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A simple bilayered chiral metamaterial is constructed by using an array of two twisted split ring resonators (SRRs). Theoretical analysis and simulated results show that chiral metamaterial can achieve strong optical activity of linearly polarized electromagnetic (EM) waves. The arbitrary polarization rotation angle can be readily realized by properly designing the twisted angle. Changing the length of the horizontal bar can tune the spectral response of the SRRs. The induced current densities and magnetic field distributions provide a good explanation to optical activity in the bilayered SRR metamaterial.
© 2014 Optical Society of America
1. Introduction
Metamaterial has been become a rapidly developing and active research field, which could be engineered to own many unique electromagnetic properties such as giant artificial chirality [1,2]. Over the past decade, many chiral metamaterials have been proposed to achieve many customized functionalities, in which the mirror symmetry of the unit-cell structure is broken either in the propagation direction or in the perpendicular plane. Following pioneered work of achieving negative refractive index by chiral metamaterials [3–5], they could readily manipulate the polarization states of electromagnetic waves and show attractive properties, such as giant gyrotropy [6, 7], circular dichroism [8–10], strong optical activity [11–18] and repulsive Casimir force [19, 20]. Thus, chiral metamaterials promise to be functionalized as polarization rotators [21, 22], linear and circular polarizers [23–27] and polarization spectrum filters with ripple-free isolated transmission peaks [28,29]. Recently, much attention has been paid to planar chiral metamaterials because of its intriguing phenomenon of so-called asymmetric transmission for both linearly and circularly polarized light [30–39]. In contrast to the nonreciprocal transmission, the asymmetric transmission in planar chiral metamaterial is reciprocal and fully compliant with the Lorentz’s reciprocity theorem. The reciprocal asymmetric transmission effect arises from reversed orthogonal polarizations conversion efficiencies for opposite directions of wave propagation. Recently, handedness switching in chiral metamaterials has been demonstrated [40], allowing electromagnetic control of the polarization of light for realizing dynamically tunable terahertz circular polarizers and polarization modulators. For electromagnetic waves, polarization is undoubtedly one of the most important fundamental characteristics. The realization of polarization state manipulation in simple structured metamaterial is attractive.
In the present paper, a simple bilayered chiral metamaterial composed of two split-ring resonators (SRRs) with a twisted angle has been proposed. Theoretical results verify that the chiral metamaterial exhibits strong optical activity and changes the polarization state of light propagating through it. The polarization rotation angle is in an excellent linear relationship with the twisted angle. The work provides significance to understand the interaction between light and the matter. Planar metamaterials provides an opportunity to manipulate polarization states of electromagnetic waves and shows significant importance of achieving ultra-thin, broadband metamaterial-based terahertz and photonic devices.
2. Theoretical analysis
Firstly, the theoretical analysis of the polarization rotation for linearly polarized EM wave has been considered. The polarization rotation can be determined from the Jones matrix (i.e. transmission matrices, T matrices) [41]. The transmission matrix relates the complex amplitudes of the incident to the transmitted field:
where Tij has been replaced by A, B, C, D for convenience. The superscript f and subscript lin indicate the forward propagation (along -z direction) and linear base (base vectors parallel to the coordinate axes). Since only reciprocal media are considered here, the transmission matrix Tb for propagation in the backward direction ( + z direction) can be derived as [33]The T matrix for a circular polarization base can be described by a change of the base vectors from linear to circular base.where for convenience T+− has been replaced by a, b, c and d, respectively. The indices + and − denote the right-handed and left-handed circularly polarized polarizations. The azimuth rotation is usually characterized by the parameter ψ, which is defined as the difference between the azimuth of the transmitted field and incident field. The azimuth rotation parameter ψ for the linear (indicated with subscript f or b and superscript x or y) is then defined as [1]It can be easily seen that generally the phase difference of a and d leads to optical activity.A simple 3D chiral metamaterial structure will be constructed and illustrated in Fig. 1(a).The meta-molecule consists of two SRR layers parallel to the x-y plane separated by a dielectric layer. The metallic resonant structure in each layer has the same pattern, however, the structure in the second layer has been rotated around the z axis with -θ. Obviously, the single-layered SRR structure with its mirror plane parallel to the y axis is anisotropic and no polarization rotation occurs for light being parallel or orthogonal to the mirror plane. For other orientations, the off-diagonal elements of the T matrix are not zero, which disappear through a proper rotation. In general case, the T matrix obeys the following form
For simplicity, we change the whole meta-molecule orientation by rotating the structure around the z axis with θ/2, shown in Fig. 1(b). The transmission matrix of the rotated metamaterial can be obtained through coordinate transformation from the T1 matrix, which can be expressed asFor a circular polarization basis, the T matrix can be expressed asSince there is an angle of θ/2 between the y axis and the mirror line of the SRR, non-zero off-diagonal element B exists. The difference T++ − T−− = 2iB is given by the off-diagonal elements in the linear polarization representation and specifies the optical rotation power.
Fig. 1 Schematics of the meta-molecule for the proposed chiral metamaterial. (a) Schematic of the unit cell in anisotropic chiral metamaterial. The back layer is rotated by –θ with respect to the front one. (b) Rotated unit cell around the z-axis by θ/2. (c) The whole metamaterial. The unit in dimensions is nanometer.
3. Simulated results and discussions
To validate the feasibility of the theory discussed above, the software package CST Microwave Studio has been adopted to simulate the chiral metamaterial. The metamaterial structure consists of an array of the SRR dimer with a twisted angle of −30° (θ = 30°) on either side of a dielectric layer. The period of the SRR dimer is p = 800nm, rendering the structures non-diffractive at normal incidence for frequencies below 375 THz. The length and width of the SRR is l = 540nm and w = 150nm. The dielectric is chosen as a photopolymer PC403 with a thickness of h = 150nm (εpc403 = 2.4), while the gold SRR is described by Drude model with the plasma frequency ωpl = 1.37 × 1016s−1 and the damping constant ωc = 4.08 × 1013s−1, and the gold thickness is t = 50nm. For other dielectrics or substrates, it is easy to obtain the good performance by optimizing geometrical structures of the SRRs. Figure 1 shows the detailed structural parameters of the chiral metamaterial. According to the analysis above, optical activity could be achieved by this anisotropic bilayered metamaterial. The periodic boundary conditions are applied along the x and y directions in the CST simulation. Normally incident x-polarized plane waves propagating in the -z direction are used for the excitation. The numerical linear transmission which is defined as is presented in Fig. 2(a).Figures 2(b) and 2(c) show the polarization rotation azimuth angle ψ and ellipticity χ, which are calculated as (φ is the phase difference between the Ey and Ex components of the transmitted wave) [42]
Fig. 2 The simulated transmission (a), polarization azimuth angle ψ (b) and ellipticity χ (c) in the SRRs metamaterial for the transmitted wave. The solid line indicates the resonant mode.
Numerical results show that the transmission reaches a maximum of 0.87 at around 147.1THz, while the polarization rotation azimuth angle ψ and ellipticity χ are around −31.4° and −5.7° marked by the black dotted line in Fig. 2. Therefore, for the linearly polarized incident wave, due to a small ellipticity, the transmitted wave is still linearly polarized but with a rotation angle of about −30° at the resonant frequency. It can be observed in Fig. 2(a) that this plasmonic metamaterial allows −30° polarization rotation of linearly polarized wave in a broad band of about 140-170 THz. Further, we calculate transmission, azimuth angle ψ and ellipticity χ at the resonant frequencies for varying θ in Fig. 3.When the twisted angle between two SRRs varies from −10° to −90°, the resonant frequency f1 undergoes a blue shift and the polarization rotation angle ψ changes from −12.98° to −77.13°, which is in an excellent linear relationship with θ. Meanwhile, the elliticity χ are always less than 6°. Consequently, it is concluded that the transmitted wave is nearly linearly polarized and the polarization plane of the incident x-polarized waves is rotated by ψ after transmission, in proportion to the twisted angle θ.
Fig. 3 Calculated resonance frequencies (a), transmission (b), polarization azimuth angle ψ (c) and ellipticity χ (d) at the resonant frequencies for different twisted angle θ.
Excitation of one of them by the incident electromagnetic wave is transferred by the coupling to the other. Induced oscillations of its charge then reemit a wave at a different polarization and with some delay, thus ensuring polarization azimuth rotation. In order to understand the mechanism of the optical activity effect, the distribution of the H fields at the resonant frequencies in the middle plane between the two SRR layers and the current density distributions around two SRRs’ surfaces have been presented. In the case of θ = 30° (i.e. the twisted angle between two SRRs is −30°), the surface currents at the resonance are illustrated in Fig. 4(a).It is noted that the current distributions on the top and bottom layers is an antisymmetric resonant mode, leading to a strong magnetic resonance in the xy plane between two layers. H1 and H2 represent the magnetic fields induced by the current pairs. When the incident wave is x-polarized, the cross-coupling between the electric and induced magnetic fields H1 and H2 leads to optical activity. When the twisted angle between two SRRs varies from −30° to −50°/-70°, the induced magnetic field H2 is rotated correspondingly, as shown in Figs. 4(b) and 4(c). Therefore, it is well understood that the polarization plane of the transmitted wave is rotated further with the increasing θ.
Fig. 4 The scattered H field distributions on the middle plane between the two metal plates and the current modes around two SRRs’ surfaces when driven by the incident E ðeld at the resonant frequencies when θ = 30° (a), 50° (b) and 70° (c).
The simulations above have demonstrated that linear polarization is rotated in direct proportion to the twisted angle between two SRRs. In order to better understand the coupling effect inside SRRs, two other bilayered metamaterials with twisted bars will be studies in Fig. 5 when the twisted angle is −30°. In comparison with Fig. 2, the similar results can be achieved from twisted double bars as well as twisted single bars, however, the resonant frequencies greatly shift. In Fig. 5(a), the twisted double bars metamaterial with l1 = 540nm is directly realized by removing the horizontal bars in the SRRs and the corresponding resonant mode occurs at about 156.6THz with ψ = −32° and χ = −6.7°. It can be concluded that the horizontal bar linked to the two vertical bars provides a coupling in the SRRs and can help tune the spectral response of the SRRs. When two vertical bars is closer as l1 = 500nm in Fig. 5(a), the strong coupling between two vertical bars happens and the corresponding resonant mode further shifts to about 165.8THz with ψ = −31° and χ = −7.3°. Figure 5(b) shows the simulation results of the twisted single bars metamaterial. The resonant mode for the twisted single bars metamaterial occurs at about 153THz with ψ = −29° and χ = −7.5°. Although the two vertical bars of the SRRs play the same role as single bars, the twisted SRRs metamaterial will be a promising candidate for well tuning and engineering the properties of the bilayered metamaterials.
Fig. 5 The simulated results of the twisted double bars and single bars metamaterials when the twisted angle is −30°. (a) Simulations of the twisted double bars metamaterial. The dashed lines indicate the resonant mode corresponding to the twisted double bars with l1 = 540nm and 500nm. The solid line indicates the resonant mode in the twisted SRRs in Fig. 2. (b) Simulations of the twisted single bars metamaterial. The dashed line indicates the resonant mode.
4. Conclusion
In summary, we have proposed a simple bilayered chiral metamaterial composed of SRRs arranged by a twisted angle. The chiral metamaterial has been theoretically demonstrated to reveal optical activity for linearly polarized waves. The polarization rotation angle can be well-engineered and is in proportion to the twist angle of two SRRs. The induced current and magnetic field distributions of the resonant mode provide an insight into the mechanism of polarization rotation. It can also be seen that the engineered electric and magnetic cross-coupling provides us a convenient way to rotate the polarization plane by changing the twisted angle and coupling strength between two vertical bars. Our findings are beneficial in designing polarization rotators and exploring polarization-controlled devices.
Acknowledgments
This work is supported by the National Science Foundation of China under Grant Nos. 61201083, 61275094, U1231201, 613111156, in part by the Natural Science Foundation of Heilongjiang Province in China under Grant No. LC201006, the China Postdoctoral Science Foundation under Grant Nos. 2012M511171 and 2013T60487, the Special Foundation for Harbin Young Scientists under Grant No. 2012RFLXG030, the Fundamental Research Funds for the Central Universities, and the 111 Project under Grant No. B13015.
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