## Abstract

We present a nonmagnetic electromagnetic transparent wall (EMTW) using the principle of total transmission and phase compensation. The device consists of two or more nonmagnetic stacked anisotropic slabs. With proper design of the constitutive tensors and relative thicknesses of each slab, EMTW is achieved which is independent of the incident angle of striking EM waves. The realization of the anisotropic slabs and furthermore EMTW in the optical range is mimicked using a metal-dielectric nano-structured system with alternating Na_{3}AlF_{6}-Ag layers. Compared to the magnetic version, the new design makes a major step forward and provides a practical path to experimental demonstration of EMTW. The proposed structure has potential applications in the antireflection coatings, microwave absorbing materials, and high-performance radomes.

©2012 Optical Society of America

## 1. Introduction

Metamaterials have sparked great attention recently due to their potentials in manipulating electromagnetic (EM) waves according to human’s expectation [1–4]. Various structures have been proposed to engineer such materials, including SRRs [5], ELCs [6], fishnet structures [7], and so on, among which one structure stands out conspicuously because of its simple explanation and easy fabrication. The structure is realized based on the effective medium theory. In details, an anisotropic material with the demanded permittivity can be effectively implemented by a layered structure of thin, alternating metal and dielectric slabs. These artificially engineered anisotropic materials have applications in many areas, such as the negative refraction [8], subwavelength imaging [9], optical hyperlenses [10,11], and carpet cloaks [12]. The structure is extremely useful for the applications at the optical spectrum due to the easy fabrication [8,13]. The oblique layered system can also serve as a universal element to build a variety of interesting functional components, including wave splitters, wave combiners, reflectionless field rotators [14], and one-dimensional (1D) cloaking devices [15]. The readers are kindly referred to some review papers for a broader perspective on metamaterials [16–18].

One of the metamaterial applications is the electromagnetic transparent wall (EMTW), a 1D structure with special material parameters. When incident waves strike the structure, EMTW behaves like a vacuum with the same thickness [19]. Two methods, one of which is based on the reflection and transmission coefficients and the other on transformation optics (TO), have been given to design the structure, which give rise to the same results [19]. Like other devices designed by TO, however, EMTW can only be realized using magnetic and anisotropic metamaterials, which is difficult for realization, particularly in the optical region.

In this paper, we propose a new EMTW design with two (or more) nonmagnetic anisotropic slabs. By properly tuning the material parameters and optical axes of the slabs [20,21], the requirement for magnetic materials can be totally alleviated. As an example for possible realization, the anisotropic slab is implemented by multi-layered structure of alternating nonmagnetic materials, one of which is silver and the other is Na_{3}AlF_{6}. Full-wave electromagnetic simulations based on the finite element method (FEM) clearly confirm the new design.

## 2. Theoretical design of EMTW

#### 2.1 Dispersion equation in anisotropic media

The basic idea of nonmagnetic EMTW is to analyze total transmissions through an anisotropic slab, which is shown in Fig. 1(a)
. As an example, a transverse magnetic (TM) wave is used, whose magnetic field is perpendicular to the *x*-*y* plane. Similar results can be obtained for transverse electric (TE) waves. For this polarization, the material parameters can be expressed as

*x*(

*y*) direction. Note that a time dependence of the form $\mathrm{exp}(\text{j}\omega t)$ is assumed for the electromagnetic field quantities. In our previous design [19], $\overline{\overline{\epsilon}}=\Lambda ({\epsilon}_{u},{\epsilon}_{\nu},{\epsilon}_{w})$ and $\mu ={\mu}_{w}$, so the optical axes of the anisotropic material slab coincide with the coordinate system, as is shown in the left part of Fig. 1(b). And imposing the transparency condition will force the material to be magnetic, i.e., ${\mu}_{33}\ne 1$, and hence a big challenge for practical realization. To remove the constraints, the optical axes are carefully rotated by an angle $\theta $, which is illustrated in the right part of Fig. 1(b). Therefore we have

#### 2.2 Conditions for total transmission in anisotropic slabs

Consider an infinite anisotropic slab with thickness *d* and parameters given in Eq. (1), as shown in Fig. 1(a), where *x* = 0 and *L* are two surfaces of the slab. A TM wave strikes the surface at *x* = 0 with an angle $\alpha $. Assume the magnetic fields in the three regions can be expressed as

*x*= 0 and

*x*=

*L*, respectively, we have

If we demand the TM wave be fully transmitted through the slab without any reflection and independent of the incidence angle, then $r=0$ in Eq. (7a) must be assured, thus$R=S$. Notice the continuity of the tangential wave number, ${k}_{x1}={k}_{0}\mathrm{cos}\alpha $and${k}_{y2}={k}_{0}\mathrm{sin}\alpha $, and substitute these equations into Eq. (4), we obtain the conditions for total transmission, i.e.,

When the rotation angle is zero, Eq. (8) simply becomes ${\epsilon}_{u}{\epsilon}_{v}=1;{\epsilon}_{u}{\mu}_{33}=1$, which is exactly the same as that in Ref [19]. Using Eq. (3), we have${\epsilon}_{11}={\mathrm{cos}}^{2}\theta {\epsilon}_{u}+{\mathrm{sin}}^{2}\theta {\epsilon}_{v}$, therefore

This equation is used to determine the rotation angle of the optical axes. Assume that ${\epsilon}_{u}=K$, ${\epsilon}_{v}=1/K$ and ${\mu}_{33}=1$for nonmagnetic materials, we havewhere*K*should be a positive number.

#### 2.3 Phase compensation

By properly designing the constitutive tensors of the slab, total transmission of the TM polarization wave is achieved. However, for the EMTW not to be detected from the outside, amplitudes and phases of the incident and transmitted waves should be completely the same with those in the background medium, which is not true at the current step. In this regard, we use stacked anisotropic slabs to construct a 1D EMTW by rendering the total phase lag equal to that of the background medium (vacuum).

For the layered structure shown in Fig. 2
, the thicknesses of the *N* anisotropic slabs are *d _{1}*,

*d*, …,

_{2}*d*, and the gaps between them are filled with air with thickness

_{N}*d→*0. Since ${\epsilon}_{0}={\mu}_{0}=1$ in free space and ${\mu}_{33}=1$for nonmagnetic materials, we have for each slab ${\epsilon}_{11}={\mathrm{cos}}^{2}{\theta}_{i}{\epsilon}_{ui}+{\mathrm{sin}}^{2}{\theta}_{i}{\epsilon}_{vi}=1$, ${\epsilon}_{ui}={K}_{i}$ and ${\epsilon}_{vi}={1/K}_{i}$ under the total transmission condition. The total phase lag for the transmitted wave is

Equations (8)-(10) along with Eq. (13) give the design for a nonmagnetic electromagnetic transparent wall. When *N* = 2, Eq. (13) simply becomes

## 3. Realization of EMTW by layered structure of homogeneous and isotropic materials

In this section, we explore realization of EMTW using the effective medium theory. As has been pointed out in many works, a uniaxial anisotropic material can be effectively realized by a two-isotropic-material multilayer structure as long as the layer thickness is much less than the working wavelength [22,23]. Suppose the permittivity and permeability of the constitutive media are $({\epsilon}_{1},{\mu}_{1})$ and $({\epsilon}_{2},{\mu}_{2})$ with${\mu}_{1}={\mu}_{2}=1$, the effective material parameters can be expressed as (See Fig. 1)

When implementing the anisotropic slab using isotropic material layers, the above two equations should always be checked for practical reasons. In Table 1 , we give detailed requirements on the two isotropic materials, which can be used to implement the EMTW. The derivation process can be found in Appendix 1.

From Table 1, it is easily seen that there are six combinations for the possible realization of the needed permittivity. However, the last three cases can be obtained by simply exchanging the two materials, and this is understandable since they are equivalent. While for the first three cases, a careful examination will show that type I is more suitable for practical realizations, especially in the infrared and visible band, because in this case, positive and negative permittivities can be easily obtained using ordinary dielectrics and metals, respectively.

In Fig. 3 , we show the calculated parameters$\eta $,$K$and $\theta $ using type I combination, where the positive and negative permittivities are chosen to be the moderate value. This figure can help determine the two isotropic materials used in the design.

Each anisotropic slab can be engineered in the same way. Then Eq. (13) or (14) can be imposed for the phase compensation, which determines the relative thickness for each slab. If two anisotropic slabs are used, a very simple design can be obtained. In this scenario, the two anisotropic slabs are engineered with the same isotropic materials, which are aligned in two perpendicular directions, as shown in Fig. 4 . Therefore ${\epsilon}_{u2}={\epsilon}_{v1},{\epsilon}_{u1}={\epsilon}_{v2}$. As a result, ${\theta}_{1}+{\theta}_{2}=\pi /2$, ${d}_{1}={d}_{2}$, i.e., optical axes for the two slabs are mirror symmetric with each other and they have equal thickness. This simple configuration is demonstrated in Fig. 4.

By properly choosing different materials and the thicknesses of them, the desired material parameters can be easily obtained. For example, at$\lambda =365nm$, we have ${\epsilon}_{N{a}_{3}Al{F}_{6}}=1.8225$ and ${\epsilon}_{Ag}=-2.4012-0.2488i$ [24,25]. According to above equations, the effective anisotropic permittivities are

## 4. Simulation results and discussions

To verify the theoretical predictions, we numerically simulate the propagation behavior of a TM-polarized Gaussian beam with the full width at half maximum of 78.43nm passing through the structure. The simulation is carried out by FEM using the commercial software COMSOL MULTIPHYSICS 3.5. In the simulation, five cases are involved: a Gaussian beam propagating in the vacuum, passing through an ideal two-layer anisotropic slab, an EMTW realized by metal and dielectric layers without loss and with losses. For all cases, the working wavelength is 365nm and all work in the TM mode.

First, we consider a Gaussian beam with an incidence angle of $\alpha \approx {19.38}^{\xb0}$ propagating in the free space. Figure 5(a)
depicts the magnetic field distribution in this case (note the fine solid lines indicate the boundaries of the materials for comparison). Then, the space between *x* = 0.4 μm to 1.28 μm is filled with two anisotropic slabs, whose widths are both 0.44 μm. In the simulation, we set K1 = 5.8341, and K2 = 1/K1. Figure 5(b) illustrates the results for this situation. It is clearly shown that outside the slab, the difference between (a) and (b) is very small. Next, we use stacked Ag-Na_{3}AlF_{6} layers to replace the bulk materials. Figure 5(c) corresponds to this case. We see a similar field distribution as that in Fig. 5(b), the wave passes through the oblique layered system with no observable reflection at the left surface and exits the slab with similar patterns. Figure 5(d) and 5(e) demonstrate similar configurations with (c) but with the loss tangent of 0.01 and 0.1036. A comparison with the field distribution in (a) - (c) clearly shows that the exit field magnitude becomes smaller due to absorption and it can be hardly seen when the loss is large. However careful observations indicate the exit wave patterns remain similar for all cases. Moreover, the reflections at the left interface are negligible, as clearly shown in the figure, which may suggest practical applications. In Fig. 5(f)-5(h), we show the magnetic field distribution when the Gaussian beam is incident from a different direction, a case similar to (c). It is very clear that the transparency characteristics are maintained because our design is angle-independent. More simulation results can be found in Appendix 2.

We also compare the field distribution of the emergent waves in the above cases. A line segment at *x* = 1.35μm is selected, and the normalized magnetic fields on that cross-section-line are extracted. The results are given in Fig. 6
. It is obvious that the magnetic fields corresponding to cases (a) and (b) in Fig. 5, represented by the black solid curve and the red circles agree exactly. For the EMTW realized by alternating layered structures, the corresponding blue curve with plus signs agrees well with that in vacuum and through the anisotropic slab. The negligible difference is mainly caused by the theoretical approximation of the effective medium theory and numerical errors in the simulation. Since loss is an inevitable problem in practical applications, we also draw the curves for this scenario, which are represented by the green dotted line and the pink stars in Fig. 6. When the loss tangent is small (0.01 in the figure), the exit wave magnitude deteriorates slightly, and the normalized waveform remains similar to the ideal one. This property is valuable for the device’s real implementation. However, when the loss tangent increases to 0.1036, the difference between the real structure and the ideal case becomes apparent, and the emergent wave can be hardly seen, as shown in Fig. 5(e). Under this condition, the equivalent parameters using practical materials based on Eq. (15) differ greatly with the lossless case and the total transparency condition cannot be satisfied. We note that this does not deny the practical applications of the proposed structure since we can use this property for the design of wide angle absorbers, in which case bigger losses are welcomed (note the incident waves are not severely affected, as shown in Fig. 5). The corresponding design and experimental setup are now underway.

Finally, we discuss EMTW for TE mode. Based on the principle of duality, it is expected that the theory works for TE waves too. What we should do is to replace permittivity with permeability, electric field with magnetic field (adding a minus sign, to be more accurate), and vice versa. Then the whole process including the dispersion equation, total transmission condition, phase compensation, and transparency condition, etc. still holds. The only problem is that we need to implement the TE structure using magnetic materials, as anisotropic magnetic permeability comes into the design formula instead of electric permittivity. Though one can find magnetic materials and continue to parallel the same process, this is not the focus of the present paper (since we focus on nonmagnetic materials). We thus conclude that the proposed nonmagnetic EMTW only works for the TM mode. However, the different EM property for TE and TM mode does have its practical applications, e.g., the EMTW may be further fine-tuned to act as an optical polarizer because it works differently for TE and TM waves [26].

## 5. Conclusions

In summary, we have proposed a new EMTW design to realize total transparency using nonmagnetic anisotropic metamaterials. The possible realization of the structure using alternating metal and dielectric layers has also been given. The performance of optical EMTW constructed by two slabs with alternating silver and Na_{3}AlF_{6} multilayer has been confirmed by the full-wave simulations at the wavelength of 365nm. The structure is easily realizable using thin layers of ordinary materials or composites, and hence leads to a simple path to experimental demonstration of EMTW.

## Appendix 1

In this appendix, we give detailed derivation process for Table 1.

For Eqs. (16) and (17) to be satisfied, the following four cases are discussed according to values of ${\epsilon}_{1}$

- For ${\epsilon}_{2}<-1$ and ${\epsilon}_{1}>1$, it is easily seen that ${\epsilon}_{1}{\epsilon}_{2}+1<0$.
- We then have the following result from Eq. (17) ${\epsilon}_{1}+{\epsilon}_{2}<0$,
- As a result, ${\epsilon}_{2}<-{\epsilon}_{1}$.
- While for $0<{\epsilon}_{2}<1$, it is easily seen that Eq. (17) is satisfied.
- So, we get the first two columns in Table 1. I: $\{\begin{array}{c}{\epsilon}_{1}>1\\ {\epsilon}_{2}<-{\epsilon}_{1}\end{array}$, II: $\{\begin{array}{c}{\epsilon}_{1}>1\\ 0<{\epsilon}_{2}<1\end{array}$

- (2) If $0<{\epsilon}_{1}<1$, then from Eq. (16), we have $-1<{\epsilon}_{2}<0$ or ${\epsilon}_{2}>1$.
- For $-1<{\epsilon}_{2}<0$ and $0<{\epsilon}_{1}<1$, we have ${\epsilon}_{1}{\epsilon}_{2}+1>0$.
- Using Eq. (17), the following result is got,${\epsilon}_{1}+{\epsilon}_{2}>0$, i.e., ${\epsilon}_{2}>-{\epsilon}_{1}$.
- While for $0<{\epsilon}_{1}<1$ and ${\epsilon}_{2}>1$, it can be easily seen that Eq. (17) is satisfied.
- So, we get another two columns in Table 1.
- III: $\{\begin{array}{c}0<{\epsilon}_{1}<1\\ -{\epsilon}_{1}<{\epsilon}_{2}<0\end{array}$, V: $\{\begin{array}{c}0<{\epsilon}_{1}<1\\ {\epsilon}_{2}>1\end{array}$

- (3) If $-1<{\epsilon}_{1}<0$, then from Eq. (16), we have${\epsilon}_{2}<-1$ or $0<{\epsilon}_{2}<1$.
- For ${\epsilon}_{2}<-1$ and $-1<{\epsilon}_{1}<0$, we have ${\epsilon}_{1}{\epsilon}_{2}+1>0$.
- Then, the above process requires that
- $\{\begin{array}{c}-1<{\epsilon}_{1}<0\\ {\epsilon}_{2}>-{\epsilon}_{1}\end{array}$,
- which cannot be satisfied.
- Wile for$-1<{\epsilon}_{1}<0$ and $0<{\epsilon}_{2}<1$, we have ${\epsilon}_{1}{\epsilon}_{2}+1>0$.
- So, we have got the following condition VI: $\{\begin{array}{c}-{\epsilon}_{2}<{\epsilon}_{1}<0\\ 0<{\epsilon}_{2}<1\end{array}$.

- For $-1<{\epsilon}_{2}<0$ and ${\epsilon}_{1}<-1$, we have ${\epsilon}_{1}{\epsilon}_{2}>1$, and ${\epsilon}_{1}{\epsilon}_{2}+1>0$.
- Then using Eq. (17), we have${\epsilon}_{1}+{\epsilon}_{2}>0$, .e., ${\epsilon}_{2}>-{\epsilon}_{1}$.
- The above process requires that $\{\begin{array}{c}-{\epsilon}_{2}<{\epsilon}_{1}<-1\\ -1<{\epsilon}_{2}<0\end{array}$,
- which cannot be satisfied.
- While for ${\epsilon}_{2}>1$ and ${\epsilon}_{1}<-1$, we have ${\epsilon}_{1}{\epsilon}_{2}+1<0$.
- So, we have the following result.
- IV: $\{\begin{array}{c}{\epsilon}_{1}<-{\epsilon}_{2}\\ {\epsilon}_{2}>1\end{array}$

## Appendx 2

As we claimed in the manuscript, the designed EMTW can realize full transparency independent of the incident angle. In this appendix, we give more simulation results to support our argument. The additional simulation is made with a different incident angles, i.e., $\alpha \approx -{32.87}^{\xb0}$. Though still other simulations are also made, they are not included here for the sake of clarity. Note that parameters in Figs. 7 -8 are the same as that in Figs. 5-6 except for the incidence angle.

## Acknowledgments

This work was supported in part by a Major Project of the National Science Foundation of China (NSFC) under Grant Nos. 60990320 and 60990324. Z. L. Mei acknowledges the Open Research Program Funds from the State Key Laboratory of Millimeter Waves (No. K201115), Natural Science Foundation of Gansu Province (No. 1107RJZA181), the Chunhui Project (No. Z2010081), and the Fundamental Research Funds for the Central Universities (No. LZUJBKY-2012-49).

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