Manipulate light polarizations with metamaterials: From microwave to visible

Abstract

Polarization is an important characteristic of electromagnetic (EM) waves, and efficient manipulations over EM wave polarizations are always desirable in practical applications. Here, we review the recent efforts in controlling light polarizations with metamaterials, at frequencies ranged from microwave to visible. We first presented a 4×4 version transfer matrix method (TMM) to study the scatterings by an anisotropic metamaterial of EM waves with arbitrary propagating directions and polarizations. With the 4×4 TMM, we discovered several amazing polarization manipulation phenomena based on the reflection geometry and proposed corresponding model metamaterial systems to realize such effects. Metamaterial samples were fabricated with the help of finite-difference-time-domain (FDTD) simulations, and experiments were performed to successfully realize these ideas at both microwave and visible frequencies. Efforts in employing metamaterials to manipulate light polarizations based on the transmission geometry are also reviewed.

Graphical abstract

Keywords

polarization / metamaterials

1 Introduction

Polarization is conventionally described by the oscillating behaviors of the \mathit{E} vectors in a wave [1–4]. If the trajectory of the end point of \mathit{E} vector forms a line (circle, ellipse) on the wave front as time varies, the wave is called linearly (circularly, elliptically) polarized. Polarized lights have important applications in different areas of scientific research, including biology, chemistry, astronomy, and physics. Practically, it is highly desirable to have full control of the wave polarizations. Several conventional methods are available to manipulate the wave polarizations. For example, by using two wave plates, one can convert the light polarization completely from one direction to its cross direction. Also, a linearly polarized light can be generated when a natural light passes through an optical grating or reflected by a medium at the Brewster angle. Other polarization-related phenomena include the optical activity, the birefringence effects, the polarization beam splitting, etc. [1–4]. While these conventional methods have their own characteristics and merits, they do suffer some limitations simultaneously. For example, these devices are typically not totally reflective or transparent so that some optical signals are inevitably lost during operation. In addition, many conventional devices are much thicker than wavelength, making the systems bulky for longer wavelength applications.

Metamaterials are artificial electromagnetic materials composed by subwavelength local resonance structures of electric and/or magnetic type and thus possess arbitrary values of permittivity \epsilon and permeability μ dictated by such resonance structures [5–12]. Many novel EM properties were predicted or discovered based on metamaterials, including the negative refraction [13–15], the superlensing effect [16–19], reversed Doppler shift [13, 20–22], abnormal Cherenkov radiation [13, 23–26], the subwavelength metamaterial cavities [27–29], etc. In particular, combining with the transformation optics method [30–32], a lot of intriguing wave-functional devices have been proposed, including the invisibility cloaks [30, 31], electromagnetic field concentrators [33], field rotators [34], cylindrical superlenses [35], and superscatterer [36].

Now, that metamaterials can in principle possess arbitrary values of \epsilon and μ, it is quite natural to ask whether they could have better controls over EM wave polarizations. Recently, there were indeed quite some attempts to employ such new materials to control EM wave polarizations [37–47], with several fascinating phenomena discovered. In what follows, we will give a detailed review on these efforts.

In the next section, we first generalize the conventional 2\times 2 transfer-matrix method (TMM) to a 4\times 4 version. This is because metamaterials are usually anisotropic [6–12], and many exciting polarization-related phenomena appear in the general situations where the incident EM wave takes arbitrary propagation direction and polarization. Theoretical calculations for such a general case are much more complicated than that for the isotropic case and for the simplified anisotropic problems (either \mathit{E} or \mathit{H} field is along one of the optical axes) [48–53]. For the latter two cases, one can always write down the wave solution as a combination of two independent modes, i.e., the s and p modes, with electric or magnetic field parallel to a prefixed planar interface. Polarizations are conserved as EM wave passes across the interface for each mode, and therefore, one can study the propagations of the two modes separately employing the standard 2\times 2 TMM [54, 55]. However, the same thing cannot be done in a general anisotropic problem, and thus, the standard 2\times 2 TMM should be generalized to a 4\times 4 version when we consider the wave transmissions/reflections in such a situation. While formalisms were previously available for an anisotropic medium with an anisotropic \overleftrightarrow{\epsilon } tensor [54], those formulas cannot be directly applied to the present metamaterial problems where both \overleftrightarrow{\epsilon } and \overleftrightarrow{\mu } could be anisotropic tensors.

In Section 3, we review our efforts in employing anisotropic metamaterials to manipulate wave polarizations under the reflection geometry. We discovered several amazing polarization manipulation phenomena in different systems and then proposed realistic metamaterial structures that can realize such effects. For example, by shining a linearly polarized EM wave on a specific metamaterial, we found that all kind of polarization states (circular, elliptic, and linear) are realizable after reflections under different conditions. We also found it possible to rotate the polarization direction of a linearly polarized EM wave by an arbitrary angle and even convert it to its cross direction completely. Several theoretical predictions were experimentally verified at both microwave and visible frequencies.

After we briefly review the efforts in employing metamaterials to manipulate light polarizations under the transmission geometry in Section 4, we conclude the paper in Section 5.

2 Computational methodology

In this section, we establish the 4\times 4 version of TMM. Consider a stratified medium consisting of homogeneous but anisotropic metamaterial layers, with the z axis chosen normal to the interfaces, as shown in Fig. 1.

2.1 The 4\times 4 transfer matrix

We start from studying the propagations of plane waves in a homogeneous anisotropic medium, with permittivity tensor and permeability tensor given by

\begin{array}{}\text{(1)}& \overleftrightarrow{\epsilon }=\left(\begin{array}{ccc}{\epsilon }_x& 0& 0\\ 0& {\epsilon }_y& 0\\ 0& 0& {\epsilon }_z\end{array}\right),\overleftrightarrow{\mu }=\left(\begin{array}{ccc}{\mu }_x& 0& 0\\ 0& {\mu }_y& 0\\ 0& 0& {\mu }_z\end{array}\right)\end{array}

According to the Maxwell equations, the \mathit{E} field inside such a medium should satisfy

\begin{array}{}\text{(2)}& \mathit{k}\times [({\overleftrightarrow{\mu }}^{-1})\cdot (\mathit{k}\times \mathit{E})]+{\displaystyle \frac{{\omega }^2}{c^2}}\overleftrightarrow{\epsilon }\cdot \mathit{E}=0\end{array}

where ω and \mathit{k} denote the angular frequency and wave vector, respectively. With k_x and k_y prefixed and applying the constrain \nabla \cdot (\overleftrightarrow{\epsilon }\cdot \mathit{E})=0, the dispersion relation between ω and k_z is found as:

\begin{array}{}\text{(3)}& {\left({\displaystyle \frac{\omega }{c}}\right)}^4{\epsilon }_x{\epsilon }_y{\epsilon }_z-\left[{\displaystyle \sum _{i\ne j\ne l}}{\epsilon }_i({\epsilon }_j{\mu }_j^{-1}+{\epsilon }_l{\mu }_l^{-1})k_i^2\right]{\left({\displaystyle \frac{\omega }{c}}\right)}^2+\left({\displaystyle \frac{1}{2}}{\displaystyle \sum _{i\ne j\ne l}}{\mu }_i^{-1}{\mu }_j^{-1}{k}_l^2\right){\displaystyle \sum _i}{\epsilon }_i{k}_i^2=0\end{array}

where i,j,l=x,y,z. Equation (3) has four roots, denoted by \{k_{z1},k_{z2}=-k_{z1},k_{z3},k_{z4}=-k_{z3}\} and corresponding to two independent modes propagating forwardly and backwardly. For each solution, the electric field distribution can be obtained through solving Eq. (2) with the solution of k_z inserted. In general, the wave solution inside the anisotropic medium must be a linear combination of these four modes, so that the EM waves can be expressed as:

\begin{array}{}\text{(4)}& \left(\begin{array}{c}{\mathit{E}}^{(n)}\\ {\mathit{H}}^{(n)}\end{array}\right)={\displaystyle \sum _{\sigma =1}^4}{E}_{\sigma }^{(n)}\left(\begin{array}{c}{\widehat{e}}_{\sigma }^{(n)}\\ {\widehat{h}}_{\sigma }^{(n)}\end{array}\right){\mathrm{e}}^{-\mathrm{i}[k_{x}x+k_{y}y+k_{z\sigma }^{(n)}(z-z_n)-\omega t]}\end{array}

where \{E_1^{(n)},E_2^{(n)},E_3^{(n)},E_4^{(n)}\} is a set of expansion coefficients to be determined, {\widehat{e}}_{\sigma }^{(n)},{\widehat{h}}_{\sigma }^{(n)},k_{z\sigma }^{(n)} are the field vectors and wave-vectors calculated in the nth layer, and z_n is the starting position of the nth layer. Imposing the condition that the tangential components of \mathit{E} and \mathit{H} are continuous at the boundary z=z_{n+1}, we found that

\begin{array}{}\text{(5)}& \begin{array}{cc}\left(\begin{array}{c}{E}_1^{(n+1)}\\ E_2^{(n+1)}\\ E_3^{(n+1)}\\ E_4^{(n+1)}\end{array}\right)\hfill & =D_{n+1}^{-1}{D}_n{P}_n\left(\begin{array}{c}{E}_1^{(n)}\\ E_2^{(n)}\\ E_3^{(n)}\\ E_4^{(n)}\end{array}\right)\hfill \\ \hfill & =M_{n+1,n}{P}_n\left(\begin{array}{c}{E}_1^{(n)}\\ E_2^{(n)}\\ E_3^{(n)}\\ E_4^{(n)}\end{array}\right)\hfill \end{array}\end{array}

with

\begin{array}{}\text{(6)}& D_n=\left(\begin{array}{cccc}{\widehat{e}}_1^{(n)}\cdot \widehat{y}& {\widehat{e}}_2^{(n)}\cdot \widehat{y}& {\widehat{e}}_3^{(n)}\cdot \widehat{y}& {\widehat{e}}_4^{(n)}\cdot \widehat{y}\\ {\widehat{h}}_1^{(n)}\cdot \widehat{x}& {\widehat{h}}_2^{(n)}\cdot \widehat{x}& {\widehat{h}}_3^{(n)}\cdot \widehat{x}& {\widehat{h}}_4^{(n)}\cdot \widehat{x}\\ {\widehat{h}}_1^{(n)}\cdot \widehat{y}& {\widehat{h}}_2^{(n)}\cdot \widehat{y}& {\widehat{h}}_3^{(n)}\cdot \widehat{y}& {\widehat{h}}_4^{(n)}\cdot \widehat{y}\\ {\widehat{e}}_1^{(n)}\cdot \widehat{x}& {\widehat{e}}_2^{(n)}\cdot \widehat{x}& {\widehat{e}}_3^{(n)}\cdot \widehat{x}& {\widehat{e}}_4^{(n)}\cdot \widehat{x}\end{array}\right)\end{array}

and

\begin{array}{}\text{(7)}& P_n=\left(\begin{array}{cccc}{\mathrm{e}}^{-\mathrm{i}{k}_{z1}^{(n)}{d}_n}& & & \\ & {\mathrm{e}}^{-\mathrm{i}{k}_{z2}^{(n)}{d}_n}& & \\ & & {\mathrm{e}}^{-\mathrm{i}{k}_{z3}^{(n)}{d}_n}& \\ & & & {\mathrm{e}}^{-\mathrm{i}{k}_{z4}^{(n)}{d}_n}\end{array}\right)\end{array}

It is noted that in the absence of magnetic anisotropy (i.e., {\mu }_x={\mu }_y={\mu }_z=\mu ), matrices D_n and P_n recover the results obtained previously [Eqs. (17) and (18) in Ref. [57]]. When both the magnetic and electric anisotropy are absent (i.e., {\mu }_x={\mu }_y={\mu }_z=\mu ,{\epsilon }_x={\epsilon }_y={\epsilon }_z=\epsilon ), the 4\times 4 transfer matrix M_{n+1,n} is reduced to two independent 2\times 2 transfer matrixes, recovering the isotropic medium case [6]. On the other hand, P_n is the usual propagation matrix to describe how EM wave propagates in the nth anisotropic layer (with a distance d_n). Therefore, expansion coefficients in an arbitrary layer (say, the nth layer) can be related to those in the first layer via applying Eq. (5) iteratively,

\begin{array}{}\text{(8)}& \begin{array}{cc}\left(\begin{array}{c}{E}_1^{(n)}\\ E_2^{(n)}\\ E_3^{(n)}\\ E_4^{(n)}\end{array}\right)=\hfill & M_{n,n-1}{P}_{n-1}{M}_{n-1,n-2}{P}_{n-2}\mathrm{\cdots }\hfill \\ \hfill & \cdot M_{2,1}{P}_1\left(\begin{array}{c}{E}_1^{(1)}\\ E_2^{(1)}\\ E_3^{(1)}\\ E_4^{(1)}\end{array}\right)\hfill \end{array}\end{array}

and the EM fields in the nth layer can be calculated from Eq. (4) after the expansion coefficients are known.

2.2 Transmission and reflection coefficients

For the system shown in Fig. 1, we label the semi-infinite reference medium on the left (right)-hand side of the system as the 0^{\mathrm{th}} [(N+1)th] layer (see Fig. 1). Assume a light is incident from the 0^{\mathrm{th}} layer, and then, the reflected waves in this layer and the transmitted wave in the last layer can be calculated with the transfer-matrix technique presented above. Apparently, the eigenmodes in the reference medium are just the conventional s and p modes that defined for the isotropic medium [38]. Let E_s^i, E_p^i, E_s^r, E_p^r, and E_s^t, E_p^t denote the complex amplitudes of the s and p modes of the incident, reflected, and transmitted waves, correspondingly. Applying the techniques described previously, the field inside the last layer (determined by the parameters E_s^t and E_p^t) can be associated with those inside the 0^{\mathrm{th}} layer (determined by the parameters E_s^i,E_s^r,E_p^i,E_p^r) by a transfer matrix as:

\begin{array}{}\text{(9)}& \begin{array}{cc}\left(\begin{array}{c}{E}_s^t\\ 0\\ E_p^t\\ 0\end{array}\right)\hfill & =Q\left(\begin{array}{c}{E}_s^i\\ E_s^r\\ E_p^i\\ E_p^r\end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cccc}{Q}_{11}& Q_{12}& Q_{13}& Q_{14}\\ Q_{21}& Q_{22}& Q_{23}& Q_{24}\\ Q_{31}& Q_{32}& Q_{33}& Q_{34}\\ Q_{41}& Q_{42}& Q_{43}& Q_{44}\end{array}\right)\left(\begin{array}{c}{E}_s^i\\ E_s^r\\ E_p^i\\ E_p^r\end{array}\right)\hfill \end{array}\end{array}

where

\begin{array}{}\text{(10)}& Q=M_{N+1,N}{P}_N{M}_{N,N-1}{P}_{N-1}\mathrm{\cdots }{M}_{2,1}{P}_1{M}_{1,0}{P}_0\end{array}

The reflection and transmission coefficients can then be calculated as:

\begin{array}{}\text{(11)}& r_{ss}={{\displaystyle \frac{E_s^r}{E_s^i}}|}_{E_p^i=0}={\displaystyle \frac{Q_{24}{Q}_{41}-Q_{21}{Q}_{44}}{Q_{22}{Q}_{44}-Q_{24}{Q}_{42}}}\end{array}

\begin{array}{}\text{(12)}& r_{sp}={{\displaystyle \frac{E_p^r}{E_s^i}}|}_{E_p^i=0}={\displaystyle \frac{Q_{21}{Q}_{42}-Q_{22}{Q}_{41}}{Q_{22}{Q}_{44}-Q_{24}{Q}_{42}}}\end{array}

\begin{array}{}\text{(13)}& t_{ss}={{\displaystyle \frac{E_s^t}{E_s^i}}|}_{E_p^i=0}=Q_{11}+{\displaystyle \frac{Q_{12}(Q_{24}{Q}_{41}-Q_{21}{Q}_{44})+Q_{14}(Q_{21}{Q}_{42}-Q_{22}{Q}_{41})}{Q_{22}{Q}_{44}-Q_{24}{Q}_{42}}}\end{array}

\begin{array}{}\text{(14)}& t_{sp}={{\displaystyle \frac{E_p^t}{E_s^i}}|}_{E_p^i=0}=Q_{31}+{\displaystyle \frac{Q_{32}(Q_{24}{Q}_{41}-Q_{21}{Q}_{44})+Q_{34}(Q_{21}{Q}_{42}-Q_{22}{Q}_{41})}{Q_{22}{Q}_{44}-Q_{24}{Q}_{42}}}\end{array}

\begin{array}{}\text{(15)}& r_{ps}={{\displaystyle \frac{E_s^r}{E_p^i}}|}_{E_s^i=0}={\displaystyle \frac{Q_{24}{Q}_{43}-Q_{23}{Q}_{44}}{Q_{22}{Q}_{44}-Q_{24}{Q}_{42}}}\end{array}

\begin{array}{}\text{(16)}& r_{pp}={{\displaystyle \frac{E_p^r}{E_p^i}}|}_{E_s^i=0}={\displaystyle \frac{Q_{23}{Q}_{42}-Q_{22}{Q}_{43}}{Q_{22}{Q}_{44}-Q_{24}{Q}_{42}}}\end{array}

\begin{array}{}\text{(17)}& t_{ps}={{\displaystyle \frac{E_s^t}{E_p^i}}|}_{E_s^i=0}=Q_{13}+{\displaystyle \frac{Q_{12}(Q_{24}{Q}_{43}-Q_{23}{Q}_{44})+Q_{14}(Q_{23}{Q}_{42}-Q_{22}{Q}_{43})}{Q_{22}{Q}_{44}-Q_{24}{Q}_{42}}}\end{array}

\begin{array}{}\text{(18)}& t_{pp}={{\displaystyle \frac{E_p^t}{E_p^i}}|}_{E_s^i=0}=Q_{33}+{\displaystyle \frac{Q_{32}(Q_{24}{Q}_{43}-Q_{23}{Q}_{44})+Q_{34}(Q_{23}{Q}_{42}-Q_{22}{Q}_{43})}{Q_{22}{Q}_{44}-Q_{24}{Q}_{42}}}\end{array}

Here, r_{ss} and r_{pp} are the direct reflection coefficients measuring the reflected waves with polarizations conserved, whereas r_{sp} and r_{ps} are the cross-reflection coefficients denoting the reflected waves with polarizations converted. Similarly, t_{ss} and t_{pp} are the direct transmission coefficients, while t_{sp} and t_{ps} denote the cross-transmission coefficients. In the absence of anisotropy where Q matrix is block-diagonalized (i.e., Q_{13}=Q_{14}=Q_{23}=Q_{24}=Q_{31}=Q_{41}=Q_{32}=Q_{42}=0), the polarization-converted terms become exactly zero, r_{sp}=r_{ps}=t_{sp}=t_{ps}=0, and the polarization-conserved coefficients recover those derived for the isotropic case [56]:

\begin{array}{}\text{(19)}& r_{ss}=-Q_{21}/Q_{22}\end{array}

\begin{array}{}\text{(20)}& t_{ss}=Q_{11}-Q_{12}{Q}_{21}/Q_{22}\end{array}

The analytical forms of the coefficients defined in Eqs. (11)–(18) are usually too complicated to derive for an arbitrary anisotropic layered system so that one needs to perform numerical computations.

3 Manipulate EM wave polarizations by metamaterials – reflection geometry

3.1 Single air/metamaterial interface

We first investigate the EM wave scatterings by an interface between air and an anisotropic metamaterial. Although the assumption of a semi-infinite sample is not quite realistic, this is the simplest model to study which yields several analytical results, from which many physical insights can be gained. Unlike the reflection by an isotropic interface that conserves the polarization, here, in the presence of anisotropy, an incident wave with a definite polarization could generate a reflected wave with another polarization because of the fact r_{sp},r_{ps}\ne 0. Define a polarization conversion ratio (PCR) as:

\begin{array}{}\text{(21)}& \text{PCR}={|r_{sp}|}^2/({|r_{ss}|}^2+{|r_{sp}|}^2)\end{array}

which measures the energy portion transformed from the original polarization, assumed as the s polarization for definiteness, to the other polarization (p polarization) after reflection. Obviously, this ratio depends on the direction (\theta ,\varphi ) of the incident wave vector {\mathit{k}}_{\mathrm{in}}, which is given by {\mathit{k}}_{\mathrm{in}}=(\omega /c)[\sin \theta \cos \varphi \widehat{x}+\sin \theta \sin \varphi \widehat{y}+\cos \theta \widehat{z}].

Consider the normal incidence case (\theta =0^{\circ }), which can be solved analytically. While \varphi is meaningless to define, the direction of {\mathit{k}}_{\mathrm{in}} in this case, it is still meaningful to differentiate two polarizations. In consistency with the \theta \ne 0^{\circ } case [57], we define that the s-polarized wave has \mathit{E}||{\widehat{e}}_s=(-\sin \varphi \widehat{x}+\cos \varphi \widehat{y}) and the p-polarized one has \mathit{E}||{\widehat{e}}_p=(\cos \varphi \widehat{x}+\sin \varphi \widehat{y}) in the case of \theta =0^{\circ }. Straightforward calculations yield that

\begin{array}{}\text{(22)}& r_{sp}={\displaystyle \frac{(\sqrt{{\epsilon }_x}\sqrt{{\mu }_x}-\sqrt{{\epsilon }_y}\sqrt{{\mu }_y})\sin (2\varphi )}{(\sqrt{{\epsilon }_y}+\sqrt{{\mu }_x})(\sqrt{{\epsilon }_x}+\sqrt{{\mu }_y})}}\end{array}

\begin{array}{}\text{(23)}& r_{ss}={\displaystyle \frac{\sqrt{{\mu }_x}\sqrt{{\mu }_y}-\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_y}+(\sqrt{{\epsilon }_x}\sqrt{{\mu }_x}-\sqrt{{\epsilon }_y}\sqrt{{\mu }_y})\cos (2\varphi )}{(\sqrt{{\epsilon }_y}+\sqrt{{\mu }_x})(\sqrt{{\epsilon }_x}+\sqrt{{\mu }_y})}}\end{array}

Several interesting conclusions were obtained through analyzing Eqs. (21)–(23). First, it is shown that all these quantities only depend on {\epsilon }_x, {\epsilon }_y{\mu }_x, and {\mu }_y but have nothing to do with {\epsilon }_z and {\mu }_z since here we only study the normal incidence situation. Second, the polarization conversion effect disappears (r_{sp}\equiv 0) in the absence of anisotropy ({\mu }_x={\mu }_y=\mu ,{\epsilon }_x={\epsilon }_y=\epsilon ), and the reflection coefficient becomes

\begin{array}{}\text{(24)}& r_{ss}={\displaystyle \frac{\sqrt{\mu }/\sqrt{\epsilon }-1}{\sqrt{\mu }/\sqrt{\epsilon }+1}}\end{array}

recovering the standard result for an isotropic case [1].

With the anisotropy tuned on, r_{sp} could take a nonzero value, but the polarization conversion effect is strongly dependent of the parameter ϕ. When \varphi =0^{\circ } or \varphi ={90}^{\circ }, one finds rigorously that r_{sp}\equiv 0, indicating the polarization conversion effect vanishes. This is reasonable since both \mathit{E} and \mathit{H} field of the incident wave are parallel to a coordinate axis so that the EM wave cannot detect the anisotropy of the medium. Meanwhile, the polarization-converted reflectance {|r_{sp}|}^2 is maximized when \varphi ={45}^{\circ }. However, a complete polarization conversion (i.e., PCR = 1) does not necessarily appear at \varphi ={45}^{\circ }, since the polarization-conserved reflectance {|r_{ss}|}^2 is also a function of ϕ. Recalling the definition of PCR in Eq. (20), we find that realizing PCR = 1 requires that r_{ss}=0, leading to the following condition:

\begin{array}{}\text{(25)}& \cos (2\varphi )={\displaystyle \frac{\sqrt{{\mu }_x}\sqrt{{\mu }_y}-\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_y}}{\sqrt{{\epsilon }_y}\sqrt{{\mu }_y}-\sqrt{{\epsilon }_x}\sqrt{{\mu }_x}}}\end{array}

Equation (25) shows that, for any value of ϕ, one can always choose appropriate material constants {\epsilon }_x, {\epsilon }_y, {\mu }_x, and {\mu }_y to achieve a complete polarization conversion effect.

Numerical calculations were performed to illustrate the above analytical results. Since all metamaterials are intrinsically frequency dispersive, the metamaterial under study was assumed to have the following typical response functions:

\begin{array}{}\text{(26)}& \begin{array}{cccccccccccccccccccc}{\mu }_x=1+{\displaystyle \frac{70}{{12.71}^2-f^2}},{\mu }_y=1+{\displaystyle \frac{22}{{6.80}^2-f^2}}\hfill & & & & & & & & & & & & & & & & & & & \\ {\epsilon }_x={\epsilon }_y={\epsilon }_z={\mu }_z=1\hfill & & & & & & & & & & & & & & & & & & & \end{array}\end{array}

where f is the linear frequency measured in GHz. The calculated results of r_{sp}^2, r_{ss}^2, and PCR are shown in Fig. 2(a), (b), and (c), as functions of frequency calculated with different values of ϕ. It is found that the reflectance (both r_{sp}^2 and r_{ss}^2) are significantly enhanced around the two resonance frequencies [6.8 GHz, 12.71 GHz, see Eq. (26); here, the resonance frequencies are belong to microwave spectrum radar bands, which can be taken on arbitrary values in principle], and the polarization-converted reflectance r_{sp}^2 is the largest at \varphi ={45}^{\circ }, in consistency with our above theoretical analysis. An interesting observation is that while the PCR can reach 1 at some particular frequencies for \varphi ={45}^{\circ },{75}^{\circ }, the same is not true for \varphi =5^{\circ },{15}^{\circ }. To understand this point, we put the frequency-dependent forms of {\epsilon }_x, {\epsilon }_y, {\mu }_x, and {\mu }_y [Eq. (26)] into Eq. (25) to calculate the value of ϕ at which a complete polarization conversion effect is realized and depicted the curve \varphi \sim f in Fig. 3. Indeed, for the model adopted here, there exists a specific regime of polarization angle ({21}^{\circ }-{76}^{\circ }) inside which a complete polarization conversion effect is realizable via adjusting frequency. For the sake of simplifications, the material losses in all the models for microwave frequency regime have been neglected [see Eqs. (26)–(28) and (31)]. It is found from numerical calculations that all qualitative conclusions drawn in this paper remain unchanged after adding losses to the materials. In particular, while the reflection coefficients r_{ss},r_{sp} are sensitive to the material losses, the PCR value is relatively robust against the losses.

We next study the oblique angle incidence case, which is much more complex. Figure 4 (a) shows the calculated PCR as a function of f and ϕ, with θ fixed as \theta ={15}^{\circ }. It is observed that the trajectory connecting those maximum PCR points looks very similar to the \varphi \sim f curve, as shown in Fig. 3, which is calculated in the normal incidence case. The physics is that both of them are dictated by the frequency dispersion of the material [Eq. (26)]. Fixing the frequency as f=6.81 GHz and 12.75 GHz, the PCR as functions of θ and ϕ were calculated and depicted the patterns in Fig. 4(b) and (c), respectively. It is noted that the PCR cannot reach 1 in oblique angle incidence case. For f=6.81 GHz, the value of ϕ to realize a maximum PCR increases toward {90}^{\circ } as θ increases, indicating that at glancing incidence, one can rotate \mathit{H} toward \widehat{y} to maximize the polarization conversion effect. An opposite behavior of ϕ was found for another resonance [see Fig. 4(c)], since the resonances at f=6.8 GHz and f=12.71 GHz belong to different directions.

3.2 Anisotropic metamaterial slabs

Although the single-interface model is intuitive, it is not quite realistic in practice. In this subsection, we study how to manipulate the EM wave polarizations by anisotropic metamaterial slabs, which are practically realizable. Two examples were given to illustrate the applications of the 4\times 4 TMM, and brute-force FDTD simulations on specifically designed samples were performed to verify all predictions based on the 4\times 4 TMM.

The first example is a 1.6-mm-thick metamaterial slab possessing magnetic responses along two in-plane directions:

\begin{array}{}\text{(27)}& \begin{array}{c}{\mu }_x=1+{\displaystyle \frac{15}{{12.49}^2-f^2}}+{\displaystyle \frac{100}{{25.73}^2-f^2}}\hfill \\ {\mu }_y=1+{\displaystyle \frac{10}{{7.11}^2-f^2}}+{\displaystyle \frac{110}{{14.8}^2-f^2}}+{\displaystyle \frac{220}{{22.12}^2-f^2}}\hfill \\ {\epsilon }_x={\epsilon }_y={\epsilon }_z={\mu }_z=1\hfill \end{array}\end{array}

With all material and geometrical parameters known, we employed the 4\times 4 TMM to study the EM wave scatterings by such a slab. Figure 5(a) shows the calculated results of r_{sp}^2 and r_{ss}^2 versus frequencies for normal incidence with \varphi ={45}^{\circ }. The corresponding PCR are shown in Fig. 5(b) denoted by solid stars. It is found that the spectrum calculated for a finite-thickness slab resembles that of a single-interface case (see Fig. 2) in many respects, although the details of the two spectra are quite different. For example, the reflectance, including both r_{sp}^2 and r_{ss}^2, reach their maximum values at the two resonance frequencies, but the PCR reach their maximums at different frequencies at which r_{ss}^2 are minimized.

We now study the second example, which is a metamaterial slab of the same thickness but possesses magnetic response only along one direction:

\begin{array}{}\text{(28)}& \begin{array}{cccccccccccccccccccc}{\mu }_y=1+{\displaystyle \frac{10}{{7.06}^2-f^2}}+{\displaystyle \frac{110}{{14.54}^2-f^2}}+{\displaystyle \frac{220}{{22.56}^2-f^2}}\hfill & & & & & & & & & & & & & & & & & & & \\ {\epsilon }_x={\epsilon }_y={\epsilon }_z={\mu }_x={\mu }_z=1\hfill & & & & & & & & & & & & & & & & & & & \end{array}\end{array}

We employed the 4\times 4 TMM to study the scatterings by such a metamaterial slab and depicted the normal-incidence spectra of r_{sp}^2, r_{ss}^2, and PCR in Fig. 6, calculated with different values of ϕ. While Fig. 6(a) and (b) show that the spectra of r_{sp}^2, r_{ss}^2 are quite similar to those calculated for sample I, here, the PCR spectrum shown in Fig. 6(c) for sample II looks quite unusual. In particular, the PCR is solely determined by the polarization angle ϕ but is independent of frequency although the metamaterial is highly dispersive [see Eq. (28)].

To understand such an intriguing phenomenon, we have derived analytical expressions for the reflection coefficients of such a slab under the normal incidence condition [38]. Although the expressions of r_{ss},r_{sp} [see Eqs. (37) and (38) in Ref. [38]] appear quite complicated, one can still gain a lot of physical insights via analyzing them. We found that r_{sp}=0 in the cases of \varphi =0^{\circ },{90}^{\circ }, due to the same reasons explained in the single-interface case. In addition, in the case of \varphi ={90}^{\circ } where the property is solely determined by {\epsilon }_x and {\mu }_y, it is easily obtained:

\begin{array}{}\text{(29)}& r_{ss}={\displaystyle \frac{\left({\mu }_y-{\epsilon }_x\right)\sin (k_{z3}d)}{-2\mathrm{i}\sqrt{{\epsilon }_x}\sqrt{{\mu }_y}\cos (k_{z3}d)+\left({\epsilon }_x+{\mu }_y\right)\sin (k_{z3}d)}}\end{array}

which again goes back to the reflectance calculated for an isotropic slab.

The most interesting result appears when the condition {\epsilon }_x={\epsilon }_y={\mu }_x\ne {\mu }_y is satisfied. In this case, while both r_{ss} and r_{sp} strongly depend on the value of {\mu }_x, {\mu }_y, and d, such dependences cancel each other in the expression of the PCR. Specifically, through straightforward calculations we found that

\begin{array}{}\text{(30)}& \text{PCR}={\displaystyle \frac{{\sin }^2(2\varphi )}{4{\sin }^4\varphi +{\sin }^2(2\varphi )}}={\cos }^2\varphi \end{array}

in the case of \varphi \ne 0^{\circ }. It is noted that the PCR cannot be defined in the case of \varphi =0^{\circ }.

This peculiar property explains the intriguing behaviors of the PCR, as shown in Fig. 6(c), since the material properties of sample II [{\epsilon }_x={\epsilon }_y={\mu }_x=1, see Eq. (28)] obviously satisfy the condition {\epsilon }_x={\epsilon }_y={\mu }_x\ne {\mu }_y. This intriguing behavior could find some applications in reality, in case that one wants to get a broad band polarization manipulation effect in practice.

FDTD simulations [38, 58] were performed to verify these theoretical predictions obtained with the 4\times 4 TMM. With the help of FDTD simulations, two realistic metallic structures have been designed that exhibit the different material responses, as shown in Eqs. (27) and (28). As depicted in Fig. 7(a) and (b), the unit cell of sample is composed of two split-ring resonators (SRR) [14, 59–61] of different sizes, which are perpendicular to each other, while that of sample II consists of only one single SRR. Repeat the unit cells along the x(y) direction with periodicity 8 mm (5 mm), and then, two 1.6-mm-thick slabs corresponding to the two samples specified by Eqs. (27) and (28) have been obtained. The EM wave normal transmission spectra through the two samples were calculated through the FDTD simulations, and the results were shown by solid lines in Fig. 7(c) and (d) for two independent polarizations (\mathit{E}||\widehat{x} and \mathit{E}||\widehat{y}). Excellent agreements are found between the spectra calculated by the FDTD simulations (solid lines) and by the TMM (open stars) based on the model systems represented by Eqs. (27) and (28). These excellent agreements demonstrated that our designed structures can indeed be well described by the effective medium models specified by Eqs. (27) and (28).

We then performed FDTD simulations to calculate the PCR spectra with the designed systems, and showed the results as open triangles in Fig. 5(b) and Fig. 6(c), respectively [38]. Compared with the PCR spectra calculated based on the 4\times 4 TMM, the agreements are generally quite satisfactory. In particular, we found that the FDTD simulations exactly reproduced the analytical results obtained for sample II [see Eq. (30)], verifying the predictions that sample II can indeed serve as a frequency-independent polarization manipulator. However, the agreement is found to be not perfect near the resonances of sample I. Such discrepancies are probably caused by the finite-size effects since we had to adopt finite-sized samples in our FDTD simulations [38].

3.3 Double-layer metamaterial reflector

From the above discussions for metamaterial slab cases, we find that while the PCR value can be as high as 1, the reflected signal is usually weak (|r_{sp}|\ne 1) indicating that the manipulation efficiency is low. The same is true for another type of metamaterial slab with \overleftrightarrow{\epsilon } and \overleftrightarrow{\mu } interchanged due to the EM symmetry. The reason is simply that such single-layer slabs are usually not totally reflective for EM waves with arbitrary incidence angles and polarizations.

In this section, we consider a double-layer metamaterial reflector [37] to remedy this problem. Such a reflector consists of a 1.3-mm-thick anisotropic meta-material layer with

\begin{array}{}\text{(31)}& \begin{array}{cccccccccccccccccccc}{\mu }_x=1+{\displaystyle \frac{70}{{12.71}^2-f^2}},{\mu }_y=1+{\displaystyle \frac{22}{{6.80}^2-f^2}}\hfill & & & & & & & & & & & & & & & & & & & \\ {\mu }_z=1\hfill & & & & & & & & & & & & & & & & & & & \\ {\epsilon }_x={\epsilon }_y={\epsilon }_z=1\hfill & & & & & & & & & & & & & & & & & & & \end{array}\end{array}

put on top of a perfect metal sheet (with {\epsilon }_m\to -\mathrm{\infty },{\mu }_m=1). The calculated results of PCR based on the 4\times 4 TMM are shown as solid lines in Fig. 8(a) for a normal incidence case with \varphi ={45}^{\circ } and in Fig. 8(b) for \theta =\varphi ={45}^{\circ }. It is observed that the PCR is strongly enhanced around two frequencies, \sim 12.7 GHz, \sim 6.8 GHz, corresponding precisely to the two resonances at which {\mu }_x or {\mu }_y tends to infinity [see Eq. (31)]. In particular, for the normal incidence case studied in Fig. 8(a), our theory predicts that \text{PCR}=100\% at the two resonance frequencies, indicating that a linearly polarized light converts its polarization completely after the reflection.

In general, both s- and p-components exist inside the reflected beam. Defining \mathrm{\Delta }{\varphi }_{sp} by r_{sp}/r_{ss}=|r_{sp}/r_{ss}|{\mathrm{e}}^{\mathrm{i}\mathrm{\Delta }{\varphi }_{sp}}, we calculated \mathrm{\Delta }{\varphi }_{sp} as the functions of frequency for the two cases studied in Fig. 8. From the results (solid lines) recorded in Fig. 9, we found that \mathrm{\Delta }{\varphi }_{sp} can take arbitrary values within [-{180}^{\circ },{180}^{\circ }] depending on the frequency (and in turn, on the material parameters), indicating that all possible polarization states (circular, linear, and elliptic) can be realized for the reflected beam. As shown in the illustrations, we picked up four frequencies to calculate the polarization patterns of the reflected beams and found that the resultant polarizations are linear (6.87 GHz), circular (7.27 GHz), and elliptic (10.9 GHz, 12.46 GHz), correspondingly [37].

The physics underlying these unusual phenomena can be understood by a simple argument. Consider the normal incidence case for simplicity. Suppose the incident wave is given by {\mathit{E}}^{\mathrm{in}}=\left(E_x\widehat{x}+E_y\widehat{y}\right){\mathrm{e}}^{\mathrm{i}(-\omega z/c+\omega t)}, then after reflection by an ordinary material, the reflected wave is usually written as {\mathit{E}}^r=r\left(E_x\widehat{x}+E_y\widehat{y}\right){\mathrm{e}}^{\mathrm{i}(\omega z/c+\omega t)} with r being the reflection coefficient. However, in the present case with anisotropy ({\mu }_x\ne {\mu }_y), the reflection coefficients, r_x,r_y are different for incident waves polarized along two directions. Thus, in general, the reflected wave should be {\mathit{E}}^r=(r_x{E}_x\widehat{x}+r_y{E}_y\widehat{y}){\mathrm{e}}^{\mathrm{i}(\omega z/c+\omega t)}, and the polarization state can be manipulated through varying r_x,r_y. For the configuration studied in Fig. 8(a) with E_x=E_y, if one tune the material parameters to yield r_x/r_y=-1, the polarization direction of the reflected wave would be -\widehat{x}+\widehat{y}, which is perpendicular to that of the original wave, \widehat{x}+\widehat{y}. A complete polarization conversion (CPC) is thus realized [37].

The key issue then is how to control r_x and r_y. With the metal sheet on the back, the entire structure is always totally reflective for EM waves, independent of their incidence angles and polarizations, i.e., |r_x|=|r_y|\equiv 1. However, the phase \mathrm{\Delta }\psi of the reflection coefficient, defined as r_{x(y)}={\mathrm{e}}^{\mathrm{i}\mathrm{\Delta }{\psi }_{x(y)}}, strongly depends on the metamaterial parameters. The calculated reflection phases (\mathrm{\Delta }{\psi }_x,\mathrm{\Delta }{\psi }_y) for the model system have been depicted in Fig. 10 as functions of frequency. In most cases where {\mu }_x({\mu }_y) is not large, we have \mathrm{\Delta }{\psi }_{y(x)}\sim \pm {180}^{\circ }, since the metamaterial layer is transparent and light can directly “see” the metal plate that is reflecting out of phase. However, at the resonances where {\mu }_x({\mu }_y)\to \pm \mathrm{\infty }, we obtained \mathrm{\Delta }{\psi }_{y(x)}=0 since light is reflected directly by the opaque metamaterial, which possesses infinite impedance [62–64]. In general, any value of \mathrm{\Delta }{\psi }_{y(x)} can be obtained through adjusting the values of {\mu }_x({\mu }_y) and d. The physics for the CPC effect is now clear. Near each resonance, one of {\mu }_x,{\mu }_y becomes very large, while another close to 1, and thus, there must be a frequency where \mathrm{\Delta }{\psi }_x-\mathrm{\Delta }{\psi }_y=\pm {180}^{\circ }, and thus, r_x/r_y=-1 (see Fig. 10).

We also studied how the effect depends on the incident angles. Depicted in Fig. 11(a) are the PCR as functions of θ and ϕ, calculated at f=6.85 GHz. Numerical computations reveal that the PCR cannot reach 1 in arbitrary incidence case, but the maximum PCR value can still approach 1. As θ increases, we found that the value of ϕ to realize a maximum PCR increases toward {90}^{\circ }, indicating that at glancing incidence, one can rotate \mathit{E} toward \widehat{x} to maximize the polarization conversion effect. An opposite behavior of ϕ was found for the resonance at f=12.7 GHz. To facilitate easy comparisons with experiments, the PCR as a function of ϕ with \theta ={45}^{\circ } and as a function of θ with \varphi ={45}^{\circ } have been shown in Fig. 5(b) and (c), respectively. It is found that the maximum polarization conversion effect now takes place at \varphi \sim {56}^{\circ } when θ is fixed as {45}^{\circ }, different from the normal incidence case. On the other hand, the PCR is a decreasing function of θ when ϕ is fixed as {45}^{\circ } and reaches 1 at \theta =0^{\circ }.

Microwave experiments and FDTD simulations were performed to demonstrate these predictions. To realize the proposed model system described by Eq. (31), we designed a frequency-selective structure, as shown in Fig. 12(a), which consists of a periodic array of planar H-shaped metallic pattern (lattice constant a = 7 mm, thickness = 0.1 mm), printed on a 1.2-mm thick PCB slab (with {\epsilon }_r=3.6) with a metal sheet on the back. Other structural parameters are fixed as b = 5 mm, w = 1.0 mm. A single H-shaped metallic structure possesses electric resonances for both polarizations [65]. When a metal plate is added, currents are induced on the metal plate surface, flowing along a direction opposite to the currents induced on the H structures. As a result, the entire structure now exhibits magnetic responses with well-defined resonance frequencies, but its electric polarization is strongly diminished since the two currents effectively cancel with each other [66, 67]. The composite material is then perfectly described by the double-layer model [see Eq. (31)], with d = 1.3 mm being the thickness of the H-pattern (0.1 mm) plus the inner dielectric layer (1.2 mm) [67]. As a demonstration, we employed FDTD simulations [37] to calculate the reflection phase spectra (\mathrm{\Delta }{\psi }_x(f),\mathrm{\Delta }{\psi }_y(f)) of the designed composite material and showed the results in Fig. 10. The good agreement with the model TMM results supports our conclusions.

A 497\text{mm}\times 497\text{mm} sample based on the designs was fabricated to perform the microwave experiments. As schematically shown in Fig. 12(c), the measurements were carried out in an anechoic chamber using a network analyzer (Agilent 8722 ES) and two linearly polarized horn antennas. The distance between the source/receiver antenna and the sample is 6.9 m. The sample was rotated appropriately to achieve a desired incidence angle (\theta ,\varphi ). Illuminating the sample by a linearly polarized signal with \mathit{E}||{\widehat{y}}_0 (i.e., s wave), we then measured the reflected signal using a {\widehat{y}}_0-polarized receiver horn (data collected as |r_{ss}{|}^2) and an {\widehat{x}}_0-polarized one (data collected as |r_{sp}{|}^2). PCR was calculated with the measured {|r_{ss}|}^2 and {|r_{sp}|}^2, and the results were drawn in Fig. 8(a) and (b) as open symbols. FDTD simulations were also performed to calculate the PCR spectra based on the designed system, and the results were shown as solid stars in the same figures. Quantitative agreements are found among the results obtained by the TMM, the FDTD simulations, and experiments. In particular, both experiments and simulations verified the CPC effects predicted by the model analysis. FDTD simulations were subsequently performed to compute the relative phase spectra (\mathrm{\Delta }{\varphi }_{sp}(f)). The FDTD results are again in good agreement with the model TMM results, as shown in Fig. 9, indicating that any desired EM wave polarization can be obtained. Finally, experiments and simulations were carried out to study the angle dependences of the polarization conversion effects, and the measured and simulated results were depicted in Fig. 11(b) and 11(c). Satisfactory agreements are noted compared with the model TMM results.

3.4 Optical activity in a metamaterial without chirality

When a linearly polarized light travels through a medium, the polarization direction is rotated by a certain angle during wave propagation. This phenomenon is known as the optical activity [2]. Conventional medium to possess optical activity include chiral media and gyrotropic media, which typically exhibit (imaginary) off-diagonal constitutional tensor elements.

In this section, we demonstrate that our double-plate reflector, which is neither chiral nor gyrotropic, possesses strong optical activity. Specifically, we found that the polarization direction of a linearly polarized incident EM wave can be rotated by an arbitrary angle after reflections at our double-layer metamaterial reflector [38]. We now review our results in some details.

Rewrite the reflection coefficients as r_{ss}=|r_{ss}|{\mathrm{e}}^{\mathrm{i}{\varphi }_s}, r_{sp}=|r_{sp}|{\mathrm{e}}^{\mathrm{i}{\varphi }_p}, the reflected waves can be rewritten as:

\begin{array}{}\text{(32)}& {\mathit{E}}_r(\mathit{r},t)=(|r_{ss}|{\mathrm{e}}^{\mathrm{i}\mathrm{\Delta }{\varphi }_{sp}}{\widehat{e}}_s+|r_{sp}|{\widehat{e}}_p){\mathrm{e}}^{\mathrm{i}{\varphi }_p}{\mathrm{e}}^{-\mathrm{i}({\mathit{k}}_r\cdot \mathit{r}-\omega t)}\end{array}

where the incident wave is assumed to be s-polarized, {\mathit{k}}_r is the wave-vector of the reflected wave, and \mathrm{\Delta }{\varphi }_{sp}={\varphi }_s-{\varphi }_p denotes the relative phase between the s- and p-polarized modes inside the reflected beam. One can easily find from Eq. (31) that the reflected wave still takes a linear polarization if the condition

\begin{array}{}\text{(33)}& \mathrm{\Delta }{\varphi }_{sp}=0\text{or}{180}^{\circ }\end{array}

is fulfilled, but the polarization direction is rotated by an angle

\begin{array}{}\text{(34)}& \mathrm{\Delta }\Theta =\arctan {\displaystyle \frac{|r_{sp}|}{|r_{ss}|}}\end{array}

as compared to that of the original wave. When the condition PCR = 1 is reached implying that |r_{ss}|=0, it is obtained that \mathrm{\Delta }\Theta ={90}^{\circ }, indicating the original s-polarization has been completely changed to the p-polarization after the reflection [4, 38].

To illustrate how the condition (33) is met, numerical calculations using the 4\times 4 TMM were performed on the model system described by Eq. (31). Again, first, consider the normal incidence case. Depicted in Fig. 13 is the calculated value of \mathrm{\Delta }{\varphi }_{sp} as functions of frequency and the azimuth angle ϕ. It is noted that the condition (\mathrm{\Delta }{\varphi }_{sp}=0\text{or}{180}^{\circ }) is satisfied at two frequencies, 6.87 GHz and 12.88 GHz, independent of the parameter ϕ. This fact indicates that the reflected wave still takes a linear polarization at these two frequencies. For the case of oblique angle of incidence (\theta \ne 0^{\circ }), extensive numerical computations were performed, and we found that the condition (33) can still be met at the vicinities of the two frequencies – 6.87 GHz and 12.88 GHz.

We chose one of those two frequencies to quantitatively study the rotation angle \mathrm{\Delta }\Theta based on Eq. (34). Setting f=6.87\text{GHz}, we calculated |r_{sp}| and \mathrm{\Delta }\Theta as functions of θ and ϕ and drew the results in Fig. 14(a) and (b). We found that the pattern for \mathrm{\Delta }\Theta resembles that of |r_{sp}| very much, and both of them are very complicated functions of the incidence angle specified by θ and ϕ. The most important observation is that an arbitrary rotation angle \mathrm{\Delta }\Theta can be obtained via adjusting the incidence direction or, equivalently speaking, via rotating the reflector appropriately. This fact indicates that one can freely rotate the polarization direction of a linearly polarized light by using our metamaterial reflector. This is remarkable. Although the similar polarization-rotation effect can be achieved based on the gyrotropic (Faraday) effect [68], a thick medium is usually required to achieve a large rotation angle, and also, the efficiency is relatively low. In contrast, such the reflector is much thinner than wavelength and the performance is remarkable.

Although the pattern of \mathrm{\Delta }\Theta \sim \theta ,\varphi appears quite complicated, for \mathrm{\Delta }\Theta, in the case of normal incidence (\theta =0^{\circ }), an analytical formula can be still derived, with which many new understandings can be gained. In such a case, the incident s-polarized EM wave is given by

\begin{array}{}\text{(35)}& \begin{array}{cccccccccccccccccccc}{\mathit{E}}_{\mathrm{in}}(\mathit{r},t)=(-\sin \varphi \widehat{x}+\cos \varphi \widehat{y}){\mathrm{e}}^{-\mathrm{i}(\omega z/c-\omega t)}\hfill & & & & & & & & & & & & & & & & & & & \\ {\mathit{H}}_{\mathrm{in}}(\mathit{r},t)=(-\cos \varphi \widehat{x}-\sin \varphi \widehat{y}){\mathrm{e}}^{-\mathrm{i}(\omega z/c-\omega t)}\hfill & & & & & & & & & & & & & & & & & & & \end{array}\end{array}

A unique property in such a situation is that the EM wave can be separated as two independent linearly polarized waves:

\begin{array}{}\text{(36)}& \begin{array}{cc}\left(\begin{array}{c}{\mathit{E}}_{\mathrm{in}}(\mathit{r},t)\\ {\mathit{H}}_{\mathrm{in}}(\mathit{r},t)\end{array}\right)=\hfill & -\sin \varphi \left(\begin{array}{c}\widehat{x}\\ \widehat{y}\end{array}\right){\mathrm{e}}^{-\mathrm{i}(\omega z/c-\omega t)}\hfill \\ \hfill & +\cos \varphi \left(\begin{array}{c}\widehat{y}\\ -\widehat{x}\end{array}\right){\mathrm{e}}^{-\mathrm{i}(\omega z/c-\omega t)}\hfill \end{array}\end{array}

For each of these two linearly polarized waves, the reflection coefficients, denoted by r_x and r_y, could take different values. Since the Maxwell equations are linear ones, the reflected wave must be

\begin{array}{}\text{(37)}& {\mathit{E}}_r(\mathit{r},t)=\left(-r_x\sin \varphi \widehat{x}+r_y\cos \varphi \widehat{y}\right){\mathrm{e}}^{\mathrm{i}(\omega z/c+\omega t)}\end{array}

With a perfect-metal slab on the back, such double-layer reflector is always totally reflecting, that is, the amplitudes of the reflection coefficients r_x and r_y are always absolutely 1, but their phases could be different. Therefore, when the condition

\begin{array}{}\text{(38)}& r_x/r_y=-1\end{array}

is satisfied, the reflected wave still takes a linear polari-zation [Note that the condition (38) is applicable only to the case of normal incidence, but the condition (33) is valid in a general situation]. As the condition (38) is satisfied, the \mathit{E} vector of the reflected wave would be {\mathit{E}}_r\sim -\sin \varphi \widehat{x}-\cos \varphi \widehat{y}. As schematically shown in Fig. 15, such an \mathit{E} vector has been rotated by an angle

with respect to that of the incident wave, {\mathit{E}}_{\mathrm{in}}\sim -\sin \varphi \widehat{x}+\cos \varphi \widehat{y}. Equation (39) thus provides us a rigorous and analytical formula to calculate the polarization rotation angle under the normal incidence condition. Indeed, the \mathrm{\Delta }\Theta \sim \varphi relation calculated by the 4\times 4 TMM is shown by solid stars in Fig. 16, which is in excellent agreement with the analytical relation Eq. (39) represented by a solid line.

Again, these theoretical predictions were successfully verified by FDTD simulations on realistic structures [38]. The calculated results of \mathrm{\Delta }\Theta are shown in Fig. 16 as a function of the polarization angle ϕ, represented by open triangles. Perfect agreements were found between the FDTD simulations and the model TMM results, as well as the analytical relation Eq. (39).

3.5 Manipulate wave polarizations at visible frequencies

Although many fascinating applications were theoretically proposed for metamaterials and some of them were successfully realized at microwave frequencies (including the polarization manipulation effects mentioned in this section), very few were experimentally confirmed at optical frequencies. This is due to the significantly enhanced challenges faced by both experiment and theory at optical frequencies. In this section, we review our recent efforts in designing and fabricating appropriate optical metamaterials and demonstrating their strong abilities to control light polarizations at visible frequencies [39].

As schematically shown in Fig. 17(a), the designed metamaterial consists of a layer of gold rod array and a continuous gold film, separated by a SiO₂ layer. This trilayer structure was fabricated layer-by-layer with electron beam evaporation of constituent materials. The 15-nm-thick gold layer was first deposited on a GaAs substrate with 1 nm Titanium as the adhesive layer and then covered by a 60 nm SiO₂ separation layer. The top gold rod array, with a thickness of 15 nm, was fabricated with standard electron-beam lithography and lift-off technique. Each gold rod is sized 240 nm × 60 nm, and the periodicities of the array are 300 nm and 150 nm along two directions. A series of samples based on the designs were fabricated with sizes fixed as 100\text{μ}\mathrm{m}\times 100\text{μ}\mathrm{m}. Figure 17(b) shows a scanning electron microscopy (SEM) picture of one typical sample.

Optical experiments were performed to measure the polarization manipulation effects. The experimental setup is schematically depicted in Fig. 17 (c). Halogen tungsten was used as the excitation white light source, which was focused on the sample surface by an objective lens. The reflected light was collected by the same lens and dispersed on the monochromator. Linear polarizers were inserted in the paths of both the incident and reflected light to control the light polarizations. In our measurements, we set the incident \mathit{E} field parallel to a fixed laboratory axis {\widehat{x}}_0 and then rotated the sample to change the incident wave polarization with respect to the sample [see Fig. 17(d)]. With a fixed incident polarization (assumed as the s polarization), we rotated the receiver to measure the reflectance for the reflected signals with the same polarization (|r_{ss}{|}^2) and that with a p polarization (|r_{sp}{|}^2). We first rotated the sample so that the incident \mathit{E} field is parallel to one of the optical axes and measured the direct reflection spectra |r_{ss}{|}^2(|r_{sp}{|}^2\equiv 0 in this case). The spectra for \varphi =0^{\circ } and \varphi ={90}^{\circ } cases were depicted in Fig. 18(a) as solid and open symbols, respectively. Both experimental data and simulation results are referenced by the reflection spectrum measured/calculated with a semi-infinite Si slab. As \lambda >750\text{nm}, the relative reflectivity is higher than 1 since our system possesses a higher reflectivity than a Si slab. The spectra clearly show that the designed metamaterial is not totally reflecting, since gold is not a perfect metal at optical frequencies. Therefore, the mechanism for microwave frequencies (Section 3.4) does not work here for the optical metamaterials.

We then rotated the sample such that \varphi ={45}^{\circ } [see Fig. 17(d)]. Since the incident wave is not polarized along one of the optical axes, both {|r_{ss}|}^2 and {|r_{sp}|}^2 are nonzero, and the measured spectra are shown in Fig. 19(a) as solid and open symbols, respectively. Strong polarization-converted reflection signal (|r_{sp}{|}^2) appear around 680 nm, which is even prominently shown in PCR spectra depicted in Fig. 19(b). The maximum PCR value is found as 83% at 680 nm wavelength. We further employed a tunable femtosecond laser as the excitation source to perform the same measurement and depicted the PCR spectra in Fig. 19(b) as solid stars. The PCR value obtained with laser input is significantly enhanced, and the maximum PCR value is as high as 96% at 685 nm. This is easy to understand, since a laser beam has a much better directionality than a focused white light beam.

FDTD simulations were performed to understand these intriguing phenomena [39]. In our simulations, the permittivity of gold is taken as \epsilon =9.0-{(1.37\times {10}^{16})}^2/[{\omega }^2+\mathrm{i}(1.0027\times {10}^{14})\omega ], the refractive index of SiO₂ is 1.5, and the permittivity of the semi-infinite GaAs substrate is 10.8. Direct reflection spectra for \varphi =0^{\circ } and \varphi ={90}^{\circ } cases were first calculated, and the calculated reflectance spectra ({|r_x|}^2 and |r_y{|}^2) were depicted in Fig. 18(b). Reasonable agreements were found when compared with the experimental spectra, and most experimental features were reproduced. Besides the reflectance, FDTD calculations also provided us the information of reflection phase. In Fig. 18(c), where the calculated reflection phase difference for two polarizations (\mathrm{\Delta }{\psi }_{yx}={\psi }_y-{\psi }_x) is shown as a function of wavelength, we found that \mathrm{\Delta }{\varphi }_{yx} approaches {180}^{\circ } within the range of 650–760 nm, coinciding with the frequency range where the PCR peak is found [see Fig. 19(b)]. In fact, by decoupling the incident wave to two independent polarizations, the PCR spectra can be calculated for arbitrary ϕ case (under normal incidence) using the following formula:

\begin{array}{}\text{(40)}& \text{PCR}={\displaystyle \frac{{[|r_x|\sin \varphi -|r_y|\cos \varphi \cos (\mathrm{\Delta }{\varphi }_{yx})]}^2}{{[|r_x|\cos \varphi +|r_y|\sin \varphi \cos (\mathrm{\Delta }{\varphi }_{yx})]}^2+{[|r_x|\sin \varphi -|r_y|\cos \varphi \cos (\mathrm{\Delta }{\varphi }_{yx})]}^2}}\end{array}

The PCR spectra thus calculated is shown in Fig. 19(c) as open squares for \varphi ={45}^{\circ }, which is in reasonable agreement with the measured spectra. A straightforward explanation of the large polarization conversion effect is that, around 680 nm wavelength, one has similar reflectance for two linear polarizations (i.e., |r_x|\approx |r_y|) but with nearly {180}^{\circ } reflection phase difference.

We found that the inherent physics behind such unusual phenomena is quite different from microwave one since the metal has a finite penetration length here. This is reinforced by Fig. 18(a) where the reflectance is not 100%. Therefore, the above double-layer effective medium model (Section 3.4) is not suitable for the present system. Instead, we found that it is better to model the present system as a single-layer effective medium with effective permittivity \tilde{\epsilon } and permeability \tilde{\mu }, since the considered wavelength \sim700 nm is much larger than the total thickness of the structure 90 nm. Effective optical parameters were retrieved [9, 69] using the FDTD simulated transmission/reflection coefficients, and the retrieved values of {\tilde{\epsilon }}_x, {\tilde{\epsilon }}_y, {\tilde{\mu }}_x, and {\tilde{\mu }}_y were plotted in Fig. 20(a)–(d). To better understand the role of resonances, we employed standard Lorentz models to fit the retrieved effective parameters and finally obtained the following analytical formulas:

\begin{array}{}\text{(41)}& \begin{array}{cc}{\epsilon }_x=\hfill & 4-{\displaystyle \frac{{656}^2}{f^2+i\cdot 2\cdot f}}+{\displaystyle \frac{{520}^2}{{205}^2-f^2-i\cdot 17.5\cdot f}}\hfill \\ \hfill & +{\displaystyle \frac{{90}^2}{{400}^2-f^2-i\cdot 17\cdot f}}\hfill \\ {\epsilon }_y=\hfill & 3.3-{\displaystyle \frac{{655}^2}{f^2+i\cdot 2\cdot f}}+{\displaystyle \frac{{180}^2}{{450}^2-f^2-i\cdot 19\cdot f}}\hfill \\ {\mu }_x=\hfill & 1+{\displaystyle \frac{{310}^2}{{350}^2-f^2-i\cdot 130\cdot f}}\hfill \\ {\mu }_y=\hfill & 1.1+{\displaystyle \frac{{135}^2}{{160}^2-f^2-i\cdot 25\cdot f}}+{\displaystyle \frac{{65}^2}{{391}^2-f^2-i\cdot 17\cdot f}}\hfill \end{array}\end{array}

where f denotes the frequency in THz. The effective parameters calculated based on Eq. (41) are shown in Fig. 4(e)–(h) as functions of frequency. It is interesting to note that, beside the common Drude-like contributions in {\epsilon }_x and {\epsilon }_y that originate from the material itself, formulas (41) imply that a series of additional EM resonances exist in such a metamaterial. The magnetic resonances are induced by an antisymmetric coupling of the currents flowing in the two metallic layers, which are similar to those found at microwave frequencies [37, 67]. For the electric resonances, the electric current in each gold rod is parallel to that flowing in the continuous gold film, forming a symmetric mode [70, 71]. In general, these EM resonances occur at different wavelengths that are dictated by the anisotropic geometric structures. Thus, our system is ideal anisotropic optical material possessing wide parameter tunability so that it can fulfill many applications with appropriate designs. The PCR spectrum was calculated employing the generalized 4\times 4 TMM with effective parameters given by Eq. (41). The results depicted in Fig. 19(c) as open stars are shown in reasonable agreements with both the direct numerical results and the experimental data.

The analytical model offers us a physical picture on the polarization manipulation effect. From the definition of PCR, it is noted that PCR=1 requires r_{ss}=0. While it is difficult to analyze the expression of r_{ss} for a metamaterial slab [Eq. (38) in Ref. [38]], we found it very helpful and intuitive to analyze r_{ss} of an air/metamaterial interface [Eq. (30) in Ref. [38]]. The condition of r_{ss}=0 rigorously leads to an analytical formula

\begin{array}{}\text{(42)}& Z_x\cdot Z_y=1\end{array}

in the case of \varphi ={45}^{\circ }. Here, Z_x=\sqrt{{\mu }_y}/\sqrt{{\epsilon }_x} and Z_y=\sqrt{{\mu }_x}/\sqrt{{\epsilon }_y} are the impedance for light polarized along x and y directions. In fact, since the normal reflection coefficients are r_x=(Z_x-1)/(Z_x+1) and r_y=(Z_y-1)/(Z_y+1) for two polarizations, when the condition (42) is met, one can easily verify that r_x=-r_y so that \text{PCR}=1 at \varphi ={45}^{\circ }. The condition Eq. (42) has a clear physical implication: the two directions must satisfy an EM reciprocal principle in order to realize a complete polarization conversion at \varphi ={45}^{\circ }. A limiting solution of Eq. (42) is that Z_x\to 0,Z_y\to \mathrm{\infty }, indicating that the system behaves as a perfect electric conductor (PEC) for x-polarized light but as a perfect magnetic conductor (PMC) for y-polarized light. This is just the situation realized in a microwave frequency regime (Section 3.4). At higher frequencies, however, metal is no longer perfect so that one cannot achieve a PEC or a PMC. However, Eq. (42) shows that those EM resonances [as shown in Eq. (41)] can be tuned to generate appropriate optical parameters that satisfy the EM symmetry. When the latter is achieved, a high polarization conversion effect can be realized. To test this picture, the real and imaginary parts of Z_x\cdot Z_y as functions of wavelength were plotted in Fig. 21(b), compared with the PCR spectra replotted in Fig. 21(a). As expected, it is found that Z_x\cdot Z_y\approx 1 at two wavelengths 628 nm and 766 nm, which directly explains the two PCR peaks observed both experimentally and theoretically in Fig. 5(a), (b), and (c). We note that the condition (42), obtained based on a single air/metamaterial interface, is not rigorously applicable to the present slab case. This also explains why the measured PCR cannot reach 1. Nevertheless, we found condition (42) is intuitive to understand the inherent physics and is helpful for future designs of similar systems.

4 Manipulate EM wave polarizations by metamaterials – transmission geometry

In aforementioned sections, we reviewed the recent efforts in employing metamaterials to control EM wave polarizations based on the reflection geometry. Similar effects can be realized based on a transmission geometry, which will be briefly reviewed in this section.

Quite recently, there have been several studies of creating metamaterial transmission polarizers [40–47]. Chin et al. proposed to use an anisotropic metamaterial slab as a transmission polarizer [40], in which the function and the transmission efficiency of the polarizer can be designed and controlled. The metamaterial is composed by electric inductance-capacitance (ELC) resonator particles, which exhibit designable effective electric permittivity. It was found that a transmission polarizer consisting of two-layer ELC can convert a linearly polarized wave to a circularly polarized one, while a four-layer ELC transmission polarizer can convert the polarization of a linearly polarized wave from one direction to it cross direction. On the other hand, Beruete et al. combined the ideas of extraordinary optical transmission and metamaterials to successfully fabricate a polarization device that can select and rotate EM wave polarizations [41]. For an input wave with arbitrary polarization, any polarization can be attained at the output by just adjusting the number of stacked plates. The most attractive feature of such a device is that the phase shift between two orthogonal eigenwaves is achieved with the negative and positive indices of refraction for two independent linearly polarized waves. T. Q. Li et al. studied the propagations of microwaves through a chiral metamaterial [42], which is an array of magnetic dimmers made of two square single split-ring resonators (SSRRs). The optical activity of such a medium is originated from the hybridization effect of magnetic resonances. T. Li et al. investigated the near infrared optical transmissions through a silver film with L-shaped holes array [43]. Besides the enhanced transmission due to the combined plasmonic excitations, nearly 45o polarization rotation was observed with relative strong transmissions for a polarized light with a specific incidence angle at the wavelength \sim 1200 nm. It is a giant optical rotation rate considering that the sample is only 80 nm thick. Plum et al. reported that very strong optical activity can be observed in artificial planar metamaterials based on an array of asymmetrically split rings without chirality at all [44]. Both microwave and optical experiments were performed to demonstrate this effect [44].

5 Conclusions

In summary, we have briefly reviewed the recent efforts in manipulating EM wave polarizations based on the metamaterial concept. We first established a generalized 4\times 4 transfer-matrix method to study the scatterings of EM waves by layered anisotropic metamaterials in a general case, and then, we employed this method to study the polarization manipulation effects under the reflection geometry with several different metamaterial systems, including an anisotropic metamaterial interface, an anisotropic metamaterial slab, a double-layer metamaterial reflector, and an optical metamaterial. We showed that several amazing polarization-related phenomena can be realized by metamaterials, including the polarization conversion, the polarization rotation and so on. We also briefly reviewed the existing efforts of polarization manipulation under the transmission geometry, which were mainly carried out in other research groups. Many of these interesting phenomena were verified by FDTD simulations and experiments at both microwave and visible frequencies. These remarkable properties have promised metamaterials a lot of technological applications, and further studies are needed to explore more fascinating new physics and applications of metamaterials with respect to the polarization control.

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