Resonance modes in stereometamaterial of square split ring resonators connected by sharing the gap

Stereometamaerials can fully utilize the 3D degrees of freedom to exploit the coupling and hybridization between multiple split ring resonators (SRRs), enabling more extraordinary resonances and properties over their planar counterparts. Here we propose and numerically study a kind of structure based on connected SRRs sharing their gap in a rotational fashion. It is shown that there are three typical resonance modes in such cage-like SRR (C-SRR) stereometamaterial in the communication and near infrared range. In the order of increasing energy, these modes can be essentially ascribed to magnetic torodial dipole, magnetic dipole, and a mixture of electric-dipole and magnetic toroidal dipole. We show that the latter two are derived from the second-order mode in the corresponding individual SRR, while the first one from the fundamental one. The highest energy mode remains relatively"dark"in an individual C-SRR due to the high-order feature and the rotational symmetry. However, they are all easily excitable in a C-SRR array, offering multiband filtering functionality.


Introduction
Artificial subwavelength metamaterials can realize a lot of peculiar electromagnetic phenomena that are not in existence or much weaker in natural material. Split ring resonator (SRR) that can be treated as a magnetic dipole atom is widely used in metamaterial design, supporting magnetic resonance with magnetic or electric excitation [1,2]. As the basic and fundamental meta-atoms for constructing metamaterial, SRRs are organized in different ways, most frequently and particularly in two-dimensional spatial arrangement. Many planar combinations of SRRs have been proposed and fascinating phenomena have been realized from the microwave and infrared to near-infrared and even the visible [3][4][5]. Environmental or structural design of SRR can result in unusual resonances, such as dark mode [5] and trapped mode [6]. Moreover, Fan et al experimentally observe toroidal dipolar response and Fano resonance in a planar metamaterial [7].
Metamaterials in planar or coplanar form are of many advantages, particularly in easy fabrication. However, they cannot fully utilize the three dimensional degrees of freedom to exploit the coupling between SRRs and the hybridization of their resonances which are supposed to be more intriguing. Stereometamaterial, as first coined and studied by Liu et al [8], has been gaining arising attention over the past years. When SRRs are allowed to be arranged in three-dimensional space in deliberate configuration, they can offer more combination possibilities and the fundamental resonance mode can couple in a more comprehensive way. This could certainly lead to more interesting controls over the electromagnetic field in subwavelength scale. For example, Tsai et al [9] present a two plasmonic metamaterial composed of four SRRs supporting toroidal dipole response at optical frequencies. And they apply the high Q-factor toroidal resonance of the plasmonic toroidal metamaterial to lasing spaser [10]. Dong et al [11] designed a feasible nanostructured metamaterial that also support toroidal dipolar response in the optical regime by asymmetric double-bar. The double-bar structure can be regarded as a special SRR in generating magnetic resonance. Fedotov et al [12] experimentally demonstrate a new kind of toroidal metamaterials that implement non-trivial non-radiating charge-current excitation based on interfering electric and toroidal dipoles which is first proposed by Afannasiev and Stepanovsky [13]. The toroidal dipolar moment violates the symmetries of both spaceinversion and time-reversal simultaneously and exhibits sub-radiating property, due to weak coupling to free space. It is essentially characterized by vortex distributions of head-to-tail magnetic dipoles and can be produced by currents flowing on the surface of a torus along its meridians [14]. Although acknowledged in atoms [15], molecules [16] and ferroelectric [17] structures, toroidal moment is hard to detect because being overwhelmed by electric and magnetic multipoles. Stereometamaterial, however, offers a rather feasible way to construct notable toroidal dipolar mode in electrodynamics [18]. In this paper, we propose and study a kind of stereometamaterial structure consisting of multiple SRRs that are connected in a rotational fashion and share the split gap. It is shown that the structure sustains hybridized resonance modes resulting from the fundamental and the second-order modes in the corresponding individual SRR. When making them in periodic array, a higher-order mode due to formation of stronger electric dipole and toroidal dipole resonance would emerge (or become easily excitable). The three resonances of the proposed stereomatamaterial respectively lie in the communication, near-infrared and visible band. We show that the visible resonance is more sensitive to the periodicity and the arrangement, corresponding to high symmetry and high-order phase relationships in the constitute SRRs. Figure 1 illustrates the geometry of the proposed stereometamaterial. The basic constitute unit is a square SRR as shown in Fig. 1(a). The design parameters can be found in the figure caption. The proposed stereometamaterial is constucted by duplicated SRRs that are rotated by 90˚, 180˚, and 270˚ degree with respect to the gap-bearing side, as shown in Fig. 1(b). Namely, four SRRs are connected by sharing the same split arm and gap. The structure can therefore be regarded as two orthogonally connecting rectangular sheets with "H"-shaped aperture, and resembles a cage. For short and simplicity let us call it as C-SRR. We study stereometamaterials of C-SRRs placed in a two-dimensional array with periodicity x P and z P .

Stereometamaterial structure and numerical analysis
The array is excited by z polarized light at normal incidence with one sheet in the xoz plane and the other parallel to the k direction [ Fig. 1(b)]. We also consider the array with the entire C-SRR unit rotated by 45˚ degree [ Fig. 1(c)]. Numerical calculations are carried out by solving the three-dimensional Maxwell equations with the finite element method (COMSOL Multiphysics). The material of the SRRs is assumed to be gold whose permittivity is taken from Johnson and Christy [19] and the background medium is set as air. Perfectly matched boundary condition and periodic boundary condition are employed to obtain the scattering cross section and the transmission spectrum in the simulation, respectively.
In order to analyze and quantify which multipole component contributes most to the resonance peaks, the radiated powers of the electric and magnetic multipoles and toroidal dipole are calculated by the induced volume current density j in the C-SRR [9,12,20], e.g., electric dipole moment: Magnetic dipole moment:   [20]. Similar rules apply to the other higher-order moments shown in Eqs. (1)- (6).
Before going directly to the two-dimensional array, we first examine single SRR and single C-SRR. Figure 2(a) shows the resonance spectra of an individual SRR (solid curve) and an isolated C-SRR (dashed curve). The SRR has the fundamental resonance 1  at 116 f  THz and its second-order resonance  Fig. 2(c)]. The C-SRR in this case is illuminated by a plane wave as shown in Fig. 1(b)  l , g and w ) from the microwave to the near-IR [3]. The C-SRR resonances can be flexibly controlled by these geometry parameters as well. More importantly, C-SRR with hetero-structured SRRs (e.g., making one SRR with shorter or longer 2 l ) can offer more resonance peaks (figure not shown here), in view of the broken 4 C rotational symmetry. Here we would like to focus on C-SRR with homo-structured SRRs.  Figure 3 shows the calculated results for C-SRR array. Firstly, we study the situation as shown in Fig. 1(b). The transmission spectrum is shown in Fig. 3(a) in which three resonances are clearly seen at 208 f  THz, 290 THz, and 380 THz for 700 x P  nm (black curve). For easy explanation, we use the notation of   Figure 4 plots both visually and schematically the current distribution for the three resonance modes observed in Fig. 3(a). Figures 4(a) Figure   4(a) shows that the current loop in each arm of the C-SRR is of the same pattern as in Fig.  2(b). Figures 4(a) and 4(c) show clearly that the current in the four arms flow from the same side of gap to the other uniformly, i.e., all of them loops in phase and are of 4 C rotational symmetry with respect to the z axis. This develops four head-to-tail magnetic dipole moments in xy plane, and collectively generates a remarkable toroidal dipole moment T .
Opposite charges are accumulated across the SRRs gap which yield an electric dipole moment P pointing reversely to the toroidal dipole moment [see Fig. 4(b)]. Notice that in this case, the current amplitudes are comparable in the four SRRs (shown in color scale). In Figs. 4(d) and 4(f), it is seen that the distribution of the induced current in each SRR arms corresponds to that in Fig. 2(c). In this case, the current in the vertical edges is in opposite direction, i.e., out-of-phase to the ones in the other two edges, preventing a complete loop in the SRR. The upper and lower edges in each SRR support anti-parallel currents which generate four magnetic dipoles that appear cancelling each other, as schematically shown in Fig. 4(e). However, due to the retardation effect, the front SRR has much stronger current than