1. Introduction

The Brewster effect, which consists in the vanishment of the reflection of TM-polarized waves at the interface between two dielectric media at a specific incidence angle [1], has a history of more than 200 years. In 1808, Malus observed that unpolarized light becomes polarized upon reflection under a particular angle at the surface of water [2]. Seven years later, Brewster experimentally showed that this angle was equal to the inverse tangent of the ratio the refractives indices of the two media [3]. Another six years later, in 1821, Fresnel completed the understanding of the phenomenon using a mechanical model of the interface system and derived the eponymic reflection and transmission coefficients [4], which embed the Brewster effect. Finally, these formulas were generalized to magneto-electric materials, which support either TM-polarization or TE-polarization Brewster transmission, with both possible only for normal incidence, by Giles and Wild [5].

The recent advent of metasurfaces has created novel opportunities to extend the Brewster effect. Metasurfaces allow indeed unprecedented manipulations of electromagnetic waves [6,7]; specifically, bianisotropic metasurfaces [8] may produce full polarization transformation [9], anomalous reflection [10] and diffractionless generalized refraction [11,12]. They have recently been shown to support Brewster-like, i.e., reflection-less, transmission when surrounded at both sides by air in planar optical silicon nanodisk configuration [13], high-refractive-index nanorod metasurfaces [14], non-planar microwave split-ring resonator configuration [15,16] and all-angle Brewster transmission in a terahertz metasurface [17]. Moreover, they have been demonstrated to allow general Brewster transmission, i.e., between two different media, in the particular case of normal incidence in a non-planar bianisotropic loop-dipole configuration [18].

Here, following our initial suggestion in [19], we present a generalization of the Brewster effect between two arbitrary different media, for arbitrary incidence angle and arbitrary polarization, using a planar bianisotropic metasurface. We derive synthesis formulas of the corresponding metasurface susceptibility tensors and demonstrate the generalized Brewster angle by full-wave electromagnetic simulation.

2. Generalized Brewster effect

Figure 1 shows the proposed metasurface-based generalized Brewster effect. Figure 1(a) illustrates the suppression of reflection for arbitrary wave incidence angle and arbitrary polarization, Figure 1(b) defines the corresponding problem in the plane of scattering, and Figure 1(c) depicts the metaparticule used in the paper as a proof of concept in the microwave regime. The metasurface is assumed to suppress reflection without altering the direction of refraction prescribed by the Snell law for the initial pair of media and without inducing any gyrotropy (polarization rotation), while being passive, lossless and reciprocal. The preservation of the Snell law for the transmitted wave implies a uniform (without phase gradient) metasurface, and hence a uniformly periodic metastructure.

 

Fig. 1. Metasurface-based generalized Brewster refraction between two media. (a) Scattering of a wave impinging on the interface under an arbitrary angle ($\theta _{\textrm {a}}$), with conventional Fresnel transmission and reflection for the case of the bare interface (left) and with reflectionless (Brewster) transmission when a properly designed metasurface is placed at the interface (right). (b) Brewster metasurface problem, for TE and TM polarizations, in the $xz$-plane scattering plane. (c) Proposed 2-layer conducting cross-potent metaparticle for a microwave proof of concept.

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3. Susceptibility tensor derivation

3.1 Field specifications

We consider the metasurface problem depicted in Figure 1(b), where a wave incident from the medium $a$ in the $xz-$plane at an arbitrary angle $\theta _{\textrm {a}}$ is fully transmitted, without reflection ($R=0$), into the medium $b$, at the Snell angle $\theta _{\textrm {b}} = \arcsin (\frac {n_{\textrm {a}}}{n_{\textrm {b}}}\sin \theta _{\textrm {a}})$. We assume the time-harmonic complex convention $\exp (-i\omega {t})$ through the paper, and we shall apply the metasurface synthesis technique described in [20,21] to determine the susceptibility tensors of the metasurface.

The first step in the synthesis is to define the desired tangential fields at both sides of the metasurface in the plane $z=0$. For a TM-polarized wave, these fields read

3.2 Transition conditions

The boundary conditions in the plane of the metasurface ($z=0$) are the generalized sheet transition conditions [7,20,21]

3.3 Homoanisotropic metasurface

We heuristically start our quest for the design described in connection with Figure 1 by considering the simplest type of metasurface, namely an homoanisotropic metasurface, which is defined as a metasurface whose only nonzero susceptibility tensors are $\overline {\overline {\chi }}_{\textrm {ee}}$ and $\overline {\overline {\chi }}_{\textrm {mm}}$. The nongyrotropy condition implies then $\chi _{\textrm {ee}}^{xy}=\chi _{\textrm {ee}}^{yx}=\chi _{\textrm {mm}}^{xy}=\chi _{\textrm {mm}}^{yx}=0$ [21], which decouples the two polarizations with $\chi _{\textrm {ee}}^{xx}$ and $\chi _{\textrm {mm}}^{yy}$ for TM and $\chi _{\textrm {ee}}^{yy}$ and $\chi _{\textrm {mm}}^{xx}$ for TE (see Figure 1(b)). Inserting the specifications (1) and (2) into the field differences and averages (5) and (6), substituting the resulting expressions into into (4), and solving for the four nonzero susceptibility components yields

The complex nature of these susceptibilities indicates the presence of loss or gain, whereas we are searching for a lossless and gainless metasurface. This attempt is therefore unsuccessful, but it demonstrates the necessity for a really bianisotropic metasurface, as will be shown next.

3.4 Bianisotropic metasurface

At this point, we can still hope that adding heteroanisotropy, corresponding to the susceptibilitity tensors $\overline {\overline {\chi }}_{\textrm {em}}$ and $\overline {\overline {\chi }}_{\textrm {me}}$, may allow to remove the loss-gain constraint of the previous design via the resulting extra degrees of freedom. Let us thus add the two heterotropic allowed pairs of nongyrotropic components, namely $\chi _{\textrm {em}}^{xy}$ and $\chi _{\textrm {me}}^{yx}$ for TM and $\chi _{\textrm {em}}^{yx}$ and $\chi _{\textrm {me}}^{xy}$ for TE. This increases the number of parameters to eight, and implies therefore the specification of an additional wave transformation for each polarization in order to make the system of Eqs. (4) full-rank and hence the synthesis problem exactly determined. Since some forms of bianisotropy can lead to nonreciprocity [22], which is here prohibited, we shall enforce reciprocity by specifying a second wave transformation corresponding to the time-reversed version of the fields in (1) and (2) [23]. The resulting system of equations can be compactly written as

4. Metaparticle design

We now need to determine the shape of a metaparticle that realizes the susceptibilities in (11) and (12). This shape can be devised upon the basis of a simple pair of conducting wires, typically folded in a dog-bone shape for higher homogeneity, as shown in Fig. 2. The direct responses, $\chi _{\textrm {ee}}^{xx}$ and $\chi _{\textrm {mm}}^{yy}$, can be realized with identical wires, as shown in Fig. 2(a). However, such a symmetric structure does not support cross responses. Such responses, specifically the required components $\chi _{\textrm {me}}^{yx}$ and $\chi _{\textrm {em}}^{xy}$, can be obtained by breaking the symmetry of the wire-pair, as shown in Fig. 2(b). The exact values of these parameters can be obtained by tuning the level of asymmetry of the structure. Repeating this reasoning for the perpendicular polarization leads to a double dog-bone structure having the form of a cross potent, also often called Jerusalem cross in the literature on metamaterials, which provides independent tuning [24] of the four other susceptibility components – $\chi _{\textrm {ee}}^{yy}$, $\chi _{\textrm {mm}}^{yy}$, $\chi _{\textrm {me}}^{xy}$ and $\chi _{\textrm {em}}^{yx}$.

 

Fig. 2. Folded wire-pair metaparticle (half of the complete cross-potent metaparticle). (a) Symmetric structure, supporting only the susceptibility components, $\chi _{\textrm {ee}}^{xx}$ and $\chi _{\textrm {mm}}^{yy}$. (b) Asymmetric structure providing the four susceptibility components required in (11). The complementary perpendicular wire pair provide the four susceptibility components required in (12). The notation $p_{\textrm {me}}^{yx}$ represents the $y$ component of the magnetic dipole response due to the $x$ component of the electric field excitation, and so on.

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The metaparticle obtained in this fashion will naturally induce some transmission phase, depending on the amounts of asymmetry, and coupling between the two layers. In the present application (Brewster transmission), this phase is not critical, and the design is therefore fully satisfactory as such. If one would wish, for some reason, to add control over the phase, an extra degree of freedom would need to be introduced in the structure. This can be typically accomplished by inserting a third conducting layer – a three-layer structure has been shown to provide full phase coverage with matching [11,12,21] – which can be designed from scattering parametric mapping [7,11,21] or using a transmission-line admittance model [12,25].

5. Results

As proofs of concept, we designed a series of two-layered metasurfaces composed of cross-potent resonators, as shown in Fig. 1(c). The metasurfaces were designed and simulated using CST Studio 2019 with periodic boundary conditions.

Fig. 3 presents the results for the metasurface design with the susceptibilities (11) and (12), which correspond to polarization-independent (TE and TM) Brewster transmission in the $xz$-plane. These results show that the specifications are perfectly realized by the designed metasurfaces for the chosen Brewster angles ($\theta _{\textrm {a}}$ of $0^{\circ }$, $30^{\circ }$ and $75^{\circ }$) in the X-band frequency range and also show the angular response of the metasurface system at the operating frequency of the metasurface.

 

Fig. 3. Full-wave simulated electric field amplitude distribution, frequency response and angular response of the reflectance and transmittance for polarization-independent $xz$-plane ($\varphi =0$) Brewster metasurfaces with the general parameters $(\epsilon _{\textrm {r,a}},\epsilon _{\textrm {r,b}})=(1,3)$ (bare-interface Brewster angle for the TM polarization at $60^{\circ }$), $\epsilon _{\textrm {r,subs}}=3$, $d=1.52$ mm, $s = 0.5$ mm and $g=0.5$ mm. (a) Brewster angle at $\theta _{\textrm {a}}=0$ (normal incidence) with $w_{y\textrm {u}}=w_{x\textrm {u}}=3.9$ mm and $w_{y\textrm {l}}=w_{x\textrm {l}}=2.75$ mm. (b) Brewster angle at $\theta _{\textrm {a}}=30^{\circ }$ with $w_{y\textrm {u}}= 3.2$ mm, $w_{x\textrm {u}} =3.3$ mm, $w_{y\textrm {l}}= 2.2$ mm and $w_{x\textrm {l}} = 2.2$ mm. (c) Brewster angle at $\theta _{\textrm {a}}=75^{\circ }$ with $w_{y\textrm {u}}= 3.35$ mm, $w_{x\textrm {u}} = 3.9$ mm, $w_{y\textrm {l}}= 2.65$ mm and $w_{x\textrm {l}} =2.45$ mm.

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The design of Fig. 3, with coinciding TM and TE Brewster angles, provides full reflection suppression for unpolarized light. However, this response is restricted to scattering in the $xz$ ($\phi =0$) plane. Indeed, according to Eqs. (11) and (12), we have $\chi _{\textrm {ee}}^{yy}\neq \chi _{\textrm {ee}}^{xx}$, $\chi _{\textrm {mm}}^{xx}\neq \chi _{\textrm {mm}}^{yy}$, $\chi _{\textrm {em}}^{xy} \neq \chi _{\textrm {em}}^{yx}$ and $\chi _{\textrm {me}}^{xy}\neq \chi _{\textrm {me}}^{yx}$, and therefore the metasurface structure is anisotropic since the rotation $(x,y)\rightarrow (y,-x)$ implies different susceptibilities and different susceptibility cannot lead to the same scattering response.

This single scattering plane restriction can be lifted with the same set of (eight) susceptibility parameters for one of the two polarizations (TM or TE) by combining the selected (TM or TE) $xz$-plane equations in (11) and (12) with the corresponding $yz$-plane equations obtained via the permutations $(x,y)\rightarrow (y,-x)$, which is in fact equivalent to making the structure isotropic ($\chi _{\textrm {ee}}^{yy}=\chi _{\textrm {ee}}^{xx}$, $\chi _{\textrm {mm}}^{xx}=\chi _{\textrm {mm}}^{yy}$, $\chi _{\textrm {em}}^{xy}=\chi _{\textrm {em}}^{yx}$ and $\chi _{\textrm {me}}^{xy}=\chi _{\textrm {me}}^{yx}$) since the same Brewster response is expected in the two planes for the selected polarization. The results for corresponding metasurfaces are presented in Fig. 4. They further confirm the accuracy of the proposed design.

 

Fig. 4. Full-wave simulated electric field amplitude distribution, frequency response and angular response of the reflectance and transmittance for azimuth-independent ($\forall \varphi$) single-polarization (TM or TE) Brewster metasurfaces with the same general parameters as in Fig. 3. (a) TM-Brewster angle at $\theta _{\textrm {a}}=30^{\circ }$ with $w_{x\textrm {u}}= w_{y\textrm {u}} = 3.15$ mm and $w_{x\textrm {l}}= w_{y\textrm {l}} = 2.1$ mm. (b) TE-Brewster angle at $\theta _{\textrm {a}}=30^{\circ }$ with $w_{y\textrm {u}}= w_{x\textrm {u}} =3.3$ mm and $w_{y\textrm {l}}= w_{x\textrm {l}} = 2.45$ mm. (c) TM-Brewster angle at $\theta _{\textrm {a}}=75^{\circ }$ with $w_{x\textrm {u}}= w_{y\textrm {u}} = 3.9$ mm and $w_{x\textrm {l}}= w_{y\textrm {l}} = 2.5$ mm. (d) TE-Brewster angle at $\theta _{\textrm {a}}=75^{\circ }$ with $w_{y\textrm {u}}= w_{x\textrm {u}} = 3.55$ mm and $w_{y\textrm {l}}= w_{x\textrm {l}} = 2.85$ mm.

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6. Discussion

Although the eight-parameter metasurfaces considered here are restricted to either single-plane or single-polarization Brewster transmission, bianisotropic metasurfaces involving a greater number of susceptibility parameters might be able to provide universal Brewster transmission. Since the possibilities of transverse ($x$ and $y$) susceptibilities have been exhausted, such metasurfaces would require resorting to normal $z$ susceptibilities. Although the related design is in principle still analytically tractable thanks to the uniformity of the metasurface [21], it is considerably more involved and will therefore be deferred to a later study.

Equations (11) and (12) do not only provide the sought after Brewster transmission design. They point to an extra fundamental capability of an interfacing bianisotropic metasuface that occurs when $\phi _{\textrm {TM}}=\phi _{\textrm {TE}}=0$, which can be achieved by phase compensation or adjustment. In this case, we have $\chi _{\textrm {ee}}^{xx}=\chi _{\textrm {mm}}^{yy}=\chi _{\textrm {ee}}^{yy}=\chi _{\textrm {mm}}^{xx}=0$, which leads to the heteroanisotropic generalized sheet transition conditions

The relations (14) have the same mathematical form as the conventional Fresnel reflection coefficients [1]. This reveals that the proposed [medium – bianisotropic metasurface – medium] system is equivalent to a [medium – effective medium] system, with the effective medium having the impedances given by Eqs. (15). Thus, inserting such a bianisotropic metasurface at the interface between two media or coating a dielectric medium exposed to free space with it can change the effective bulk impedance of the transmission medium, which enriches the design possibilities of existing materials.

Although the proof of concept systems in Figs. 3 and 4 pertain to the microwave regime, where bianisotropic metasurfaces (surrounded by air) have been well documented, bianisotropic metasurfaces have also been recently demonstrated in all-dielectric configuration [26,27]. Therefore, the proposed concepts of metasurface-based generalized Brewster effective refractive medium can be readily extended to the optical regime, where they may be particularly beneficial in terms of reducing the insertion loss associated to impedance mismatch in many common components.

7. Conclusion

In summary, we have shown that a properly designed bianisotropic metasurface placed at the interface between two dielectric media or coating a dielectric medium exposed to the air provides Brewster transmission at arbitrary angles and for both the TM and TE polarizations. We have presented a rigorous derivation of the corresponding surface susceptibility tensors and demonstrated the system by microwave proof-of-concept designs. Moreover, we have noted that such a system leads to the concept of effective refractive media with tailorable impedances. The proposed bianisotropic metasurfaces offer deeply subwavelength matching solutions for initially mismatched media, and alternatively lead to the possibility of on-demand manipulation of the conventional Fresnel coefficients. They represent thus a fundamental advance in optical science and posses a considerable potential for novel technological developments.