable the anomalous transmission of energy through very narrow metallic channels, independent of the specific dimensions of the channel (zero-order resonance). Finally, we discussed some key properties of anisotropic ENZ metamaterials made of an array of microstructured wires.