Figure 2 (a) − 2 (f) shows the transmission spectra of the split-ring metamaterial structures with different periods of 35, 40, 45, 50, 55, and 60 µm, respectively. As shown in Fig. 2 (a) and 2 (b), when the lattice period varies between 35 and 40 µm, the resonance valley at low frequency is a surface plasmon mode with the resonance bandwidth being wide and the resonance frequency being maintained at about 1.8 THz, while the resonance valleys at high frequencies are lattice modes. When the lattice period changes at 45–50 µm, all resonance frequencies show a red shift, and the lattice mode gradually approaches the surface plasmon mode, and the two modes are coupled with each other, resulting in energy transfer. At this time, the bandwidth of the low-frequency resonance valley is narrowed, and the lattice-induced transparency peaks with narrow bandwidths are generated, as shown in Figs. 2 (c) and 2 (d). The coupling works best when the period P = 45 µm. When the lattice period changes between 55 and 60 µm, with the continuous increase of the period, the lattice mode moves to the low frequency position, and the interaction between the lattice mode and the surface plasmon mode gradually weakens, as shown in Fig. 2(e) and 2 (f).

With the increase of lattice period, the lattice modes of split-ring metamaterials gradually approach the surface plasmon modes and are coupled with each other, which leads to significant changes in the width and resonance strength of the resonance lines at low frequency, high frequency and lattice induced transparency peaks. Figure 3 shows the Q-factor and factor of merit (FOM) of low frequency and high frequency resonant valleys of split-ring metamaterials with respect to lattice periods along *x*-axis, Px. Q-factor is a good parameter to evaluate the resonant quality, which can be defined as: Q = fres/FWHM, where fres is the resonant frequency, FWHM is the full width at half maximum. As the lattice period increases from 35 µm to 60 µm, the lattice mode gradually moves to low frequency, the resonance valley bandwidth at low frequency gradually narrows, the loss becomes smaller, and the Q-factor increases gradually from 8 to 110, as shown in Fig. 3(a). The growth of the resonant valley’s Q-factor at low frequencies is due to the gradual increase in the coupling of lattice modes to surface plasmon modes, which confines the electromagnetic energy in the metamaterial array. However, at high frequencies, the resonance valley bandwidth gradually widens, and the Q-factor decreases from 79 to 22, as shown in Fig. 3(b). The FOM is utilized to quantize the trade-off between Q-factor and resonant strength, which has been defined as: FOM = Q×∆T, where ∆T is the amplitude change of resonant transmission dip. Figure 3(c) and 3(d) show the FOM of resonance at low and high frequencies, which reach a maximum value of 57 and 60, respectively. Since the resonance amplitude does not change much with the change of period, the maximum value of FOM is relatively large.

To gain more insight the coupling mechanism between lattice modes and surface plasmon modes, the 2D field plots are shown in Fig.4, which displays the surface current density for the split-ring metamaterial structure. When P=40 µm, the lattice mode is far away from the surface plasmon mode, and there is no significant coupling between the two. Figure4(a) and 4(d) show the resonance spectrum of the split-ring metamaterial structure. The low- and high-frequency surface currents flow in the same direction, and the modes at both low and high frequencies are simple dipole resonance modes. With the increase of the period, the lattice mode gradually approaches the surface plasmon mode, as shown in Fig.4(b). When P=45 µm, the convection in the opposite direction appears in the surface current map at low frequency. This implies that higher-order quadrupole modes appear at low frequencies, which derive from the strong coupling between the lattice mode and the surface plasmon mode. In this case, the electric field is strongly confined in the metamaterial owing to the generation of quadrupole modes, which results in the higher Q-factor of the resonant spectral lines at low frequencies. As shown in Fig.4(e), the mode at high frequency is still a dipole mode, and the dipole mode acting as the bright mode that interacts with the quadrupole mode (dark mode), which produces lattice-induced transparency. Figures4(c) and (f) show surface currents at low and high frequencies when P=60 µm. Convection occurs at low frequencies, which proves the existence of a quadrupole mode. At this time, the highest Q-factor of 110 is achieved. The current at the high frequency is unidirectional, manifestation of the dipole mode in nature.

Figure 5. The cross-free diagram of the resonance spectrum of the split-ring metamaterial structure affected by the period, and the period changes are 35, 40, 45, 50 and 55 µm. Anti-crossing behavior is seen at the intersection of the dipole mode (solid blue line) at 1.8 THz and the lattice mode (dashed blue line) calculated from the Eq. (1).

To further investigate the phenomenon of strong coupling and mode splitting induced by the first-order (0,1) lattice mode, P is increased from 35 µm to 50 µm, it is clear that period-dependent lattice mode and bright dipole mode cross around 1.8 THz. The lattice modes cause the coupling of the bright and dark modes, bringing about the observed anti-crossing. Therefore, such anti-crossing rows are characteristic of strongly coupled systems, and as the period increases, this mode splitting will eventually disappear.

Figure 6. (a)-(e) Simulated transmission amplitudes of double-split-ring metamaterial arrays with different lattice periods P = 50, 55, 60, 65 and 70 µm, and the microstructural parameters are consistent with those in Fig. 2. (f) The cross-free diagram of the resonance spectrum of the split-ring metamaterial structure affected by the period. Anti-crossing behavior can be seen at the intersection of the dipole mode at 1.52 THz (blue dashed line) and the lattice mode (red dashed line) calculated from the Eq. 1.

We also propose a double-slit ring metamaterial structure, and the transmission amplitude of that is shown in Fig. 6. When the lattice period varies from 50 to 70 µm, the lattice mode gradually approaches the surface plasmon mode, and the interaction between them gradually becomes stronger. Meanwhile, the low frequency resonance valley becomes sharper. Figure 6(f) shows the cross-free diagram of the resonance spectrum of the double-slit-ring metamaterial structure varying with period. Anti-crossing behavior can be seen at the intersection of the dipole mode (blue dashed line) and the lattice mode (red dashed line) at 1.52 THz, which demonstrates that the induced transparency stems from the coupling of lattice modes and eigenmodes of the metamaterial structure.

We obtained an asymmetric double split-ring structure by translating the microstructure on the basis of the double slit-ring microstructure. By translating the two split ring structures in reverse, the asymmetry b = a/L is introduced, where a is the center distance of the translated microstructure (a = 4 µm), and L is the side length of the microstructure. In this case, the symmetry of the microstructure is broken, and the Fano resonance (asymmetric model) is excited. The interaction between the eigenmodes and lattice modes of metamaterial structures leads to the emergence of resonance lines, while asymmetric modes also involved, as is shown in Fig. 7. Here, we set lattice period P to be 50, 55, 60, 65, 70 and 75 µm, respectively. As shown in Fig. 7(a), when the lattice period P = 50 µm and the asymmetry b = 0 (black line), there is only mutual coupling between the bright eigenmode and the lattice mode (dark mode), therefore only two transmission valleys can be observed. By breaking the symmetry of the microstrcutre with asymmetry parameter of b = 0.2 (red line), the Fano resonance can be excited. At this time, the eigenmodes, lattice modes and asymmetric modes of the metamaterial structure are coupled with each other, resulting in three transmission valleys, as shown in Fig. 7(a). With the increase of the period, when P = 55 µm, the lattice mode approaches Fano resonance. At this condition, the high-frequency resonance valley and the middle-frequency resonance valley generated by Fano resonance are almost merged together, with only a tiny split, as shown in Fig. 7(b). When the period is tuned in the range of 60–75 µm, the resonances regulated by the lattice modes gradually shift to low frequencies and interact with the resonances caused by the Fano resonance, resulting in splitting at high frequencies. Due to the energy transfer between the different modes, the resonance valley becomes sharp and the amplitude becomes large, as shown in Fig. 7(c)-7(f). Lastly, when P = 70 µm, the resonance spectral lines produced by the collective effect of metamaterial lattice modes, eigenmodes and Fano resonances are the sharpest and the narrowest.

In order to theoretically study the strong coupling between the eigenmodes, lattice modes and asymmetric modes, the coupled three harmonic oscillators are used to fit the period dependent transmission, the eigenmode is represented by the harmonic oscillator b, the lattice modes and the asymmetric modes are represented by harmonic oscillators d1, d2, respectively. The charges ab, ad1 and ad2 in the harmonic oscillator satisfy the following coupled equations [31, 32]:

$$\left\{\begin{array}{c}{\ddot{a}}_{b}\left(t\right)+{\gamma }_{b}{\dot{a}}_{b}\left(t\right)+{{\omega }^{2}a}_{b}\left(t\right)+{k}_{d1}{\dot{a}}_{d1}\left(t\right)+{k}_{d2}{\dot{a}}_{d2}\left(t\right)={\text{g}}_{0}E\left(t\right) \\ {\ddot{a}}_{d1}\left(t\right)+{\gamma }_{d1}{\dot{a}}_{d1}\left(t\right)+{{\omega }^{2}a}_{d1}\left(t\right)-{k}_{d1}{\dot{a}}_{b}\left(t\right)=0 \\ {\ddot{a}}_{d2}\left(t\right)+{\gamma }_{d2}{\dot{a}}_{d2}\left(t\right)+{{\omega }^{2}a}_{d2}\left(t\right)-{k}_{d2}{\dot{a}}_{b}\left(t\right)=0 \end{array} \right.$$

2

By solving the Eq. (2) with the approximation ωb ≈ ωd1 ≈ ωd2 ≈ ω, ω2-\({{\omega }}_{\text{b}}^{2}\)≈(ω-ωb)\(\bullet\)2ω and \(\text{g}\)=g0/(2ω), the energy dissipation of the proposed system is obtained as:

\(\text{p}\left({\omega }\right)\propto \frac{1}{\text{A}}\) (\({{\omega }}_{\text{d}1}-\omega -j\frac{{\gamma }_{d1}}{2}\))(\({{\omega }}_{\text{d}2}-\omega -j\frac{{\gamma }_{d2}}{2}\)) (3)

A is given by:

\(\text{A}=\) (\({{\omega }}_{\text{d}1}-\omega -j\frac{{\gamma }_{d1}}{2}\))(\({{\omega }}_{b}-\omega -j\frac{{\gamma }_{b}}{2}\))(\({{\omega }}_{\text{d}2}-\omega -j\frac{{\gamma }_{d2}}{2}\)) (4)

\(-\frac{{\text{k}}_{\text{d}1}^{2}}{4}\) (\({{\omega }}_{\text{d}2}-\omega -j\frac{{\gamma }_{d2}}{2}\))\(-\frac{{\text{k}}_{\text{d}2}^{2}}{4}\)(\({{\omega }}_{\text{d}1}-\omega -j\frac{{\gamma }_{d1}}{2}\))

Where (ωd1, γd1), (ωd2, γd2) and (ωb, γb) are the resonance frequencies and damping factors of oscillators d1, d2 and b, respectively. kd1 and kd2 are the coupling coefficients between oscillators b and d1, and b and d2, respectively. By varying the actual parameters of the lattice mode, we plotted the transmission spectrum, as shown in Fig. 8. It is worth mentioning that the theoretical model does not consider the near-field effect due to the inter-unit cell coupling of different periods. Therefore, there are slight differences in the linewidths of the simulated results and the theoretical curves, but in other respects, they show good agreement.

Figure 9. The variations of the fitting parameters, γd1, γb, γd2, kd1 and kd2 with respect to lattice period used in the three-oscillator model for the fitting transmission spectra in Fig. 8.

Figure 9 shows the fitting parameter values that vary with the lattice period during the calculation process based on the Eq. 5. During the dynamic modulation process, with the increase of the lattice period, the damping coefficients γd1 and γd2 remain basically unchanged, and the coupling coefficients γb and kd2 has a significant decreasing trend. The dependence between the fitting parameters and the lattice period shows that the damping and coupling coefficients play a decisive role in the dynamic modulation of the resonance spectrum of the microstructure.