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Abstract
Brewster effect has attracted extensive interests through microwave to optical regime. However, previous work mainly focused on the cases with single linear polarization. We demonstrate in this paper an equal Brewster’s angle for both TE (transverse-electric) and TM (transverse-magnetic) waves. Tunable Brewster effect is furthermore achieved. For practical implementation of this effect, we propose an anisotropic metagrating composed of modified split ring resonators (MSRR) as a proof-of-concept in X-band. Measurement results of the manufactured prototypes validate our method. The proposed metagrating featuring Brewster’s angle manipulation potentially paves the way for many applications throughout the spectrum, e.g., angular selectivity.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Well-studied since Sir David Brewster and Malus, traditional Brewster effect has various applications in both optical and microwave regimes: polarizer [1], Brewster angle microscopy (BAM) [2], dielectric properties quantifying [3], and extraordinary optical transmission (EOT) [4,5]. These classical applications are yet limited to transverse-magnetic (TM, or p-polarized) electromagnetic (EM) waves. For example, in the most common laser cavity, where the Brewster window acts as a polarizer, the angular selectivity is always entangled with TM polarization. This restriction is caused by the fact that the classic Brewster angle for transverse-electric (TE, or s-polarized) waves only exists when the waves impinge on the surface of magnetic media [1], but the magnetic response in natural media is usually tenuous, especially in the optical regime. Therefore, it is of practical significance to find a medium with an identical Brewster’s angle that is not sensitive to the polarization. Another possible application of such a medium is an angle-filter promoting the signal-noise-ratio (SNR) in the receivers of the oriented wireless communication, especially for unknown polarization cases. More importantly, it would inspire many novel applications spanning the entire spectrum, e.g., information encryption with geometric angles, and efficient energy harvesting and conversion [6].
The confines abovementioned have been overcome with the advances in metamaterials and metasurfaces. The generalized Brewster effect—realizations of Brewster’s angle for TE waves—were recently reported in these artificially composite media. For the visible frequencies, Ryosuke et al. realized non-reflection only for TE waves at a certain angle (Brewster’s angle), using stratified metal-dielectric metamaterial [7]. Paniagua-Domínguez et al. applied the first Kerker’s condition [8] and the Mie theory [9] to achieve scattering cancellation, realizing the generalized Brewster effect in all-dielectric metasurfaces (silicon nanodisks) [10]. Similarly, high-refractive index (HRI) nanophotonic structures were properly described by coupled electric and magnetic dipole (CEMD) analytical formulation, and were further numerically confirmed to achieve the generalized Brewster effect [11]. Besides, the optical angular selectivity was proposed by Y. Shen [12,13], where the broadband total transmission at a narrow-angle range was enabled by hetero-structured photonic crystals.
The reported meta-atoms in the optical or terahertz regimes, e.g., all-dielectric disks, spheres, cylinders, stacks, graphene, or other Mie scatters and gratings [10-18], feature geometrically simple structures and thus their electrical and magnetic polarizabilities can be described with analytical expressions, which in turn simplifies yet constrains the applicable range of the design method. Moreover, these three-dimensional structures are inconvenient to be miniaturized or integrated if applied to lower frequencies. Thus, the generalized Brewster effect with classical microwave fabrication techniques remains relatively elusive. Yasuhiro Tamayama experimentally observed TE Brewster effect in a bulk metamaterial [19], and theoretically proposed bi-anisotropic metafilms with 45° Brewster’s angle for only TE waves [20]. Recently, Guillaume Lavigne and Christophe Caloz extended the Brewster effect to arbitrary angles and polarizations in bi-anisotropic metasurfaces [21] and analyzed it with generalized sheet transition conditions (GSTC) [22,23]. Unfortunately, their proposed metasurface consisted of purely bi-anisotropic scattering particle tend to be difficult to implement [24]. Despite the provided possible solutions, onerous susceptibility derivation could not be avoided [25].
In this paper, we generalize angularly tunable Brewster effect for both TE and TM polarized electromagnetic waves impinging on our metagrating. Two main advances are made in our work in comparison with the previous research in microwave and optical regime. For one thing, the aforementioned polarization restriction is overcome by a properly designed thin metasurface with identical Brewster’s angle for TE and TM waves. For the other, the angular manipulation can be realized through changing either the grating space or the grating tilt angle. These two key phenomena can be explained by the anisotropic EM characteristic of the proposed metagrating. Therefore, we will first analytically demonstrate the derivation of the generalized Brewster effect conditions in anisotropic media. In accordance to these conditions, we then adopt the modified split ring resonator (MSRR) structure [26] with simplified EM parameters. Based on its simulated scattering parameters, the permittivity and permeability tensors will be retrieved to confirm the Brewster effect by eliminating the possibility of Fabry-Pérot-type of total transmission. The circularly polarized waves are also applied in the simulation and experiment to substantiate polarization arbitrariness. In the last section, the manipulation of the Brewster’s angle is demonstrated. The proposed two facile manipulating methods make it possible to achieve applications involving angular selectivity. Experimental results of the manufactured prototypes further validate our design method.
2. Brewster effect in anisotropic media
Considering the most general media, one can describe their EM characteristics with four dyadic tensors: , , , . The constitutive relations should be written as
where c0, ε0, μ0 denote the speed of light, permittivity, and permeability in the vacuum. In a time-harmonic field, Maxwell’s equations can be applied to get the wave equation for the electric fieldThe wave vector k must satisfy Eq. (3) with a non-trivial electric field, which leads towhere “det” means the determinant of its following matrix.The Eq. (4) is one of the dispersion equations for any plane wave in the most general form. However, many hidden difficulties make these equations unpractical to apply. Chief among them is that the plane wave is hard to decompose to TE and TM waves in the most bi-anisotropic cases. In addition, a worse scenario lies in four undetermined dyadic tensors with 36 unknown parameters. To simplify this problem for further design, we assume that
where “diag” denotes the diagonal matrix. When the TE polarized waves in the free space impinge on the surface of an anisotropic medium at the Brewster’s angle (θB), Brewster effect happens, so there should be no reflection. Regarding the xOz plane as the incident plane, the dispersion equation for the TE polarized wave takes the form as the follows,where kx = |k|sinθt and kz = |k|cosθt, and θt is the refraction angle. Meanwhile, the no-reflection assumption can be used to simplify the EM boundary conditions, by which one getsSubstituting Eq. (7) to Eq. (6), the Brewster’s angle for the TE polarized wave in the simplified anisotropic media can be solved, which reads,The EM duality principle also applies here, so the Brewster’s angle for the TM polarized wave reads,3. Practical realization
3.1 Unit cell design of the metagrating
In the microwave regime, any non-magnetic dielectric generates the non-unity relative permittivity (εxx, εyy, εzz ≠ 1), which guarantees the prevalence of the Brewster effect for TM polarized waves. As for the TE Brewster’s angle, Eq. (8) requires at least non-unity relative permeability (μxxμzz ≠ 1). Thus, only in the media with permittivity and permeability both non-unity and properly designed, can the Brewster’s angle exist for arbitrary polarized waves.
Looking back to the advance in metamaterial and metasurface, the classic artificially composed medium that fits this condition is called double negative material (DNG) proposed by D. R. Smith [27,28]. We similarly take the SRR structure in [19,20] and [27,28] into account (as shown in Fig. 1(a)). Unfortunately, the traditional SRR unavoidably behaves as a bi-anisotropic particle with its electric and magnetic moments shown as [26]
According to Eq. (10), an external magnetic field along the x axis would generate the electric moment along the y axis. As shown in Fig. 1(c), this phenomenon can be verified by the normalized surface current distribution extracted from the CST Microwave Studio simulation. The currents on the inner and outer rings are in the opposite directions, accumulating the net electric charges in the way shown in Fig. 1(c) and producing the electric moment pm sketched in Fig. 1(a).
Fig. 1 The prototypes and normalized surface current distributions of the traditional SRR (for (a) and (c)) and the Modified SRR ((for (b) and (d))). The mx and pm (or pmf, pmb) in (a) and (b) denote the magnetic moment and electric moment respectively, which are driven by the external magnetic field along the x axis. In (c) and (d), the normalized surface currents extracted from CST simulation are illustrated by the arrows with stratified colors.
Therefore, we then replace one of the rings by another one on the opposite side of the substrate with the split orientating to the opposite direction [26]. Our modified SRR (MSRR) unit cell and its simulated current distribution are shown in Figs. 1(b) and 1(d). The EM behaviors of the MSRR should be similar to those of traditional SRR, e.g., the negligible non-diagonal components of the electric permittivity and the magnetic permeability due to high polarization isolations, i.e., εij, μij with i ≠ j. They behave disparately, however, with regard to the bi-anisotropic effect. In fact, as shown in Fig. 1(d), the currents on both rings afflux in the same direction. The oppositely-placed splits and exactly equal dimensions enable the cancellation of electric moments along the y axis caused by the external magnetic field (pmf = −pmb), in that the electric charges on the front ring are just opposite to those on the back. Hence, the cross-coupling susceptibility χem in Eq. (10) and Eq. (11) would be minimized to zero. To put it another way, the magnetic moment is purely contributed by the external magnetic field, while the electric moment totally comes from the external electric field.
3.2 Brewster effect for arbitrary polarized EM waves
After cancelling the cross-coupling susceptibility, we can reasonably apply Eq. (8) in our design of metagrating and verification for the generalized Brewster effect. The proposed metagrating is sketched in Fig. 2.
Fig. 2 Brewster effect when the TE or TM polarized wave impinges on the proposed metagrating. The grating space is Px, and the substrate thickness ts. The unit cell dimensions are defined by Py and d, split width s, as well as the outer radius Rout and inner radius Rin of the metal ring. The red arrows (Ey, Hx, Hz) denote electric and magnetic fields in the TE case, while the purple arrows (Hy, Ex, Ez) denote those in the TM case. The incident planes, both 3D and cross-section, marked with the coordinate axis are shown in blue dashed boxes, where the incident angle (Brewster’s angle)θB and the refraction angle θt are labled. The effective EM parameter tensors are also shown on the cross-section of the incident plane.
When the TE polarized wave impinges on the metagrating, the electric and magnetic field can be illustrated as the red arrows in Fig. 2. The electric field along the y axis Ey and the existence of the substrate are able to produce non-unity εyy, while the non-unity μxx can be realized by magnetic field along the x axis Hx. Meanwhile, the periodical repetition of the unit cells provides more freedom for the parameters in Eq. (8). In this case, Brewster’s angle for the TE polarized wave can be achieved by our metagrating, and it is also true for the TM case with similar analysis. Besides, it is clear that the dimension of the ring and the space between the gratings could influence the EM response of the metagrating, and then make it possible to manipulate the Brewster’s angle.
Here, we choose F4B as the substrate with relative permittivity εr = 2.2 and its thickness ts = 0.8 mm. In addition, we set the grating space Px = 8 mm, unit cell dimension Py = d = 20 mm, the outer radius of the ring Rout = 9 mm, the inner radius Rin = 3 mm, and the split width s = 3.5 mm.
As shown in Fig. 3(a), when the TE or TM wave in the free space impinges on the metagrating, the reflection index can be retrieved from the simulated scattering parameters (Sij, i, j = 1, 2), which can be written as
where z is the generalized equivalent impedance of the metagrating and derives from [29]Fig. 3 (a) The cross-section of the incident plane on which scattering parameters are distinguished from the reflection index. (b) Retrieved reflection index for the TE polarized wave, varying with the frequency and the incident angle. (c) Retrieved reflection index for the TM polarized wave, varying with the frequency and the incident angle. The white dash line demonstrates the Brewster’s angles, while the black dash line shows the critical angle. The identical Brewster’s angle (45°) is marked out by two cross markers.
Based on the simulated scattering parameters, the retrieved reflection indices for TE and TM polarized waves are shown in Figs. 3(b) and 3(c), respectively. The Brewster’s angles are further outlined by the white dash lines. It is interesting to note that the critical angle, above which the waves would be totally reflected, can also be witnessed in Fig. 3. Yet we care much more about the case when TE and TM polarized waves have the same Brewster’s angle, as marked by the white crosses in Figs. 3(b) and 3(c). It can be witnessed that TE and TM polarized waves share 45° Brewster’s angle at 10.3 GHz.
However, the Brewster effect should be verified by retrieving the EM parameters of the metagrating and excluding the Fabry-Pérot type of total transmission. Retrieving from the simulated scattering parameters, one can usually calculate the equivalent EM parameters of a structure, but it should be noticed that the traditional retrieval method is not accurate here due to the oblique incidence—the retrieved EM parameters inherently vary with the incidence angle [30]. Hence, we apply a modified retrieval method in which scattering parameters in the normal incidence case are taken as assisted conditions. As shown in Fig. 3(a), one can apply the EM boundary conditions here and readily write the scattering parameters with the reflection index, which reads,
where R can be represented by Eq. (12) and Q = exp(jnTE(θt)k0dcosθt). nTE(θt) is the refraction index at the angle of θt. To conveniently utilize the available retrieval formulation developed in the normal incidence case, we define the generalized refraction index nTE’ as follows,In this form, it can readily be retrieved by Eq. (14)-(15).The Snell’s law can be further applied as
Considering the TE polarized wave case, nTE(θt) also conforms toIn addition, the reflection index for the TE polarized wave is written asComparing it with Eq. (12), one can get the generalized normalized impedance for the oblique incidence which can also be retrieved from the scattering parameters.In the normal incidence case (θt = 0), εyy can be solved from Eq. (18) and Eq. (20), which reads,It should be noted that to determine anisotropic components of these electromagnetic parameters is not straightforward since the number of unknown parameters (six in total) is larger than the conditions even with two polarizations applied (four in total). Nevertheless, since the electric field of TE waves is always along the y axis as shown in Fig. 2, εyy should remain the same value despite of the variation of incident angle. It is also true for μyy in the TM case. Such conditions help us find a convenient and feasible approach to determine all the anisotropic components uniquely. Therefore, εyy calculated by Eq. (21) can be then used to determine the other two EM parameters through Eq. (17)-(20) with retrieved nTE’ in the TE polarized case. They are, We again adopt the EM duality principle here, and get the following equations for the TM case, which read, After several mathematical calculations, the retrieved permittivity and permeability dyadic tensors are written as Substituting them to Eq. (8) and Eq. (9), one should get the retrieved Brewster’s angle: 43.60° for the TE wave and 44.98° for the TM wave, which are well consistent with the simulated one, i.e., 45°. Furthermore, the Fabry-Pérot type of total transmission can be excluded by |1−exp(j2n(θt)k0dcosθt)| ≠ 0.To substantiate that 45°-Brewster’s angle applies to arbitrary polarized EM waves, we take TE, TM, and circularly polarized waves as excitations, separately. The experiment setup photograph and schematic are shown in Figs. 4(a) and 4(b), respectively. In order to measure the return loss (S11) of the metagrating, two standard gain horn antennas (XB-GH90-20N with the BJ-100 waveguide) are worked as the source (Tx) and the receiver (Rx) in our experiments. In addition, the zoom-in photographs of the experimental prototype of the metagrating (24 × 9 units, around 6.32λ0 × 6.18λ0 at 10.3 GHz) and two sides (marked as A and B separately) of its single strip are demonstrated in Figs. 4(c) and 4(d).
Fig. 4 (a) The photograph of the experiment setup. (b) The schematic of the experiment setup, where θ1-θ4 indicate the different incident angles. (c) The zoom-in photograph of the manufactured prototype of the metagarating. (d) The photograph of the two sides labeled with A and B of a single grating strip. (e) The simulated and measured return loss (S11) versus the incident angle. (d) The simulated and measured return loss (S11) versus the frequency.
Adopting the experiment setup abovementioned, the simulated and measured return loss (S11) are illustrated in Figs. 4(e) and 4(f), where good agreement can be found between the simulation and the experiment results. When the circularly polarized waves are implemented with cone horns, the measured axial ratio of transmitted waves is as low as 0.78 dB (after calibration) at 10.3 GHz and 45° incident angle, which is also well consistent with the simulated 0.11 dB.
3.3 Brewster’s Angle Manipulation
As aforementioned, many structural parameters of the metagrating play a notable role in determining Brewster’s angle and its corresponding frequency. Specifically, the average radius of MSRR would mainly determine at which frequency the Brewster effect occurs. On the other hand, the slot width (s) and the grating space (Px) can both be changed to manipulate the Brewster’s angle. Yet once the MSRR structure is printed, the dimension of the metagrating units will be fixed, so the only possible way to manipulate the Brewster’s angle is to change the grating space or its tilt angle. The metagratings with different grating space (as shown in Fig. 5(a)) are manufactured and applied to our experiments. In Figs. 5 (b) and 5(c), we demonstrate the simulated and measured TE and TM Brewster’s angles at different frequencies when the transvers space Px varies from 4 mm to 10 mm. Both of them cover a broad angle range (TE: 15~80°, TM: 20~80°) in the X-band, which proves the efficiency of this manipulating method.
Fig. 5 (a) Intercepted photographs of the metagratings with different grating spaces Px. (b) The simulated and measured frequency versus the corresponding Brewster’s angle for the TE polarized wave, manipulated by changing the grating space Px. (c) The simulated and measured frequency versus the corresponding Brewster’s angle for the TM polarized wave, manipulated by the grating space Px varies. (d) The sketch of the Brewster’s angle manipulation process when the grating’s tilt angle Θ. (e) The simulated tunable Brewster’s angle for the TE polarized wave versus the tilt angle Θ at different frequencies. (f) The simulated tunable Brewster’s angle for the TM polarized wave versus the tilt angle Θ at different frequencies.
The intrinsic reason for the aforementioned manipulation is to change the equivalent EM parameters of the meta-grating by tuning the space between the strips. From the microscopic perspective, the underlying mechanism is based on coupled electric/magnetic dipole theories. Accordingly, we could also mechanically achieve the similar effect by adjusting the tilt angle of strips (Θ) as shown in Fig. 5(d). The simulated variation of Brewster’s angle at different tilt angles for several frequencies is shown in Figs. 5(e) and 5(f). By tuning the tilt angle of the metagrating, the Brewster’s angle covers the range from 0° through 80°. The tunability of EM properties achieved by tilting the meta-atoms’ axes with respect to the metasurface plane has also been reported by Sanchez-Gil’s group [31]. Overall, these two facile manipulation methods for Brewster effect are indeed of practical application.
4. Conclusion
To summarize, we have both theoretically investigated and experimentally achieved the generalized Brewster effect in anisotropic media. To this aim, the analytical formulations of the Brewster’s angles for both TE and TM polarized electromagnetic waves have been firstly derived. Although these two formulations have been properly simplified from the most general forms, they are still suitable for many occasions throughout all frequency bands. Most importantly, their explicit forms make it possible to instruct the practical designing process, especially when more sophisticated structures, e.g., metal and dielectric combined, are required.
Based on the theoretical analysis, a metagrating composed of modified split ring resonator (MSRR) has been proposed, featuring the identical and tunable Brewster’s angle for arbitrary polarized electromagnetic waves. To justify its commitment to the mentioned simplification, the discrepancies between the MSRR and the traditional split ring resonator (SRR) have been circumspectly analyzed. As a case study, 45°-Brewster’s-angle metagrating has been verified by full wave simulation, experiments and numerical retrieval of electromagnetic parameters. Furthermore, the manipulation of the Brewster’s angle for both two linear polarized waves have also been proved efficient enough.
All the simulated, measured and relating retrieved results are satisfactorily congruous with one another. The proof-of-concept prototype of our metagrating validates our design, which could illuminate further research on Brewster-effect-related topics. For instance, two-dimensional (2D) grating might also be able to achieve generalized Brewster effect with broader bandwidth. Once achieved, it will release the frequency-polarization entanglement in the traditional Brewster effect, as well as be easier to be miniaturized and integrated on a chip in possibly any frequency regime.
Funding
National Natural Science Foundation of China (NSFC) (61671178, 61301013).
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