The Role of Asymmetry on the Resonances of Conjoined Split-Ring Resonators

Abstract

The conjoined split-ring resonator (Co-SRR) is proposed as the unit cell to construct terahertz (THz) metamaterial. The size and position of the gaps on both sides of the structure were adjusted, and the impact on the electromagnetic response to the incident THz wave was investigated via simulation. Results show that by properly controlling the structural asymmetry, the resonances can be tuned simultaneously or independently. The devices exhibit frequency shifts of up to 510 GHz, a tuning range of free spectral range (FSR) as wide as 613 GHz, and a high modulation depth (MD) of 93.4%. Additionally, a wide range of amplitude modulation can occur across multiple frequencies. Incorporating spatial asymmetry further enhances the performance, resulting in a high quality factor (Q) of 44.8 and a figure of merit (FOM) of 40.1. The impressive characteristics prove that Co-SRR-based metamaterial is a great candidate for applications in optical sensing, switching, filtering and programming devices.

1. Introduction

In recent years, terahertz (THz) waves have garnered significant attention for their exceptional physical and chemical properties among electromagnetic (EM) waves, leading to flourishing in-depth research. Within this realm, THz metamaterials [1,2] stand out due to their advantageous optical behaviors in the frequency range, playing a pivotal role in THz wave studies. By thoughtfully designing their microstructures and arrangements, metamaterials can exert unique control over EM waves. Their diverse applications span cloaking, superlenses, antenna technology, imaging, sensing, and numerous other fields [3,4,5,6,7]. Split-ring resonators (SRRs) have been a popular component in constructing THz metamaterials due to their resonant properties and ease of fabrication [8,9,10].

Meanwhile, in the pursuit of flexible adjustment of optical properties and enhanced performance, the design of tunable metamaterials [11,12,13] has attracted great attention for potential applications in biomedical sensing [14], EM absorption and shielding [15], information processing [16], and so on. Innovative modifications have been made to the SRR structure to achieve tunability [17,18,19,20]. For example, Zheng and Lin introduced a tunable double split-disk resonator (SDR) that toggles between on and off states by adjusting the SDR’s rotation angle and the distance between the layers [17]. Manjappa et al. presented an asymmetric metal split-ring resonator [18] enabling active Fano resonance [19] control by altering the photoconductivity of the substrate. Lin’s group designed a three-dimensional double split-ring resonator (DSRR) [20] offering tunability through modifications in geometric parameters.

However, despite these advancements, each study has focused on improving only one or two aspects of performance, such as tuning range, modulation depth (MD), and quality factor (Q). No single unified structure has been systematically adjusted to produce multiple interesting EM phenomena, including Fano resonances, electromagnetic-induced transparency (EIT), as well as multiple-band modulation, while simultaneously exhibiting a broad tuning range, large MD, and high Q.

Among various means to tweak structural parameters, asymmetry in resonating structures has been known to effectively boost their optical performance [21,22,23]. In this study, we propose a tunable metamaterial design featuring a conjoined split-ring resonator (Co-SRR) as its unit structure. By altering its asymmetry, its resonances can be effectively tuned and modulated. We studied the asymmetry from several aspects, including controlling the gap size and gap location. Additionally, the structure was divided into two parts to introduce spatial asymmetry, and its influence on the resonances was analyzed. With such measures, the resonances can be tuned simultaneously or individually. The maximum resonant-frequency shift reaches 510 GHz, the tuning range of free spectral range (FSR) is as large as 613 GHz, the maximum MD is 93.4%, and the best Q and figure of merit (FOM) are 44.8 and 40.1, respectively. Given these extraordinary characteristics, Co-SRR exhibits huge potential in sensor, filter, switch, and programmable device applications.

2. Materials and Methods

The unit structure utilized in this study, the Co-SRR, is depicted in Figure 1. This design can be visualized as two symmetrically positioned split-ring resonators (SRRs) converging at the connecting bar located centrally within the Co-SRR, aligned along the y-direction. This component will be referred to as “the middle bar” in subsequent text. The gaps on either side of the middle bar divide the outer frame into two sections, designated as “the upper frame” and “the lower frame” hereinafter. The incoming THz lightwave is polarized in the x-direction and propagates along the negative z-axis, striking the device’s surface perpendicularly, as exemplified in Figure 1a. Figure 1b defines the parameters of the unit structure. The length (l) and width (s) of the outer frame are 64 µm and 56 µm, respectively. The line width (w) is 4 µm. The periods of the unit structure in both the x-direction (P_x) and y-direction (P_y) are both 100 µm. These parameters remain constant throughout the study. Prior to any changes being introduced, the gap size (g) is set to be 4 µm and centered at y = 0, resulting in widths of 30 µm for both the upper (b) and lower (a) frames. The simulation software, 3D finite-difference time-domain (FDTD) solutions, was used to obtain the transmission spectra of the proposed structure. It was also used to analyze the electrical (E) and magnetic (H) field distributions within the structure at resonant frequencies. To achieve this, a unit structure was designed within the software, with periodic boundary conditions applied in the x-y plane and perfectly matched layer (PML) boundaries utilized along the z-direction. The substrate material chosen was silicon dioxide (SiO₂ (Glass)–Palik model from the software’s material database), with a thickness of 50 µm, while the metal resonator was made of aluminum (Al-CRC model from the software’s material database), with a thickness of 0.5 µm. The chosen materials have been well characterized and widely used in the field of research. Such metamaterial devices can be prepared through mature micro-fabrication processes, including steps of lithography, metal deposition, and lift-off [24,25]. The transmission spectra were measured at the bottom of the substrate, and the resonant frequencies were determined by identifying the frequencies at which local minima in transmission amplitude occurred. The E-field and H-field distribution maps were observed at the interface between the metal layer and the substrate.

According to the Drude–Lorentz model, the resonant frequency of the proposed Co-SRR is closely related to its refractive index [26], which is given by

$$n_{EM}=\sqrt{\mu (\omega )\epsilon (\omega )}$$

(1)

$$\mu (\omega )=1-{\displaystyle \frac{F{\omega }_{pm}^2}{{\omega }^2-{\omega }_{0m}^2}}$$

(2)

$$\epsilon (\omega )=1-{\displaystyle \frac{F{\omega }_{pe}^2}{{\omega }^2-{\omega }_{0m}^2}}$$

(3)

where ω_p is the plasma frequency, ω₀ is the resonant frequency, and F is a dimensionless quantity, while subscripts e and m refer to electric and magnetic response, respectively.

• The transmittance of EM wave can be expressed as follows [27,28]:

$$T={\displaystyle \frac{4n_{air}{n}_{sub}{n}_{EM}^2}{{(n_{air}{n}_{sub}+n_{EM}^2)}^2}}$$

(4)

where n_air and n_sub are the refractive index of air and the substrate, respectively.

3. Results and Discussion

3.1. Symmetric Changes in Both Sides

Symmetric changes were introduced into the Co-SRR. This included widening both gaps while maintaining their left–right symmetry and positioning them off-center in a symmetrical manner.

First, the size of both gaps increased from 4 μm to 56 μm at a step of 8 μm, while fixing their centers at y = 0. The transmission spectra for devices with such a change are shown in Figure 2a. Consequently, the resonant frequencies shifted from 1.006 THz to 1.516 THz, resulting in a 510 GHz frequency shift. The resonant frequency fits into a linear relationship with the gap size, as shown in Figure 2b. The fitting equation is expressed as follows:

y = 0.01034x + 0.92641,

(5)

with an R-squared value of 0.98823.

The mechanism of the SRR can be understood by the analogy to an LC oscillating circuit [29]. Under the influence of the polarized electric field of the incident light, electrons oscillate within the metallic wires of the SRR, giving rise to oscillatory surface currents. As a result, there will be periodic accumulation and depletion of charges at the gap of the SRR. Figure 2c,d show the E-field and H-field distribution maps for devices with g = 4 μm and g = 52 μm, respectively, at their respective resonant frequencies. The x- and y-span both range from −50 to 50 μm. The color bar extends linearly from minimum to the maximum value. The E-field and H-field distribution maps revealed that at resonance, surface currents oscillate simultaneously and symmetrically along the upper and lower frames (indicated by the large H-field along the frames), causing charges accumulated or depleted at the two ends of both frames, resulting in large E-field at those positions. Comparing Figure 2c,d, it is evident that as the gap enlarges, the current path reduces. Thus, the surface current can oscillate at a higher frequency, causing a blue shift in the resonance. Note that the x- and y-span as well as the linear color bar settings are kept the same in all the following E-field and H-field distribution images.

Figure 3 investigates the impact of gap location on the transmission spectra. The offset of the gap centers (Δ) varied from 0 to 26 μm. The spectra for devices with such off-center gaps are presented in Figure 3a. As the gaps start to deviate from the center, starting from Δ = 4 μm, a Fano resonance develops in the lower frequency region (the first resonance). This resonance intensifies as the degree of asymmetry increases and exhibits a red shift of 205 GHz. On the other hand, the second resonance, which is the only resonance present in a symmetrical device, undergoes a blue shift of 423 GHz. When the degree of asymmetry reaches a significant level, starting from Δ = 18 μm, a third resonance emerges in the higher frequency range and becomes more pronounced as the asymmetry intensifies. However, the resonant frequency of this third resonance was maintained as almost constant. The frequency shift is only 26 GHz.

The resonant frequencies were extracted from the spectra and plotted against the center offset (Δ) in Figure 3b. For gap offsets exceeding 8 μm, a linear trend is observed for the second resonant frequency. The fitting equation is given by the following:

y = 0.02027x + 0.8965,

(6)

with an R-squared value of 0.99839, indicating a high degree of linear correlation. As the asymmetry intensifies, due to the red shift in the first resonance and the blue shift in the second resonance, the free spectral range (FSR) between the two exhibits a broad tuning range. Specifically, the FSR starts at 232 GHz for Δ = 4 μm and gradually increases to 845 GHz for Δ = 26 μm, yielding a tuning range of 613 GHz.

The spectra exhibited attractive amplitude modulation (AM) characteristics of these devices. Due to the large shifts in the resonances at each specific resonant frequency, passband and a resonant dip can be observed at the same time. For example, consider the black curve (Δ = 0) and the purple curve (Δ = 26 μm) in Figure 3a. At 1.006 THz, it is within the passband of the purple curve but corresponds to the resonant dip of the black curve. Such amplitude modulation suggests that, at this frequency, the transmission of incoming THz lightwave can be effectively switched on and off, and the devices can function as an optical switch to control the passage level of the light or blocking it entirely by controlling the gap center offset. The modulation depth (MD) is defined as follows:

MD = (A_max − A_min)/A_max × 100%,

(7)

where A_max and A_min are the maximum and minimum transmission amplitude at resonant frequency. Specifically, the largest amplitude variation range was observed at 1.006 THz, which corresponds to the second resonant frequency for the device with Δ = 0 μm. The modulation depth (MD) at this frequency was calculated to be 93.4%, indicating a significant modulation effect. Additionally, notable MD values were also observed at 1.429 THz and 1.517 THz, which are the second and third resonant frequencies, respectively, for the device with Δ = 26 μm. The MDs at these frequencies were 87.4% and 69.5%, respectively, showing that the devices exhibited strong modulation characteristics at multiple resonant frequencies.

Taking into consideration that the resonance around 1 THz shifts to higher frequencies with the increase in gap size (Figure 2a,b) as well as the increase in gap offset (Figure 3), it is possible to increase gap size while decreasing gap offset simultaneously to achieve a balance where the resonance in question remains unshifted. Meanwhile, due to the decreased gap offset, the resonance to its left (the first resonance in Figure 3) should shift to higher frequencies and be tuned independently. The specific combinations of gap size (g) and offset (Δ) that yield such results have been identified. The corresponding transmission spectra are presented in Figure 4a. As predicted, the second resonance maintained stability at around 1.398 THz with a minimal frequency shift of less than +/−7 GHz. On the contrary, the first resonance shifted significantly from 0.637 THz (with Δ = 22 μm and g = 8 μm) to 1.063 THz (with Δ = 6 μm and g = 38 μm), resulting in a frequency shift of 426 GHz. This number could be increased even more, but as Δ further decreases, the device becomes more symmetrical, leading to attenuation in the first resonance. Such attenuation becomes evident in the spectra of the devices with Δ = 8 μm and Δ = 6 μm.

To gain deeper insights into these resonances, we examined the E-field and H-field distribution maps for each resonance. A typical device with Δ = 10 μm and g = 31 μm was selected, and its E-field and H-field distribution maps are presented in Figure 4b,c, respectively. It becomes obvious that at different resonant frequencies, the upper frame and the lower frame are excited one at a time, independent of each other. The first resonance can be primarily attributed to the lower frame alone, and the second to the upper. From the perspective of an analogous LC circuit, at the lower resonant frequency, electrons are excited to oscillate along the lower frame, which has a longer current path length and a correspondingly longer “round-trip time”. Conversely, at the higher resonant frequency, electrons oscillate along the shorter frame, resulting in a reduced travel period. Therefore, we tabulated the widths of the upper (b) and lower (a) frames in

Table 1. The combination of gap offset (Δ) and gap size (g) depicted in Figure 4 can be expressed by a set of widths of the upper frame (b) and lower frame (a).

Δ	g	a	b
22	8	50	6
20	12	46	6
16	20	38	6
12	28	30	6
10	31	26.5	6.5
8	35	22.5	6.5
6	38	19	7

for the devices used in Figure 4a. Not surprisingly, the widths of the upper frames remained almost constant at about 6 μm, whereas that of the upper varied significantly from 50 μm to 19 μm. The slight increment in b for larger gap sizes compensates for the diminishing coupling between the upper and lower frames.

With the same principle, to achieve independent tuning of the second resonance, the width of the lower frame (a) was kept almost constant at about 30 μm, with slight variations to compensate for the diminishing coupling effect between the frames, whereas that of the upper frame (b) varied from 30 to 6 μm. The resulting transmission spectra are shown in Figure 5. The first resonance remained fixed at 0.850 THz, despite its increased intensity attributed to increased asymmetry. In contrast, the second resonance shifted from 1.006 THz (b = 30 μm) to 1.398 THz (b = 6 μm), resulting in a frequency shift of 392 GHz.

It is noteworthy that Figure 4a and Figure 5 also exhibit significant amplitude modulation. By modifying either structural parameter a or b, the transmission state can be switched on or off at the first or second resonance with substantial MDs. Specifically, MDs at the first resonant frequencies in Figure 4a averaged 73.3% (SD = 0.0581), while those at the second resonances in Figure 5 averaged 91.2% (SD = 0.0199).

3.2. Asymmetric Changes in Both Sides

Next, we investigated the impact of asymmetric changes in both sides of the Co-SRR. First, the gaps on both sides were offset in opposite directions, with the offset increasing from 0 to 24 μm at a step of 4 μm. The gap size was kept constant at 4 μm throughout all simulations. The transmission spectra obtained are presented in Figure 6a. Three resonant dips were observed. The resonant frequencies and corresponding transmission amplitudes were extracted from the spectra and plotted against the gap offset, as shown in Figure 6b and Figure 6c, respectively.

The first resonance occurs around 0.5 THz. As the offset increases, this resonance becomes more pronounced and exhibits a red shift. Specifically, the resonant frequency shifts from 0.519 THz at 8 μm offset to 0.446 THz at 24 μm offset. The resulting frequency shift is 73 GHz. Notably, the first resonance was too weak in the devices with offset 0 and 4 μm. The resonant frequencies presented in Figure 6b were estimated at 0.519 THz, marked by asterisks, based on the value observed for the device with 8 μm offset. As the resonance intensifies, the transmission amplitude at resonance decreases from 0.782 to 0.647, yielding an MD of 17.3%.

The second resonance exhibits attenuation as the offset increases and displays a blue shift. The resonant frequency shifts from 1.006 THz at zero offset to 1.178 THz at 24 μm offset. The resulting frequency shift is 172 GHz. The transmission amplitude at resonance increases from 0.053 to 0.455, resulting in an MD as large as 88.4%.

The third resonance was only prominent in devices with offsets of 20 and 24 μm. The frequency shift between these two devices was only 4 GHz. Transmission amplitudes for all devices were measured at 1.528 THz, except for the device with 24 μm offset, which was measured at its own resonant frequency of 1.524 THz. As gap offset increases, the transmission amplitude at resonance decreases from 0.765 to 0.500, resulting in an MD of 34.6%.

Then, we fixed both gap centers at y = 0 and increased the gap size on one side of the Co-SRR from 4 μm to 52 μm at a step of 4 μm, while keeping the gap size on the other side fixed at 4 µm. The impact of this asymmetric gap size variation was then investigated.

The transmission spectra for such devices are presented in Figure 7a. Two resonant dips can be observed in the spectra. The E-field and H-field distributions at each resonance for devices with enlarged gap sizes of 20 µm and 52 µm are shown in Figure 7b–e, respectively. Judging by the frequency and intensity, the first resonance in Figure 7a bears a resemblance to that presented in Figure 2a. When the varying gap is small (20 μm), the E-field and H-field distributions in Figure 7b also mimic those in Figure 2c,d. Specifically, the incident light wave excites both the upper and lower frames, while negligible E and H fields are distributed around the middle bar. However, as the asymmetry intensifies, in Figure 7c, the H-field becomes increasingly concentrated to the right of the Co-SRR. The surface current flows in “П”-shaped path formed by the upper half of the middle bar and the right half of the upper frame, and also in the mirroring path at the lower half of the structure. Comparing this “П”-shaped path to the path along the upper and lower frames in Figure 7b, it becomes evident that the electrons oscillate over a shorter distance in each cycle, resulting in an increased resonant frequency.

The second resonance becomes prominent only when the asymmetry is significant, denoted by a large value of g. When the device exhibits a high degree of asymmetry with g = 52 μm, at its resonant frequency of 1.516 THz, the transmission amplitude measures 0.316. In contrast, for a symmetric device with g = 4 μm, the transmission amplitude at this frequency is 0.758. This results in an MD of 58.3%. However, it is noteworthy that, even when the gap size is small, as shown in Figure 7d, the middle bar is excited due to the asymmetry. As g increases, comparing Figure 7d with Figure 7e, the H-field becomes increasingly concentrated to the left of the Co-SRR. The surface current flows in a “┐”-shaped path formed by the upper half of the middle bar and the left half of the upper frame, as well as in the mirroring path in the lower half of the structure. Due to the same reason that causes the blue shift in the first resonance, the length of the current path decreases as asymmetry increases. Consequently, this second resonance also exhibits a blue shift.

3.3. Spatial Asymmetry in Co-SRR

Next, spatial asymmetry was created in the Co-SRR and its influence on the resonances was analyzed. To do so, a gap of 4 μm was introduced into the center of the middle bar such that the structure broke into two face-to-face “E”-shaped resonators. Then, the upper “E” was elevated to a plane that is at a height of h above the lower part (see Figure 8a). h was defined to be the distance between the top surface of the upper “E” and the top surface of the lower “E”. The width of the outer frame of the lower “E” was defined as a; and that of the upper “E” was defined as b (see Figure 8b). Such structures can be realized via a MEMS technique [20,30]. In the following studies, all three gaps are kept at 4 μm wide and the other parameters remain the same as the planar structures mentioned previously.

The impact of the distance between the two parts was investigated, and the transmission spectra for devices with different h values are presented in Figure 9. Although the Co-SRR maintained the symmetric layout in planar view, the added spatial asymmetry split the single resonance spectra of the planar device (h = 0) into two resonances. The first resonance becomes prominent when the distance reaches 1 μm and above. It first shows a slight blue shift with increasing h, then stabilized at 0.85 THz for h ≥ 6 μm. The second resonant frequency shifts significantly from 1.006 THz to 1.353 THz, resulting in a shift of 347 GHz. Moreover, sharp resonance dips were observed at first resonance for h = 1 and 2 μm. Q and FOM were calculated using the formulas

Q = f_r/FWHM

(8)

and

FOM = Q ∗ (1 − A),

(9)

where f_r represents the resonant frequency, FWHM is the full width at half maxima, and A stands for the transmission amplitude at resonance. The largest Q and FOM are 22.8 and 22.0, respectively, for the resonant dip associated with h = 1 μm.

Moreover, an EIT phenomenon can be spotted in the plot. The Q and FOM for the EIT peaks were also calculated. The formula for Q stays unchanged, while FOM is modified as follows:

FOM = Q ∗ A

(10)

Again, the sharpest EIT peak occurs when h equals to 1 μm. The Q and FOM are 10.3 and 9.4, respectively.

Lastly, the planar symmetry of the two parts was broken, and the impact was investigated while the spatial distance between them (h) was kept constant at 2 μm. By simultaneously increasing b and decreasing a in steps of 4 µm, the gap size (g) was maintained at 4 μm. Notably, for devices with specific combinations of b and a (b = 34 μm, a = 26 μm and b = 38 μm, a = 22 μm), the simulated EM field within the structure could not reach a stable state, rendering the data unreliable and, therefore, excluded. The simulated transmission spectra, grouped by b < a and b > a, are presented in Figure 10a. The color lines in the plot vary from black (least symmetric) to purple (most symmetric, where b = a). The spectra indicate that as symmetry increases, the first resonance undergoes a blue shift, while the second resonance undergoes a red shift. Consequently, the FSR decreases from 818 GHz (at b = 14 μm) to 298 GHz (at b = 42 μm), and then increases to 664 GHz (at b = 54 μm) again. Such tuning of the FSR is depicted in Figure 10b, showing a tuning range of 520 GHz.

Another thing to note is the Q-factor of the first resonance dip. The dips become shaper with increased symmetry. The Q and FOM for the device with b = 30 μm are 15.98 and 15.38, respectively. They are 44.8 and 40.1 for the device with b = 42 μm. The Q and FOM values for spectra displayed in Figure 9 and Figure 10a are tabulated in

Table 2. Exemplary Q-Factors (Q) and Figures of Merit (FOM) for resonant dips and the electromagnetically induced transparency (EIT) peak in three structures presented in Figure 9 and Figure 10.

	Resonant Dips			EIT Peak
h (μm)	1	2	2	1
b (μm)	30	30	42	30
frequency (THz)	0.820	0.831	0.852	0.835
Q	22.8	15.98	44.8	10.3
FOM	22.0	15.38	40.1	9.4

. It is obvious that when the width of the upper frame (b) is slightly larger than that of the lower frame (a), as exemplified by b = 42 μm, the largest Q and FOM can be obtained. This can be seen as a compensation for the asymmetry caused by the lack of substrate beneath the upper “E” structure, achieved by increasing the width of the upper frame (b), thereby restoring a more balanced overall symmetry. In other words, when b = 42 μm, the upper and lower structure approaches the most symmetric layout considering the single-sided substrate.

To gain a deeper understanding of the dual resonant dips in the transmission spectra, we examined the E-field and H-field distributions across four typical devices. For consistency with previous work on planar devices, the maps were also obtained at the interface between the lower “E” pattern and the substrate. These devices represent the highest and lowest levels of symmetry within the two groups categorized by b ≤ a and b > a with h = 2 μm, respectively. They will be referred to as Device A (b = 14 μm, a = 46 μm), B (b = 30 μm, a = 30 μm), C (b = 42 μm, a = 18 μm), and D (b = 54 μm, a = 6 μm) hereinafter for ease of description. The E and H distribution maps are presented in Figure 11.

Two resonant modes can be identified from Figure 11. One is associated with the excitement of lower “E”, whereas the other is with the excitement of the upper “E”. They will be referred to as L-mode and U-mode for simplicity. For Device A and B, where b ≤ a, L-mode contributes to the first resonance and U-mode to the second. For Device C and D, where b > a, the reverse is observed. This can be understood by linking the resonant frequency with the current path. In all cases, at the lower resonant frequency, induced surface current oscillates in the longer frame and at the higher resonant frequency in the shorter frame.

Another thing to note is the coupling between the upper and lower “E”. In U-mode, while large current flows along the upper frame, non-negligible current is also induced in the lower frame. On the contrary, in L-mode, only when symmetry level is high, as in Device B and C, can the minimal current be induced in the upper frame. Such coupling gives rise to the interlayer capacitance, which is a determining factor for resonant frequency according to the LC analogy of SRR [29]. In LC circuits, the resonant frequency is given by the following:

ω = (LC)^−1/2

(11)

where L and C are the inductance and capacitance, respectively. Here, as the interlayer distance, h, increases, and the capacitance effectively decreases, which should lead to a blue shift in both resonant frequencies. This is confirmed by the shift in the second resonance in Figure 9. As for the first resonance, the coupling is inherently weak in L-mode for Device B and becomes even weaker when h increases; therefore, the blue shift is not obvious and becomes negligible when h is above 6 μm.

With the understanding of the coupling in L-mode and U-mode, we further verified the relationship between resonances and h for Device C and D. The transmission spectra for these two devices are presented in Figure 12a and Figure 12b, respectively. For both Devices, the first resonance, which is attributed by the U-mode with strong coupling between the layers, shows significant blue shift with increasing h. For Device C, due to the low asymmetry in structure, resonance hybridization can be spotted as the first resonance shifts to close proximity of the second resonance (h ≥ 4 μm). Such hybridization effect can also be spotted in Device B with a similarly highly symmetric layout, when h is at 0.5 μm (see Figure 9). Blue shifts in both resonances in Device C and the first resonance in Device D, as shown in Figure 12, confirmed the existence of interlayer capacitance. The blue shift is large for the first resonance, and small for the second resonance, due to the variation in coupling strength in U-mode and L-mode. Similarly to Device B, the weak coupling in L-mode gradually disappear at higher h, resulting in less and less shift in the second resonance of Device C. Lastly, the second resonance is almost stable for Device D, because as asymmetry level increases, the coupling in L-mode is too weak to make a significant impact on the second resonance.

3.4. Comparison with Previously Reported THz Metamaterial Modulators and Sensors

To better evaluate the performance of the metamaterial we propose, which utilizes Co-SRR as its unit structure, we compare it against the background of THz metamaterials reported in recent years.

Table 3. Comparison of frequency shift and tuning range of FSR between recently reported THz metamaterials and our work.

References	[17]	[20]	[31]	[32]	[33]	Our Work
Frequency shift (GHz)	200	400	318	108	109	510 1
FSR tuning range (GHz)	240	550	-	-	-	613 2

¹ Achieved when symmetrically varying both gap sizes, g, from 4 to 56 μm. ² Achieved when symmetrically offsetting both gaps from center by Δ = 4 to 26 μm.

,

Table 4. Comparison of modulation depth (MD) between recently reported THz metamaterials and our work.

References	[17]	[34]	[35]	[36]	[37]	[38]	Our Work
MD (%)	75.6	71	45	50	25	23	93.4 3

³ Achieved when symmetrically offsetting both gaps from center by Δ = 0 to 26 μm. Generally, an MD over 70% can be achieved from most of the spectra presented in this work.

and

Table 5. Comparison of Q-factor between recently reported THz metamaterials and our work.

References	[6]	[17]	[39]	[40]	[41]	Our Work
Q-factor	22.1	14.5	30.5	38.9	3	44.8 4

⁴ Achieved when spatial asymmetry was introduced into the structure with h = 2 μm and b = 42 μm.

summarize the performance of our device along with data published in recent literature. Note that FOM often lacks a unified definition for different device applications; therefore, a comprehensive table-based comparison is not conducted here. Nevertheless, it is evident that our device exhibits significant advantages in terms of frequency shift, tuning range of FSR, modulation depth, and Q-factor.

4. Conclusions

In conclusion, we have introduced a metamaterial design, known as the Co-SRR, which exhibits exceptional EM characteristics. Although the direct verification of the effective permittivity and permeability of the structure remains pending, the intriguing transmission spectra that evidence unique interactions with THz lightwaves suggest that the Co-SRR holds promise as a unit structure for constructing tunable THz metamaterials. By adjusting the size and position of the gaps on both sides of the structure, we can tune the resonances simultaneously or independently, achieving significant frequency shifts as large as 510 GHz. Furthermore, this design boasts remarkable amplitude modulation properties, including high MD of up to 93.4% and the ability to modulate at multiple frequencies. By incorporating spatial asymmetry into the structure, the metamaterial generates resonant dips and EIT peaks with excellent Q and FOM values of 44.8 and 40.1, respectively. Lastly, by managing the degree of structural asymmetry, we can adjust the FSR over a broad tuning range of as wide as 613 GHz. Given these impressive features, the Co-SRR holds great potential for applications in reconfigurable metasurfaces, utilizing FSR tunability for THz filtering, asymmetric-resonance-based sensors exploiting high-Q resonance for material characterization, and optical switches leveraging high MD, enabling dynamic control of light transmission in optical communication systems.