1. Introduction

Perfect absorption, characterized by complete suppression of scattered waves, is necessary in the design of engineering applications such as photodetectors and photovoltaics. In addition to traditional methods to achieve perfect absorption that utilize coupling between one-port incident wave and optical resonators [1–6], a recently proposed concept, coherent perfect absorption (CPA), has drawn much attention because of its unusual mechanism [7]. CPA relies on the interference of two counter-propagating coherent and monochromatic waves: the reflected waves from one direction exactly cancel the transmitted waves coming from the other direction and vice versa. This absorption mechanism was originally proposed in optics as the time-reversed process of lasing and was demonstrated to be a general and robust phenomenon arising from the properties of the scattering matrix [7]. When scattering is completely suppressed, perfect absorption is achieved.

Since this concept was described [5], much research effort has been devoted to the area of CPA [8–29]. For instance, a two-dimensional CPA was demonstrated with a specially designed cylinder absorber [20]. A one-dimensional metamaterial with PT-symmetric defects was suggested as a type of coherent perfect absorber whose operation frequencies can be tuned [21]. The tunability was also explored in graphene-based systems [23,24]. An equivalent realization of CPA under single beam illumination was demonstrated by introducing a perfect magnetic conductor as a mirror boundary [25]. Broadband absorption was reported in radio/micro [26] and near-infrared regime [27]. Waveguide structures were used to explore the feasibility to realize CPA [28,29]. The concept of CPA was also generalized into the scenarios of coherent divergent beams [30]. In experimental realizations, multiple platforms based on thin silicon slices [8], passive resonators coupled to microwave transmission line [22], conductive films [25,26] and silicon racetrack resonators [28] were explored.

Most of the progress thus far in this field has been based on studying the CPA for given materials and structures. Thus, there is no much freedom in selecting the incident waves with different parameters, such as operating frequency and incident angle [27]. A more practical inverse problem, in which the materials and structures need to be determined with the knowledge of incident waves, is more challenging because the required material parameters are sometimes stringent and might not be easily fulfilled by naturally occurring materials. Fortunately, the problem may be tackled with the notion of metamaterials, whose key properties arise from their artificial building blocks rather than from their chemical composition [31–36]. These key properties are characterized by effective medium parameters, which are determined by proper effective medium theories [37–45], rendering the rich possibilities of designing building blocks that give rise to the desired material properties required by CPA.

In this work, we propose a unified scheme to achieve CPA of electromagnetic (EM) waves in anisotropic metamaterials. The scheme is based on a study of the condition for CPA and an anisotropic effective medium theory in conjunction with a numerical retrieval method. It describes an inverse design process in determining the material and geometric properties of a practical absorber to perfectly absorb two coherent incident waves. The scheme is scalable to wavelengths and applicable to incident wave over a wide range of incident angles. This unified scheme, though designed for CPA, also applies to more general cases with arbitrarily given amplitudes and relative phase of the incident waves. The effectiveness of the scheme is demonstrated through numerical calculations, in which CPA of oblique incident waves is achieved by the designed metamaterials.

The remainder of this paper is organized as follows. The conditions of CPA in a homogeneous lossy slab are analyzed based on the scattering matrix in Section 2. The inverse design of the anisotropic metamaterials that achieve CPA, is described in Section 3 where two examples considering different incident angles are demonstrated. In Section 4, the performance of the designed metamaterial absorbers is numerically studied. By integrating these absorbers, we further propose a coherent point source absorber in Section 5. Conclusions are drawn in the last section.

2. Conditions for CPA

A schematic of the perfect absorber considered in our study is shown in Fig. 1, where the absorber, a slab of lossy medium with thickness $L_{\text{2}}$ (layer 2), is sandwiched between two layers of a background medium, denoted as layers 1 and 3, which, for simplicity, are chosen to be air. Two coherent and monochromatic waves propagate towards the absorber and are absorbed. Without loss of generality, we consider a TE-(transverse electric) polarized EM wave, ($\overrightarrow{E}=(0,0,E_z)$).The electric fields in different domains can be expressed as

 

Fig. 1 Schematic representation of the CPA. The absorber is made of a lossy slab (layer 2) whose thickness is $L_{\text{2}}$. Two coherent and monochromatic waves with the same incident angle, $\theta$, are illuminated from the background medium (layers 1 and 3). They propagate towards the absorber and are absorbed.

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The continuity conditions relate the complex amplitudes of the scattered waves,${\beta }_{\text{1}}$ and ${\alpha }_{\text{3}}$, and those of the incident waves, ${\alpha }_{\text{1}}$and ${\beta }_{\text{3}}$, through

As an example, we present in the following the design of coherent perfect absorbers, covering a wide range of incident angles. We choose the operating frequency as $\tilde{\omega }=\omega l/2\pi c_{air}=0.55$, amplitudes of $\left|{\alpha }_1\right|=1$ and $\left|{\beta }_3\right|=1$, and the phase difference $\pi$. Specifically, we change $\theta$ and look for the proper thickness and material parameters of the absorber such that Eq. (5) is satisfied. In Fig. 2, a possible combination of the material parameters and the corresponding wave vectors (calculated from Eq. (6)) in the designed absorbers as functions of the incident angle are plotted. The thickness of the designed absorbers is fixed as $L_{\text{2}}=3l$. We do this to benefit the future integration process, which will be demonstrated in Section 5. Here we also introduce a length unit $l$, which normalizes the frequency and thickness, therefore making our design easy to be scaled. For example, if we consider a frequency in the terahertz regime (10 THz), the length unit, $l$, is about 10 microns (0.0165 mm), such that $\tilde{\omega }=\omega l/2\pi c_{air}=0.55$. Unless otherwise stated, the above parameters are used in the following simulations and discussions.

 

Fig. 2 Permittivity ${\epsilon }_{\text{2}}$ (black curves with circles), x-component (red curves with squares) and y-component (blue curves with diamonds) of the permeability tensor, ${\overleftrightarrow{\mu }}_{\text{2}}$, as functions of the incident angle, $\theta$, solved from Eqs. (3), (5) and (6). The superscripts in the legends distinguish the real parts (a) and imaginary parts (b) of the quantities. The moduli of $k_{2x}$, $k_{2y}$ and $k_2$ as functions of $\theta$ are also respectively presented in (c) by using purple curves with up-triangles, pink curves with down-triangles and green curves with stars.

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3. Design of the metamaterial absorbers

In Fig. 2, we notice that the parameters of the absorber required by CPA exhibit unusual behaviors. For example, at $\theta =5^{\circ }$, the permittivity and permeabilities of the slab are${\epsilon }_{\text{2}}=0.0364+i0.0289$, ${\mu }_{2x}=-0.6555+i0.3881$ and ${\mu }_{2y}=0.0192+i0.2045$. These stringent parametrical requirements are difficult to be satisfied with naturally occurring materials, but they may be achieved by using metamaterials. In this section, we introduce a two-step process for designing proper building blocks of the metamaterials to realize CPA. The first step is to determine the building blocks of the metamaterial absorber by inversely applying an anisotropic effective medium theory (AEMT) developed by the authors [43]; and the second step is to optimize the building blocks from a parameter retrieval method. The detailed procedure of this two-step process is demonstrated in the following two representative examples.

In one example, the incident angle is small ($\theta =5^{°}$). At this angle, according to Fig. 2, the wave vector in the desired absorber is small ($\left|k_{\text{2}}\right|=0.2670<<1$), satisfying the applicable condition of the previously derived AEMT [43] for a rectangular array of elliptical cylinders in a dielectric matrix, whose detailed derivations are described in [43]. AEMT is written as

To further affirm the results and to enhance the performance of the absorbing layer, we introduce an optimization process based on a parameter retrieval method [35,42,46–49]. In this method, we numerically compute the transmission, $T(\theta )$, and reflection, $R(\theta )$, coefficients of the metamaterial absorber to retrieve the effective refractive index, $n_{eff}$, and the effective impedance, $Z_{eff}$, of the corresponding homogeneous slab. Then, we compare the retrieved effective medium parameters with the desired ones. If the two sets of parameters are identical, the optimization process finishes. If they are not identical, we slightly tune the material and geometric parameters of the elliptical cylinders until the retrieved effective medium parameters converge to the desired ones. This process is demonstrated by the flowchart in Fig. 3(a) where the red (thicker) arrows indicate the optimization loop.

 

Fig. 3 The flowcharts of the design process for (a) $\theta \le {20}^{°}$ where $\left|k_{\text{2}}\right|$ is small and (b) $\theta >{20}^{°}$ where $\left|k_{\text{2}}\right|$ is large. In the former, the optimization starts with the metamaterial predicted by AEMT; in the latter, the optimization starts with the optimized metamaterial absorbers obtained at a smaller angle, $\theta \text{'}<\theta$.

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The standard formulas for $T(\theta )$ and $R(\theta )$ for a homogeneous slab are respectively written as [49]

The retrieved quantities, ${\epsilon }_{eff}$, ${\mu }_{eff,x}$ and ${\mu }_{eff,y}$, are then compared with those of the desired parameters ${\epsilon }_{\text{2}}$, ${\mu }_{2x}$ and ${\mu }_{2y}$, respectively. For $\theta =5^{°}$, we find that the retrieved effective medium parameters of the metamaterial slab are close to the desired values. We further optimize the metamaterial by slightly adjusting the cylinders until the retrieved parameters overlap with the desired ones. We obtain ${\epsilon }_s^{opt}=13.61+i1.03$, $a_s^{opt}=0.202l$ and $b_s^{opt}=0.153l$. Note that these quantities are very close to the values predicted by AEMT, i.e., ${\epsilon }_s=13.82+i1.03$, $a_s=0.2l$ and $b_s=0.154l$, affirming the validity of AEMT.

The above-described process is effective in finding the real structures of coherent perfect absorbers when the incident angle, $\theta$, is small, because the magnitude of the wave vector inside the absorber, $\left|k_{\text{2}}\right|$, is small and AEMT works well. However, as the incident angle increases, $\left|k_{\text{2}}\right|$ also increases because of the Snell’s law. Figure 2 (c) shows that when the incident angle exceeds ${20}^{\circ }$, $\left|k_{\text{2}}\right|$ becomes larger than unity, deeming AEMT inaccurate.

To design a real metamaterial perfect absorber for waves with large incident angles (where $\left|k_{\text{2}}\right|$ is large), we adopt a similar optimization process to the one presented in Fig. 3(a), but with a different starting point. Instead of using brute force to get ${\epsilon }_s{|}_{\theta }$, $a_s{|}_{\theta }$ and $b_s{|}_{\theta }$ from Eq. (7) and then finding the optimized ${\epsilon }_s^{opt}{|}_{\theta }$, $a_s^{opt}{|}_{\theta }$ and $b_s^{opt}{|}_{\theta }$, we use a metamaterial that works for a smaller incident angle, ${\theta }^{\prime }$ (close to $\theta$), for which the parameters of the inclusions have already been determined as ${\epsilon }_s^{opt}{|}_{{\theta }^{\prime }}$, $a_s^{opt}{|}_{{\theta }^{\prime }}$ and $b_s^{opt}{|}_{{\theta }^{\prime }}$, as the starting point to perform the optimization process. Such an extrapolation process is based on the premise that the physical system is smoothly continuous without going through any abrupt changes. The flowchart of this process is shown in Fig. 3(b) with blue (thicker) arrows. As an example, we consider $\theta ={25}^{\circ }$ where $\left|k_{\text{2}}\right|=1.4926>1$. Under this condition, AEMT cannot provide an accurate prediction of the material’s parameters. As described, we choose ${\theta }^{\prime }={20}^{\circ }$ ($\left|k_{\text{2}}\right|=0.9713<1$) as the initial state in the optimization process. We have already obtained the required parameters following the flowchart shown in Fig. 3(a) for the metamaterial that can perfectly absorb the coherent incident waves at this angle. The results are ${\epsilon }_s^{opt}{|}_{{20}^{\circ }}=13.25+i1.02$, $a_s^{opt}{|}_{{20}^{\circ }}=0.207l$ and $b_s^{opt}{|}_{{20}^{\circ }}=0.156l$. Since ${\theta }^{\prime }={20}^{\circ }$ is close to $\theta ={25}^{\circ }$, we expect that the quantities obtained at $\theta$ should not deviate much from those for ${\theta }^{\prime }$. We follow the flowchart shown in Fig. 3(b) and obtain the desired parameters as ${\epsilon }_s^{opt}{|}_{{25}^{\circ }}=12.59+i1.39$, $a_s^{opt}{|}_{{25}^{\circ }}=0.22l$ and $b_s^{opt}{|}_{{25}^{\circ }}=0.161l$, which give effective medium parameters of ${\epsilon }_{eff}=0.0827+i0.0328$, ${\mu }_{eff,x}=-0.3528+i0.3420$ and ${\mu }_{eff,y}=0.3887+i0.2285$ from the retrieval method. These values indeed coincide with the required material parameters of the perfect absorbing layer presented in Fig. 2, indicating that the proposed process is applicable to finding the metamaterial absorbers that can absorb coherent waves with large incident angles.

Knowing the configuration of the metamaterial at $\theta$, we apply the same strategy to find the configuration of the metamaterial at $\theta +\Delta \theta$ and eventually we succeed in realizing CPA with anisotropic metamaterials under a large range of incident angles, i.e., $(0^{°},{60}^{°})$_. The corresponding parameters are presented in Fig. 4, with dielectric constants, ${\epsilon }_s$ and ${\epsilon }_s^{opt}$, plotted in (a), semi-major axes, $a_s$ ($a_s^{opt}$), and semi-minor axes, $b_s$ ($b_s^{opt}$), plotted in (b). Note that for $\theta >{20}^{°}$ where $\left|k_{\text{2}}\right|$ is large, only ${\epsilon }_s^{opt}$, $a_s^{opt}$ and $b_s^{opt}$ are presented because AEMT does not apply in this regime. From Fig. 4, we can clearly observe that when the incident angle is small ($\theta \le {20}^{°}$),where $\left|k_{\text{2}}\right|$ is also small, the results for the metamaterial absorbers predicted by AEMT are in good agreement with those for the optimized metamaterials.

 

Fig. 4 The designed metamaterial absorbers at different incident angles, $\theta$, featured with (a) the real and imaginary parts of the dielectric constants, ${\epsilon }_s$ (red curves with solid circles and squares) and ${\epsilon }_s^{opt}$ (black curves with hollow circles and squares), and (b) the semi-major axes, $a_s$ (red curve with solid circles) and $a_s^{opt}$(black curve with hollow circles) and the semi-minor axes, $b_s$ (red curve with solid squares) and $b_s^{opt}$(black curve with hollow squares).

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Here, we would like to point out that we used the AEMT presented in [43] in our inverse design process, but the scheme is not limited to that AEMT. Other effective medium theories may also be used here if a more complex structure is desired. Moreover, the scheme is scalable to EM frequencies. Changing the material and geometric parameters would give rise to CPA at other frequencies of interest.

4. Performance of the metamaterial absorbers

In this section, we use COMSOL Multiphysics (a commercial software based on the finite element method) to conduct numerical experiments to examine the performance of our metamaterial absorbers.

Figure 5(a) shows the electric field distributions of two out-of-phase coherent and monochromatic oblique incident waves with incident angle $\theta =5^{\circ }$, impinging on a uniform slab with ${\epsilon }_{\text{2}}=0.0364+i0.0289$, ${\mu }_{2x}=-0.6555+i0.3881$ and ${\mu }_{2y}=0.0192+i0.2045$. The upper and lower boundaries are imposed by periodic boundary conditions. Similar results are shown in Fig. 5(b) with the uniform slab replaced by a slab of metamaterial with building blocks of ${\epsilon }_s^{opt}=13.61+i1.03$, $a_s^{opt}=0.202l$ and $b_s^{opt}=0.153l$. From Figs. 5(a) and 5(b) (and Visualization 1 and Visualization 2), we do not find any distortion on the wave fronts, indicating the scattered field indeed vanishes and the incident waves are completely absorbed. More evidence can be found in Figs. 5(c) and 5(d), where the uniform distributions on the intensity of the electric fields in air (layers 1 and 3) are observed.

 

Fig. 5 Performance of the metamaterial absorbers. (a) (Corresponding to Visualization 1) Simulated electric field distribution for the uniform absorber slab with parameters of ${\epsilon }_{\text{2}}=0.0364+i0.0289$, ${\mu }_{2x}=-0.6555+i0.3881$ and ${\mu }_{2y}=0.0192+i0.2045$ at the incident angle of $\theta =5^{°}$. Clearly observed is the perfect preservation of the plane wave fronts, indicating that the scattering is completely suppressed and the incident energy is totally absorbed. (b) (Corresponding to Visualization 2) The same as (a), but with the uniform slab replaced by the metamaterial absorber with parameters ${\epsilon }_s^{opt}=13.61+i1.03$, $a_s^{opt}=0.202l$ and $b_s^{opt}=0.153l$. Good agreement between (a) and (b) is seen. Further evidence is demonstrated in (c) and (d) where the uniform distributions of $|E_z{|}^2$ are seen in both layers 1 and 3, implying total absorption. (e)-(h) Respectively the same as (a)-(d), but with the absorbers corresponding to those at $\theta ={25}^{°}$, whose parameters are ${\epsilon }_{eff}=0.0827+i0.0328$, ${\mu }_{eff,x}=-0.3528+i0.3420$ and ${\mu }_{eff,y}=0.3887+i0.2285$ for the uniform slab and${\epsilon }_s^{opt}=12.59+i1.39$, $a_s^{opt}=0.220l$ and $b_s^{opt}=0.161l$ for the metamaterial slab. Total absorption is observed again. (e) and (f) correspond to Visualization 3 and Visualization 4, respectively. (i) Quantitative study of the coherent absorption, $A_c$, as functions of the incident angle, $\theta$. Curves and symbols indicate results of uniform and metamaterial absorbers, respectively. The absorption at different incident angles is calculated with different absorbers whose parameters are shown in Fig. 4. To further demonstrate the absorption performance, $1-A_c$ is also plotted in logarithmic scale.

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Similar results are plotted in Figs. 5(e) and 5(f) (Visualization 3 and Visualization 4) for the case of $\theta ={25}^{\circ }$. These results again demonstrate no field distortion on both the uniform slab and the real metamaterial slab. Their respective intensity distributions, shown in Figs. 5(g) and 5(h), affirm that perfect absorption is reached.

To better quantify the absorption performance of the metamaterial absorbers, we plot the coherent absorption $A_c$, calculated from Eq. (4), and $1-A_c$ as a function of the incident angles in Fig. 5(i). Here absorption at different incident angles is calculated in different absorbers whose parameters are shown in Fig. 4. Perfect absorption is clearly seen in both cases with uniform absorber slabs (curves) and the metamaterial absorbers (symbols). It is indeed remarkable that the metamaterial absorbers can cover such a wide range of incident angles, as the higher-order diffraction is usually unavoidable when the incident angle becomes large. We also notice that the building blocks of metamaterial absorbers presented in Fig. 4 have dielectric constants close to that of alumina (99.6%) around a frequency of 10 THz [50], suggesting that these metamaterial absorbers are feasible in experiments.

5. A coherent point source absorber

We have demonstrated that slabs with proper anisotropic parameters can perfectly absorb oblique incident coherent waves at different incident angles, suggesting that if we carefully integrate these slabs together to form a gradient metamaterial absorber, then it is possible to absorb the incident waves emanating from two coherent point sources incident onto it, because the waves coming from a point source incident onto a slab have continuously varying incident angles on the surface of the slab. Note that the strategy used here is different from that in [51] where the authors proposed a design of a non-local material by transversely homogeneous multilayered planar slabs to achieve all-angle CPA.

A schematic of our coherent point source absorber is presented in Fig. 6(a). Two coherent point sources are located at a distance of $d=13b\times \cot {30}^{°}$ and $d=22.5170b$ away from the left and right edges of the absorber, respectively. The absorber in the middle comprises 13 blocks. Each block can absorb the incident wave energy that shines on it. The corresponding incidence angle from $0^{°}$ to ${30}^{°}$ with a step of $5^{°}$ is marked on each block in Fig. 6(a). To best utilize the design we proposed, each block covers the height of $2b$. The building blocks and corresponding effective medium parameters of the gradient metamaterial absorbers used here are listed in the Appendix B.

 

Fig. 6 Demonstration of a coherent point source absorber integrated by the metamaterial absorbers that cover the angles up to ${30}^{\circ }$. (a) The setup of the point source absorber. Thirteen absorbers designed at different incident angles are integrated to form the point source absorber. Two coherent point sources radiate at the distance of $d=22.5170b$ away from the left and right edges of the absorber. For comparison, the electric field distribution of the two point sources radiating in air is shown in (b). Strong interference pattern is observed. (c) (Corresponding to Visualization 5) The same as (b), but with the presence of the metamaterial absorber. It is clearly seen that the cylindrical wave fronts are perfectly restored, suggesting that the energy emanating on the absorber has been totally absorbed. (d) (Corresponding to Visualization 6) The same as (c), but with the uniform absorber replacing the metamaterial absorber. Good agreement between (c) and (d) is seen.

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For comparison, we first plot in Fig. 6(b) the electric field distribution of two out-of-phase coherent point sources in air. Clearly seen is the strong interference pattern. However, when the absorber is placed in the middle of these two sources, as shown in Fig. 6(c) (Visualization 5), almost perfect cylindrical wave fronts are restored, indicating that almost all the energy that impinges upon the absorber is absorbed. We also present in Fig. 6(d) (Visualization 6) the same radiation pattern as that in Fig. 6(c) but with the metamaterial absorbers replaced with uniform absorbers. Good agreement is seen.

Even though we only present perfect absorption of the wave energy radiated from two coherent point sources onto the surface of the absorber, it is possible to enhance the absorption by increasing the size of the absorber, i.e. increasing the number of blocks shown in Fig. 6(a) and placing a mirror just behind the point source to reflect back the energy radiated away from the absorber [30].

6. Conclusions

We have developed a unified and efficient scheme based on scattering matrix analysis and effective medium theory predictions to realize CPA of EM waves in anisotropic metamaterials. The scattering matrix is used to determine the CPA conditions. AEMT in conjunction with a parameter retrieval method is employed: AEMT is first used to predict the initial configurations of the metamaterials; the parameter retrieval method is then used to optimize the metamaterials such that they can exhibit the best absorption performance. The proposed scheme is scalable to wavelengths and covers a wide range of incident angles. Beyond these advantages, the scheme is also applicable to a more general case with arbitrary relative amplitude and phase of the incident waves, which might pave the way of designing perfect absorbers.

As a demonstration, we employed our scheme in the design of absorbers in anisotropic metamaterials to perfectly absorb coherent incident waves with different incident angles and integrated them to form a coherent point source absorber. Numerical simulations for those metamaterial absorbers show complete suppression of the scattered fields, validating our scheme. Our design with anisotropic metamaterials brings more degrees of freedom in realizing CPA and its good performance suggests promising applications. We also found that the dielectric constants of the designed metamaterial absorbers at different incident angles are close to the dielectric constant of the alumina (99.6%) around a frequency of 10 THz [50], suggesting that experimental realizations are possible.

It is worth mentioning that although the derivations and demonstrations presented here are for TE-polarizations, our scheme is applicable to transverse-magnetic (TM)-polarizations. For TM-polarized coherent incident waves, we may choose another solution satisfying the CPA condition, i.e., solutions to Eqs. (3), (5) and (6), and apply the AEMT to find the proper material and geometric parameters of the dielectric cylinders in the design of real absorbers.

Appendix A Derivation of the scattering matrix

In Section 2, we use a scattering matrix to analyze the CPA conditions of the absorber illustrated in Fig. 1. The scattering matrix of a slab describes the relation between the scattered waves (out-going waves) and the incident waves (incoming waves). Its matrix elements are determined by the boundary conditions on the two interfaces of the slab. For the TE-polarized wave considered in this work, the electric and magnetic fields follow the expressions of Eqs. (1) and (2), respectively. The continuities of $E_z$ and $H_y$ at the interfaces between the slab and air, i.e., $x=0$and $L_2$, yield,

(11)$$\begin{array}{l}{\alpha }_{\text{1}}{e}^{i{k}_{1y}y}+{\beta }_{\text{1}}{e}^{i{k}_{1y}y}={\alpha }_{\text{2}}{e}^{i{k}_{2y}y}+{\beta }_{\text{2}}{e}^{i{k}_{2y}y},\\ \frac{k_{1x}}{\omega {\mu }_1}\left({\alpha }_{\text{1}}{e}^{i{k}_{1y}y}-{\beta }_{\text{1}}{e}^{i{k}_{1y}y}\right)=\frac{k_{2x}}{\omega {\mu }_{\text{2y}}}\left({\alpha }_{\text{2}}{e}^{i{k}_{2y}y}-{\beta }_{\text{2}}{e}^{i{k}_{2y}y}\right),\end{array}$$

and

(12)$$\begin{array}{l}{\alpha }_{\text{2}}{e}^{i{k}_{2x}{L}_{\text{2}}}{e}^{i{k}_{2y}y}+{\beta }_{\text{2}}{e}^{-i{k}_{2x}{L}_{\text{2}}}{e}^{i{k}_{2y}y}={\alpha }_{\text{3}}{e}^{i{k}_{3y}y}+{\beta }_{\text{3}}{e}^{i{k}_{3y}y},\\ \frac{k_{2x}}{\omega {\mu }_{2y}}\left({\alpha }_{\text{2}}{e}^{i{k}_{2x}{L}_{\text{2}}}{e}^{i{k}_{2y}y}-{\beta }_{\text{2}}{e}^{-i{k}_{2x}{L}_{\text{2}}}{e}^{i{k}_{2y}y}\right)=\frac{k_{3x}}{\omega {\mu }_3}\left({\alpha }_{\text{3}}{e}^{i{k}_{3y}y}-{\beta }_{\text{3}}{e}^{i{k}_{3y}y}\right),\end{array}$$

where ${\alpha }_{\text{1}}$ and ${\beta }_{\text{3}}$ denote the amplitudes of the incident waves in layers 1 and 3, respectively, and ${\beta }_{\text{1}}$ and ${\alpha }_{\text{3}}$ represent those of the corresponding scattered waves, respectively. They are related through Eq. (3), i.e.,

$$\left(\begin{array}{l}{\beta }_{\text{1}}\\ {\alpha }_{\text{3}}\end{array}\right)=S\left(\begin{array}{l}{\alpha }_{\text{1}}\\ {\beta }_{\text{3}}\end{array}\right),$$

where $S$ is the scattering matrix. Combing Eqs. (11), (12) and Eq. (3) and invoking Snell’s law ($k_{1y}=k_{2y}=k_{3y}$), we obtain the following expressions for the matrix elements of $S$:

$$S_{11}=-\frac{\left(k_{3x}{\mu }_{2y}-k_{2x}{\mu }_3\right)\left(k_{2x}{\mu }_1+k_{1x}{\mu }_{2y}\right)e^{2i{k}_{2x}{L}_{\text{2}}}+\left(k_{3x}{\mu }_{2y}+k_{2x}{\mu }_3\right)\left(k_{2x}{\mu }_1-k_{1x}{\mu }_{2y}\right)}{\left(k_{3x}{\mu }_{2y}-k_{2x}{\mu }_3\right)\left(k_{2x}{\mu }_1-k_{1x}{\mu }_{2y}\right)e^{2i{k}_{2x}{L}_{\text{2}}}+\left(k_{3x}{\mu }_{2y}+k_{2x}{\mu }_3\right)\left(k_{2x}{\mu }_1+k_{1x}{\mu }_{2y}\right)},$$

$$S_{21}=\frac{4k_{3x}{k}_{2x}{\mu }_{2y}{\mu }_1{e}^{i{k}_{2x}{L}_{\text{2}}}}{\left(k_{3x}{\mu }_{2y}-k_{2x}{\mu }_3\right)\left(k_{2x}{\mu }_1-k_{1x}{\mu }_{2y}\right)e^{2i{k}_{2x}{L}_{\text{2}}}+\left(k_{3x}{\mu }_{2y}+k_{2x}{\mu }_3\right)\left(k_{2x}{\mu }_1+k_{1x}{\mu }_{2y}\right)},$$

$$S_{21}=S_{12},$$

and

$$S_{22}=S_{11}.$$

Appendix B The material and geometric parameters of the metamaterial absorbers and their effective medium parameters

The material and geometric parameters of the metamaterial absorbers described in Section 5 and their corresponding effective medium parameters with different incident angles are listed in Table 1.

Table 1. The material parameters (${\epsilon }_s^{opt}$) and geometric parameters ($a_s^{opt}$ and $b_s^{opt}$) of the metamaterial absorbers at different incident angles. The corresponding effective medium parameters (${\epsilon }_{eff}$, ${\mu }_{eff,x}$ and ${\mu }_{eff,y}$) are also listed.

View Table

Funding

King Abdullah University of Science and Technology Baseline Research Fund.

Acknowledgments

The authors would like to thank Ping Bai and Zeguo Chen for stimulating discussions, and Dr. Virginia Unkefer for editorial work on this manuscript.

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