Total transmission and super reflection realized by anisotropic zero-index materials

We demonstrate theoretically and experimentally total transmission and super reflection which are realized by anisotropic zero-index materials (AZIMs). We show that total transmission will be observed when using a rectangular perfectly magnetic conductor (PMC)-coated object sandwiched by two AZIM slabs, which has the properties of μrx = 0 (in the normal direction to the wavefronts) and μry = εrz = 1 for transverse-electric polarized incident waves. When the object is coated with a perfectly electric conductor (PEC), however, the incident waves will be totally reflected by the finite-sized object in the way of an infinite PEC plane, generating a super reflection. Closed-form formulas are derived to explain the physical mechanisms of total transmission and super reflection, which agree with the full-wave numerical simulations perfectly. Experimental samples of AZIM and PMC are designed, fabricated and measured in the microwave frequency, which show good transparent and super-reflecting effects.


Introduction
Metamaterials have the ability to control the propagation of lights or waves; a lot of fantastic devices have been designed based on the knowledge of the negative refractive index [1], transformation optics [2][3][4] and conformal mapping [5][6][7][8][9], etc. Besides the above devices realized by negative or positive refractive-index materials, zero-index materials (ZIMs), whose permittivity and (or) permeability equal(s) zero, are another hot research topic. In theory, there has been a lot of work based on ZIMs: enhancing the directivity of the embedded source [10,11], tunnelling the electromagnetic waves through a narrow channel [12][13][14][15][16], realizing total transmission and reflection [17][18][19][20] and tailoring the radiation phase patterns [21]. Particularly, it was predicted theoretically [17] that an object coated with a perfectly magnetic conductor (PMC) or perfectly electric conductor (PEC) in a waveguide can conceal or block the in-coming waves completely when the object is surrounded by isotropic ZIMs, with both the permittivity and permeability being zero. However, low-loss and well-matching ZIMs are not easy to realize and hence only a few experiments on ZIMs have been reported, e.g., the realization of tunnelling effects [13,16] and Dirac cones [22]. More recently, anisotropic zeroindex materials (AZIMs) have been presented to generate high-directivity radiation [23][24][25][26], spatial-power combinations [27] and low-loss wave bending [28], in which only one component of the permittivity or permeability is zero. Since it is easier for AZIMs to make impedance matching, more experiments have been conducted [24][25][26][27] on AZIMs, and some of them have the very good performances needed to reach practical applications [25,26].
In this paper, we propose an AZIM-based transparent and super-reflecting scheme for objects in a waveguide system and make experimental verifications. We show theoretically that total transmission will be observed when a rectangular PMC-coated object in a waveguide is sandwiched by two AZIM slabs, while the same object coated with a PEC will generate super reflection under the incidence of transverse-electric (TE) polarized waves. We fabricate the desired AZIM and PMC using split ring resonators (SRRs) with metal patches in the microwave frequency, the electric-field distributions of experimental samples are measured using a twodimensional (2D) near-field scanning apparatus, all of which are in good agreement with the theoretical calculations and numerical simulations.

Theoretical design
Consider a 2D waveguide with PMC boundaries, which consists of six regions (0-5), as illustrated in figure 1. The semi-infinite regions 0 and 5 are air, serving as the input and output ports. Two AZIM slabs (regions 1 and 4) form an empty space, in which a rectangular object (region 3, coated with either PMC or PEC) is embedded with surrounding air (region 2). A TEpolarized electromagnetic wave is incident from region 0 in the +x direction. The proposed AZIM has the property of µ r x = 0 (in the normal direction to the wavefronts, i.e. the +x direction), µ r y = 1 and ε r z = 1. The detailed dimensions of the model are shown in figure 1, in which d 0 < d 1 < d 1 + d < d 2 and h 2 < h 1 . In the following discussions, we will show that the two AZIM slabs make the PMC-coated object have total transparency or the PEC-coated object block the incident waves completely (i.e. making the incident waves have total reflection).
The dispersion relation of AZIMs in regions 1 and 4 can be easily derived from the Maxwell equations of the 2D TE-polarized version, which is written as If µ r x = 0, then k y must be zero, and the wave number in the x direction will be k x = k 0 under the conditions of ε r z = 1 and µ r y = 1. As a consequence, the waves in such AZIMs only propagate along the x direction and have invariant electric fields along the y-axis. Assume that the electric field of the incident TE-polarized plane wave (see figure 1) is given by where E i and x s represent the magnitude and zero-phase surface of the incident electric field, respectively, and k 0 is the wave number in free space. The wave impedance along the x-axis (i.e. propagating direction) is defined as µ r y /ε r z . Then the impedance of regions 1 and 4 will be perfectly matched to those in free space (regions 0, 2 and 5) when ε r z = 1 and µ r y = 1. However, the incident waves will be scattered when they illuminate the object (region 3). From the above analysis, the AZIM slabs will force the scattered waves to propagate only along the x direction. Hence the whole system can be treated as a problem of multilayer transmissions and reflections. The electromagnetic fields in all regions can be written as Since H x = 0 in all regions, there is only the H y component: in which R and T are the reflection and transmission coefficients of the electric field and A and B are the total summations of the transmission and reflection coefficients in region 2.
If the object is coated with a PMC, the boundary condition Applying the Ampere-Maxwell law H · dl = ∫ ∂ D ∂t · ds in regions 1 and 4, we have From equations (9)-(12), the reflection coefficient R is derived as Once R is known, the other parameters T, A and B will be achieved. According to equation (13), the reflection coefficient R will be zero when d satisfies the following relation: The above equation shows that the incident waves will be totally transmitted when the length (d) of the object along the x direction is an integer times half a wavelength, which is independent of the object height (h 1 − h 2 ) along the y direction.
We calculate the transmission (T) and reflection (R) coefficients using equations (12) and (13), as shown in figure 2(a), in which we choose h 1 = 2h 2 . We notice that |R| 2 + |T | 2 is always equal to 1, implying that the summation of the reflected and transmitted powers equals the incident power. The calculated results show the total transmissions (T = 1) when d satisfies equation (14). We also observe the maximum reflection (|R| 2 = 0.36) and the minimum transmission (|T | 2 = 0.64) when the length (d) of the object along the x direction is an odd multiple of a quarter wavelength. Figures 2(b) and (c) illustrate the full-wave simulation results of electric fields inside the waveguide at 10 and 15 GHz, respectively, in which  In order to show the transparent ability for thin or lossy AZIM slabs, four different cases have been further considered: thick lossless slabs, thick lossy slabs, thin lossless slabs and thin lossy slabs. The thickness of the slabs is expressed as We suppose that d s = 5 mm (the same as in figure 2) represents thick slabs and d s = 0.1 mm represents thin slabs. The loss tangent of the permittivity for lossy slabs is set as 0.2 in our simulations, i.e., ε z = 1 + i0.2. Figure 3 illustrates the simulation results of electric fields along the x direction with y = 20 mm at 10 GHz for all four cases above, which shows that the incident waves can be totally transmitted for thick and thin lossless slabs and nearly totally transmitted for thin lossy slabs. However, for thick lossy slabs, there will be some reflection and the associated transmission is about 80%. Hence, thin AZIM slabs can be chosen for designing the transparent system, because the incident waves are nearly totally transmitted even for thin lossy slabs.
When the object is coated with PEC, the boundary condition ensures that the electric fields are zero at x = d 1 and d 1 + d. From equation (3), we then obtain the following relation: Clearly when |R| ≡ 1 total reflection always occurs. Figure 4 shows the full-wave simulation results of the electric fields inside the waveguide at 10 GHz. We observe that all incident waves are totally reflected by the finite-size PEC object (h 1 − h 2 = 30 mm) with the help of two AZIM slabs, which only happens to the full-size PEC object (h 1 − h 2 = 60 mm) if AZIM slabs are not used. Hence AZIM makes the small PEC object a super reflector.

Experimental fabrication and measurement
To demonstrate the total transmission and super reflection experimentally, we have designed the required PMC using artificial magnetic conductors (AMCs) [29] and AZIM using metamaterials in the microwave frequency. To make the experiments easier, we also occupy region 2 by AZIM, which has been proven to reach the same transparency and super reflecting effects. Hence, in our experiments, the background material used is AZIM with the AMC boundaries, which has the property of µ r x = 0, and a square object is placed in the AZIM, as shown in figure 5(a).  We realize AZIM using conventional split ring resonators (SRRs), as illustrated in figure 5(b), whose substrate is a commercially printed circuit board (PCB), F4B, with a thickness of 0.5 mm, relative permittivity of 2.65 and loss tangent of 0.001. The SRR dimensions are given in figure 5(d), in which a = 3 mm, b = 2.3 mm, w = 0.2 mm, g = 0.2 mm and l s = 0.95 mm. The effective permeability of SRR is also given in figure 5(d), showing that µ r x is nearly zero (µ r x = 0.021 + i0.006) around 10 GHz (from 9.6 to 10.4 GHz). Correspondingly, the relative permittivity ε r z and permeability µ r y of SRR are 3 and 1 at 10 GHz. The other important part, AMC, is designed using periodic metal patches, which are also fabricated on F4B with a thickness of 0.25 mm, as shown in figure 5(c). The dimensions of the square AMC unit are given in figure 5(e), which include two layers of F4Bs with a distance d 2 = 3 mm and periodicity a 1 = 3 mm. The bottom layer is etched by copper completely, while the top layer is printed with a copper square with the thickness of 0.018 mm and side length b 1 = 2.65 mm. The excitation port is set d 1 = 10 mm away from the top layer, as demonstrated in figure 5(e). According to the boundary condition, when the plane waves are incident on the PEC, the phases of the incident electric field 1 and reflected electric field 2 must satisfy | 1 − 2 | = π to guarantee a zero tangent electric field on the PEC. Similarly, if a surface makes the phases satisfy | 1 − 2 | = 0, then it can be treated as a PMC based on the duality principle. The phase of S11 of the designed AMC is shown in figure 5(e), from which we observe that the phase is zero at 10 GHz, simulating a PMC.
The measurement results of the transparency and super-reflection effects are illustrated in figure 6 when the object is covered by PMC and PEC. From figure 6(a), we observe that the incident electromagnetic waves can pass the PMC-coated object through the AZIM channels and return to their original paths, showing total transmission. When the object is coated by PEC, however, nearly total reflections of the incident waves are observed, as shown in figure 6(b), showing the super-reflection ability of AZIM. Although small distortions for both the transmitted and reflected waves exist, the measured results show good transparency and super-reflection effects as per the theoretical prediction. Such distortions come from the imperfect zero permeability of µ r x and the relatively large component of ε r z , which makes the impedance not well matched. In addition, the waveguides with AMC boundaries in our experiment are filled with AZIM, as shown in figure 5(a), but the simulations are based on the free space background, as shown in figures 5(d) and (e). Hence, in order to verify if the AMC is influenced by the filled AZIM or not, some additional simulations have been conducted, as illustrated in figure 7. It is known that, when the TEM-mode incident waves illuminate the PMC, the phase shift between the incident and reflected waves is zero, i.e., | 1 − 2 | = 0, and the tangential magnetic field on the PMC vanishes. This can be directly observed from figure 5(e) (zero phase shift at 10 GHz) and figure 7(a) (zero tangential magnetic field). Figure 7(b) gives the three-dimensional (3D) magnetic-field distributions for the designed AMC structure at 10 GHz, which clearly show that the tangential magnetic field on the surface of the AMC is zero, similar to that of PMC. According to figures 5(e), 7(a) and (b), we conclude that the AMC we designed actually behaves as a PMC coating with the vertical incident waves in the free-space background at 10 GHz. However, when the incident angle and dielectric environment are changed, the response frequency of the AMC will be shifted, as shown in figure 7(c). Suppose that the angle between the incident waves and the x-axis is θ , we then notice that the zero-reflected phase has been shifted from 10 to 11 GHz when the incident angle increases from 0 • to 45 • , as illustrated in figure 7(c). When the relative permittivity of the background dielectric environment is increased from 1 (free space) to 3, the zero-reflected phase shifts from 10 to 8.3 GHz. The phases of S11 reflected by AMC with different incident wave angles and dielectric environments.

Conclusions
In conclusion, we have proposed, realized and experimentally demonstrated the transparency and super-reflection effects of AZIM. In our design, the SRRs are utilized to design the AZIM and metal patches are utilized to design the PMC boundaries. The fabricated prototypes have been measured by 2D near-field scanning apparatus; the measurement results are in good agreement with the theoretical predictions and numerical simulations.