1. Introduction

Terahertz (THz) radiation has received tremendous attentions owing to its unique properties and potential applications in many frontier fields [1–4]. THz technology has made significant advance, especially THz radiation sources, detectors and application research in sensing and imaging. However, various THz functional devices are still in the exploratory stage. The emergence of metamaterials has greatly broadened the possibility of manipulating THz wave and realizing field-matter interaction [5,6]. A variety of THz metamaterials were suggested, exhibiting interesting physical phenomena by plasmonic resonant elements [7–10]. To achieve sharp resonance with high quality factor (Q-factor) has always been one of the important objectives of metamaterials design, because of many potential applications in narrow-band filters, ultra-sensitive sensors and slow light devices [11–15]. Those applications usually require photonic metamaterials to produce ultra-sharp spectral response with strong field concentration at sub-wavelength scale. A prime example is that high Q-factor resonance with extremely narrow linewidths can provide a favorable approach for achieving ultra-sensitive real-time chemical and biomolecular noninvasive sensing [16–18]. The sharp resonance with a large modulation depth is helpful to improve the sensing sensitivity. Many efforts have been made for achieving sharp resonances and further optimizing spectra fineness and modulation depth. However, due to inherent ohmic damping and strong radiation losses, the key Q-factor of traditional metamaterials and plasmonic resonators face serious restrictions, hardly more than the order of 10², hindering their realistic application of sensors and filters [19–21].

Since metals have excellent conductivity at THz frequencies, the ohmic losses of most metallic metamaterials are relatively quite low, so radiation losses play a dominant role in loss mechanism of THz plasmonic metamaterials. The key to achieving high Q-factor resonance is to fundamentally reduce dipole radiation losses [22]. Several strategies were proposed to reduce radiation losses and support the high Q-factor resonance of metamaterials [20,22–28]. One of the most popular approaches is to break the symmetry of metamaterial resonator [20–24], for instance, typically asymmetric split ring resonators (SRRs). The narrow resonance can arise due to the excitation of trapped and octupolar modes or Fano mode, with low radiation losses and high Q-factor of 10² in orders of magnitude [16–18]. The second approach is to exploit the weak coupling of dark modes excited by two adjacent SRRs or sub-radiation mode excited by a structural unit composed of four SRRs to produce the suppressed radiation loss and high Q-factor of about 41 and 24.2 [25,26]. The third approach is to utilize the diffraction coupling of two mirror-symmetric metamaterials to reduce radiation losses and excite ultra-sharp trapped-mode resonances with a Q-factor of approaching 60 [27]. In addition, toroidal dipolar mode has been shown as a new route to support high Q-factor of about 42.5 in planar toroidal metamaterials [28]. However, those design schemes are difficult in THz regime to obtain extremely high Q-factor resonance comparable with photonic resonance in visible-infrared regime. The Q-factor at THz spectra is much lower by several orders of magnitude than that at optical bands. To explore the ultra-high Q-factor resonance of THz metamaterials remains a basic challenge.

In the paper, we propose a unique metasurface scheme to produce extremely high Q-factor Fano resonance in THz regime. The scheme of metamaterial array composed of single vertically symmetric SRR can support extremely high Q-factor beyond the 10⁴ level with a large modulation depth. The underlying mechanism derives from the nearly perfect reconstructive coherent mode between each individual SRR and the adjacent SRRs. The electromagnetic radiation components of the current elements can be completely eliminated through destructive interference, which dramatically increases radiative damping and enhances Q-factor in THz regime. Manipulating the Q-factor of the coherent mode is achievable by adjusting the geometric parameter of the metamaterial. Such metasurface scheme can be applied for ultra-sensitive sensors, filters, and strong field-matter interaction.

2. Metasurface design and simulation

The schematic diagram of the proposed THz metasurface resonator is illustrated in Fig. 1. The periodic array of U-shaped SRR is deposited on SiO₂ substrate, where the thickness of the substrate is 20 µm and the refractive index is 1.5 [29]. The origin of the structural element coincides with the center of SRR, as shown by the purple dotted line of Fig. 1(b). In the simulations, the optimized geometric parameters of the unit cell include the backbone of a = 50 µm, two arms of b = 30 µm, the split distance g = 40 µm, the uniform width w = 5 µm, the array period P = 160 µm and the thickness of SRR is 5 µm. The parameters remain unchanged unless otherwise specified. The metallic SRR is selected as gold, and its relative permittivity is expressed by the typical Drude model as ɛ = 1-ω_p²/(ω²+iγ_cω), with plasma frequency ω_p = 1.37×10¹⁶ rad/s and collision frequency γ_c = 4.08×10¹³ rad/s, and the model works well at THz frequencies [22,30,31]. The electric field E, magnetic field H, and wave vector k of the incident plane wave is along the x, y and z axes, respectively. The three-dimensional model is carried out by finite-element method (FEM) to solve the Maxwell’s equations using COMSOL software. The transmission T(ω) = |S₂₁|² can be obtained from the S-parameters realized by the frequency domain analysis of RF module. Period boundary conditions are set in the x and y directions of the unit structure, and perfectly matched layers conditions are constructed along the z-direction.

 

Fig. 1. (a) Schematic diagram of the THz metasurface resonator design. (b) Front view of a unit cell.

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3. Results and discussions

The transmission spectrum with x-polarization incidence is shown in Fig. 2. It is observed that a sharp Fano resonance response with distinct narrow linewidth is excited at 1.826 THz. The Q-factor is defined as Q = 2πf₀P_S/P_L = f₀/Δf [22], where P_S is stored energy, P_L is dissipated energy, f₀ is the resonant frequency and Δf is the full width at half maximum. There is an ultra-sharp resonance dip (its Δf is 40 MHz) with extremely high Q-factor as high as 4.565 × 10⁴, and a large modulation depth [(T_max−T_min)/(T_max+T_min) × 100%] for the resonance is up to 86%. As proposed by miroshnichenko et al. [26], the Fano resonance can be illustrated by the coupled mode theory with normalized Fano profile, which is expressed by the following formula [32–34],

 

Fig. 2. (a) The transmission spectra calculated by FEM simulation and normalized Fano profile theory. (b) The simulation results of two different simulation methods.

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Note that previously, to experimentally obtain the high Q-factor resonances usually requires the introducing of structural asymmetries [20–22], but the unit structure in this work is only a simple symmetric SRR. The distinct mechanism is closely associated with the coupling between the localized resonance of individual SRRs and the Wood-Rayleigh anomaly of periodic array [35–37]. Additionally, we further performed the simulations with different numerical technique of finite integration method (FIM) supported by the full-wave simulation software CST Microwave Studio. The results are illustrated in the black line of Fig. 2(b), showing good agreement with the FEM simulation.

The lattice period is of great importance to the electromagnetic response of a metamaterial structure. Here the influence of the lattice period P on the spectral response behavior is investigated. Figure 3(a) shows the transmission spectra with periodicity P varying from 130 µm to 170 µm. As the periodicity increases, the resonant peaks becomes sharper, and the resonance shows a distinctly redshift due to the decrease of effective permittivity of resonance response caused by the geometric scale. As shown in Fig. 3(b), when the period increase from 130 µm to 170 µm, the Q-factor exhibits an interesting exponential growth behavior versus the lattice period P, but the modulation depth decreases from nearly 100% to 83%. The dependence of the Q-factor on the lattice period is concluded as Q_f1 = Aexp[B/(P + D)], where the extracted numerical fitting coefficients are A = 4.073×10⁶, B = −1774.557, D =−30.469, and the fitting R-square is 0.9999, which is very close to unity. The exponentially increasing Q-factor rapidly even beyond the order of magnitude of 10⁴ because the destructive interference of the radiation field gradually appears between the unit cells. The Q-factor can break through a record level of over 10⁵ in orders of magnitude when P = 170 µm, which is comparable with photonic resonance in visible-infrared regime [38]. It can be seen that the Fano resonance with extremely high Q-factor and large absorption depth mostly depends on the suitable lattice period.

 

Fig. 3. (a) The variation of transmission spectra with periodicity P. (b) The variation of modulation depth and Q-factor with periodicity P.

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In essence, the arrangement of metallic subwavelength SRRs into a periodic lattice can produce an in-plane propagating collective lattice surface mode. This leads to abrupt changes in transmission near the Wood-Rayleigh anomaly [35–37]. It can be found that the ultrasharp resonance shown in Fig. 2 is around the (1, 0)_air order of the Rayleigh anomaly (∼1.88 THz). It is worth noting that when the calculated spectral range is extended, one may find the corresponding localized dipole resonance around the high-energy spectral range, and another high Q-factor surface lattice resonance that related to the (1, 0)_substrate diffraction order [35,37]. To deeply understand the physical mechanism of Fano resonance, the electric field intensity |E|, the contour plot of E-vector, the magnetic field H_z and the surface current distribution at resonance frequency are given in Fig. 4. The physical origins of ultra-high Q-factor resonance can be qualitatively interpreted by the basic multipole theory of electromagnetic scattering and radiation [39]. Here assuming that $\vec{J}(\vec{r^{\prime}})$ is the current density of the current-element at the source point $\vec{r^{\prime}}$, the relevant vector potential $\vec{A}$ at any point r can be described as

 

Fig. 4. The simulations display (a) electric field |E| distribution, (b) contour plot of E-vector, (c) magnetic field H_z distribution and (d) surface current distribution at resonance frequency, respectively. White arrows in (b), (c) and pyramidal in (d) indicate their vector directions. The color bar represents intensity distribution.

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In far filed radiation zone, its asymptotic form is adopted as follows

The time-averaged power radiated per unit solid angle by the oscillation dipole moment $\vec{p}$ is

Because of the unique design of the metasurface, the electric field is symmetrically localized around the SRR and the four corners of the lattice unit about the y-axis, showing an interesting elliptical distribution, as seen in Fig. 4(a). The electric field intensity of the SRR is obviously stronger than that of the corner of the lattice unit, and the electric field intensity on the backbone of the SRR is clearly higher than that of the arm. According to the contour plot of the E-distribution, i.e., Fig. 4(b), where the white arrows indicate the direction of the vector field, it can be found that the direction of the electric field propagates from the SRR to all directions, typically showing radial radiation characteristics. This outstanding feature indicates that the direction of the radial field of each individual SRR is opposite to that of the adjacent SRRs, including vertical SRRs, horizontal SRRs and diagonal SRRs, which can induce reconstructive coherent effects between unit cells and cause destructive interference of the radiation field. From the magnetic field H_z distribution shown in Fig. 4(c), it can be observed that the plasmonic SRR resonator generates magnetic dipole resonance, and maintains spatially regular magnetization. The vortex magnetic field circulates around the concentrated radial electric field, orthogonal to the E vector in the plane. The surface current distribution is calculated as shown in Fig. 4(d). Due to the special symmetrical design, the electric current elements in the resonator oscillate out of phase (point in opposite directions, all flow to the middle of backbone of SRR) when excited with an electric field perpendicular to the gap, resulting in the cancellation of net dipole moment, accompanied with destructive interference of the radiated fields. The results indicate that the generation of high Q-factor Fano resonances is caused by the excitation of surface lattice resonances, which is related to the coupling between the localized resonance of individual SRRs and the Rayleigh anomaly of the array. Under the light incidence at Rayleigh cutoff wavelength, the diffraction wave induced by the array can propagate along the metasurface and interact with surrounding SRR structures [35,37]. When the localized plasmon resonance wavelength of an individual SRR approaches Rayleigh cutoff wavelength, the effective energy transfer can occur from the incident beam into localized plasmon modes near the Wood anomaly. Consequently, the sharp plasmon resonance is induced [35,37]. In our work, under the premise that the localized surface plasmon frequency of each SRR is close to the Wood-Rayleigh anomaly frequency, the propagation directions of the diffraction waves from an individual SRR and its adjacent SRRs are always opposite in the surface. This leads to reconstructive coherent coupling effect. Based on the above theoretical analysis, it can be known that each SRR and the neighboring SRRs can produce a reconstructive coherent coupling mechanism, and almost all electric and magnetic dipole radiation components of current elements induced in the system can be completely eliminated in spatial directions through destructive interference. Thus, the radiation loss is greatly suppressed and the photon energy is strongly confined into the metastructures, which ultimately produces reconstructive coherent mode resonance with an extremely high Q-factor over the order of magnitude of 10⁴ and a large modulation depth.

Furthermore, the influences of key geometric parameters and the refractive index n_s of substrate on the tunability of reconstructive coherent mode are discussed. Transmission responses with varying the backbone a, the arm b, the split gap g, the thickness d of the SRR and the refractive index n_s of substrate are shown in Fig. 5. Table 1 summarizes the variation of Q-factor (in units of 10⁴) with the change of geometric parameters and the refractive index n_s of substrate. As shown in Fig. 5(a), with the increase of the backbone size a, the transmission spectra exhibits a distinct redshift, accompanied by a broadening resonance and decreasing transmittance. From Table 1, the corresponding Q-factor gradually decreases due to the increase of radiation damping caused by the size mismatch of reconstructive coherent mode, but still remains at high level. Interestingly, the Q value reaches about 7.939×10⁴ when a is 45 µm. When the arm b gradually increases, as shown in Fig. 5(b), the redshift behavior also appears, and the resonance linewidth gradually reduces and the resonance peak gradually decreases. From Table 1, the Q value is rapidly enhanced to be 5.218×10⁴ when b is 25 µm.

 

Fig. 5. (a)–(e) The change of transmission spectra with varying geometric parameters a, b, d, g and the refractive index n_s of substrate.

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Table 1. The Variation of Q (in units of 10⁴) with Geometric Parameters (µm) and the Refractive Index n_s of Substrate.

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As shown in Fig. 5(c), with the increases of d, the resonant frequency also shows a redshift and the transmittance rises slightly, maintaining extremely high Q-factor. The increase of Q-factor is related to the increase of photon energy confined on the metasurface. As shown in Fig. 5(d), the resonance does not occur when the split gap g is zero. The resonance response appears and a blue shift is observed with increasing the gap g. The transmission peak gradually decreases. Obviously, the split gap is critical for the generation of reconstructive coherent mode. The Q-factor decreases first and then increases from Table 1, which may be related to the change of gap capacitance. As the refractive index n_s gradually increases, as seen in Fig. 5(e), the resonance peak exhibits a redshift, and the value of the peak decreases slightly. The increase of Q-factor is due to the strong diffraction coupling induced by the decreasing refractive index, suppressing the radiation losses. The results suggest that extremely high Q-factor of the reconstructive coherent mode can be manipulated by adjusting the geometric parameters of SRR, which has good tunability. The manipulating of the sharp resonances by adjusting the geometry parameters can be attributed to the variations of the localized dipole resonance and the Rayleigh anomaly [35,37].

4. Conclusions

In conclusion, we theoretically demonstrated a metasurface scheme to produce reconstructive coherent mode with extremely high Q-factor and large modulation depth by utilizing a periodic array consisting of single vertically symmetric SRR. The reconstructive coherent coupling interaction of each SRR and the adjacent SRRs induces the radiation field to undergo destructive interference, which greatly suppresses radiation losses, and makes the Q-factor exceed 10⁴ with large modulation depth at THz regime. The simulation results are analyzed by normalized Fano profile theory and it fitted well with the theory. The Q-factor exhibits an interesting exponential growth with the increase of array periodicity. In addition, the Q-factor can be manipulated by varying the geometric parameters and the refractive index of the substrate, remaining high level throughout. Such approach of metasurface design and reconstructive interference mechanism can greatly enhance the Q-factor of THz resonance. It is as a reliable technique to optimize metamaterial designs for suppression of radiative losses and to produce ultrahigh Q-factor resonance for practical THz applications, such as ultra-sensitive sensors, ultra-narrow-band filters, low threshold lasing spasers and strong field-matter interaction applications.