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Abstract
Previous reflectionless metasurfaces based on balanced electric and magnetic responses in engineered resonant meta-atoms become ineffective at oblique incident angles and usually have strong reflection at grazing incidence, where the impedance becomes near-zero or divergent. Here, by introducing the concept of anomalous generalized Brewster effect to metasurfaces, we demonstrate an exceptional resonance-free Brewster metasurface that exhibits ultrabroadband zero reflection at grazing incidence. The anomalous generalized Brewster effect is obtained via combining the mechanisms of the generalized Brewster effect and the anomalous Brewster effect, which are both resonance-free and thus enable ultrabroadband functionalities. As a practical application, Brewster metasurfaces exhibiting ultrabroadband reflectionless perfect absorption at grazing incident angles are constructed and demonstrated by full-wave simulations and microwave experiments. Our work could enable reflectionless wave manipulation at grazing incidence with an ultrawide working bandwidth.
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1. INTRODUCTION
Reflectionless manipulation of electromagnetic waves is of great importance in numerous electromagnetic and photonic applications, such as beamforming and beam-steering antennas, focusing lenses, perfect absorbers, and so on. But such an on-demand wave control is not an easy task because it requires the systematic consideration of multiple wave channels including reflection, transmission, absorption, and even scattering. Notably, artificial electromagnetic materials denoted as metamaterials and metasurfaces [1–5], which go far beyond the physical limit of natural materials, provide an accessible platform for wave control in an almost arbitrary way. Previously, through tuning electric and magnetic responses with meta-atoms, reflectionless wave manipulation becomes possible [6–42]. In particular, Huygens’ metasurfaces [19–42], accomplished through engineering surface electric impedance and magnetic admittance simultaneously, have demonstrated unprecedented capabilities of controlling electromagnetic waves, including wavefront [19–26], polarization [27,28], and absorption [29–37]. Nevertheless, such an engineering process usually requires complicated meta-atoms and becomes much more challenging in the pursuit of ultrabroad operating bandwidth [29–38], especially for waves at grazing incidence, where the impedance/admittance becomes near-zero or divergent [43,44].
In this work, we demonstrate a class of reflectionless metasurfaces capable of wave control at grazing incident angles over an ultrabroad frequency range. The mechanism is completely different from the Huygens’ metasurfaces. The proposed metasurfaces are free of resonances and do not require unusual magnetic responses, rendering it possible to achieve ultrabroad operating bandwidth with simpler meta-atoms. The underlying physics lies in the anomalous generalized Brewster effect (AGBE), as the combination of the generalized Brewster effect (GBE) [43–53] and the anomalous Brewster effect (ABE) [54–56]. The proposed AGBE can prohibit reflections and simultaneously allow manipulation of wave propagation and absorption at arbitrarily large angles over an ultrawide frequency range, which is beyond the classical Brewster effect [57–59]. At grazing incidence, the classical Brewster effect requires extremely high-index dielectrics [Fig. 1(a)]. Moreover, the introduction of loss in permittivity would break the Brewster condition, leading to unavoidable reflections [Fig. 1(b)]. Interestingly, based on the principle of AGBE, ultrabroadband reflectionless metasurfaces, hereby termed as Brewster metasurfaces, are demonstrated to exhibit ultrabroadband reflectionless absorption of near-perfect efficiency at grazing incidence, without requiring high-index dielectrics [Fig. 1(c)]. By constructing such Brewster metasurfaces, we have verified the AGBE via full-wave simulations and microwave experiments. Our results demonstrate a mechanism of Brewster effect that is capable of realizing ultrabroadband near-perfect absorption in grazing incidence.
Fig. 1. (a) In the classical Brewster effect, the required relative permittivity ${\varepsilon _d}$ of the lossless dielectric material increases dramatically and diverges when the Brewster’s angle ${\theta _{\rm B}}$ is in the grazing region. (b) The classical Brewster effect breaks down when material loss is introduced to the permittivity of the dielectric material, ${\varepsilon _d}$, causing inevitable reflection. (c) Schematic graph of an AGBE-based Brewster metasurface exhibiting ultrabroadband reflectionless absorption of TM-polarized waves at grazing incidence, without requiring high-index dielectrics.
2. AGBE AND ITS UNDERLYING PHYSICS
We begin with the wave phenomena at an air–dielectric interface. We know that when a transverse magnetic (TM)-polarized wave (magnetic field parallel to the interface) is incident onto the interface, the reflection vanishes only at the classical Brewster’s angle ${\theta _{\rm B}} = {\tan ^{- 1}}\sqrt {{\varepsilon _d}}$, where ${\varepsilon _d}$ is the relative permittivity of the dielectric material. Such a non-reflection Brewster condition is solely determined by ${\varepsilon _d}$, and it requires extremely large ${\varepsilon _d}$ at grazing incidence, i.e., ${\theta _{\rm B}} \to 90^\circ$. To make the non-reflection condition flexible and relieve the requirement on ${\varepsilon _d}$, an ultrathin nonmagnetic film coating is added to the interface as illustrated in Fig. 2(a). It was found that when the film coating possesses an appropriate surface conductivity or sheet resistance, the non-reflection behavior could be observed for any angle and polarization of choice [47,48]. Such a reflectionless phenomenon with angular and polarization flexibility is termed as the GBE [43–53]. The condition of the GBE can be derived by imposing the reflection coefficient at the interface to be zero. Here, we assume that the thickness of the film coating ${t_c}$ is much smaller than the free-space wavelength ${\lambda _0}$. In this situation, we find that the required sheet resistance ${R_{s,c}}$, defined as $1/({{\sigma _c}{t_c}})$ [60,61] with ${\sigma _c}$ being the conductivity, for the realization of GBE for TM polarization is (see Supplement 1)
Fig. 2. (a) The schematic diagram of GBE at an air–dielectric interface, where an ultrathin CF coating is added. Reflection vanishes under the GBA ${\theta _{{\rm GB}}}$. (b) The required normalized sheet resistance ${R_{s,c}}/{Z_0}$ of the CF coating for the realization of GBE as a function of ${\theta _i}$ and ${\varepsilon _d}$. (c) Anisotropy is added to the dielectric substrate in (a) in a way such that the zero reflection under the GBA ${\theta _{{\rm GB}}}$ is maintained. (d) The required rotation angle $\alpha$ of the optical axis of the anisotropic substrate as a function of ${\theta _i}$ and ${\varepsilon _d}$ to maintain the GBE and zero reflection. (e) The AGBE realized by utilizing the reciprocity principle to the GBE model in (c). The AGBE allows ${\varepsilon _{{y^\prime}}}$-independent zero reflection and ${\varepsilon _{{y^\prime}}}$-controllable refraction under the AGBA ${\theta _{{\rm AGB}}} \equiv - {\theta _{{\rm GB}}}$. (f) Reflectance $R$ on a log scale of the model in (e) as a function of ${\theta _i}$. The anisotropic dielectric substrate has the fixed parameters of ${\varepsilon _{{x^\prime}}} = {\varepsilon _d} = 4$ and $\alpha = 28.88^\circ$. The CF coating has a sheet resistance of ${R_{s,c}} = 0.63{Z_0}$ and a thickness of ${t_c} = {\lambda _0}/1000$. (g) Simulated distributions of normalized magnetic field ${H_z}/{H_0}$ (color) and group velocity (arrows) of the four examples in (f) with different values of ${\varepsilon _{{y^\prime}}}$ at ${\theta _i} = - 75^\circ$.
We note that such a unique transition behavior for TM polarization was rarely noticed and reported before. For transverse-electric (TE) polarization, a gain medium is always needed when ${\varepsilon _d} \gt 1$. Therefore, to make the implementation practical, the permittivity of the substrate is usually considered to be lower than the that of the incident medium, in which case the CFs work [47].
With the CF-assisted GBE, the reflections can be completely eliminated under the generalized Brewster’s angle (GBA) ${\theta _{{\rm GB}}}$, which can be any chosen incident angle ${\theta _i} \gt {\theta _{\rm B}}$. However, there still lack degrees of freedom in controlling the refracted waves. Interestingly, inspired by the ABE [54–56], we find that material anisotropy can provide extra degrees of freedom without affecting the GBE condition. Figure 2(c) illustrates the geometrical interpretation. The dielectric substrate is assumed to be anisotropic with a relative permittivity tensor of $({\begin{array}{*{20}{c}}{{\varepsilon _{{x^\prime}}}}&{}\\{}&{{\varepsilon _{{y^\prime}}}}\end{array}})$ in the $x^\prime -y^\prime$ coordinate, which is rotated by an angle of $\alpha$ with respect to the $x-y$ coordinate. In order not to affect the GBE, the anisotropic substrate is assumed to satisfy the following conditions:
Clearly, Eq. (2) is irrelevant to ${\varepsilon _{{y^\prime}}}$. Such a ${\varepsilon _{{y^\prime}}}$ independence brings an extra degree of freedom that will not affect the GBE—in other words, the ${\varepsilon _{{y^\prime}}}$-independent zero reflection can be obtained under the GBA ${\theta _{{\rm GB}}}$. Nevertheless, the ${\varepsilon _{{y^\prime}}}$ cannot control the refracted wave because its direction is perpendicular to the electric field of the refracted wave. Strikingly, interesting things happen when the reciprocity principle [65] is applied. In Fig. 2(e), we consider flipping the incident wave from the left side (${\theta _i} \gt 0$) to the right side (${\theta _i} \lt 0$). According to the reciprocity principle, the reflection coefficients are exactly the same under incident angles of ${\pm}{\theta _i}$, that is, $R({- {\theta _i}}) = R({{\theta _i}})$ [65]. This leads to an intriguing conclusion that the ${\varepsilon _{{y^\prime}}}$-independent zero reflection can be extended from the ${\theta _{{\rm GB}}}$ to ${-}{\theta _{{\rm GB}}}$, that is,
For numerical verification, we set ${\varepsilon _{{x^\prime}}} = {\varepsilon _d} = 4$, $\alpha = 28.88^\circ$, ${t_c} = {\lambda _0}/1000$, and ${R_{s,c}} = 0.63{Z_0}$ based on Eqs. (1) and (2), so as to obtain a GBA at $75^\circ$. Figure 2(f) shows the calculated reflectance $R$ on a log scale as a function of the incident angle ${\theta _i}$ for different values of ${\varepsilon _{{y^\prime}}}$. As expected, zero reflection occurs at ${\theta _i} = \pm {\theta _{{\rm GB}}} = \pm 75^\circ$, irrespective of whether ${\varepsilon _{{y^\prime}}}$ is a positive, negative, or complex value. Here we note that the zero-reflection angle can be flexibly chosen through adjusting ${R_{s,c}}$ and $\alpha$ according to Eqs. (1) and (2), provided that its absolute value is larger than the classical Brewster’s angle ${\theta _{\rm B}} = 63.43^\circ$. Thus, the ${\varepsilon _{{y^\prime}}}$-independent reflectionless phenomena can be observed near grazing incidence, i.e., $| {{\theta _i}} | \to 90^\circ$. More examples are shown in Supplement 1.
Here we would like to emphasize that although the reflections in both cases of ${\theta _i} = \pm {\theta _{{\rm GB}}}$ disappear in the same way, the refractive behaviors are fundamentally different. As discussed in the GBE model [Fig. 2(c)], the refracted wave under ${\theta _i} = {\theta _{{\rm GB}}}$ cannot “see” ${\varepsilon _{{y^\prime}}}$, and therefore cannot be influenced by ${\varepsilon _{{y^\prime}}}$ in any way. Such a GBE model hence cannot provide the flexible manipulation of the refracted wave through engineering ${\varepsilon _{{y^\prime}}}$. However, when flipping the incident wave to the right side with ${\theta _i} = - {\theta _{{\rm GB}}}$, the electric field of refracted wave will no longer be normal to the direction of ${\varepsilon _{{y^\prime}}}$, bestowing tunable refractive behaviors by changing ${\varepsilon _{{y^\prime}}}$, whereas zero reflection is guaranteed by reciprocity at the same time. This remarkable property is numerically demonstrated in Fig. 2(g) using the software COMSOL Multiphysics. The models are the same as those in Fig. 2(f). The color and arrows denote, respectively, the magnetic field distribution and group velocity under ${\theta _i} = - 75^\circ$. It is seen that the refractive behaviors vary significantly with ${\varepsilon _{{y^\prime}}}$, while the reflection remains zero. The angle of refraction can be flexibly tuned from positive (${\varepsilon _{{y^\prime}}} = 4$) to negative (${\varepsilon _{{y^\prime}}} = - 10$). Reflectionless absorption can be achieved when the ${\varepsilon _{{y^\prime}}}$ has an imaginary part (${\varepsilon _{{y^\prime}}} = 4 + 10i$ or ${-}10 + 10i$). These results clearly manifest the remarkable ability of reflectionless wave manipulation including the refraction angle and absorption through changing ${\varepsilon _{{y^\prime}}}$ under ${\theta _i} = - {\theta _{{\rm GB}}}$, completely different from the GBE model under ${\theta _i} = {\theta _{{\rm GB}}}$. In this sense, we term the unique reflectionless phenomenon under ${\theta _i} = - {\theta _{{\rm GB}}}$ as AGBE, and the angle ${\theta _{{\rm AGB}}} \equiv - {\theta _{{\rm GB}}}$ as anomalous generalized Brewster’s angle (AGBA).
In short, the AGBE, as a new extension of the classical Brewster effect, is capable of reflectionless wave manipulation including independently tunable refraction and absorption, and simultaneously is flexible in choosing non-reflection angles, even near grazing incidence without requiring high-index dielectrics. The conditions of AGBE are elaborated in Eqs. (1)–(3), which can also be derived straightforwardly based on the anisotropic model in Fig. 2(e) in three steps. The first step is to derive the reflection coefficient by taking into account both the anisotropic substrate and the ultrathin film coating. The second step is to impose the reflection coefficient to be zero and to find out the zero-reflection condition. Finally, upon imposing the zero-reflection condition to be ${\varepsilon _{y{\rm ^\prime}}}$ independent, we can arrive at the AGBE condition [i.e., Eqs. (1)–(3)]. The analytical proof is summarized in Supplement 1.
Fig. 3. (a) The normalized decay rate $| {{\rm Im}({{k_y}})} |/{k_0}$ as a function of ${\rm Re}({{\varepsilon _{{y^\prime}}}})$ and ${\rm Im}({{\varepsilon _{{y^\prime}}}})$ for various ${\varepsilon _{x{\rm ^\prime}}}$ ($= {\varepsilon _d}$) and ${\theta _i}$ ($= {\theta _{{\rm AGB}}}$). The rotation angle $\alpha$ is changed according to Eq. (2), so that the AGBE condition is always satisfied. The ${\theta _i}$ is fixed at ${-}70^\circ$ in the upper three panel graphs, while the ${\varepsilon _{x{\rm ^\prime}}}$ is fixed at 10 in the right three panel graphs. (b) Upper panel graph: real (left) and imaginary (right) parts of ${k_y}/{k_0}$ based on the dispersion relation of the anisotropic dielectric with ${\varepsilon _{x{\rm ^\prime}}} = 4$, ${\varepsilon _{{y^\prime}}} = - 1.13 + 0.05i,$ and $\alpha = 28.02^\circ$. The solid and dashed lines denote the forward- and backward-propagating modes, respectively. The black dotted lines correspond to the position of ${k_x}/{k_0} = \sin ({- 70^\circ})$. Lower panel graph: simulated magnetic-field distribution for a 0.06 ${\lambda _0}$-thick Brewster metasurface suspended in air under the illumination of a TM-polarized Gaussian beam with ${\theta _i} = - 70^\circ$. The Brewster metasurface comprises a 0.01 ${\lambda _0}$-thick CF coating with ${R_{s,c}} = 1.52{Z_0}$ and a 0.05 ${\lambda _0}$-thick anisotropic dielectric slab with ${\varepsilon _{x{\rm ^\prime}}} = 4$, ${\varepsilon _{{y^\prime}}} = - 1.13 + 0.05i,$ and $\alpha = 28.02^\circ$.
3. BREWSTER METASURFACES FOR HIGH-EFFICIENCY ABSORPTION: EXTRAORDINARILY LARGE DECAY AND ULTRABROADBAND REFLECTIONLESS ABSORPTION
As an important consequence of the AGBE, we can harness complex-valued ${\varepsilon _{{y^\prime}}}$ to arbitrarily control wave absorption with no reflection at large incident angles. This suggests that by introducing the concept of AGBE to metasurfaces, termed as Brewster metasurfaces here, the rare property of reflectionless characteristics and extra degrees of freedom to control transmission waves could be obtained. In the following, we will demonstrate two types of Brewster metasurfaces capable of high-efficiency absorption near grazing incidence, including the extraordinarily large decay and ultrabroadband reflectionless absorption.
Before going to actual meta-atoms of the Brewster metasurfaces, we first investigate the absorption efficiency of the anisotropic dielectric substrate with a complex-valued ${\varepsilon _{{y^\prime}}}$ based on the effective-medium model in Fig. 2(e). For a TM-polarized plane wave, the magnetic field inside the anisotropic substrate can be expressed as ${H_z} = {H_0}{e^{i({{k_x}x + {k_y}y}) - i\omega t}}$, where ${H_0}$ is the magnetic-field amplitude of incidence, and $\omega$ is the angular frequency. ${k_x}$ and ${k_y}$ are, respectively, the $x$ and $y$ components of the wave vector inside the anisotropic substrate, which are governed by the following dispersion relation:
When considering a wave incident from free space at ${\theta _i} = {\theta _{{\rm AGB}}}$, we have ${k_x} = {k_0}\sin {\theta _{{\rm AGB}}}$. From Eq. (4), a complex-valued ${k_y}$ would be induced if the ${\varepsilon _{{y^\prime}}}$ is a complex value. Here we assume that the imaginary part of ${\varepsilon _{{y^\prime}}}$ is positive, i.e., ${\rm Im}({{\varepsilon _{{y^\prime}}}}) \gt 0$, which indicates that the anisotropic dielectric substrate is purely lossy. In this situation, the wave inside the substrate would decay exponentially at a rate of $| {{\rm Im}({{k_y}})} |$ (the absolute value of the imaginary part of ${k_y}$), and therefore this quantity is defined as the decay rate here. Since the reflection is prohibited by the AGBE, a larger decay rate indicates the higher absorption efficiency. To obtain high-efficiency absorption, the decay rate should be as large as possible.
According to Eqs. (2–4), the decay rate $| {{\rm Im}({{k_y}})} |$ is found to rely on four independent variables, i.e., ${\theta _i}$ ($= {\theta _{{\rm AGB}}}$), ${\varepsilon _{{x^\prime}}}$ ($= {\varepsilon _d}$), ${\rm Re}({{\varepsilon _{{y^\prime}}}}),$ and ${\rm Im}({{\varepsilon _{{y^\prime}}}})$. To identify the parameter combinations that yield the maximal decay rate, we plot the normalized decay rate $| {{\rm Im}({{k_y}})} |/{k_0}$ as a function of the four variables in Fig. 3(a). The rotation angle $\alpha$ is changed according to Eq. (2) so that the AGBE condition is always satisfied. The ${\theta _i}$ (or ${\theta _{{\rm AGB}}}$) is fixed at ${-}70^\circ$ in the upper three panel graphs, while the ${\varepsilon _{{x^\prime}}}$ (or ${\varepsilon _d}$) is fixed at 10 in the right three panel graphs. In each panel graph, we can find out a point of $| {{\rm Im}({{k_y}})} |$ approaching infinity occurring for one particular combination of ${\rm Re}({{\varepsilon _{{y^\prime}}}})$ and ${\rm Im}({{\varepsilon _{{y^\prime}}}})$ (white dot), which is found to be very close to the line of ${\rm Im}({{\varepsilon _{{y^\prime}}}}) = 0$. This indicates an extraordinary phenomenon—the decay rate $| {{\rm Im}({{k_y}})} |$ increases with decreasing ${\rm Im}({{\varepsilon _{{y^\prime}}}})$ and tends to be infinitely large when ${\rm Im}({{\varepsilon _{{y^\prime}}}}) \to 0$. These results violate “common sense” that in materials with negligibly small losses the wave dissipation is negligible.
Fig. 4. (a) Schematic diagram of a Brewster metasurface exhibiting ultrabroadband reflectionless absorption. The Brewster metasurface consists of a CF coating (sheet resistance ${R_{s,c}}$) and a tilted CF array (sheet resistance ${R_{s,a}}$) embedded in a dielectric host (relative permittivity ${\varepsilon _d}$). The tilted CF array can be homogenized as a uniform effective anisotropic dielectric. (b) The normalized decay rate $| {{\rm Im}({{k_y}})} |/{k_0}$ of the wave inside the tilted CF array as the function of ${\theta _i}$ ($= {\theta _{{\rm AGB}}}$) and ${R_{s,a}}/{Z_0}$ based on the effective-medium model at 10 GHz. The relevant parameters are ${\varepsilon _d} = 2.55$, $a = 5{\rm mm},$ and ${t_a} = 0.1\;{\rm mm} $. The rotation angle $\alpha$ is adjusted with varying ${\theta _i}$ according to Eq. (2), so that the AGBE condition is satisfied everywhere. The star marks denote the optimal ${R_{s,a}}$ that yields the maximal $| {{\rm Im}({{k_y}})} |$ at ${-}70^\circ$, ${-}80^\circ ,$ and ${-}85^\circ$. (c) Reflectance $R$ (red) and absorptance $A$ (blue) as the function of the working frequency for three examples of Brewster metasurfaces chosen from (b) as marked by stars. The lines and symbols denote the results based on the effective-medium models and the actual metasurfaces, respectively. The three Brewster metasurfaces have the same thickness of 15 mm. (d) Evaluation of contribution of the CF coating and the tilted CF array to the absorption of incident energy for the three Brewster metasurfaces.
The physical origin behind such an extraordinary phenomenon could be explained as follows. Based on Eq. (4), the expression of ${k_y}$ can be derived, which has a denominator of ${\varepsilon _{{yy}}}$, i.e., ${\varepsilon _{{x^\prime}}}{\sin ^2}\alpha + {\varepsilon _{{y^\prime}}}{\cos ^2}\alpha$ or $({1 - {\varepsilon _{{y^\prime}}}/{\varepsilon _{{x^\prime}}}}){\sin ^2}{\theta _{{\rm AGB}}} + {\varepsilon _{{y^\prime}}}$. Infinitely large ${k_y}$ will occur when the denominator equals zero, that is,
Such an extraordinarily large decay rate can endow Brewster metasurfaces with the exceptional ability of extremely high-efficiency absorption. For a numerical demonstration, we consider a 0.06 ${\lambda _0}$-thick Brewster metasurface comprising a 0.01 ${\lambda _0}$-thick CF coating and a 0.05 ${\lambda _0}$-thick anisotropic dielectric slab with ${\varepsilon _{{x^\prime}}} = 4$ and ${\varepsilon _{{y^\prime}}} = - 1.13 + 0.05i$. According to Eqs. (1) and (2), we set the sheet resistance of the CF coating to be $1.52{Z_0}$ and the rotation angle of optical axis of the anisotropic dielectric to be $28.02^\circ$, so as to obtain an AGBA at ${-}70^\circ$. The lower panel graph in Fig. 3(b) presents the simulated magnetic-field distribution for the Brewster metasurface suspended in air under the illumination of a TM-polarized Gaussian beam with ${\theta _i} = - 70^\circ$. As expected, the incident wave is totally absorbed by the Brewster metasurface, with no reflection or transmission. We note that the skin depth of wave in the anisotropic dielectric slab is exceptionally small ($\lt\! 0.01{\lambda _0}$, see the inset) due to the extraordinarily large decay rate. These results clearly manifest the extremely high-efficiency absorption by the Brewster metasurface with deep-subwavelength thickness at large incident angles.
The extraordinarily large decay requires appropriately tilted hyperbolic dispersion, and therefore, the operating bandwidth of the Brewster metasurfaces based on this principle is inherently narrow. In the following, we would like to show a kind of simply designed Brewster metasurfaces exhibiting ultrabroadband reflectionless absorption at large incident angles.
Figure 4(a) illustrates a Brewster metasurface comprising a CF coating and a titled CF array embedded in an isotropic dielectric host (relative permittivity ${\varepsilon _d}$). The rotation angle is $\alpha$, and the sheet resistance of the tilted CFs is ${R_{s,a}}$, which generally differs from that of the coating CF. This is because the coating CF is responsible for realizing the GBE to eliminate reflection, satisfying Eq. (1), while the tilted CFs are exploited to introduce effective anisotropy and achieve the maximal decay rate at the AGBA based on Eq. (4). The separation distance between two adjacent titled CFs is $a$, which is much larger than the of the CF thickness ${t_a}$, but smaller than the wavelength in the dielectric host to avoid diffraction. Under such circumstances, the tilted CF array can be approximately homogenized as an effective anisotropic dielectric with ${\varepsilon _{{x^\prime}}} \approx {\varepsilon _d}$ and ${\varepsilon _{{y^\prime}}} = {\varepsilon _d} + i\gamma$, which are, respectively, the effective permittivities normal and parallel to the tilted CFs [54]. Here, $\gamma = ({{Z_0}/{R_{s,a}}})/({{k_0}a\cos \alpha})$. Obviously, the effective parameter ${\varepsilon _{{x^\prime}}}$ is frequency independent, and therefore, the required ${R_{s,c}}$ and $\alpha$ for the realization of AGBE are also frequency independent according to Eqs. (1–3). Such frequency independence endows the Brewster metasurface with an ultrabroad working bandwidth. We note that any negative angle whose absolute value is larger than ${\tan ^{- 1}}\sqrt {{\varepsilon _d}}$ can be chosen as the AGBA ${\theta _{{\rm ABG}}}$ of the Brewster metasurface.
Fig. 5. (a) Schematic graph of a microwave Brewster metasurface consisting of ITO films and a PMMA slab. (b) Photograph of the fabricated Brewster metasurface. (c) Calculated reflectance on a log scale ${\rm log}(R)$ at the air–metasurface interface as a function of the working frequency and incident angle. The inset illustrates the numerical setup for the reflection computation. The white dashed line denotes the location of AGBA ${\theta _{{\rm AGB}}} = - 70^\circ$. (d) Photograph of the experimental setup for the measurement of far-field power radiation patterns. (e) Measured normalized far-field radiation patterns in free space (dashed lines) and on the metasurface sample (solid lines) at 8 GHz (red), 10 GHz (blue), and 12 GHz (green). (f) Simulated (lines) and experimentally measured (symbols) absorptance $A$ (red) and reflectance $R$ (blue) of the Brewster metasurface under the AGBA ${-}70^\circ$. (g) Photograph of the experimental setup for the measurement of near-field electric fields. (h) Upper panel graphs: measured electric-field distributions at 8 GHz (left), 10 GHz (middle), and 12 GHz (right) under the AGBA ${-}70^\circ$. Lower panel graphs show the corresponding electric-field distributions in simulations.
To gain high-efficiency absorption, we first numerically explore the absorption efficiency of the tilted CF array. Here we consider positive ${R_{s,a}}$, thus we have $\gamma \gt 0$, indicating that the effective anisotropic dielectric is purely lossy. We set ${\varepsilon _d} = 2.55$, $a = 5\;{\rm mm} $, ${t_a} = 0.1\;{\rm mm} $, and calculate the normalized decay rate $| {{\rm Im}({{k_y}})} |/{k_0}$ as a function of ${\theta _i}$ ($= {\theta _{{\rm ABG}}}$) and ${R_{s,a}}/{Z_0}$ at 10 GHz based on the effective-medium model as plotted in Fig. 4(b). The rotation angle $\alpha$ is adjusted with varying ${\theta _i}$ according to Eq. (2), so that the AGBE condition is satisfied everywhere. We see that at any angle of choice, there exists an optimal sheet resistance ${R_{s,a}}$ that yields the maximal decay rate. The optimal sheet resistance is found to be around $0.3{Z_0}$, insensitive to the angle. Possessing the optimal sheet resistance, the Brewster metasurface can realize ultrabroadband reflectionless absorption with high efficiency. For verification, we choose three examples of Brewster metasurfaces from Fig. 4(b) (marked by stars), which are designed to possess the AGBA at ${-}70^\circ$, ${-}80^\circ ,$ and ${-}85^\circ$, respectively. From Fig. 4(b), the optimal sheet resistance of the tilted CFs is found to be $0.3{Z_0}$, $0.29{Z_0},$ and $0.29{Z_0}$, respectively. Figure 4(c) shows the reflectance $R$ (red) and absorptance $A$ (blue) as the function of the working frequency for the three Brewster metasurfaces having the same thickness of 15 mm. The lines and symbols denote the results based on the effective-medium models and the actual metasurfaces, respectively, showing excellent agreement. We see that zero reflection and near-complete absorption are obtained within the frequency range of 5–15 GHz for all the three Brewster metasurfaces, confirming the ultrabroadband reflectionless absorption with near-perfect efficiency. We note that the absorption is slightly decreased at low frequencies because of the finite thickness of the Brewster metasurface. Since the reflection is prohibited by the AGBE, the absorption performance at low frequencies can be greatly improved through increasing the metasurface thickness.
The high-efficiency absorption performance originates from not only the tilted CF array but also the CF coating. In the three Brewster metasurfaces, the sheet resistance of the CF coating ${R_{s,c}}$ is set to be $1.05{Z_0}$, $0.27{Z_0}$, and $0.11{Z_0}$ according to Eq. (1). Figure 4(d) shows the evaluation of contribution of the CF coating and tilted CF array. We see that the incident energy is mostly dissipated inside the tilted CF array at relatively small angles, while is mainly absorbed by the CF coating for angles approaching ${-}90^\circ$.
4. EXPERIMENTAL VERIFICATION OF AGBE AND MICROWAVE BREWSTER METASURFACE
In the following, we demonstrate experimental realization of the Brewster metasurface exhibiting ultrabroadband reflectionless high-efficiency absorption in the microwave regime. Figure 5(a) illustrates the microwave Brewster metasurface. An ITO film is coated on the upper surface of a slab of polymethyl methacrylate (PMMA, relative permittivity ${\varepsilon _d} = 2.55$), inside which a periodic array of tilted ITO films (lattice constant $a = 5\;{\rm mm} $) are inserted. The ITO films are very thin (thickness ${\sim}0.1\;{\rm mm} $) and can serve as CFs at microwaves. In experiments, a metasurface sample having a length of 1.2 m, a height of 0.2 m, and a thickness of 15 mm is fabricated [Fig. 5(g)]. The photograph and the detailed parameters of the sample are shown in Fig. 5(b). The microwave Brewster metasurface is designed to possess an AGBA ${\theta _{{\rm AGB}}}$ at ${-}70^\circ$. According to Eq. (1), we find that the required sheet resistance of the coating CF should be ${R_{s,c}} = 1.05{Z_0} \approx 396{\Omega}$, and therefore an ITO film with ${R_{s,c}} = 400{\Omega}$ is utilized as the coating ITO film in the experiment. According to Eq. (2), we set the rotation angle $\alpha$ to be $36^\circ$. Based on the analysis in Fig. 4(b), the optimal sheet resistance of the titled ITO films is around $0.3{Z_0} \approx 113{\Omega}$ at 10 GHz. In the experiment, ITO films with ${R_{s,a}} = 130{\Omega}$ are exploited as the tilted ITO films [Fig. 5(b)].
To confirm that the fabricated Brewster metasurface indeed possesses the AGBE-induced ultrabroadband reflectionless characteristic, we first calculate the reflectance on a log scale ${\rm log}(R)$ at the air–metasurface interface as a function of the working frequency and incident angle as plotted in Fig. 5(c). The inset illustrates the numerical setup for the reflection computation. Clearly, zero reflection emerges at the predetermined AGBA ${\theta _{{\rm AGB}}} = - 70^\circ$ from dc to gigahertz (GHz) frequencies, confirming the occurrence of AGBE and its ultrabroadband reflectionless characteristic. We note that there is no lower limit for the working frequency.
In experiments, we first measured the far-field radiation patterns using the experimental setup shown in Fig. 5(d). An emitting horn antenna is placed 1 m away from the metasurface sample to generate the incident waves. A receiving horn antenna placed at the same distance is used to measure the radiation pattern. The receiving horn antenna can be freely moved around the sample so that we could receive scattering signals in all directions. Both the emitting and receiving horn antennas are connected to a vector network analyzer (KEYSIGHT PNA Network Analyzer N5224B) for data acquisition. The measured normalized radiation patterns are plotted in Fig. 5(e). The solid lines denote the case in the presence of the metasurface sample at 8 GHz (red), 10 GHz (blue), and 12 GHz (green) under the AGBE (i.e., ${\theta _i} = - 70^\circ$). The dashed lines denote the reference patterns in free space (i.e., in the absence of sample). Clearly, the scatterings by the metasurface sample in all directions, including the directions of specular reflection and transmission, are very weak for all the three frequencies (see the insets), indicating the broadband high absorption of the metasurface. We note that disordered scattering patterns on the metasurface sample may be caused by fabrication errors and random background noises. Based on the radiation patterns, we further evaluate the reflectance $R$ (blue symbols) and absorptance $A$ (red symbols) of the metasurface sample as plotted in Fig. 5(f). The reflectance is evaluated through integrating far-field power for all directions on the source side, and the evaluation of absorptance takes into account the far-field power for all directions on both the source and transmission sides. For comparison, the $R$ and $A$ are also calculated in simulations as denoted by the solid lines. We see that both simulation and experimental results show near-zero reflection and high absorption over a broad frequency band, confirming the AGBE-induced broadband high absorption in the microwave Brewster metasurface. We note that the deviation between the simulation and experimental results mainly attributes to the beam spread along the propagation path in experiments.
Furthermore, we measured the near-field electric fields before and after the metasurface sample using the experimental setup shown in Fig. 5(g). A probing antenna is exploited to probe the near-field electric fields on scanning areas ($80 \times 20\,{{\rm cm}^2}$ each), which are located on the central plane of the sample. The measured electric-field distributions at 8 GHz, 10 GHz, and 12 GHz under ${\theta _i} = - 70^\circ$ presented in upper panel graphs in Fig. 5(h) coincide well with the simulation results (lower panel graphs). We see that both the reflection and transmission are very weak at all the three frequencies. These results further demonstrate the AGBE-induced broadband reflectionless high-efficiency absorption of the fabricated microwave Brewster metasurface.
Fig. 6. (a) Schematic layout of the microwave Brewster metasurface integrated with a PEC backreflector on the bottom surface. (b) Photograph of the fabricated Brewster metasurface. An aluminum film is exploited as the PEC backreflector. (c) Simulated absorptance as the function of the working frequency and incident angle. The area bounded by the black dashed lines denotes the region with $A \gt 0.9$. The white dashed lines denote the angles of ${\pm}70^\circ$. (d) Measured normalized far-field radiation patterns on an aluminum film (dashed lines) and on the metasurface sample (solid lines) on the source side at 8 GHz (red), 10 GHz (blue), 12 GHz (green) under the GBA $70^\circ$ (left) and AGBA ${-}70^\circ$ (right). (e) Simulated (lines) and experimentally measured (symbols) absorptance of the Brewster metasurface under the GBA $70^\circ$ (red) and AGBA ${-}70^\circ$ (blue). (f) Measured (left) and simulated (right) electric-field distributions under the GBA $70^\circ$ (upper) and AGBA ${-}70^\circ$ (lower) at 10 GHz.
We note that the absorption performance of the microwave Brewster metasurface is angularly asymmetric because of the coexistence of the GBE and AGBE. The GBE occurs at positive angles, in which case the titled ITO films will not contribute to the absorption as the electric field of the wave inside the metasurface is normal to these ITO films. Consequently, the wave absorption caused by the GBE at positive angles is much lower than that induced by the AGBE at negative angles, thus leading to the angular asymmetry (see more results in Supplement 1).
Interestingly, we find out a simple approach to extend the high-efficiency absorption at the AGBA ${\theta _{{\rm AGB}}}$ to the GBA ${\theta _{{\rm GB}}}$ to greatly improve the absorption performance. As illustrated in Fig. 6(a), a perfect electric conductor (PEC) backreflector is integrated on the bottom surface of the Brewster metasurface. When a wave is incident under the GBA ${\theta _{{\rm GB}}}$, the transmitted wave will be reflected back by the PEC backreflector and then absorbed by the tilted ITO films. In the experiment, an aluminum film [Fig. 6(b)] is utilized to serve as the PEC backreflector. Figure 6(c) presents the simulated absorptance as the function of the working frequency and incident angle. The area bounded by the black dashed lines denotes the region with $A \gt 0.9$. We see that the angular asymmetry in absorption is removed. Near-perfect absorption occurs at both the GBA ${\theta _{{\rm GB}}} = 70^\circ$ and the AGBA ${\theta _{{\rm AGB}}} = - 70^\circ$ in the frequency range of 5–15 GHz (white dashed lines). Moreover, wide-angle (${-}79^\circ -79^\circ$) high absorption ($A \gt 0.9$) is observed around 8.2 GHz. The high absorption at small incident angles mainly attributes to the approximate satisfaction of the Dallenbach absorber condition [66], in which case the waves reflected from the upper surface and the bottom PEC are nearly $\pi$ out of phase due to the presence of the PMMA slab. Consequently, the reflected waves destructively interfere with each other, and almost all the incident waves are absorbed by the ITO films inside the PMMA slab.
In experiments, the far-field radiation patterns of the metasurface sample (solid lines) on the source side are measured under the GBA $70^\circ$ (left) and AGBA ${-}70^\circ$ (right) at 8 GHz (red), 10 GHz (blue), and 12 GHz (green), as plotted by solid lines in Fig. 6(d). These cases are marked by stars in Fig. 6(c). Compared with the reference patterns on an aluminum film (dashed lines), the radiation patterns on the metasurface sample are very weak at both the GBA and AGBA for all the three frequencies (see the insets), demonstrating the high-efficiency absorption of the metasurface sample. Figure 6(e) further compares the simulated (lines) and experimentally measured (symbols) absorptance of the metasurface sample at $70^\circ$ (red) and ${-}70^\circ$ (blue), showing high absorption in the frequency range of 5–15 GHz at both angles. Such high absorption is further confirmed by the measured (left) and simulated (right) electric-field distributions at 10 GHz as shown in Fig. 6(f). The upper and lower panel graphs correspond to the cases of ${\theta _i} = {\theta _{{\rm GB}}} = 70^\circ$ and ${\theta _i} = {\theta _{{\rm AGB}}} = - 70^\circ$, respectively. Since there is no transmission, the total absorption is associated with a null of reflection. Figure 6(f) shows almost no patterns induced by the interference of reflected and incident waves, indicating that nearly all the incident waves are absorbed by the Brewster metasurface. These results demonstrate that the PEC backreflector can remove the angular asymmetry and greatly improve the absorption performance of the microwave Brewster metasurface.
We would like to note that besides the PEC backreflector, other strategies can also be employed to greatly improve the absorption performance of the microwave Brewster metasurface, such as the utilization of more suitable materials and optimal structures. Actually, the Brewster metasurface is quite flexible in the material selection. Any other material possessing low chromatic dispersion and low material loss can be used as the dielectric host, and other conductive materials like graphite can be used as CFs to achieve absorption. Therefore, it is possible to obtain better absorption performance in thinner microwave Brewster metasurfaces.
5. OPTICAL BREWSTER METASURFACE
The principle of AGBE and the AGBE-based Brewster metasurfaces is universal. Besides microwave frequencies, it is possible to extend to higher frequency regimes, such as terahertz, infrared, and even visible regions. In the following, we show an example of optical Brewster metasurface composed of Cr films and a silica (${\rm SiO}_2 $) slab, as illustrated in Fig. 7(a). A 18 nm Cr film is coated on the upper surface of a 400 nm ${\rm SiO}_2 $ slab, inside which tilted Cr films ($t = 7\;{\rm nm} $, $\alpha = 42.2^\circ$, $a = 125\;{\rm nm} $) are embedded. The coating Cr film is responsible for realizing the GBE to eliminate reflection, and the tilted Cr films are exploited to introduce effective anisotropy and achieve the maximal decay rate at the AGBA. These parameters are optimized based on the AGBE condition [i.e., Eqs. (1)–(3)], such that the optical Brewster metasurface exhibits an AGBA at ${\theta _{{\rm AGB}}} = - 80^\circ$. For verification, in Fig. 7(b) we simulate the TM-polarized light of ${\lambda _0} = 1000\;{\rm nm}$ incident onto the metasurface suspended in air under ${\theta _i} = {\theta _{{\rm AGB}}} = - 80^\circ$, showing very weak reflection and transmission. The absorptance is found to be higher than 0.95, manifesting the high-efficiency absorption of the optical Brewster metasurface.
Moreover, the absorptance of the optical Brewster metasurface as the function of the incident angle and working wavelength is plotted in Fig. 7(c). The dispersive parameters of the Cr and ${\rm SiO}_2 $ are adopted from [67,68], respectively. Near-unity absorptance (${\geq} 0.9$) from 515 to 1600 nm is observed nearby the AGBA ${-}80^\circ$. The absorption performance can be further improved when the metasurface is placed on a Cr substrate as shown in Fig. 7(d). This is because the transmitted light under the GBA $80^\circ$ would be reflected back by the Cr substrate and then be absorbed by the metasurface. As a result, the absorption becomes symmetric with respect to the ${\theta _i}$ and is significantly improved. Figure 7(d) shows near-unity absorptance (${\geq} 0.9$) for angles around ${\pm}80^\circ$ in the spectrum from 515 to 1600 nm, demonstrating the broadband high-efficiency absorption of the optical Brewster metasurface for light near grazing incidence.
6. DISCUSSION AND CONCLUSION
We note that the mechanism of reflectionless metasurfaces based on the AGBE is fundamentally different from that of previously proposed reflectionless metasurfaces. Traditionally, simultaneous engineering of electric and magnetic responses is required to achieve impedance matching to eliminate reflections on metasurfaces, which usually requires complicated resonant meta-atoms such as multilayer metastructures [7–15,23,39,40], vertical split-ring-resonators or helixes [19,26,28–33,38] and dielectric particles with engineered electric and magnetic responses [34–37]. Consequently, such resonance-assisted reflectionless metasurfaces usually work only for normally incident waves within a limited operating frequency range. In contrast, the proposed Brewster metasurfaces are free of resonances and unusual magnetic responses for TM polarization. This advantage renders it possible to realize reflectionless wave manipulation with an ultrawide working bandwidth, based on simpler and fabrication-friendly meta-atoms. Moreover, the Brewster metasurface is flexible in choosing the angle of non-reflection. It can realize ultrabroadband zero reflection near grazing incidence, which is quite challenging with traditional approaches because the impedance becomes near-zero or divergent when incident angles approach ${\pm}90^\circ$.
Fig. 7. (a) Schematic graph of an optical Brewster metasurface consisting of Cr films and a ${\rm SiO}_2 $ slab. The relevant parameters are $d = 400\;{\rm nm} $, $\alpha = 42.2^\circ ,$ and $a = 125\;{\rm nm} $. The thicknesses of the coating Cr film and the tilted Cr films are 18 nm and 7 nm, respectively. (b) Simulated intensity of TM-polarized light incident onto the optical Brewster metasurface suspended in air. The wavelength is 1000 nm, and the incident angle is ${-}80^\circ$. [(c) and (d)] Simulated absorptance as the function of the incident angle and working wavelength when the metasurface is (c) suspended in air, (d) placed on a Cr substrate.
We would like to emphasize that the AGBE has significant advantages over GBE and ABE under the circumstance of grazing incidence. The GBE [43–53] lacks the capability of introducing efficient loss as well as the degrees of freedom to independently tune the refracted waves without causing reflection as we have shown in Figs. 2(a) and 2(c). The ABE [54–56], on the other hand, require dielectrics with extremely large refractive index when approaching the grazing incidence. Interestingly, as the integration of the previous two effects, the AGBE shows a way to overcome the disadvantages of both. The non-reflection angle, i.e., the AGBA, can be arbitrarily large, and the system does not need high-index dielectrics. Moreover, the AGBE-based absorbers are more efficient because both the coating and tilted CFs contribute to absorption. We also note that the ultrabroadband reflectionless high-efficiency absorption of the AGBE-based absorbers cannot be realized with the classical Brewster effect [57–59] although it is intrinsically broadband. The introduction of loss in permittivity would break the Brewster condition, and reflection unavoidably appears and increases dramatically with the loss. In addition, extremely high-index dielectrics are required at grazing incidence, thus hindering its practical applications in high-efficiency absorbers. The detailed comparison is elaborated in Supplement 1. These reasons explain well why broadband reflectionless absorption at grazing incidence has not been reported before in the literature, to the best of our knowledge.
In summary, we unveil a principle of AGBE via the integration of GBE and ABE. This principle bestows Brewster metasurfaces exhibiting ultrabroadband reflectionless absorption with near-perfect absorption at grazing incident angles, which have been substantiated by full-wave simulations and microwave experiments. Our findings show a path toward high-efficiency wave manipulation near grazing incidence with an ultrawide working bandwidth.
Funding
National Key Research and Development Program of China (2020YFA0211300, 2017YFA0303702); National Natural Science Foundation of China (11974176, 12174188, 12274314, 92050104); China Postdoctoral Science Foundation (2021M702401, SYG202039); Natural Science Foundation of Jiangsu Province (BK20221240, BK20221354).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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