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Abstract
We propose a method for controlling the polarization dependence of the complex transmission spectrum using the Brewster effect in a two-dimensional array of meta-atoms with finite thickness, which we refer to as a metafilm. We show that the complex transmission spectra of the orthogonal linear polarization components can be controlled independently without reflection by stacking multiple metafilms that exhibit only an electric dipole resonance. As a proof-of-concept numerical demonstration, we design several broadband waveplates with high transmission efficiency based on simple design principles. The proposed method would enable us to easily design anisotropic metamaterials with various complex transmission spectra.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
There has been an extensive effort to investigate methods for controlling electromagnetic waves using metasurfaces, the two-dimensional counterpart of metamaterials. Arbitrary transmittance and reflectance cannot be realized by a single surface with only an electric response [1,2], so multilayer metasurfaces and thick metasurfaces have been introduced to circumvent this restriction. For example, arbitrary control of the complex reflectance over a broad frequency range can be achieved by introducing a ground plane to a metasurface [3–5]. Alternatively, Huygens metasurfaces have been designed such that radiation from the induced electric and magnetic dipoles destructively interferes only in the propagation direction of the reflected wave, resulting in arbitrary complex transmittance [6–9].
It is more difficult to suppress reflections in Huygens metasurfaces than to suppress transmission using a ground plane; however, broadband Huygens metasurfaces have been developed by tuning the electric and magnetic responses to have the same frequency dependences [10–17]. Huygens metasurfaces with bianisotropy [18,19] and multipolar responses [20–23] have also been investigated to suppress the dependence of their transmission properties on the propagation direction of the electromagnetic wave [18,23] or to control multiband/broadband electromagnetic waves [19–22]. In addition, tunable Huygens metasurfaces have been demonstrated for the active control of electromagnetic waves [24–28].
Although it is important to develop methods for controlling electromagnetic waves using two-dimensional structures such as Huygens metasurfaces, techniques for fabricating three-dimensional metamaterial structures are also being developed [29–32]. Metamaterials with three-dimensional structures have the potential to explore exotic electromagnetic wave phenomena that are difficult to realize using metasurfaces [33,34]. Previously, we have shown that single-layer metamaterials with the same electromagnetic responses as broadband Huygens metasurfaces can be realized by three-dimensionally arranging meta-atoms that exhibit only an electric dipole resonance [35]. This phenomenon is based on the Brewster effect in a two-dimensional array of meta-atoms with finite thickness, which is referred to as a metafilm hereafter. When the oscillation direction of the induced electric dipoles in the meta-atoms coincides with the propagation direction of the reflected wave, vanishing reflectance can in principle be achieved independent of frequency. Therefore, complicated transmission spectra may be designed by simply stacking multiple metafilms where the Brewster effect occurs. As a demonstration of controllable transmission spectra using the Brewster effect in metafilms, we numerically and experimentally showed that broadband control of the group delay can be achieved by stacking metafilms with different resonance frequencies [36]. Although broadband control of the group delay has also been demonstrated using metasurfaces [37,38], our theory for controlling the frequency dependence of the group delay is much simpler than the theory in these studies.
In our previous study [36], we focused on control of the transmission spectrum of only one linear polarization component. However, methods for controlling both the polarization dependence and the frequency dependence of the complex transmittance are essential for the development of applications for electromagnetic waves. In this study, we extend the concept of controlling electromagnetic waves using the Brewster effect in metafilms to the polarization dependence of the complex transmission spectrum. As a proof-of-concept numerical study of the proposed method, we show that several broadband waveplates with high transmission efficiency can be designed simply using the Brewster effect in metafilms.
To date, broadband transmissive waveplates have already been realized using metasurfaces comprising either thin [39–42] or thick planar structures [43–45]. The thick planar structures have a higher transmission efficiency. In particular, Ref. [44] showed that a broadband quarter-wave plate with nearly perfect transmission can be achieved using a metasurface comprising metal-stripe structures embedded in a dielectric layer. Broadband transmissive waveplates have also been realized using Huygens metasurfaces in multiple ways, for example using three-layer structures with parameters optimized using a genetic algorithm [15,16], or using structures that exhibit multipolar responses [20,22].
We emphasize that the purpose of this study is to show that the complex transmission spectra of the orthogonal linear polarization components can be controlled independently without reflection by simply stacking multiple metafilms with only an electric response. For this purpose, we design broadband waveplates with high transmission efficiency as examples of anisotropic broadband electromagnetic media.
2. Theory
We briefly review the transmission properties of a metafilm designed so the Brewster effect occurs (hereafter referred to as a Brewster metafilm) before describing a method for controlling the polarization dependence of the complex transmission spectrum using the Brewster effect in metafilms. Let us assume that an electromagnetic wave is incident on a metafilm that exhibits only an electric dipole resonance. When the oscillation direction of the induced electric dipole in the metafilm coincides with the propagation direction of the reflected wave, the reflectance vanishes because the electric dipole cannot radiate in its oscillation direction, which is similar to the physical mechanism of the Brewster effect at a planar boundary between a dielectric medium and a vacuum. Because we assume that only the electric dipole can be induced in the Brewster metafilm, the metafilm can only be excited by the p-polarization. Based on coupled mode theory [46–48], the complex transmittance of the Brewster metafilm for a p-polarized electromagnetic wave is calculated as follows:
When multiple Brewster metafilms are stacked as shown in Fig. 1(a), the total complex transmittance of the stacked metafilms is the product of the transmittances of the constituent metafilms. This is because the incident wave is not reflected at each metafilm and multiple reflections do not occur. In this study, the angle between the surface normal of the metafilms and the propagation direction of the incident wave is fixed at $45^{\circ }$ for simplicity. However, the angle does not need to be $45^{\circ }$ as long as the oscillation direction of the electric dipole coincides with the propagation direction of the reflected wave [35]. The complex transmittance of the Brewster metafilm for the p-polarized incident electromagnetic wave is given by Eq. (1), while that for the s-polarization is unity because the electric field of the s-polarized electromagnetic wave is orthogonal to the oscillation direction of the induced electric dipole. Therefore, the total complex transmittance of the stacked Brewster metafilms for the p-polarization is the product of the complex transmittances of the constituent Brewster metafilms, as given by Eq. (1), and that for the s-polarization is unity.
Fig. 1. (a) Propagation of electromagnetic waves in a metamaterial comprising $N$ layers of Brewster metafilm. The orientation of the incident electromagnetic field is shown at left. The constituent Brewster metafilms are shown as dashed diagonal lines, where $\boldsymbol {p}_i$ indicates the electric dipole induced in the $i$th layer. The transmitted wave is shown at right. The complex transmittance of the $N$th Brewster metafilm for p-polarized electromagnetic waves is defined as $t_{\mathrm {p}i}$ ($i=1,2, \ldots, N$). (b) Independent control of the $x$- and $y$-polarized components of the incident electromagnetic wave using metamaterials 1 and 2, both Brewster metafilms. The top (middle) row shows the view in $zx$-plane ($yz$-plane). Dark (light) vectors indicate that the electric dipoles are located at the near (far) side. The complex transmittances of metamaterials 1 and 2 for p-polarized electromagnetic waves are defined as $t_x$ and $t_y$, respectively. The bottom row shows the three-dimensional view. Gray represents the planes of metafilms in metamaterials 1 and 2. Although the electric dipoles are periodically arranged in each gray plane, only one electric dipole is drawn to simplify the figure.
Based on the above discussion, we now propose a method for controlling the polarization dependence of the complex transmission spectrum using the Brewster effect in metafilms. Let us assume that an electromagnetic wave is incident on two stacked metamaterials as shown in Fig. 1(b). The incident electromagnetic wave propagates along the $z$ direction and only the electric dipoles in the $x$ direction ($y$ direction) can be induced in metamaterial 1 (metamaterial 2), each comprising at least one Brewster metafilm. The meta-atoms are arranged so the vector obtained by rotating the surface normal vector $\boldsymbol {n}_1$ of the Brewster metafilm(s) in metamaterial 1 by $90^{\circ }$ about the $z$-axis agrees with the surface normal vector $\boldsymbol {n}_2$ of the Brewster metafilm(s) in metamaterial 2. In this configuration, the $x$-polarization ($y$-polarization) corresponds to the p-polarization when the electromagnetic wave is incident on metamaterial 1 (metamaterial 2). As described in the previous paragraph, the complex transmittance of the p-polarization is given by Eq. (1) and that of the s-polarization is unity. Therefore, the complex transmittances of the $x$- and $y$-polarization components of the incident wave can be controlled independently without reflection using the two stacked metamaterials. Note that this simple method for controlling the polarization dependence of the complex transmittance in the linear polarization basis is enabled by the fact that the Brewster metafilm responds to only one linear polarization component and does not reflect the incident electromagnetic wave independent of frequency.
As a proof-of-concept study for controlling the polarization dependence of the complex transmission spectrum in the linear polarization basis using the Brewster effect in metafilms, we designed several broadband waveplates with high transmission efficiency based on two different design methods, described below. We exploited the fact that the frequency derivative of the transmission phase of a Brewster metafilm is small in the off-resonance region in the first method, which we refer to as the off-resonance method. Figure 2(a) shows the theoretical frequency dependences of the arguments of the complex transmittances $t_x$ and $t_y$ for the $x$- and $y$-polarization components when metamaterial 1 (metamaterial 2) is a Brewster metafilm with resonance angular frequency $\omega _1$ ($\omega _2$). The frequency derivative of $\arg {(t_x)}-\arg {(t_y)}$ vanishes at a certain angular frequency $\omega _0$ [$\min {(\omega _1 , \omega _2)} < \omega _0 < \max {(\omega _1 , \omega _2)}$]. As the difference between the resonance angular frequencies $|\omega _1 - \omega _2|$ decreases, the transmission phase difference $|\arg {(t_x)}-\arg {(t_y)}|$ at $\omega = \omega _0$ increases and the frequency bandwidth where the transmission phase difference can be regarded as a constant decreases because the difference between $\omega _0$ and $\omega _{1,2}$ becomes smaller and the frequency dispersion at $\omega = \omega _0$ becomes stronger [Figs. 2(a) and 2(b)]. When metamaterial 1 (metamaterial 2) is $N$-layers of Brewster metafilm with resonance angular frequency $\omega _1$ ($\omega _2$), the transmission phases [$\arg {(t_x)}$ and $\arg {(t_y)}$] increase proportionally with $N$ because reflection does not occur at Brewster metafilms. This implies that the transmission phase difference increases proportionally with $N$ and that the frequency bandwidth is independent of $N$ [Figs. 2(a) and 2(c)]. Therefore, in the off-resonance method, a waveplate with the desired transmission phase difference and frequency bandwidth may be designed by varying $|\omega _1 - \omega _2 |$ and the number of metafilm layers $N$ within each metamaterial.
Fig. 2. Theoretically calculated arguments of the complex transmittances $t_x$ (red) and $t_y$ (blue) for the $x$- and $y$-polarization components in (a)-(c) the off-resonance and (d)-(f) the on-resonance methods. The value of $\omega _2 - \omega _1$ in (a) is three times as large as that in (b) and the same as that in (c). The value of $\varDelta \omega$ in (d) is half that in (e) and the same as that in (f). The value of $N$ is shown in each panel. The transmission phase difference $\varDelta \phi$ can be regarded as a constant in the frequency ranges highlighted in yellow. The transmission phase is wrapped in the range $-\pi$ to $\pi$.
In the second method, we exploited the fact that the frequency derivative of the transmission phase can be constant over a certain frequency range in a metamaterial comprising multiple Brewster metafilms when the resonance angular frequencies of the constituents are suitably designed [36]. We refer to this method as the on-resonance method. Figure 2(d) shows the theoretically calculated transmission phase when metamaterial 1 comprises three distinct layers of Brewster metafilm with resonance angular frequencies $\omega _1$, $\omega _2$, and $\omega _3$. The frequency derivative of the transmission phase is qualitatively confirmed to be constant over a broad frequency range. Roughly speaking, this situation can be achieved when the separation of neighboring resonance angular frequencies is smaller than about the resonance linewidth. This figure also shows that the frequency dependence of the transmission phase of metamaterial 2 is the same as that of metamaterial 1 shifted by $\varDelta \omega$ when metamaterial 2 comprises three distinct layers of Brewster metafilm with resonance angular frequencies $\omega _1 + \varDelta \omega$, $\omega _2 + \varDelta \omega$, and $\omega _3 + \varDelta \omega$. In the frequency range where the frequency derivatives of the transmission phases are constant values, the transmission phase difference $|\arg {(t_x)}-\arg {(t_y)}|$ is also constant and increases with increasing $\varDelta \omega$ [Figs. 2(d) and 2(e)]. The frequency range where the frequency derivatives of the transmission phases are constant can be increased by adding Brewster metafilms with resonance angular frequencies $\omega _4$, $\omega _5$, $\cdots$, and $\omega _N$ ($\omega _4 + \varDelta \omega$, $\omega _5 + \varDelta \omega$, $\cdots$, and $\omega _N + \varDelta \omega$) to metamaterial 1 (metamaterial 2) [Figs. 2(d) and 2(f)]. Therefore, in the on-resonance method, a waveplate with the desired transmission phase difference and frequency bandwidth may be designed by varying $\varDelta \omega$ and the number of metafilm layers $N$ within each metamaterial.
3. Methods
We verified that it is possible to control the polarization dependence of the complex transmission spectrum in the linear polarization basis using the Brewster effect in metafilms by numerically designing several waveplates based on the above simple design principles. We used dipole resonators for a higher resonance frequency and meander line resonators for a lower resonance frequency, shown in Fig. 3(a), as the meta-atoms of the Brewster metafilms because these resonators exhibit an electric dipole resonance. Electric dipole oscillations induced in these resonators are oriented longitudinally [horizontally in Fig. 3(a)].
Fig. 3. (a) Unit structures of the simulated Brewster metafilms. The dipole resonator (left panel) and the meander line resonator (right panel) are used in Brewster metafilms with higher and lower resonance frequencies, respectively. Orange represents the metal and white represents the dielectric substrate. (b) Schematic diagram of the simulated system. In panels (a) and (b), the geometrical parameters are $w=0.5\,{\textrm{mm}}$, $g=0.5\,{\textrm{mm}}$, $l_{\textrm {m}}=17.5\,{\textrm{mm}}$, $w_{\textrm {s}}=19.5\,{\textrm{mm}}$, $d=20.0\,{\textrm{mm}}$, and $L_{\textrm {h}}=16.0\,{\textrm{mm}}$. The value of $L_{\textrm {v}}$, which is the same as the size of the simulated system orthogonal to the plane of paper, is $27.0\,{\textrm{mm}}$ in Sec. 4.1 and $15.0\,{\textrm{mm}}$ in Sec. 4.2. This change is merely for altering the radiative loss of the metafilms. The values of $l_{\textrm {d}}$ and $w_{\textrm {m}}$ vary and are stated in the main text. Note that the orientation of the $xyz$-axes depends on whether the electromagnetic response is being analyzed for metamaterial 1 or 2.
We investigated the transmission properties of these Brewster metafilms using COMSOL Multiphysics. A schematic of the simulated system is shown in Fig. 3(b). The dipole and meander line resonators were assumed to be fabricated using printed circuit boards. The metal of the printed circuit board was assumed to be a perfect electric conductor with vanishing thickness, and the dielectric substrate was assumed to be polyphenylene ether with relative permittivity $\varepsilon _{\textrm {r}}= 3.3( 1+ \textrm {i} 0.005)$ and thickness $t=0.8\,{\textrm{mm}}$. The perfectly matched layer boundary condition was applied to the boundaries represented by bold lines. The periodic boundary condition was applied to the other boundaries to model periodically arranged meta-atoms. The complex transmittance was calculated as the ratio of the complex amplitudes of the transmitted electric fields with and without the metafilm(s) between the transmitting and receiving ports. This implies that we ignored the thickness of the metafilm(s) in the calculation. Because it is difficult to define the thickness of metafilms without ambiguity, it is preferable to define the complex transmittance without taking the thickness into account. In addition, whether the thickness is taken into account only affects the transmission phase. This additional transmission phase is the same in all the elements of the transmission matrix defined below [Eq. (2)]. Therefore, this additional phase is not physically important and we ignored the thickness in the calculation of the complex transmittance.
Because the periodicity of metamaterial 1 is different from that of metamaterial 2, the electromagnetic response of a metamaterial that includes both cannot be analyzed using periodic boundary conditions. Thus, we separately analyzed the electromagnetic responses of metamaterials 1 and 2 using COMSOL and calculated the total transmittance of the stack based on the following theory.
The reflectances of metamaterials 1 and 2 do not strictly vanish in real systems, but multiple reflection between the metamaterials does not occur because the reflected wave at metamaterial 2 is not incident on metamaterial 1. Therefore, the total transmittance is determined only by the transmittances of the two metamaterials. Let us define the transmittances of metamaterials 1 and 2 for the p- and s-polarizations as follows:
4. Results
4.1 Design of waveplates based on the off-resonance method
We designed several waveplates based on the off-resonance method schematically shown in Figs. 2(a)–2(c) to demonstrate control of the polarization dependence of the complex transmission spectrum using the Brewster effect in metafilms. Figures 4(a) and 4(b) show the absolute value and argument of the complex transmittance of a Brewster metafilm comprising the meander line resonator with $w_{\textrm {m}}=1.7\,{\textrm{mm}}$, labeled $t_1$. The frequency derivative of $\arg {(t_{\textrm {1pp}})}$ reaches a maximum at the fundamental resonance frequency $5.5\,{\textrm{GHz}}$. The absolute values of the transmittances satisfy $|t_{\textrm {1pp}}| \approx 1$, $|t_{\textrm {1ss}}| \approx 1$, $|t_{\textrm {1sp}}| \approx 0$, and $|t_{\textrm {1ps}}| \approx 0$ in the analyzed frequency region, except at $10.1\,{\textrm{GHz}}$, where the value of $|t_{\textrm {1pp}}|$ decreases from unity because of the second-order resonance, which was confirmed by the current distribution analyzed using COMSOL (not shown). The radiative loss of the second-order resonance is lower than that of the fundamental resonance; thus, it is found from Eq. (1) that $|t_{\textrm {1pp}}|$ at the second-order resonance frequency is less than that at the fundamental resonance frequency. Because the absolute value of the complex transmittance should be as high as possible in waveplates, we assume in this study that the Brewster metafilms are operated below the second-order resonance frequency. Because of this assumption, we did not calculate the transmission spectrum with a sufficiently small frequency step to clearly observe the second-order resonance line.
Fig. 4. Frequency dependence of the complex transmittance when metamaterial 1 is a Brewster metafilm comprising the meander line resonator with $w_{\textrm {m}}=1.7\,{\textrm{mm}}$ and metamaterial 2 is a Brewster metafilm comprising the dipole resonator with $l_{\textrm {d}}=10.5\,{\textrm{mm}}$. (a) Absolute value of the complex transmittance of metamaterial 1. (b) Transmission phases of metamaterials 1 and 2. (c) Total complex transmittance when metamaterials 1 and 2 are stacked. The transmission phase and the transmission phase difference are wrapped in the range $-\pi$ to $\pi$. The horizontal dashed lines indicate the region where the deviation of the transmission phase difference from $-\pi /4$ is less than $10{\%}$. The vertical dashed lines indicate the corresponding frequency range. Note that the red circles and blue squares in (a) and (c) [the black and cyan diamonds in (a)-(c)] almost overlap.
Figure 4(b) also shows the transmission phase of a Brewster metafilm comprising the dipole resonator with $l_{\textrm {d}}=10.5\,{\textrm{mm}}$, labeled $t_2$. The resonance frequency of this metafilm is $9.3\,{\textrm{GHz}}$. The frequency dependence of $\arg {(t_{\textrm {2ss}})}$ is almost the same as that of $\arg {(t_{\textrm {1ss}})}$. Although the values of $\arg {(t_{\textrm {1ss}})}$ and $\arg {(t_{\textrm {2ss}})}$ do not vanish, the difference between $\arg {(t_{\textrm {1ss}})}$ and $\arg {(t_{\textrm {2ss}})}$ is small. This implies that $t_{\{ 1,2 \} \mathrm {ss}}$ do not affect the transmission phase difference in the configuration shown in Fig. 1(b), which is substantially the same situation as that assumed in the theory.
Figure 4(c) shows the total complex transmittance calculated using Eq. (5) when metamaterial 1 is a Brewster metafilm comprising the meander line resonator with $w_{\textrm {m}}=1.7\,{\textrm{mm}}$ and metamaterial 2 is a Brewster metafilm comprising the dipole resonator with $l_{\textrm {d}}=10.5\,{\textrm{mm}}$. Halfway between the resonance frequencies, the transmission phase difference $\arg {(t_{xx})}-\arg {(t_{yy})}$ is $-\pi /4$, and it does not vary much with frequency. In this paper, let us define the operating frequency range of a waveplate as the frequency range where the deviation of the transmission phase difference from the desired value is less than $10{\%}$. Then the operating frequency range of this $\lambda /8$ plate is from $6.79\,{\textrm{GHz}}$ to $8.32\,{\textrm{GHz}}$ (relative bandwidth: 20%). In this frequency range, the conditions $|t_{xx}| \approx 1$, $|t_{yy}| \approx 1$, $|t_{xy}| \approx 0$, and $|t_{yx}| \approx 0$ are satisfied. Therefore, an efficient $\lambda / 8$ plate can be realized by simply stacking metamaterials 1 and 2 comprising Brewster metafilms based on the off-resonance method.
Next, we showed that waveplates with different transmission phase differences can be produced by changing the difference between the resonance frequencies of metamaterials 1 and 2. Figure 5(a) shows the transmission phases of metamaterials 1 and 2 and the total complex transmittance when metamaterial 1 is a Brewster metafilm comprising the meander line resonator with $w_{\textrm {m}}=0.8\,{\textrm{mm}}$ and metamaterial 2 is a Brewster metafilm comprising the dipole resonator with $l_{\textrm {d}}=12.0\,{\textrm{mm}}$. From the transmission phase, the resonance frequencies of metamaterials 1 and 2 are found to be $6.4\,{\textrm{GHz}}$ and $8.5\,{\textrm{GHz}}$, respectively. The resonance frequency difference is smaller than that in Fig. 4. The transmission phase difference $\arg {(t_{xx})}-\arg {(t_{yy})}$ halfway between the resonance frequencies is $-\pi /2$, and the operating frequency range of this $\lambda /4$ plate is from $7.02\,{\textrm{GHz}}$ to $8.02\,{\textrm{GHz}}$ (relative bandwidth: 13%). The absolute values of $t_{xx}$ and $t_{yy}$ are almost unity in this frequency range; therefore, this design realizes an efficient $\lambda /4$ plate.
Fig. 5. Frequency dependences of the complex transmittances when metamaterial 1 (metamaterial 2) is a Brewster metafilm comprising (a) the meander line resonator with $w_{\textrm {m}}=0.8\,{\textrm{mm}}$ (the dipole resonator with $l_{\textrm {d}}=12.0\,{\textrm{mm}}$) and (b) the dipole resonator with $l_{\textrm {d}}=15.0\,{\textrm{mm}}$ ($l_{\textrm {d}}=13.0\,{\textrm{mm}}$). The upper row shows the transmission phases for each metamaterial, and the lower row shows the total complex transmittances. The transmission phase and the transmission phase difference are wrapped in the range $-\pi$ to $\pi$ except for $\arg {(t_{xx})}-\arg {(t_{yy})}$ in (b), which is wrapped in the range 0 to $2\pi$ for better visualization. In the lower panels, the horizontal dashed lines indicate the region where the transmission phase difference deviates by less than $10{\%}$ from $-\pi /2$ in (a) or from $\pi$ in (b). The vertical dashed lines indicate the corresponding frequency ranges. Note that the red circles and blue squares in the lower panels [the black and cyan diamonds in all the panels] almost overlap.
Figure 5(b) shows the transmission phases of metamaterials 1 and 2 and the total complex transmittance when metamaterial 1 is a Brewster metafilm comprising the dipole resonator with $l_{\textrm {d}}=15.0\,{\textrm{mm}}$ and metamaterial 2 is a Brewster metafilm comprising the dipole resonator with $l_{\textrm {d}}=13.0\,{\textrm{mm}}$. The resonance frequencies of metamaterials 1 and 2 are $7.2\,{\textrm{GHz}}$ and $8.0\,{\textrm{GHz}}$, respectively. In this case, the resonance frequency difference is further decreased from that in Fig. 5(a). The transmission phase difference halfway between the resonance frequencies is $\pi$, and the operating frequency range of this $\lambda /2$ plate is from $7.36\,{\textrm{GHz}}$ to $7.86\,{\textrm{GHz}}$ (relative bandwidth: 7%). Because the resonance frequencies are close to the operating frequency range in this case, the decreases in $|t_{xx}|$ and $|t_{yy}|$ from unity are larger than in the other cases. However, $|t_{xx}|$ and $|t_{yy}|$ are still high because the radiative loss is much larger than the non-radiative loss ($\gamma _{\textrm {r}} \gg \gamma _{\textrm {nr}}$). These results confirm that decreasing the resonance frequency difference increases the transmission phase difference while decreasing the operating frequency bandwidth.
As described in Sec. 2, the transmission phase difference between the orthogonal linear polarization components can be increased without decreasing the resonance frequency difference. It can also be increased proportionally by increasing the number of layers of Brewster metafilm in metamaterials 1 and 2 because reflection does not occur at Brewster metafilms. We analyzed the total complex transmittance when metamaterial 1 (metamaterial 2) comprises two layers of Brewster metafilm with $w_{\textrm {m}}=1.7\,{\textrm{mm}}$ ($l_{\textrm {d}}=10.5\,{\textrm{mm}}$), which are the same dimensions as in Fig. 4. The calculated total complex transmittance is shown in Fig. 6. The transmission phase difference $\arg {(t_{xx})}-\arg {(t_{yy})}$ is $-\pi /2$ halfway between the resonance frequencies, and the operating frequency range of this $\lambda / 4$ plate is from $6.76\,{\textrm{GHz}}$ to $8.31\,{\textrm{GHz}}$ (relative bandwidth: 21%). The transmission phase difference halfway between the resonance frequencies is twice as large as in Fig. 4, and the relative bandwidth is almost the same between the two cases, demonstrating that the transmission phase difference can be proportionally increased by increasing the number of Brewster metafilm layers in metamaterials 1 and 2 while maintaining the same operating bandwidth.
Fig. 6. Frequency dependence of the total complex transmittance when metamaterials 1 and 2 comprise two layers of Brewster metafilm each with $w_{\textrm {m}}=1.7\,{\textrm{mm}}$ and $l_{\textrm {d}}=10.5\,{\textrm{mm}}$, respectively. The transmission phase and transmission phase difference are wrapped in the range $-\pi$ to $\pi$. The horizontal dashed lines indicate the region where the deviation of the transmission phase difference from $-\pi /2$ is less than $10{\%}$, and the vertical dashed lines indicate the corresponding frequency range. Note that the red circles and blue squares (the black and cyan diamonds) almost overlap.
These results show that the transmission phase difference and the operating bandwidth of the waveplate can be selected by tuning the resonance frequency difference and number of Brewster metafilm layers in metamaterials 1 and 2. The operating bandwidth of a waveplate with a fixed transmission phase difference can be increased by increasing both the resonance frequency difference and the number of layers. However, the operating bandwidth cannot be increased indefinitely in real systems because second- and higher-order resonances occur in meta-atoms, as described above and shown in Fig. 4. Transmission spectrum designs that consider second- and higher-order resonances are possible, and such design methods may be developed in future studies.
4.2 Design of waveplates based on the on-resonance method
In this section, we design waveplates based on the on-resonance method schematically shown in Figs. 2(d)–2(f). We numerically analyzed the transmission properties of various metamaterials comprising three distinct layers of Brewster metafilm to find parameters that realized a linear increase in the transmission phase with frequency over a broad frequency range. Figures 7(a) and 7(b) show the transmittance of metamaterial 1, comprising three distinct layers of Brewster metafilm with $l_{\textrm {d}} =17.0$, $12.8$, and $10.5\,{\textrm{mm}}$. The transmission phase for the p-polarization, $\arg {(t_{\textrm {1pp}})}$, increases almost linearly with frequency between $7\,{\textrm{GHz}}$ and $10\,{\textrm{GHz}}$. The decrease of $|t_{\textrm {1pp}}|$ in this frequency range is larger than that in the operating frequency range of the waveplates designed based on the off-resonance method, shown in the previous section, for two reasons. First, the resonance frequency of each constituent metafilm is now within the operating frequency range, and second, the transmittance at each resonance frequency decreases from unity through dielectric loss in the substrate. The decrease in $|t_{\textrm {1ss}}|$ from unity is also larger than that in the waveplates designed based on the off-resonance method. This is because the number of layers of the dielectric substrate has increased, which increases the reflectance of the s-polarization. [Note that the longitudinal direction of the dielectric substrate, i.e., the vertical direction in Fig. 3(a), is parallel to the electric field of the s-polarization component.] The values of $|t_{\textrm {1sp}}|$ and $|t_{\textrm {1ps}}|$ are also negligibly small in this case.
Fig. 7. Frequency dependence of the complex transmittance when metamaterial 1 (metamaterial 2) comprises three distinct layers of Brewster metafilm with $l_{\textrm {d}} =17.0$, $12.8$, and $10.5\,{\textrm{mm}}$ ($l_{\textrm {d}} =15.1$, $12.0$, and $9.8\,{\textrm{mm}}$). (a) Absolute value of the complex transmittance of metamaterial 1. (b) Transmission phases of metamaterials 1 and 2. (c) Total complex transmittance when metamaterials 1 and 2 are stacked. The horizontal dashed lines indicate the region where the deviation of the transmission phase difference from $\pi /2$ is less than $10{\%}$, and the vertical dashed lines indicate the corresponding frequency range. Note that the red circles and blue squares in (a) and (c) [the black and cyan diamonds in (a)-(c)] almost overlap.
Based on the above results, we searched for parameters for metamaterial 2, also comprising three distinct layers of Brewster metafilm, with a $\varDelta \omega$ suitable for realizing a $\lambda /4$ plate. Figure 7(b) shows the transmission phase of metamaterial 2 with layer dimensions $l_{\textrm {d}} =15.1$, $12.0$, and $9.8\,{\textrm{mm}}$, respectively, which are smaller than the values of $l_{\textrm {d}}$ for metamaterial 1. Metamaterial 2 has a frequency range over which $\arg {(t_{\textrm {2pp}})}$ increases almost linearly with frequency, and the resonance frequencies are higher than those of metamaterial 1. The frequency dependences of $\arg {(t_{\mathrm {\{ 1,2 \} ss}})}$ for metamaterials 1 and 2 are almost the same.
Figure 7(c) shows the total complex transmittance when metamaterials 1 and 2 are stacked, calculated using Eq. (5). The deviation of the transmission phase difference $\arg {(t_{xx})}-\arg {(t_{yy})}$ from $\pi /2$ is less than $10{\%}$ in the frequency range $7.02\,{\textrm{GHz}}$ to $10.23\,{\textrm{GHz}}$ (relative bandwidth: 37%). In this frequency range, decreases in $|t_{xx}|$ and $|t_{yy}|$ are not large. Therefore, a broadband $\lambda /4$ plate with high transmission efficiency can also be designed based on the on-resonance method.
Finally, we show that a $\lambda /2$ plate can be realized by increasing $\varDelta \omega$, i.e., further decreasing the values of $l_{\textrm {d}}$ for metamaterial 2. Figure 8(a) shows the transmission phase of metamaterial 1 (metamaterial 2) comprising three distinct layers of Brewster metafilm with $l_{\textrm {d}} =17.0$, $12.4$, and $10.5\,{\textrm{mm}}$ ($l_{\textrm {d}} =13.5$, $11.2$, and $9.0\,{\textrm{mm}}$). The resonance angular frequency difference $\varDelta \omega$ has increased compared with $\varDelta \omega$ of the $\lambda /4$ plate in Fig. 7.
Fig. 8. Frequency dependence of the complex transmittance when metamaterial 1 (metamaterial 2) comprises three distinct layers of Brewster metafilm with $l_{\textrm {d}} =17.0$, $12.4$, and $10.5\,{\textrm{mm}}$ ($l_{\textrm {d}} =13.5$, $11.2$, and $9.0\,{\textrm{mm}}$). (a) Transmission phases of metamaterials 1 and 2. (b) Total complex transmittance when metamaterials 1 and 2 are stacked. The transmission phase (transmission phase difference) is wrapped in the range $-\pi$ to $\pi$ ($0$ to $2\pi$). The horizontal dashed lines indicate the region where the deviation of the transmission phase difference from $\pi$ is less than $10{\%}$, and the vertical dashed lines indicate the corresponding frequency range. Note that the red circles and blue squares in (b) [the black and cyan diamonds in (a) and (b)] almost overlap.
Figure 8(b) shows the total complex transmittance when metamaterials 1 and 2 are stacked. The deviation of the transmission phase difference $\arg {(t_{xx})}-\arg {(t_{yy})}$ from $\pi$ is less than $10{\%}$ between $7.39\,{\textrm{GHz}}$ and $10.42\,{\textrm{GHz}}$ (relative bandwidth: 34%). In this frequency range, the absolute values of transmittances $|t_{xx}|$ and $|t_{yy}|$ are comparable with those of the $\lambda /4$ plate shown in Fig. 7. Therefore, a broadband $\lambda /2$ plate with high transmission efficiency can be realized by increasing $\varDelta \omega$ relative to the $\lambda /4$ plate.
These results confirm that an efficient waveplate with a desired transmission phase difference and operating bandwidth can be designed by varying $\varDelta \omega$ and the number of metafilm layers. We note that even if the layer numbers of metamaterials 1 and 2 are increased, the operating bandwidth cannot increase indefinitely because of the second and higher-order resonance of the meta-atoms, similar to the limitations of the off-resonance method.
5. Discussion
We discuss the difference between Brewster metafilms and Huygens metasurfaces in this section. The independent control of the orthogonal linear polarization components shown in this study may also be realized using broadband Huygens metasurfaces in principle. However, it is not easy to design broadband Huygens metasurfaces that respond to only one linear polarization component because Huygens metasurfaces exhibit electric dipole, magnetic dipole, and other multipolar responses as described in Sec. 1. Therefore, to control the polarization dependence of the complex transmission spectrum using Huygens metasurfaces, it is necessary in practice to design the structure of metasurfaces by simultaneously taking into account the frequency dependences of the electromagnetic responses for the orthogonal linear polarization components.
In contrast to Huygens metasurfaces, Brewster metafilms can be composed of meta-atoms that exhibit only an electric dipole response, thus, broadband reflectionless metamaterials that respond to only one linear polarization component can easily be designed using Brewster metafilms. The orthogonal linear polarization components can be controlled independently by simply stacking Brewster metafilms, therefore, anisotropic metamaterials with various complex transmission spectra may be designed without difficulty. In fact, we showed in this study that broadband waveplates with high transmission efficiency could be designed based on two different methods. The waveplates designed based on these two methods both have a constant transmission phase difference over a broad frequency range but differ in the frequency derivative of the transmission phase, i.e., the group delay. This implies that the frequency and polarization dependences of the transmittance, the transmission phase, the group delay, and even the group delay dispersion may be designed together at will over a broad frequency range using Brewster metafilms. In addition, when structures that respond to either of the circular polarizations [49] are used as meta-atoms of Brewster metafilms, the complex transmission spectrum may be controlled in the circular polarization basis.
The structures of Brewster metafilms are non-planar, thus, the fabrication of Brewster metafilms is more difficult than Huygens metasurfaces. We think that teraherz (optical) Brewster metafilms may be fabricated using microelectromechanical systems [29] (3D laser lithography [30] and 3D nanoprinting techniques [32]). However, in the optical region, Brewster metafilms may need to be designed using dielectric meta-atoms when the value of $\gamma _{\textrm {nr}} / \gamma _{\textrm {r}}$ has to be small.
6. Conclusion
We have presented a proof-of-concept numerical study demonstrating control of the polarization dependence of the complex transmission spectrum using the Brewster effect in metafilms. When structures that exhibit only an electric dipole response are used as meta-atoms of Brewster metafilms, the orthogonal linear polarization components of an incident electromagnetic wave can be controlled independently by simply stacking two metamaterials, both Brewster metafilms. To verify this theory, we showed that broadband waveplates with high transmission efficiency could be designed based on two different methods. The results imply that the complicated frequency and polarization dependence of the complex transmittance may be simply designed over a broad frequency range using Brewster metafilms. Further development of Brewster metafilms will enable us to design electromagnetic media with properties that are difficult to realize using metasurfaces.
Funding
Japan Society for the Promotion of Science (JP21K04192); Japan Science and Technology Agency (CREST, JPMJCR2101).
Disclosures
The author declares no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.
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