2.1.

Modeling of Double-Layered Closed-Ring Resonator Arrays

In our model, we assume that periodic arrays of CRR made of aluminum are placed on quartz substrate. Here, we chose aluminum in accordance with a previous report by Gu et al.²⁷ We also did similar simulation with copper and gold, however, not much difference was seen in the obtained spectra (data are not shown). The unit cell of designed CRR arrays and the polarization direction of the incident EM field are shown in Fig. 2(a). The side length $L$ and width $W$ of each CRR are 80 and $5\mu \mathrm{m}$, respectively, and they are periodically arranged with the spacing $A$ between adjacent CRRs in the same array to be $25\mu \mathrm{m}$ in both $x$ and $y$ directions. The thickness of aluminum ($t_{\mathrm{Al}}$) is 200 nm and its conductivity is $3.56\times 10{}^7\mathrm{S}/\mathrm{cm}$. The thickness of the quartz substrate ($t_{\mathrm{d}}$) is $10\mu \mathrm{m}$ and its dielectric constant is 3.75. Two such substrates with CRR arrays are placed closely with the gap $G$ of $22\mu \mathrm{m}$ making both of CRR arrays side face inside. The geometric parameters of CRR array are summarized in Table 1.

Fig. 2

Schematic diagram of the analysis model. (a) Unit cell of a double-layered CRR, including definition of the parameters (see Table 1) and direction and polarization of incident EM wave. The front substrate is omitted for clarity. (b) Effective circuit model of the unit cell of double-layered CRR arrays. (c) Unit cell model with periodic boundary conditions.

Table 1

Geometric parameters of double-layered CRR arrays.

In our simulation, the EM wave is incident from the normal direction. In such a situation, the resonant modes between the two sides of the paired CRRs with gap $G$ and the near-field coupling between adjacent CRRs on the same substrate can be excited at their resonant frequency. These resonant modes can be approximately modeled by a composite LC-resonance circuit, as shown in Fig. 2(b). The effective values of capacitance and inductance of such an LC circuit are quite sensitive to the geometrical parameters of the CRR array. Therefore, slight displacement of the relative position of CRR arrays and change of the dielectric constant of the dielectrics inserted between CRR arrays mainly alter effective capacitance, and thus our operative principle is possible.

2.2.

Numerical Simulation

To model the periodic array of CRR, periodic boundary conditions are applied in two ($x$ and $y$) directions, as schematically shown in Fig. 2(c). THz plane waves, whose electric and magnetic fields are uniform over the unit cell and polarized in $y$ and $x$ directions, respectively, are sent to the device from one port in normal ($z$) direction to the CRR arrays and we evaluate the transmission and reflection coefficients (or complex scattering parameters $S_{21}$ and $S_{11}$), as shown in Fig. 2(c).

As discussed above, the resonant response of our device is quite sensitive to the geometric parameters. We examined modulation of the resonant responses by changing the lateral shift of relative position of the double-layered CRR arrays. The initial arrangement where CRR arrays on opposing substrates overlap with no relative displacement looking from normal direction is the defined origin ($\mathrm{\Delta }x=\mathrm{\Delta }y=0\mu \mathrm{m}$). One of the arrays was laterally moved up to $25\mu \mathrm{m}$ in either $x$ or $y$ direction and resonant responses were evaluated. Tuning of resonant responses of layered-metamaterials by lateral shift in micrometer range was experimentally demonstrated,^22,23 therefore, our models are experimentally feasible. We also examined the modulation of the resonant responses by changing the dielectric constant of the dielectric spacer layer (${ϵ}_{\mathrm{d}}$) inserted between the double-layered CRR arrays from 2.0 to 4.0 assuming the insertion of organic functional materials.

2.3.

Parameter Retrieval

The impedance $Z$ and refractive index $n$ of a metamaterial slab can be retrieved from its complex transmission ($S_{21}$) and reflection ($S_{11}$) coefficients.^33,34 The impedance $Z$ for normal incidence is expressed as a function of scattering parameters $S_{11}$ and $S_{21}$:

The refractive index $n$ of a homogeneous passive slab with thickness $d$ is

3.1.

Index-Tuning by Shifting the Relative Position of the Double-Layered Closed-Ring Resonator Arrays

A single-layer CRR array may have a relatively wide stop band, as shown in dotted lines in Fig. 3(a). On the contrary, a relatively narrow transmission peak appears within the stop band at around 0.69 THz in a double-layered structure with no displacement ($\mathrm{\Delta }x=\mathrm{\Delta }y=0\mu \mathrm{m}$), as can be seen in solid lines in Fig. 3(a). The resonant frequency of this transmission peak can be adjusted to particular frequency of interest by modifying the side length $L$ and width $W$ of each CRR accordingly by a scaling rule.^25–27 It is clarified by Gu et al.²⁷ that this narrow line transmission peak is attributed to two factors, the magnetic response at the two sides of the closest stacked CRR pair spaced by $G$, and the near-field interaction between adjacent CRRs separated by $A$ in the same array. In Fig. 3(c), 3-D plots of the simulated distribution of the magnetic field at 0.577 and 0.694 THz on the cutting plane placed at the middle of each CRRs are shown. It can be visually recognized that closely placed CRRs strongly interact and strong magnetic resonance are induced at around the resonant frequency (0.694 THz). On the contrary, at 0.577 THz, in which the frequency is off from the resonant frequency, only one of the layered CRRs interact with EM wave and show relatively weak resonance. Such resonant responses can be modeled by an equivalent circuit [Fig. 2(b)]. The magnetic response is modeled by a microstrip loop across the two sides of the opposing CRRs. The near-field interaction is modeled by a coplanar line loop across adjacent CRRs in the same array. We previously reported that these resonant responses were quite sensitive to the geometric parameters of the CRR array and showed tuning capability of resonant frequency by giving relative displacement to the CRR arrays or changing dielectric constant of dielectric spacer layer inserted between CRR arrays.^25,26 Here, we further show that the effective refractive index can also be tuned by similar procedures. Figure 3(d) summarizes the effective refractive index of the double-layered CRR array retrieved from complex scattering parameters $S_{21}$ and $S_{11}$ shown in Figs. 3(a) and 3(b). At around narrow band resonant transmission peak, negative index of refraction is observed, similar to a previous report.²⁷ In the following, we focus on the change in the effective indices by giving relative displacement to the CRR arrays or changing dielectric constant of dielectric media inserted between CRR arrays.

Fig. 3

Summary of the THz response of double-layered CRR arrays without relative displacement. (a) Magnitude and (b) phase of the simulated scattering parameters $S_{11}$ (reflection) and $S_{21}$ (transmission). (c) Simulated distribution of the magnetic field at 0.577 and 0.694 THz on the cutting plane placed at the middle of each CRRs. (d) Refractive index retrieved from $S_{11}$ and $S_{21}$.

By giving lateral displacement to one of the double-layered CRR arrays in the $x$ direction, the narrow-line resonant transmission peak monotonically shifted to lower frequency, as summarized in Figs. 4(a) and 4(b). The frequency shift reached 17.5 GHz for $\mathrm{\Delta }x=25\mu \mathrm{m}$. In Figs. 4(c) and 4(d), 3-D plots of the simulated distribution of the magnetic field at 0.577 and 0.694 THz on the cutting plane placed at the middle of each CRRs are shown. The obtained responses are quite similar to those shown in Fig. 3(c), in which no lateral displacement was given to the layered CRRs. This implies that the basic principle of interaction with EM waves is unchanged under slight displacement of relative position of CRRs. This can also be understood by the fact that the overall spectral shape does not largely change and slight shift of resonant frequency is obtained.

Fig. 4

Summary of the THz response of double-layered CRR arrays with relative displacement $\mathrm{\Delta }x$. (a) Transmission spectra of double-layered CRR arrays with relative displacement $\mathrm{\Delta }x$. (b) Relative displacement $\mathrm{\Delta }x$ dependence of resonant transmission peak frequency of double-layered CRR arrays. (c) and (d) Simulated distribution of the magnetic field at 0.577 and 0.694 THz on the cutting plane placed at the middle of each CRRs.

By introducing a lateral displacement, the effective refractive index can also be tuned. Figures 5(a) and 5(b) summarize the real and imaginary part of the retrieved effective refractive indices, $n_{\mathrm{r}}$ and $n_{\mathrm{i}}$, respectively, for each relative displacement $\mathrm{\Delta }x$ as a heat map. It can be seen that the refractive index greatly changes from a positive to a negative value at around 0.6 THz. Above 0.7 THz, the refractive index is small and near zero. For $\mathrm{\Delta }x$ larger than $5\mu \mathrm{m}$, however, the value of the refractive index increases either positive or negative at around 0.8 THz. Figures 5(c) and 5(d) summarize the change in the real part of refractive index ($n_{\mathrm{r}}$) and FOM, respectively, with respect to the displacement of the relative position $\mathrm{\Delta }x$ at several frequencies of 0.6250, 0.7050, 0.8025, and 0.8325 THz. These results imply that various ways of tuning of the refractive index can be realized. At 0.6250 THz, the refractive index abruptly changes from a positive high value of $\sim 5.7$ to a negative value of approximately $-5.7$ for $\mathrm{\Delta }x=2\mu \mathrm{m}$ and keeps almost the same value for lager $\mathrm{\Delta }x$. The FOM keeps moderately high value of $\sim 3$. At 0.7050 THz, the refractive index shows negative value and its absolute value continuously decreases; the refractive index changes from $-2.9$ to $-1.2$ by increasing the relative displacement $\mathrm{\Delta }x$ up to $25\mu \mathrm{m}$. The FOM shows a high value of 5.8 for $\mathrm{\Delta }x=0$, however, continuously decreases by increasing $\mathrm{\Delta }x$. At 0.8025 THz, the refractive index similarly shows a negative value but shows opposite trend, that is, its absolute value continuously increases; the refractive index changes from $-0.6$ to $-4.4$ by increasing $\mathrm{\Delta }x$ up to $25\mu \mathrm{m}$. When $\mathrm{\Delta }x$ is between 5 and $10\mu \mathrm{m}$, relatively steep change in the refractive index can be obtained. The FOM, however, shows a low value. At 0.8325 THz, near zero index can be achieved for $\mathrm{\Delta }x$ smaller than $6\mu \mathrm{m}$, and abruptly changes to a positive high value of $\sim 4.0$ at $\mathrm{\Delta }x=7\mu \mathrm{m}$ and keeps almost the same value for $\mathrm{\Delta }x$ up to $25\mu \mathrm{m}$. The FOM is low. These varieties of index-tuning capabilities in the double-layered CRR arrays can be utilized to develop various types of THz optical materials, components, and devices.

Fig. 5

Summary of relative displacement $\mathrm{\Delta }x$ dependence of the effective refractive indices. The heat maps represent the change in (a) real and (b) imaginary part of the refractive index with respect to the relative displacement $\mathrm{\Delta }x$. Summary of relative displacement $\mathrm{\Delta }x$ dependence of (c) real part of the refractive index and (d) FOM of double-layered CRR arrays at specific frequencies of 0.6250, 0.7050, 0.8025, and 0.8325 THz.

We have also investigated similar relative displacement dependence, giving lateral displacement in $y$-direction. In this case, the narrow-line resonant transmission peak monotonically shifted to the opposite direction, to a higher frequency, as summarized in Figs. 6(a) and 6(b). The amount of frequency shift is larger and reached 27.5 GHz for $\mathrm{\Delta }y=25\mu \mathrm{m}$. Therefore, in our device, the resonance frequency can be shifted to both lower and higher frequency according to the direction of the relative displacement ($\mathrm{\Delta }x$, $\mathrm{\Delta }y$) given to the double-layered CRR arrays.^25,26 In Figs. 6(c) and 6(d), 3-D plots of the simulated distribution of the magnetic field at 0.577 and 0.694 THz on the cutting plane placed at the middle of each CRRs are shown. The obtained responses in this case are also similar to those shown in Figs. 3(c), 4(c), and 4(d).

Fig. 6

Summary of the THz response of double-layered CRR arrays with relative displacement $\mathrm{\Delta }y$. (a) Transmission spectra of double-layered CRR arrays with relative displacement $\mathrm{\Delta }y$. (b) Relative displacement $\mathrm{\Delta }y$ dependence of resonant transmission peak frequency of double-layered CRR arrays. (c) and (d) Simulated distribution of the magnetic field at 0.577 and 0.694 THz on the cutting plane placed at the middle of each CRRs.

The heat maps in Figs. 7(a) and 7(b) show real and imaginary part of the retrieved effective refractive index, $n_{\mathrm{r}}$ and $n_{\mathrm{i}}$, respectively, for each relative displacement $\mathrm{\Delta }y$. It can be seen that the band around 0.6 to 0.7 THz in which the refractive index shows negative value shifts to higher frequency together with a narrow-line resonant transmission peak by increasing $\mathrm{\Delta }y$. Figures 7(c) and 7(d) summarize the change in the real part of the refractive index ($n_{\mathrm{r}}$) and FOM, respectively, with respect to the displacement of the relative position $\mathrm{\Delta }y$ at 0.6250 and 0.7225 THz. At 0.6250 THz, the refractive index shows a negative high value of $-5.7$ for $\mathrm{\Delta }y=0$ and abruptly changes to a positive high value of 5.7 by giving small displacement $\mathrm{\Delta }y$ of only $1\mu \mathrm{m}$ and almost unchanged for $\mathrm{\Delta }y$ up to $25\mu \mathrm{m}$. The FOM keeps a moderately high value of $\sim 3$. At 0.7225 THz, the refractive index shows a negative value, and its absolute value continuously increases; the refractive index changes from $-1.0$ to $-3.1$ by increasing the relative displacement $\mathrm{\Delta }y$ up to $25\mu \mathrm{m}$. The FOM continuously increases by increasing $\mathrm{\Delta }y$.

Fig. 7

Summary of relative displacement $\mathrm{\Delta }y$ dependence of the effective refractive indices. The heat maps represent the change in (a) real and (b) imaginary part of the refractive index with respect to the relative displacement $\mathrm{\Delta }y$. Summary of relative displacement $\mathrm{\Delta }y$ dependence of (c) real part of the refractive index and (d) FOM of double-layered CRR arrays at specific frequencies of 0.6250 and 0.7225 THz.

3.2.

Index-Tuning by Changing the Dielectric Constant of the Dielectric Media Inserted Between the Double-Layered Closed-Ring Resonator Arrays

We have also investigated the index-tuning capability of the double-layered CRR arrays by changing the dielectric constant ${ϵ}_{\mathrm{d}}$ of spacer inserted between the double-layered CRR arrays. In this case, no lateral displacement was given to the arrays ($\mathrm{\Delta }x=\mathrm{\Delta }y=0$). The narrow-line resonant transmission peak monotonically shifted to lower frequency by changing the dielectric constant ${ϵ}_{\mathrm{d}}$ from 2.0 to 4.0, as shown in Figs. 8(a) and 8(b). The resonant frequency shifted by 110 GHz. The heat maps in Figs. 8(c) and 8(d) show real and imaginary part of the retrieved effective refractive index, $n_{\mathrm{r}}$ and $n_{\mathrm{i}}$, respectively, for each dielectric constant ${ϵ}_{\mathrm{d}}$. In comparison with the index-tuning by giving the relative displacement $(\mathrm{\Delta }x,\mathrm{\Delta }y)$, the change in the effective refractive index due to the change in the dielectric constant ${ϵ}_{\mathrm{d}}$ is remarkable. Figures 8(e) and 8(f) summarize the change in the real part of the refractive index ($n_{\mathrm{r}}$) and FOM, respectively, with respect to the dielectric constant ${ϵ}_{\mathrm{d}}$ of a dielectrics inserted between the double-layered CRR arrays at 0.5075, 0.5650, and 0.6500 THz. At 0.5075 THz, the refractive index shows a positive high value of $\sim 7.0$ for ${ϵ}_{\mathrm{d}}$ less than 2.6 and abruptly changes its sign to negative and keeps almost the same value of $-7.0$ by further increasing ${ϵ}_{\mathrm{d}}$. The FOM keeps a moderately high value, larger than $\sim 3.5$. At 0.5650 THz; the refractive index shows a negative value, and its absolute value continuously decreased; the refractive index changes from $-6.3$ to $-1.4$ by increasing the dielectric constant ${ϵ}_{\mathrm{d}}$ of the dielectric from 2.0 to 3.2, however, returned to almost the same value by further increasing ${ϵ}_{\mathrm{d}}$ up to 4.0. The FOM shows moderately high value for ${ϵ}_{\mathrm{d}}$ less than 3.0, however, drops to less than 1 by further increasing ${ϵ}_{\mathrm{d}}$. At 0.6500 THz, the refractive index shows negative value, and its absolute value increased; the refractive index changes from $-0.9$ to $-5.5$ by increasing the dielectric constant ${ϵ}_{\mathrm{d}}$ of the dielectric from 2.0 to 3.0, however, abruptly changes its sign to positive and keeps almost the same value of $\sim 5.4$ by further increasing the dielectric constant ${ϵ}_{\mathrm{d}}$. The FOM is low. Changing the refractive index of dielectrics does not require mechanical displacement of the layers and can be more easily implemented by liquid crystals, etc.

Fig. 8

Summary of the THz response of double-layered CRR arrays by changing the dielectric constant ${ϵ}_{\mathrm{d}}$ of dielectric spacer inserted between double-layered CRR arrays. (a) Transmission spectra of double-layered CRR arrays for several values of dielectric constant ${ϵ}_{\mathrm{d}}$. (b) Transmission peak frequency of double-layered CRR arrays as a function of dielectric constant ${ϵ}_{\mathrm{d}}$. Heat maps represent the change in (c) real and (d) imaginary part of the refractive index as a function of dielectric constant ${ϵ}_{\mathrm{d}}$. Summary of (e) real part of the refractive index and (f) FOM of double-layered CRR arrays at specific frequencies of 0.5075, 0.5650, and 0.6500 THz as a function of the dielectric constant ${ϵ}_{\mathrm{d}}$ of dielectric inserted between double-layered CRR arrays.