Issue 
EPJ Appl. Metamat.
Volume 4, 2017
Artificial materials for advanced applications in electromagnetics and mechanics



Article Number  2  
Number of page(s)  7  
DOI  https://doi.org/10.1051/epjam/2016016  
Published online  13 January 2017 
https://doi.org/10.1051/epjam/2016016
Research Article
Theoretical study on dynamical planarchirality switching in checkerboardlike metasurfaces
^{1}
Department of Electronic Science and Engineering, Kyoto University, Kyoto
6158510, Japan
^{2}
Center for Energy and Environmental Science, Shinshu University, 4171 Wakasato, Nagano
3808553, Japan
^{*} email: urade@giga.kuee.kyotou.ac.jp
Received:
10
September
2016
Accepted:
15
November
2016
Published online: 13 January 2017
In this paper, we show that the handedness of a planar chiral checkerboardlike metasurface can be dynamically switched by modulating the local sheet impedance of the metasurface structure. We propose a metasurface design to realize the handedness switching and theoretically analyze its electromagnetic characteristic based on Babinet’s principle. Numerical simulations of the proposed metasurface are performed to validate the theoretical analysis. It is demonstrated that the polarity of asymmetric transmission for circularly polarized waves, which is determined by the planar chirality of the metasurface, is inverted by switching the sheet impedance at the interconnection points of the checkerboardlike structure. The physical origin of the asymmetric transmission is also discussed in terms of the surface current and charge distributions on the metasurface.
Key words: Planar chirality / Asymmetric transmission / Tunable metamaterials / Babinet’s principle
© Y. Urade et al., Published by EDP Sciences, 2017
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Chiral structures have been one of the most interesting subjects in metamaterials research [1, 2]. In analogy to the chirality of threedimensional structures, a planar structure is said to be planar chiral if it has no line of mirror symmetry in its structure plane (e.g., an Archimedean spiral) [3, 4]. In other words, a planar chiral structure cannot be superimposed on its inplane mirror image by inplane rotations and/or translations. Note that the handedness of a planar chiral structure is reversed if it is observed from the opposite side of its structure plane. Polarization effects related to the planar chiral structures for electromagnetic waves have been studied in terms of optical activity and reciprocity for scattered light [5–7]. In addition, it has been identified that the total transmission intensity of circularly polarized waves normally incident on planar chiral structures depends on the direction of the wave propagation, provided that the planar chiral structures are anisotropic and lossy [8]. This phenomenon is called asymmetric transmission for circularly polarized waves and is consistent with the Lorentz reciprocity theorem, although it is seemingly nonreciprocal. Asymmetric transmission of planar chiral metamaterials has been experimentally reported in from microwave [8, 9] and terahertz [10] to optical frequency regions [11]. Moreover, it has also been extended to linearly polarized waves [12].
Recently, metamaterials and metasurfaces with dynamically tunable electromagnetic properties have been extensively studied towards applications to the advanced control of electromagnetic wave propagation [13, 14]. The recent progress has enabled the dynamical tuning of optical activity resulting from the chirality of artificial structures by employing photoexcitation of semiconductors [15–19], microelectromechanical systems [20, 21], and phase change materials [22]. On the other hand, the possibility of dynamical tuning of planar chirality and the associated asymmetric transmission for circular polarizations remains unexplored in literature.
In this paper, in order to realize dynamical planarchirality switching, we utilize the singular characteristic of metallic checkerboardlike structures, which is critically dependent on the electrical connectivity at the interconnection points of their metallic patches [23–32]. We have shown that their characteristics can be understood in terms of Babinet’s principle, which is extended to screens with finite sheet impedances [30, 31]. According to Babinet’s principle, by switching the local sheet impedance at the interconnection points, the responses of the checkerboardlike structures can dynamically alternate between two contrasting behaviors, such as capacitiveinductive responses [33] and anisotropic transmission characteristics for orthogonal linear polarizations [34]. Here, we apply such dynamicalcheckerboard designs to dynamical planarchirality switching. In contrast to the previous designs [33, 34], a new design contains resonant planar chiral structures, which enable planarchirality switching of the whole structure.
This paper is organized as follows. First, in Section 2, we present the design of a metasurface that enables dynamical planarchirality switching, and theoretically demonstrate its capability of planarchirality switching based on Babinet’s principle. In Section 3, we show the results of numerical simulations of the presented metasurface using the finite element method and discuss the physical origin of planarchirality switching. Finally, the conclusions are drawn in Section 4.
2 Design and operation principle
2.1 Babinet’s principle for the incidence of circularly polarized waves
Before proceeding to a theoretical analysis of planar chiral metasurfaces based on Babinet’s principle, we review Babinet’s principle in particular for the incidence of circular polarization. Babinet’s principle relates scattering problems of metasurfaces to those of their complementary structures. We assume that a metasurface in a vacuum has a spatially varying sheet impedance of Z_{s}(x, y), where (x, y) is the coordinate on the metasurface. Then, the sheet impedance of its complementary metasurface is given by(1)where Z_{0} is the impedance of a vacuum. This transformation is called the impedance inversion [30]. The electromagnetic fields in the two problems are related to each other if the incident fields in the complementary scattering problem satisfy , where (E_{in}, H_{in}) is the incident electromagnetic field in the original problem. In particular, for the incidence of linearly polarized plane waves, this interchange of E and H is equivalent to the polarization rotation by π/2. On the other hand, for the incidence of circularly polarized plane waves, the incident polarization state in the complementary problem is equal to that in the original problem, up to a phase factor of exp(±iπ/2), which depends on the handedness of the incident circular polarization [30]. For example, let us consider a problem shown in Figure 1a. Its complementary problem is depicted in Figure 1b. For the incidence of left circularly polarized (LCP) plane waves, we can derive the following equations via Babinet’s principle [30]:(2) (3)where and (i, j = L, R) denote the complex amplitude transmission coefficients from incident j circular polarization to i circular polarization in the original and complementary problems, respectively. For the incidence of right circularly polarized (RCP) waves, L and R in equations (2) and (3) are interchanged.
Figure 1. Illustration of Babinet’s principle for the incidence of LCP plane waves. (a) Original problem; (b) complementary problem. The symbol k denotes the wavevector of the plane waves. This figure is based on Figure 2 of reference [31]. 
2.2 Design of the metasurface for planar chirality switching
Let us consider a planar chiral checkerboardlike metasurface in a vacuum, as shown in Figure 2a. We assume it to be infinitely periodic and infinitely thin. At the interconnection points of the perfectelectricconductor (PEC) patches, there are variable impedance sheets with a sheet impedance of Z_{s}. Deformed gammadions (hereafter, RΓs) [35, 36] and their complementary inplane mirror images (CLΓs) are embedded in the holes and the PEC patches of a plain checkerboardlike structure so that the metasurface with Z_{s} = Z_{0}/2 is congruent with the inplane mirror image of its complementary structure. The handedness of the planar chiral metasurface is determined by the handedness of RΓs and CLΓs, and by the electrical connectivity of the PEC patches. As mentioned above, electromagnetic responses of metallic checkerboardlike structures critically depend on the electrical connectivity at the interconnection points of their PEC patches. Here, we initially assume . In other words, the PEC patches are virtually disconnected.
Figure 2. (a) Highimpedance structure of the planar chiral checkerboardlike metasurface; (b) complementary structure obtained through the impedance inversion of that shown in (a); (c) lowimpedance structure of the metasurface. 
To evaluate the asymmetric transmission of the metasurface, we define the totaltransmissionintensity difference ΔT_{L} [8] for the LCP waves by(4)where power transmittance and the symbol → (←) indicate normal incidence from z > 0 (z < 0). Here, we assume that scattering into diffraction modes is negligible. In other words, the periodicity of the metasurface is smaller than the incident wavelength. Note that is satisfied owing to the reciprocity. From the mirror symmetry with respect to the z = 0 plane, we can show ΔT_{R} = −ΔT_{L} for the normal incidence of the RCP waves. Thus, we will omit the subscript as ΔT ≔ ΔT_{L} for simplicity below. The sign of ΔT is determined by the handedness of the planar chiral metasurface observed from z > 0. We note that ΔT can also be written as because we have from the reciprocity. Thus, the asymmetric transmission for circularly polarized waves can be regarded as dichroism with respect to circular polarization conversion.
2.3 Theoretical analysis of planarchirality switching based on Babinet’s principle
Figure 2b shows the complementary structure of Figure 2a, the electromagnetic responses of which are related through Babinet’s principle to those of the original one. In the complementary structure, the original PEC (hole) areas are changed to holes (PEC), and the impedance sheets with finite sheet impedance are transformed into impedance sheets with via equation (1). According to Babinet’s principle, equation (3), the polarization conversion efficiency of the complementary structure is equal to that of the original one, i.e., and hold, where the superscripts “(c)” and “(high)” represent quantities of the complementary and highimpedance structures, respectively. Therefore, we have(5)
On the other hand, by switching the sheet impedance at the interconnection points from to , the metasurface shown in Figure 2a changes to the structure shown in Figure 2c. If is satisfied, the lowimpedance structure shown in Figure 2c is congruent with the inplane mirror image of the complementary structure shown in Figure 2b. In other words, Figure 2c is the observation of Figure 2b from the opposite (z < 0) side. Thus, we have and , where the superscript “(low)” represents the lowimpedance structure. Finally, we obtain the following equations:(6)where equation (5) is used. This equation indicates that the sign of ΔT is inverted just by switching the sheet impedance at the corners. Therefore, the handedness of the planar chiral metasurface is dynamically switchable by using variable impedance sheets. Note that we have in the limit of and , where the metasurface is lossless. This is because loss is essential for the asymmetric transmission in planar chiral structures [8]. We also note that ΔT ^{(c)} = −ΔT ^{(high)} = −ΔT ^{(low)} = 0 if .
3 Numerical simulation
3.1 Planar chiral checkerboardlike metasurface in a vacuum
In order to validate the theoretical analysis in Section 2, we numerically simulated the electromagnetic responses of the proposed metasurface using the finite element method solver (ANSYS HFSS). Figure 3a shows the unit cell of the simulated metasurface, which was placed at z = 0 in the computational domain of a vacuum. The geometrical parameters in Figure 3a are: a = 300 μm, d = 30 μm, l = 180 μm, p = 80 μm, and w = 10 μm. The CLΓs have the same parameters as the RΓs. There are sheets with variable sheet impedances Z_{s} at the interconnection points of the PEC patches. We imposed periodic boundary conditions on the side boundaries of the computational domain to simulate an infinitely periodic system. In the simulation, the complex amplitude transmission coefficients for the normal incidence of x and y linearly polarized waves were calculated, and then, the transmission coefficients were transformed into those in the circularpolarization basis.
Figure 3. (a) Unit cell of the metasurface in the numerical simulation. The geometric parameters are as follows: a = 300 μm, d = 30 μm, l = 180 μm, p = 80 μm, and w = 10 μm; (b) calculated ΔT spectra for Z_{s}/Z_{0} = 25 and 0.01; (c) calculated spectra of and for Z_{s}/Z_{0} = 25 and 0.01. 
The calculated ΔT spectra for Z_{s}/Z_{0} = 25 and 0.01 are shown in Figure 3b. Note that Z_{s}/Z_{0} = 25 and 0.01 are complementary to each other via the impedance inversion. We observe ΔT is resonantly enhanced around 0.504 THz. In addition, it is clearly confirmed that the polarity of the transmission asymmetry is inverted by switching the sheet impedance Z_{s}, which agrees well with equation (6). In other words, the handedness of the planar chiral metasurface is dynamically switchable by modulating its local sheet impedance Z_{s}. Figure 3c shows the calculated spectra of T_{RL} for incidence on the front of the metasurface (z > 0) or back (z < 0). Because the problems of Z_{s}/Z_{0} = 25 are related to the problems of Z_{s}/Z_{0} = 0.01 with the opposite incident direction via Babinet’s principle, as expected from equation (3), two types of T_{RL} spectra, one of which shows a dip around 0.504 THz, are interchanged if the incident direction is reversed or Z_{s} is switched.
To understand the resonant character of ΔT around 0.504 THz, we simulated the distributions of the surface current J_{surf} at z = 0 and the z component of the electric flux density D_{z} at z = 1 μm, which approximately corresponds to the surface charge density on the metasurface, for the normal incidence of the LCP waves at 0.504 THz. Figure 4 shows the snapshots of the distributions at the phase when J_{surf} takes its maximum value. For clarity, the phase of D_{z} is shifted by π/2 relative to J_{surf}.
Figure 4. Distributions of the electric flux density D_{z} at z = 1 μm and surface current J_{surf} on the metasurface for the incidence of the LCP waves from the front (z > 0) or back (z < 0) sides of the metasurface structure at 0.504 THz: (a) Z_{s}/Z_{0} = 25 with frontside incidence; (b) Z_{s}/Z_{0} = 25 with backside incidence; (c) Z_{s}/Z_{0} = 0.01 with frontside incidence; (d) Z_{s}/Z_{0} = 0.01 with backside incidence. The color maps and arrows in green indicate D_{z} and the surface current vector, respectively. For clarity, the phase of D_{z} is shifted by π/2 relative to J_{surf}. 
In the case of Z_{s}/Z_{0} = 25 with frontside incidence of the LCP waves shown in Figure 4a, a dipolar current along the y direction is induced on RΓs, which causes a local magnetic field in the x direction around the bars of RΓs. Generally, such a dipolar current contributes to circular polarization conversion because its ypolarized emission is a superposition of the LCP and RCP waves [9]. However, the calculated T_{RL} spectrum shown in Figure 3c shows a dip around 0.504 THz. Therefore, the emission from the resonance mode in RΓs destructively interferes with that from the charge accumulated at the corners of the checkerboard structure (see Appendix). As mentioned above, the case of Z_{s}/Z_{0} = 0.01 with backside incidence shown in Figure 4d is the complementary problem of the case shown in Figure 4a. In Figure 4d, we confirm a loop current induced around the arms of CLΓs and charge accumulation along the slits of CLΓs in the y direction, which causes a local electric field E_{loc} in the x direction in the slits of CLΓs. This resonance mode characterized by E_{loc} is complementary to the one characterized by H_{loc} in Figure 4a. We note that the polarization of the waves emitted from the complementary mode of CLΓs is rotated by π/2 compared to that from the RΓs due to the interchange of E and H in Babinet’s principle. The dip in the T_{RL} spectrum can be explained in the same way as above.
On the other hand, in the cases of Z_{s}/Z_{0} = 25 with backside incidence (Figure 4b) and Z_{s}/Z_{0} = 0.01 with frontside incidence (Figure 4c), both the resonance modes of RΓs and CLΓs are excited. The phase maximizing E_{loc} in CLΓs is delayed by approximately π/2, relative to that maximizing H_{loc} in RΓs. Here, note that D_{z} is depicted with a π/2 phase shift relative to J_{surf} in Figure 4. Hence, the RCP components are dominantly emitted from RΓs and CLΓs in the k directions of Figures 4b and 4c, and the corresponding T_{RL} exhibits a relatively large value around 0.504 THz, as shown in Figure 3c.
From an application perspective, continuous tunability of ΔT is also a desired feature. Figure 5 shows the calculated Z_{s} dependence of ΔT at 0.504 THz. It is confirmed that ΔT can be continuously tuned by controlling Z_{s} around Z_{0}/2. As predicted in Section 2, ΔT tends to zero when Z_{s} → ∞ or 0 due to the disappearance of lossy elements. ΔT is also nearly zero at Z_{s} = Z_{0}/2. The maximum value of ΔT is about 0.22 in the present case. There is little room for further increasing ΔT by optimizing the geometry of the deformed gammadions as T_{RL} is theoretically bound by 0.25 for electric singlelayer structures [37].
Figure 5. Calculated Z_{s} dependence of ΔT at 0.504 THz. 
3.2 Planar chiral checkerboardlike metasurface on a dielectric substrate
To actually materialize the proposed metasurface, there must be a dielectric substrate supporting the thin metasurface structure. The existence of the substrate breaks the mirror symmetry with respect to z = 0 and thereby invalidates Babinet’s principle and equation (6). This difficulty can be approximately circumvented by just sandwiching the metasurface structure between the same dielectric plates [31]. In addition, it is reasonable to expect that similar responses are obtained if the metasurface is placed on a dielectric substrate with a moderate refractive index. Here, we will numerically demonstrate that sign inversion of ΔT can occur for a metasurface placed on a dielectric substrate. We simulated a similar model to Figure 3a with the computational domain in z < 0 filled by a lossless dielectric material having a refractive index of n = 3.4, which corresponds to that of highresistivity silicon in the terahertz regime [38]. To reflect the wavelengthshortening effect in the substrate, all the geometrical parameters were divided by n. Figure 6 shows the calculated ΔT spectra for Z_{s}/Z_{d} = 25 and 0.01, where Z_{d} ≔ Z_{0}/n. We can confirm a sign inversion of ΔT, although the spectral shapes of ΔT are not identical between the two cases. This is attributed to the broken mirror symmetry with respect to z = 0 caused by the substrate.
Figure 6. Calculated ΔT spectra of the planar chiral checkerboardlike metasurface on a dielectric substrate for Z_{s}/Z_{d} = 0.01 and 25. 
4 Conclusion
In conclusion, we investigated the dynamical handedness switching in a planar chiral checkerboardlike metasurface loaded with switchable impedance sheets. We theoretically and numerically showed that the polarity of the asymmetric transmission in the proposed planar chiral metasurface can be inverted by modulating the sheet impedance at the interconnection points of the checkerboardlike structure. Such impedance switching can be experimentally implemented in the terahertz regime by exploiting, for example, photoexcitation of semiconductors [16, 17, 39] and metalinsulator transitions of vanadium dioxide [33, 34]. We explained the behaviors of the metasurface from the microscopic point of view, i.e., the field distributions. Note that we can also discuss them in terms of macroscopic parameters such as averaged sheet admittance tensors [40, 41], which facilitate design of metasurfaces with desired transmission and reflection characteristics. Finally, it should be noted that although the simulation was performed in the terahertz regime, the theory discussed in this paper is also applicable to the other frequency ranges, where metals exhibit high conductivity and, thus, Babinet’s principle is valid.
Acknowledgments
This research was partially supported by JSPS KAKENHI Grant Numbers JP15J07603 and JP25790065 and by grant from the Murata Science Foundation. One of the authors (Y.U.) was supported by the JSPS Research Fellowships for Young Scientists.
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Appendix
Origin of the dip in the crosspolarized transmission spectra
To understand the origin of the dip around 0.504 THz in the T_{RL} spectra, we calculated T_{RL} spectra of the metasurface without gammadions of one handedness. Figure 7 shows the spectra of the metasurface with and without RΓs for Z_{s}/Z_{0} = 25. While the spectrum is relatively flat in the case without RΓs, the case with RΓs shows an acute spectral change around 0.5 THz. In addition, we can confirm the enhancement of conversion around 0.46 THz when RΓs are added to the structure. These observations can be understood as in the case of Fano resonance [42] by considering constructive and destructive interference between the resonance of RΓs and the broader background spectrum of the checkerboard metasurface without RΓs, which is caused by the charge accumulated at the corners of the checkerboard structure.
Figure 7. Calculated spectra of the metasurface with and without RΓs for Z_{s}/Z_{0} = 25. 
Cite this article as: Urade Y, Nakata Y, Nakanishi T & Kitano M: Theoretical study on dynamical planarchirality switching in checkerboardlike metasurfaces. EPJ Appl. Metamat. 2017, 4, 2.
All Figures
Figure 1. Illustration of Babinet’s principle for the incidence of LCP plane waves. (a) Original problem; (b) complementary problem. The symbol k denotes the wavevector of the plane waves. This figure is based on Figure 2 of reference [31]. 

In the text 
Figure 2. (a) Highimpedance structure of the planar chiral checkerboardlike metasurface; (b) complementary structure obtained through the impedance inversion of that shown in (a); (c) lowimpedance structure of the metasurface. 

In the text 
Figure 3. (a) Unit cell of the metasurface in the numerical simulation. The geometric parameters are as follows: a = 300 μm, d = 30 μm, l = 180 μm, p = 80 μm, and w = 10 μm; (b) calculated ΔT spectra for Z_{s}/Z_{0} = 25 and 0.01; (c) calculated spectra of and for Z_{s}/Z_{0} = 25 and 0.01. 

In the text 
Figure 4. Distributions of the electric flux density D_{z} at z = 1 μm and surface current J_{surf} on the metasurface for the incidence of the LCP waves from the front (z > 0) or back (z < 0) sides of the metasurface structure at 0.504 THz: (a) Z_{s}/Z_{0} = 25 with frontside incidence; (b) Z_{s}/Z_{0} = 25 with backside incidence; (c) Z_{s}/Z_{0} = 0.01 with frontside incidence; (d) Z_{s}/Z_{0} = 0.01 with backside incidence. The color maps and arrows in green indicate D_{z} and the surface current vector, respectively. For clarity, the phase of D_{z} is shifted by π/2 relative to J_{surf}. 

In the text 
Figure 5. Calculated Z_{s} dependence of ΔT at 0.504 THz. 

In the text 
Figure 6. Calculated ΔT spectra of the planar chiral checkerboardlike metasurface on a dielectric substrate for Z_{s}/Z_{d} = 0.01 and 25. 

In the text 
Figure 7. Calculated spectra of the metasurface with and without RΓs for Z_{s}/Z_{0} = 25. 

In the text 
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