1. Introduction

Control of the coherent coupling in plasmonic resonances is an important means to tailor the resonance spectral shape, which naturally extends to metamaterials. By controlling the coupling between dipolar and quadrupolar excitations in 3-D metamaterial, electromagnetic induced transparency (EIT) has been demonstrated [1–3]. A coherent coupling between the superradiant and subradiant plasmon modes allowed the observation of Fano resonances in a symmetry-broken concentric disk-ring and dolmen-style slab metamaterials consisting of metallic nanostructures [4, 5]. By modifying the near-field interaction geometrically in a plasmonic lattice, the Fano interference is controlled through a control of coherent interlayer coupling [6]. Furthermore, following the scheme of frequency-selective surface design, [7, 8] a dissymmetric metaparticle structure was employed to excite a closed mode resonance possessing a high Q-factor as an example of coherent coupling [9].

Another approach to control coherent coupling is to introduce a superlattice structure of metaparticles, where the in-plane coherent coupling between metaparticles is exploited to pursue novel properties. Examples include a dual-band metamaterial superlattice composed of two different sized symmetric split ring resonators, [10] a wave-length dependent optical wave plate based on asymmetric superlattice, [11] a resonance sharpening in supercell with four split ring resonators, [12] and an EIT-like induced transparency in the two asymmetric split ring resonator superlattice [13]. In the above and other examples, the plasmonic structures are geometrically altered in order to achieve a coherent coupling control.

Among the novel features resulting from a coherent coupling in metamaterials, the closed mode excitation possessing a high Q-factor drew a research interest since the sharp spectral response allows for application to the efficient sensor and frequency filtering [14]. Therefore we focus our study on the closed mode excitation as an important example of coherent coupling in metamaterials.

In a 2-D planar metamaterial made of a symmetrically oriented square array of asymmetric double split-ring resonators (DSRR), the dissymmetric structure in the individual DSRR meta-particle is essential to giving rise to the closed mode excitation. Asymmetric current flows in the dissymmetric structure are composed of two oppositely directed symmetric currents and one remnant uncanceled current, and the two symmetric current cancels coherently resulting in an enhanced Q-factor of the closed mode through a weak free-space coupling [9, 14].

Now we put the question how the closed mode excitation will behave when a dissymmetry resides not in the individual structure of DSRR but in the orientation of DSRR. In other words, we look at the metatmaterial superlattice where the dissymmetry is brought in such that symmetric DSSRs are asymmetrically orientated in a square lattice, in contrast with the metamaterial where asymmetric DSRRs are symmetrically oriented in a square lattice.

For this purpose, we construct a 2-D planar superlattice structure and examine the coherent coupling among the nearest neighboring symmetric DSSR metaparticles. Since the dissymmetry is present in the orientation of symmetric DSRR metaparticles, we expect that cancelation of oppositely directed symmetric currents among the nearest neighboring DSRR will excite a closed mode through coherent coupling in the superlattice. However, one important characteristic feature of a superlattice is that the contribution of orientational dissymmetry to the closed mode excitation depends on how the electric field, i.e., polarization direction, of an incident electromagnetic wave is oriented with respect to the superlattice structure. Hence we closely monitor the behavior of closed mode excitations as a function of polarization direction of the normally incident linearly-polarized electromagnetic wave in the 2-D planar superlattice structure.

In Sec. 2, the micron-sized metamaterial fabrication and THz transmission spectra measurement are presented. Section 3 describes the experimental results and theoretical analysis of closed mode excitation by use of Lorentzian resonance curve analysis. In Sec. 4, a finite-difference time-domain numerical simulation is conducted to clarify the nature of coherent coupling in metamaterial superlattice. Section 5 summarizes the major finding of this study.

2. Sample fabrication and THz measurement

The metaparticle for construction of superlattice is a DSRR, with inner radius 14μm and outer radius 18μm. The width of circle line 4μm. The symmetric gap openings of 20° arc occur between -10° and +10° and between +170° and +190°. The geometric structure of DSRR metaparticles is highly symmetric, possessing the point group D _2h symmetry. Figure 1 shows the optical micrographs of two metamaterials. Inspection by an optical microscope ensured the uniformity of metaparticle cells over the entire metamaterial sample.

 

Fig. 1. Optical micrographs of (a) reference metamaterial and (b) sample metamaterial superlattice are shown.

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Reference metamaterial is a square array (lattice constant 50μm) of the DSRR metaparticle with the gap orientation fixed along x-axis. See Fig. 2(a). Sample metamaterial is a superlattice structure (the same lattice constant 50μm) of the DSRR metaparticle with the gap orientation alternating along x-axis (0°) and along 135° axis, which are termed as 0°-particle and 135°-particle. See Fig. 2(b). A standard photolithography and lift-off process of Au with thickness of 200nm on top of adhesion layer of 10nm thick titanium on p-doped silicon wafer yielded a metamaterial superlattice with the size of 8mm×8mm containing total number of 160×160 = 25,600 DSRRs.

 

Fig. 2. Microscope pictures of (a) reference metamaterial and (b) sample metamaterial superlattice are shown along with (c) polarization angle of E-field. THz transmission spectra of (d) reference metamaterial and (e)&(f) sample metamaterial superlattice are shown for different polarization angles.

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Time-domain terahertz transmission measurements were carried out with a TeraView TPS Spectra 3000 Spectrometer [15] at a resolution of 1.2 cm⁻¹ at room temperature in vacuum. The time-domain pulse duration is about 2 ps, leading to the accessible spectral range of 0.1–3 THz (3–100cm⁻¹). THz dipolar radiation from emitter is highly linear-polarized due to the very design of Auston switch with less than 0.5% depolarization, and the receiver is a twin copy of the emitter, which further renders the polarization in the parallel direction. Therefore the transmission is measured in the same polarization as the incident wave. The linearly polarized THz dipolar radiation is normally incident on the sample through an open aperture of diameter 5 mm, which is the standard aperture size to ensure that the pico-second pulse is nearly a plane wave. The layout of sample measurement is a generic one, as can be found in Fig. 1.8 of Ref. [16].

Figure 2(c) defines the polarization angle of the polarization direction of the normally incident beam. Polarization angle 0° (90°) stands for x- (y-) polarization, corresponding to the E-field parallel (perpendicular) to the gap orientation of DSRR metaparticle shown in Fig. 2(a).

3. Experimental results and analysis

In Fig. 2(d)–2(f), the time-domain THz transmission spectra of reference and sample metamaterial are shown for different polarization directions.

The reference metamaterial spectra exhibited a mirror symmetry with respect to the y-axis, that is, the spectra for 90°~180° is a mirror image of that for 0°~90°, which is from the symmetric distribution of 0° -particles. As shown in Fig. 2(d), there are two low Q-factor resonances located near 50cm⁻¹ and 80cm⁻¹. As the polarization angle increases from 0° to 90°, the magnitude of low (high) frequency resonance decreases (increases). As seen at 22.5°, 45°, and 67.5°, the spectra has a dual-band structure, [10] a linear sum of two low Q resonances with weight determined by the polarization direction. Differently from single-split ring resonator, no magneto-electric coupling takes place, and the electric fields along the symmetric axes, x- and y-axis, excite electric resonances with low (50cm⁻¹) and high (80cm⁻¹) frequency resonances, respectively.

On the other hand, in the spectra of sample metamaterial, three distinct resonances are identified, as shown in Figs. 2(e) and 2(f). Two low Q-factor resonances ω^L_O=45cm⁻¹ and ω^H_O = 76~80cm⁻¹ correspond to low (50cm⁻¹) and high (80cm⁻¹) frequency resonances of reference metamaterial. A notable and important feature is the appearance of an enhanced Q-factor resonance (ω_C= 40cm⁻¹. That is, an enhanced Q-factor resonance develops as the polarization angle approaches to 67.5°. Owing to the presence of an enhanced Q-factor resonance ω_C, a strong anisotropy is observed in the transmission spectra between 0° ~90° and 90° ~ 180°.

Changes in Q = ω ₀/Δω are closely examined as shown in Fig. 3. In order to identify the detailed feature of the spectra, the transmission is converted to the absorbance, defined as absorbance = - log₁₀ T, which is also called the optical density [17]. The angular domains are designated as ①, ②, ③, and ④. Resonances appearing in the transmission spectra originate from the excitation of Drude-type currents in the patterns with the resonance frequency determined by the structural dimension. By treating each resonance of ω_C, ω^L_O, and ω^H_O as Lorentzian oscillators, the least-squared fit was performed to obtain the Q-values of each resonance.

In the domain ① (0°~45°), the low frequency resonance ω^L_O transits to ω_C as the polarization angle increases from 0°, while in the domain ③ (90°~135°) the low frequency resonance transits to ω_C as the polarization angle decreases from 135°. The highest Q ≈15 takes place for the polarization angle 67.5° as shown in Fig. 3(d). Resonance in domain ② belongs to the enhanced Q-factor resonance, and no closed mode is observed for the polarization angle between 135° and 180° in domain ④.

The spectrum at the polarization angle 67.5° (domain ②) contrasts the most strikingly with that at the polarization angle 157.5° (domain ④), where the polarization angle differs by 90°. In both the gaps of 0°- and 135°-particle are symmetrically oriented with respect to the E-field. The inclination angles are 22.5° and 67.5° for the polarization angles 157.5° and 67.5°, respectively. Despite the symmetric orientations, the low Q-factor resonance ω^L_O is dominant at 157.5° without ω^H_O excitation, while the highest Q-factor resonance ω_C appears with ω^H_O excitation accompanied at 67.5°.

This is in contrast with the transmission spectra observed in a metamaterial made of asymmetric DSRRs in a symmetrically orientated square lattice. As shown in Fig. 2 and Fig. 3 of Ref. [9], the closed mode is excited for the polarization direction perpendicular to the symmetry axis of asymmetric DSRR and no closed mode is excited for the polarization direction parallel to the symmetry axis. This is because asymmetric current flows take place only for the polarization direction perpendicular to the symmetry axis. The situation is quite different in the case of the metamaterial superlattice. At the polarization angle 67.5° (domain ②) and 157.5° (domain ④), 0°- and 135°-particles’ orientations are mirror-symmetric each other with respect to the vertical plane containing the polarization direction of electric field. However, the behaviors of closed mode excitation are completely different for the polarization angles 67.5° (domain ②) and 157.5° (domain ④). When 0°- and 135°-particles make the inclination angles of ±67.5° with respect to the electric field corresponding to the polarization angle 67.5° (domain ②), the closed mode is excited in the most pronounced way. On the other hand, when both 0°- and 135°-particles make the inclination angle of ±22.5° with respect to the electric field corresponding to the polarization angle 157.5° (domain ④), no closed mode is excited at all. That is, the cancelation of oppositely directed currents, responsible for the closed mode excitation, is dependent on the polarization angle, and the coherent coupling takes place among the nearest neighboring DSSRs. As mentioned in the introduction, contribution of the structural dissymmetry, present in the metamaterial superlattice inherently, to the closed mode excitation is controllable by tuning the polarization angle.

 

Fig. 3. (a)–(d) Absorbance plots of THz transmission spectra of sample metamaterial superlattice are shown for four angular domains ① (0°~45°), ② (45°~90°), ② (90°~135°), and ④ (135°~180°), defined in (e). (f) The quality factor Q is plotted as a function of polarization angle for each resonance, black circle for ω_C, blue inverted triangle for ω^L_O, and red upright triangle for ω^H_O.

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Fig. 4. (a)&(c) Schematics of metamaterials and E-field polarization direction. (b)&(d) Absorbance plots of transmission spectra with Lorentzian resonance fits.

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In order to understand the distinct structure of the spectrum at the polarization angle 67.5°, we performed a spectrum curve analysis by employing the Lorenztian oscillators. Figure 4(a) shows the schematics of metamaterial superlattice and E-field polarization direction corresponding to the polarization angle 67.5° and Fig. 4(b) shows the absorbance plot of transmission spectra with Lorentzian resonances fits. We find that 3 Lorentzian oscillators are identified with a pronounced ω_C resonance at the polarization angle 67.5°.

Now we examine how the symmetry breaking in metametarial superlattice is related to the appearance of ω_C resonance. In the case of reference metamaterials, owing to the mirror-symmetry with respect to x- and y-axes, transmission spectra are identical for polarization angles of 67.5° and 112.5° as seen in Fig. 2(d), where no coherent couplings are present among nearest neighboring DSSRs as evidenced by the absence of closed mode excitations. Now, in order to find out what the transmission spectra of metamaterial superlattice would look like if no coherent couplings were present among nearest neighboring DSSRs, we rotate each 135°-particle in Fig. 4(a) by 45° counter-clockwise with the electric field attached to the DSSR, which is shown in Fig. 4(c). In other words, the orientation of E-field polarization on the reference metamaterial is altered along 67.5° and 112.5° on the array of 0°-particle, which is what we would get when each 0°- and 135°-particles in the sample metamaterial superlattice are excited independently by E-field with the polarization angle 67.5°. Again, Fig. 4(c) is not the experimental configuration but the configuration mimicking Fig. 4(a) when the coherent coupling is assumed to be absent. Figure 4(d) is the spectra taken from Fig. 2(d), which is what we would get for the configuration shown in schematics of Fig. 4(c). Only 2 Lorentzian oscillators are identified, namely, ω^L_O and ω^H_O.

When Figs. 4(b) and 4(d) are compared, we find that Fig. 4(b) is not the weighted linear sum or linear superposition of two independent ω^L_O and ω^H_O resonances. It is clear that the enhanced Q-factor resonance ω_C originates from the symmetry-broken superlattice structure of sample metamaterial, indicating that a coherent coupling among the nearest neighboring metaparticles plays a key role in providing the distinct structure of the spectrum at the polarization angle 67.5°. The important feature of the metamaterial superlattice is that the symmetry broken structure permits a polarization angle controllable coherent coupling among the nearest neighbors. In the current superlattice composed of 0°- and 135°-particles, the polarization angle parallel to the bisecting line of two metaparticle gap orientations permits the highest Q-value ω_C, which corresponds to the maximum coherent coupling.

4. Finite-Difference-Time-Domain numerical simulation

 

Fig. 5. For the polarization angle of 67.5° and resonance frequency ω_C= 40cm⁻¹, a detailed distribution of current densities obtained by a finite difference time domain simulation is plotted. (a) and (b) correspond to the J_x and J_y, and (c) is the schematic diagram of current densities, respectively. The blue oblique line refers to E-field direction.

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In order to clarify the nature of coherent coupling among the nearest neighbors, a detailed distribution of the current densities in the metamaterial superlattice is obtained by a finite difference time domain simulation (FDTD).

 

Fig. 6. For the polarization angle of 137.5° and resonance frequency ω^L_O= 45cm⁻¹, a detailed distribution of current densities obtained by a finite difference time domain simulation is plotted. (a) and (b) correspond to the J_x and J_y, and (c) is the schematic diagram of current densities, respectively. The blue oblique line refers to E-field direction.

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All the FDTD simulation is conducted with the commercial software FDTD solution Lumerical [18]. We simulated a 2×2 cubic unit cell as shown in Fig. 5 and Fig. 6. The mesh size was 1.0μm×1.0μm×1.0μm in dx, dy, and dz. The refractive index of p-doped silicon substrate is obtained from the measurement by TDS as n_Si = 3.44 in the range between 0.1 THz and 3.0 THz. The refractive index (real and imaginary) dispersions of Au and Ti are obtained from Lumerical material data sheet which is based on CRC handbook [19]. For boundary conditions, the periodic boundary conditions are adopted for x- and y-directions, while the perfectly matched layer boundary condition is adopted for z-direction. We defined the source wave as linearly polarized plane wave with the spectral range from 0.1 THz to 3.0 THz. In the Lumerical FDTD solution, the frequency domain information is calculated by a Fourier transform of the results obtained in the time domain, hence we took the simulation time at least as twice as long as the time EM wave takes to travel across the entire simulation area with the highest index of refraction, i.e., t > 2nL/c with n the highest refractive index of gold, L the thickness along the z-direction [18]. With the time step taken as 0.0002ps, the steady state has been achieved after completing around 75,000 iterations.

In order to find out how the current flow directions in 0°- and 135°-particles are related to the closed mode excitation, we closely examine the case when the electric field makes with the x-axis the angle 67.5° (domain ②) and 157.5° (domain ④). In Fig. 5 and Fig. 6 the calculated current densities are plotted for the closed mode excitation (67.5°) and the open mode excitation (157.5°).

For the polarization angle of 67.5° and resonance frequency ω_C, Figs. 5(a) and 5(b) show that the current densities J_x and J_y of 0°- and 135°-particles are in the opposite direction. As shown in the schematics of current density, Fig. 5(c), the current densities inside the DSRR are in the opposite direction among the nearest-neighbors. The counter-flowing current densities cancel each other resulting in the excitation of the closed mode. For the polarization angle of 157.5° and resonance frequency ω^L_O, Figs. 6(a) and 6(b) show that the current densities J_x and J_y of 0°- and 135°-particles are in the same direction. As shown in the schematics of current density, Fig. 6(c), the current densities inside the DSRR are in the same direction among the nearest-neighbors. The co-flowing current densities do not cancel each other resulting in the excitation of the open mode.

Even though the underlying mechanism for closed mode excitation, i.e., current flow cancelation, is identical to what happens in the metamaterial where asymmetric DSSRRs are symmetrically oriented in a square lattice, the current flow cancelation takes place among the nearest neighbors in the metamaterial superlattice, not inside the individual metaparticles. Compare Fig. 5 and Fig. 6 with Fig. 2 and Fig. 3 of Ref. [9].

The damping of a plasmon resonance, γ, consists of the Drude damping, γ_D, and the radiative damping, γ_R. That is, γ = γ_D +γ_R [9, 14, 20]. While the Drude damping is determined intrinsically by the composition and substrate of metamaterial, the radiative damping is a function of the polarization angle in the metamaterial superlattice. At the polarization angle 67.5°, the maximum cancelation of anti-symmetric current flows takes place, leading to the minimum radiative damping resulting in the highest Q-factor. In the open modes ω^H_O and ω^L_O, the oscillator couples to the free-space decay channel resulting in a low Q resonance, while a coherent coupling among nearest-neighboring 0°- and 135°-particles significantly suppresses the free-space decay in the closed mode ω_C with an enhanced Q-factor. This is reminiscent of a recent report on the suppression of radiation losses in periodic arrays in coherent metamaterials [21].

In the work of Al-Naib et al. [22], miniaturized asymmetric single split resonators were introduced to enable asymmetric current flows at the individual asymmetric resonator in the GHz regime, enhancing Q-factor compared to symmetric single split resonators. In contrast, the excitation of closed mode in the metamaterial superlattice reported here is from the cancelation of current flows among the nearest neighboring double-split ring resonators through a coherent coupling which can be controlled by the incident polarization angle.

5. Summary

In summary, a polarization angle tuning of coherent coupling is demonstrated in the metamaterial superlattice by resorting to the closed mode excitation. Alternately oriented double-spit ring resonators superlattice structure permits the angular control of the amount of anti-symmetric current flows among the nearest-neighboring metaparticles. The metamaterial superlattice will be a useful tool for angle-selective enhancement of local fields, for example, in the study of the structure edges of graphenes. The future works will include the investigation of the polarization angle control of EIT in metamaterial superlattices.