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Abstract
In this paper, we propose what we believe to be a novel coupling mechanism for generating quasi-bound states in the continuum (quasi-BIC) in symmetrical metasurface structures. We demonstrate for the first time in theoretical predictions that supercell coupling can induce quasi-BIC(s). We utilize the coupled mode theory (CMT) to analyze the physical mechanism for the generation of quasi-bound states in such symmetrical structures, which result from our investigation of the coupling between sub-cells that are separated from supercells. We verify our theory by using both full-wave simulations and experiments.
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1. Introduction
Bound states in the continuum (BICs) are waves that exist in the continuum radiation spectrum but remain localized [1,2]. Although BICs are first proposed in quantum mechanics [3,4], it is a general wave phenomenon that has been studied in recent years in electromagnetic waves [5], acoustic waves in air [6], and elastic waves in solids [7]. The concept of BIC has been introduced into metamaterial systems and it has generated significant interest among researchers. Due to its high-quality factor characteristics, it is widely used in sensing [8,9], active modulation [10], topological optics [11–13], nonlinear optics [14–16] and other fields.
There are many ways to generate quasi-BIC, such as symmetrical breaking or anisotropic [17–19], modes coupled [20], and inverse construction [21]. Studies have shown that fully symmetric structures can also generate BICs or quasi-BIC due to the coupling between different modes [22–24]. Recently, there are various theories to explain BIC in photonic systems, such as coupled mode theory (CMT) [25–27], multipole theory [28], Fano resonance theory [29–31], and topology temporal [12,32].
The coupled-mode theory is initially used for the study of coupling between two or more electromagnetic wave modes, often used in the study of waveguide mode coupling [33–35]. It is later introduced into the study of coupling between metasurfaces. A novel study has found that if two structures in a unit cell have the same resonance frequency and phase, the coupling between them can produce a BIC. In contrast, if two structures have different resonance frequencies and phases, their coupling acquire a quasi-BIC [25]. Another remarkable work proves theoretically and experimentally that when the frequency and phase of the transmission spectrum of a single structure are similar with each other, a metasurface unit cell composed of two completely different structures can also produce a BIC or quasi-BIC resonance [27].
A unit cell containing two sub-structures can be regarded as a supercell, where coupling between identical sub-structures can generate a BIC, and coupling between different sub-structures can generate a quasi-BIC. Additionally, each sub-structure can be perceived as a sub-cell. It remains us that a supercell containing multiple identical single structures can be decomposed into sub-cells with the same number of identical single structures to produce a BIC, or divided into sub-cells with different numbers of identical single structures to generate a quasi-BIC. Based on the coupled-mode theory, we propose a mechanism for generating quasi-BIC induced by supercells coupling. A supercell structure can be split into asymmetric sub-cell structures, and the quasi-BIC peaks of supercells can be predicted by the coupling between sub-cell structures.
To demonstrate our theory, we employ two different supercell structures, cut wires (CWs) and split ring resonators (SRRs). As shown in Fig. 1, we demonstrate a periodic structure composed of supercells containing three identical metal CWs. The terahertz beam is normally incident along the z-direction above the metasurface, and the transmission spectrum can be obtained after the terahertz wave passes through the metasurface structure. Due to coupling, three identical CWs or SRRs in the supercells can produce one quasi-BIC peak. It can also be analyzed by another mechanism, for example, in a system of three metal CWs, the middle metal CW produces a current that is opposite to that of the two metal CWs on both sides, resulting in two opposite ring dipoles, which can be used Multipole scattering theory explains the generation of the quasi-BIC phenomenon [36,37].
By changing the number of structures in the supercell, we find that four identical structures in one supercell can produce one quasi-BIC peak, due to the coupling effect. In comparison, due to the coupling effect, five identical structures in one supercell can produce two quasi-BIC peaks. The generation of quasi-BICs from a supercell composed of different structures can be explained by both coupled mode theory and breaking the symmetry of the structure. Moreover, the supercell structures composed of three, four, or five identical structures are completely symmetrical, and the quasi-BICs generated by these structures can be explained by coupled mode theory, but not by breaking the symmetry of the structure.
2. Theory
We propose a universal coupled theory to describe the quasi-BICs in supercells of metasurfaces. When there is coupling between the structures in the supercell, it can be analyzed by the coupled mode theory. We find that by splitting the supercell contains more than three identical structures into a combination of two different sub-cells, and quasi-BICs can be generated through the coupling between the sub-cells. The phenomenon of quasi-BICs generated by completely symmetrical structures can be explained by coupled mode theory. The physical parameters in the coupling theory are obtained from the basic physical parameters of the sub-cell structures, such as the resonant frequency $\omega _A$, $\omega _B$, the losses $\gamma _A$, $\gamma _B$, the phases $\phi _A$, $\phi _B$, and the coupling strength $g$ between the sub-cell structures.
We take the $|a|^{2}$ and $|b|^{2}$ as the energies in each sub-cell structure. According to the coupled mode theory, we obtain our universal coupled theory by adding the phase difference between sub-cells. The equation is written as follows,
By solving Eq. (1), we can obtain expressions for the energy amplitudes of each sub-cell $a$, $b$ as follows.
The effective electric susceptibility is the linear superposition of the energy amplitudes $|a|^{2}$ and $|b|^{2}$, which can be written as [25]
Finally, we obtain the transmission spectrum with the expression $T=1-Im(\chi _\text {eff})$ [38], as follows,
For better demonstration, we take examples of supercells containing different numbers of identical metal cut wires, as shown in Fig. 2. We denote the split sub-structures as A and B. When the supercell contains three identical single structures, there is only one way to split the supercell into two different sub-cells, as shown in Fig. 2(a). Therefore, it should have only one quasi-BIC peak. The array with three CWs in the supercell can be divided into a sub-cell with one CW and another with two CWs. Moreover, the sub-cell with two CWs can also be split into two sub-cells with one CW. However, it becomes the three identical sub-cells with a single CW, the sub-cells with identical CW have the same resonant frequency, loss, and phase to obtain the perfect BIC which is infinite high Q Fano resonance. Therefore, we can not measure this infinite high Q Fano resonance in simulation and experiments. Thus, splitting three identical CWs into three sub-cells is pointless. Furthermore, based on our idea, the array with "n" CWs in the unit cell can be divided into "n" sub-cells with one CW. However, in coupled-mode theory, the coupling strength between sub-cells is a more critical physical parameter. Therefore, it is impossible to develop a coupled theory for "n" particles only as a function of the resonant frequency, loss, and phase of the sub-cell with only one CW. When the supercell contains four identical single structures, there are two ways to split the supercell into two sub-cells. When this supercell is divided into two identical sub-cells, a perfect BIC can be generated instead of a quasi-BIC. Therefore, a supercell composed of four identical single structures can only be divided into sub-cells comprising one CW and three identical CWs, as shown in Fig. 2(b), and it should also have one quasi-BIC peak. A supercell consisting of five identical single structure sub-cells has two split methods, which can be split into a single CW and four CWs sub-cells or can be split into two CWs and three CWs sub-cells shown in Fig. 2(c) and Fig. 2(d). According to the theory, when there are n identical structures in a supercell, (n-1)/2 quasi-BIC peaks can be generated when n is odd, and (n-2)/2 quasi-BIC peaks can be generated when n is even.
Fig. 2. The schematic figure of the supercell structures contains different numbers of identical metal CWs. (a) The supercell structure contains three identical metal CWs. Sub-cell A contains one metal CW. Sub-cell B contains two identical metal CWs. (b) The supercell structure contains four identical metal CWs. Sub-cell A contains one metal CW. Sub-cell B contains three identical metal CWs. (c) The schematic figure of the supercell structure contains five identical metal CWs. Sub-cell $A_1$ contains one metal CW. Sub-cell $B_1$ contains four identical metal CWs. (d) The schematic figure of the supercell structure contains five identical metal CWs. Sub-cell $A_2$ contains two metal CWs. Sub-cell $B_2$ contains three identical metal CWs.
Although this method can be extended to a supercell consisting of n single structures, as the number of single structures increases, the distance between sub-cells increases and the coupling between sub-cells becomes weaker. Therefore, the quasi-BIC peaks generated by the coupling between sub-cells are weakened and cannot be observed. We only discuss the case of $n\leq 5$ in this paper. Furthermore, when n is less than 3, the supercell cannot be split into two sub-cells with different numbers of single structure, so the case where n is less than 3 is not considered.
3. Full-wave simulations
We use the CST Microwave Studio to obtain the transmission spectra of the supercell structures in simulations. The solver we use in the simulations is a frequency-domain solver. Unit cell boundary conditions are set in both $x$ and $y$ directions, the open boundary condition is set in $Z_{min}$ (incidence direction), and the open (add space) boundary condition is set in $Z_{max}$ (exit direction). The number of modes used in the simulation is 60. The polarization direction of the incident terahertz wave is $y$ polarization. The meshes we use in simulations are tetrahedral, and the meshes are refined for more accurate results. The mesh settings for the model and background were 15 cells and 1 cell per max model box edge, respectively.
Based on our theoretical predictions, we demonstrate the coupling mechanism of supercells consisting of three, four, and five identical metal CWs, respectively. As shown in Fig. 2(a), the supercell contains three identical metal CWs. The period in the $x$ direction is $P_x=250\mu m$, and the period in the $y$ direction is $P_y=200\mu m$. The lengths of the metal cut wires are $L=180\mu m$, the width $w=30\mu m$, and the distance between the metal CWs are $d=20\mu m$. In the cases of supercells containing four, or five identical CWs, as shown in Fig. 2(b), and Fig. 2(c), the geometrical parameters of CWs are the same as the previous case, except that $P_x=310\mu m$ and $P_x=380\mu m$ respectively. When the supercell contains three CWs, the distance between the closest CWs in adjacent supercells is 120$\mu m$, while in supercells containing four and five CWs, the distance between the closest CWs in adjacent supercells are 130$\mu m$ and 150$\mu m$, respectively. To ensure that the coupling within the supercell is dominant, the minimum distances between CWs in adjacent supercells are greater than the distances between adjacent CWs within a supercell.
Based on our theory, the quasi-BIC can come from the coupling of sub-cells. The supercell, which contains three identical CWs only can deconstruct as sub-cells A and B. We use the full-wave simulations of sub-cells A and B to obtain the resonant frequency ( $\omega _{A}$, $\omega _{B}$), the loss ($\gamma _A$, $\gamma _B$) and phase with each resonant frequency ($\phi _A$, $\phi _B$) for each sub-cell structures, as shown in Fig. 3(a) and Fig. 3(b). The transmission spectrum of the supercell composed of three metal CWs is shown in Fig. 3(c), in which a quasi-BIC peak appears near the frequency of 0.46 THz. According to the calculation, the positions of the lower-order lattice modes are marked in Fig. 3(c), and the higher-order ones are located in the higher frequency band, all to the right of the eigen-peak. The quasi-BIC spectra obtained from the coupling theory of sub-cell and the quasi-BIC spectra obtained from the electromagnetic simulation are shown in Fig. 3(d), where the red dashed line indicates the theoretical results and the solid black line indicates the simulation results.
Fig. 3. (a) The transmission spectrum and phase of sub-cell A. (b) The transmission spectrum and phase of sub-cell B. (c) The transmission spectrum of the three CWs supercell structure. (The frequencies of the (1,0), (0,1), and (1,1) order lattice modes are marked in the figure). (d) The simulation and theory transmission spectrum of BIC in three CWs supercell structure.
From the simulation results of three CWs, we can obtain the resonance frequency $\omega _{A}$= 0.464 THz, $\omega _{B}$= 0.490 THz, the loss $\gamma _A$= 0.091 THz, $\gamma _B$= 0.131 THz and phase with each resonance frequency $\phi _A$=0.519 $\text {rad}$, $\phi _B$= 0.701 $\text {rad}$ for the sub-cells contain one and two identical structures. Therefore, $\omega _{a}$=0.464 THz, $\omega _{b}$=0.490 THz, $\gamma _a$= 0.091 THz, $\gamma _b$= -0.02 THz and $\phi$= -0.182 $\text {rad}$. The corresponding quasi-BIC transmission spectrum in theory can be obtained by substituting these parameters (frequencies, losses, and phases) into the coupled-mode equation (Eq. (1)) and then fitting Eq. (5) (see the red dashed line of Fig. 3(d)). When the similarity of the frequencies and phases between the two sub-cells is higher, the Q factor of the quasi-BIC peak generated by the coupling between the sub-cells is larger. To verify this theory (see Fig. 9 in Sec. 5 for the related demonstration) and compare the Q factors of quasi-BIC peaks generated by different supercells, we calculated the Q factor of the quasi-BIC peaks. From the result of quasi-BIC (as shown in the black line of Fig. 3(d)), we observe the quasi-BIC phenomenon with Fano resonance transmission spectrum with $Q_\text {1} = w_{BIC}/\Delta w$ = 24.30, where $\Delta w$ is the full width at half maximum of Fano resonance.
When the supercell contains four identical CWs there is only one method to split it into sub-cell A and sub-cell B. We use the full-wave simulations of sub-cells A and B to obtain the resonant frequency ( $\omega _{A}$, $\omega _{B}$), the loss ($\gamma _A$, $\gamma _B$) and phase with each resonant frequency ($\phi _A$, $\phi _B$) for each sub-cell structure, as shown in Fig. 4(a) and Fig. 4(b). The transmission spectrum of the full structure as shown in Fig. 4(c), and the fundamental lattice modes are marked on it.
Fig. 4. (a) The transmission spectrum and phase of sub-cell A. (b) The transmission spectrum and phase of sub-cell B. (c) The transmission spectrum of four CWs supercell structures. (The frequencies of the (1,0),(0,1) order lattice modes are marked in the figure). (d) The simulation and theory transmission spectrum of quasi-BIC in four CWs supercell structure.
Similarly, for the quasi-BIC generated by a supercell composed of four identical single structures shown in Fig. 4, we can obtain the resonance frequency $\omega _{A}$= 0.426 THz, $\omega _{B}$= 0.492 THz, the loss $\gamma _A$= 0.057 THz, $\gamma _B$= 0.053 THz and phase with each resonance frequency $\phi _A$=-0.438 $\text {rad}$, $\phi _B$= 0.198 $\text {rad}$ for each sub-supercell structure by simulation. Therefore, $\omega _{a}$=0.464 THz, $\omega _{b}$=0.490 THz, $\gamma _a$= 0.057 THz, $\gamma _b$= 0.002 THz and $\phi$= -0.681 $\text {rad}$. The Q factor of the quasi-BIC is $Q_\text {2}$ = 16.62. The corresponding quasi-BIC transmission spectrum can be obtained by substituting these parameters into the coupled-mode equation (Eq. (1)) and fitting Eq. (5) (see the red dashed line of Fig. 4(d)). Compared to the $Q_\text {1}$, we find that $Q_\text {1}>Q_\text {2}$. The resonant frequencies and phases of sub-cells A and B, which are divided by the supercell containing three identical structures, are more similar to each other than those of sub-cells A and B, which are divided by the supercell containing four identical structures. Therefore, the Q factor of the quasi-BIC generated by the supercell containing three identical structures is greater than the Q factor of the quasi-BIC generated by the supercell containing four identical structures. This result is consistent with the theory that the Q factor of quasi-BIC transmission spectrum increases as the difference between two structures within a supercell decreases.
When a supercell contains five identical CWs, there are two methods to split it into sub-cells. The sub-cell containing one CW obtained by the first splitting method is denoted as $A_\text {1}$, while the sub-cell containing four CWs is denoted as $B_\text {1}$. Similarly, the sub-cell containing two CWs obtained by the second splitting method is represented by $A_\text {2}$, while the sub-cell containing three CWs is represented by $B_\text {2}$. According to theory, this type of supercell structure can generate two quasi-BIC peaks. In analogy to the above methods, we can analyze the phenomenon of two quasi-BIC peaks in the transmission spectrum of a supercell containing five identical structures. For the first quasi-BIC peak in Fig. 5(e), we can obtain the resonance frequency $\omega _{A1}$= 0.384 THz, $\omega _{B1}$= 0.476 THz, the loss $\gamma _{A1}$= 0.028 THz, $\gamma _{B1}$= 0.199 THz and phase with each resonance frequency $\phi _{A1}$= 0.413 $\text {rad}$, $\phi _{B1}$= -0.394 $\text {rad}$ for each sub-cell structure as shown in Fig. 5(a) and (b) by simulating the sub-cells $A_\text {1}$ and $B_\text {1}$. Therefore, $\omega _{a}$=0.384 THz, $\omega _{b}$=0.476 THz, $\gamma _a$= 0.028 THz, $\gamma _b$= -0.086 THz and $\phi$= 0.807 $\text {rad}$. The Q factor of the first quasi-BIC is $Q_\text {3}$ = 13.39.
Fig. 5. (a) The transmission spectrum and phase of sub-cell $A_\text {1}$. (b) The transmission spectrum and phase of sub-cell $B_\text {1}$. (c) The transmission spectrum and phase of sub-cell $A_\text {2}$. (d) The transmission spectrum and phase of sub-cell $B_\text {2}$. (e) The transmission spectrum of five CWs supercell structure.(The frequencies of the (1,0),(0,1) order lattice modes are marked in the figure). The simulation and theory transmission spectra of two quasi-BIC peaks in five CWs supercell structures(insert figures in (e)).
For the second quasi-BIC peak in Fig. 5(e), according to the simulation results of the sub-cells $A_\text {2}$ and $B_\text {2}$, we can obtain the resonance frequency $\omega _{A2}$= 0.379 THz, $\omega _{B2}$= 0.471 THz, the loss $\gamma _{A2}$= 0.049 THz, $\gamma _{B2}$= 0.021 THz and phase with each resonance frequency $\phi _{A2}$= -0.931 $\text {rad}$, $\phi _{B2}$= 0.146 $\text {rad}$ for each sub-cell structure as shown in Fig. 5(c) and (d). Therefore, $\omega _{a}$=0.379 THz, $\omega _{b}$=0.471 THz, $\gamma _a$= 0.049 THz, $\gamma _b$= 0.014 THz and $\phi$= -1.077 $\text {rad}$. The Q factor of the second quasi-BIC is $Q_\text {4}$ = 52.87.
According to the calculate, we find that $Q_\text {4}>Q_\text {3}$. The first quasi-BIC peak with $Q_\text {3}$ = 13.39 is generated by the coupling between sub-cell $A_\text {1}$ (containing one CW) and sub-cell $B_\text {1}$ (containing four CWs). Similarly, the second quasi-BIC peak with $Q_\text {4}$ = 52.87 is generated by the coupling between sub-cell $A_\text {2}$ (containing two CWs) and sub-cell $B_\text {2}$ (containing three CWs). The resonant frequencies and phases of sub-cells $A_\text {2}$ and $B_\text {2}$, which are divided by the supercell containing five identical structures, are more similar to each other than those of sub-cells $A_\text {1}$ and $B_\text {1}$, which are splitted by the other splitting method. Therefore, the result that $Q_\text {4}>Q_\text {3}$ is consistent with the theory.
In order to demonstrate the universality of our theory, we demonstrate the SRRs coupling to construct quasi-BICs. As shown in Fig. 6(a), the supercell, which is composed of three identical single SRRs, can also generate quasi-BIC peak. A supercell composed of four identical single SRRs can also produce quasi-BIC, as shown in Fig. 6(b). In Fig. 6(c), the supercell consisting of five identical single SRRs can also produce two quasi-BIC peaks. The quasi-BIC phenomenon produced by supercells composed of different numbers of identical SRRs is similar to that produced by metal cut wires. The quasi-BIC peaks are produced by the coupling between sub-cells composed of different numbers of single structures. Through both theoretical and simulation approaches, we validate that completely symmetrical structures can generate quasi-BICs.
Fig. 6. (a)The transmission spectrum of the three SRRs supercell structure. (The frequencies of the (1,0) lattice modes are marked in the figure). (b) The transmission spectrum of the four SRRs supercell structure. (The frequencies of the (1,0) and (2,0) lattice modes are marked in the figure). (c) The transmission spectrum of the five SRRs supercell structure. (The frequencies of the (1,0) and (2,0) lattice modes are marked in the figure). The schematic figure of the supercell structures contains different numbers of identical SRRs. (d) The supercell structure contains three identical SRRs. (e) The supercell structure contains four identical SRRs. (f) The supercell structure contains five identical SRRs.
The transmission spectrum of the supercell contains three identical single SRRs in Fig. 7(a), the (1,0) lattice mode is marked in the figure and other lattice modes are beyond the range of our focus frequency. The quasi-BIC spectra obtained from the coupling theory of sub-cells and the electromagnetic simulation are shown in Fig. 7(b), where the red dashed line indicates the theoretical results and the solid black line indicates the simulation results. The generation of quasi-BIC here is due to the mutual coupling between ring dipoles in the SRRs. The geometric parameters corresponding to the structures in Fig. 6 are $S=60\mu m$, $S_1=25\mu m$, $S_2=8\mu m$, $S_3=17.5\mu m$, and $d=10\mu m$. The transverse periods($P_x$) of Fig. 6(a), (b) and (c) are 250$\mu m$, 400$\mu m$ and 420$\mu m$ respectively, and the longitudinal periods($P_y$) are all 80$\mu m$. When the supercells contain three, four, and five SRRs, the distances between the closest SRRs in adjacent supercells are 50$\mu m$, 130$\mu m$, and 80$\mu m$, respectively. It also ensures that the coupling between the SRRs in the supercell is greater than the coupling between the adjacent supercells, that is, the distance between the SRRs in the supercell is smaller than the minimum distance between the SRRs in the adjacent supercells.
Fig. 7. (a) The transmission spectrum of three SRRs supercell structure. (The frequency of the (1,0) order lattice modes is marked in the figure). (b) The simulation and theory transmission spectra of quasi-BIC in three SRRs supercell structures.
4. Experimental
According to the simulation results, we design and process the samples, and then test them with a terahertz time-domain spectroscopy system (THz-TDS). The samples are fabricated by standard lithography techniques. Firstly, a positive photoresist is spin-coated and patterned using standard photolithography on a 4-inch clean quartz wafer. Then, an aluminum film with a thickness of 200 $nm$ is deposited by magnetron sputtering and the photoresist is lifted off subsequently. The microscope images of the fabricated samples are presented in the middle part of Fig. 8.
Fig. 8. (a) The experimental and simulation transmission spectra of a supercell are composed of three metal CWs (left). The solid line shows the simulation results and the hollow circle shows the experimental results. Electron micrograph of the sample consisting of three metal CWs (middle). The schematic diagram of the structure correspondingly (right). (b) The experimental(solid line) and simulation(hollow circle) transmission spectra of a supercell composed of four metal CWs (left). The electron micrograph of the sample consists of four metal CWs(middle). The schematic diagram of the structure correspondingly (right). (c) The experimental (solid line) and simulation(hollow circle) transmission spectra of a supercell composed of five metal CWs(left). The electron micrograph of the sample consists of five metal CWs (middle). The schematic diagram of the structure correspondingly. (d)The experimental(solid line) and simulation(hollow circle) transmission spectra of a supercell consisting of three SRRs(left). The electron micrograph of three SRRs(middle). The schematic diagram of the structure correspondingly(right).
A standard THz-TDS is used to measure the transmission data of samples under normal incidence. The transmission spectra are defined as $T(\omega ) = E_\text {sam}(\omega )/E_\text {ref}(\omega )$, where $E_\text {sam}(\omega )$ and $E_\text {ref}(\omega )$ are the Fourier transforms of the time-domain signals transmit through the samples and the reference (bare quartz substrate), respectively. During the experiment, we keep the temperature at 25$^\circ C$, maintain the humidity in the testing environment below 10$\%$ (theoretically 0$\%$) by using a dryer, and set the pulse cutoff time to 40$\text {ps}$. The spectral resolution of the measurement is 25GHz by performing a 40$\text {ps}$ time-domain terahertz pulse scan.
The left part of Fig. 8(a)-(c) demonstrates the experimental and simulation results of transmission spectra of supercells containing different numbers of metal CWs, in which the solid lines represent the simulation results and the hollow circle represents the experimental results. The middle part of Fig. 8 shows the electron micrographs of the corresponding samples. The structural parameters of the samples are given in the simulation section and marked in the structural schematic diagram at the right part of Fig. 8.
Figure 8(a)-(c) shows that the experimental and simulation results are in good agreement. The experiment proves that completely symmetric structures can produce quasi-BICs. In the experiment, the thickness of the quartz substrate is 2 $mm$, and the permittivity of quartz is smaller compared to silicon or sapphire. When the THz time-domain spectroscopy system scans to approximately 30 $\text {ps}$, the secondary reflection peaks are reached. At this time, the signal passing through the sample is not completely attenuated, which causes a slight oscillation before and after the resonant peak in the transmission spectra. Due to the limitation of the resolution of the experimental system, the second quasi-BIC resonant peak with a higher Q factor (greater than 50) and smaller amplitude (less than 0.2) in Fig. 8(c) is not clearly displayed in the experimental transmission spectrum.
The experimental(solid line) and simulation(hollow circle) transmission spectra of a supercell composed of three SRRs are shown in the left part of Fig. 8(d). The electron micrograph and the structural schematic diagram of the sample composed of three SRRs are shown in the middle part and right part of Fig. 8(d), correspondingly. The fit between the experimental and simulation results of SRRs seems to be inferior to that of CWs which can be attributed to the following aspects. Firstly, the dimension of the SRRs is smaller and the structure is more complicated than CWs when considering the fabrication process. Due to the limitation of machining precision, there may be some discrepancies between the size and material loss of the device used in the experiment and the settings in the simulation. Moreover, the resolution of the terahertz measurements is 0.02ps in the time domain, and the spectral resolution of it is 25GHz. Furthermore, the Q factor of the three SRRs transmission spectrum is 35.77, which is larger than the Q factors of three CWs ($Q_\text {1}$=24.30), four CWs ($Q_\text {2}$=16.62), and five CWs ($Q_\text {3}$=13.39). Larger Q factors correspond to narrower resonant linewidths, which makes it more challenging to obtain accurate results. Therefore, due to the limitations of the measurement system and the existence of fabrication error, the intensity of the experimentally measured quasi-BIC resonant peak is weaker than that by simulation.
5. Discussion
In the aforementioned results, we observe the presence of Fano resonance peaks in the transmission spectra of supercells except for the quasi-BICs peaks we discuss. We analyze the physical mechanisms responsible for the generation of these peaks. It is found in previous studies that lattice modes can also induce BIC (or quasi-BIC) [39–41]. Lattice modes (sometimes called Rayleigh anomaly) originate from the Wood anomaly and exist in periodic metasurface structures. The frequency of lattice mode is defined as follows [42],
where $i$, $j$ are a pair of integers that define lattice mode orders, $P$ is the period of the structure, $\theta$ is the incident angle of terahertz waves, $\varepsilon _{e}$ is the effective permittivity of the metal-dielectric interface, $\varepsilon _{m}$ and $\varepsilon _{d}$ are the permittivities of the metal and substrate, respectively. In the terahertz band, metals can be regarded as perfect electrical conductors (PEC). That is, $\varepsilon _{m}$ is much larger than $\varepsilon _{d}$. Therefore, the effective permittivity is approximately equal to the permittivity of the substrate quartz ($\varepsilon _{e}$ $\approx$ $\varepsilon _{d}$=3.75). When the periods of the unit structure in the $x$ direction and the $y$ direction are different, the calculation method of lattice mode can refer to references [43,44].According to the theory of lattice modes, we calculate the frequencies of the lattice modes and label them in the transmission spectrum of each supercell structure, respectively. As shown in Fig. 3(c), we calculate the fundamental lattice mode (1,0),(0,1), and the higher-order lattice mode (1,1), which are annotated in the figure. Since the frequency of the higher-order lattice mode is beyond the frequency band of our interest, it will not be discussed. In Fig. 4(c), the lattice modes (0,1) and (1,0) are located on both sides of the eigen-peak. In Fig. 5(e), the lattice mode (0,1) is located between two quasi-BIC peaks, and the lattice mode (1,0) is located on the right side of the eigen peak. In Fig. 6(a), the lattice mode (1,0) is located between the quasi-BIC peaks and eigen peak, and other lattice modes are located away from both the quasi-BIC peak and eigen peak. In Fig. 6(b), the lattice mode (1,0) is located on the left side of the eigen peak, and the lattice mode (2,0) is located on the right side of the quasi-BIC peak. In Fig. 6(c), the lattice mode (1,0) is located on the left side of the eigen peak, and the lattice mode (2,0) is located on the right side of the second quasi-BIC peak. In Fig. 7(a), the lattice mode (1,0) is located between quasi-BIC peak and eigen peak.
In the previous sections, we verify that the quasi-BICs at these positions are generated by the coupling between the sub-cells split from the supercells. The calculated frequencies of the lattice modes precisely match the positions of the Fano resonances in transmission spectra, except for the quasi-BIC peaks discussed in this paper. Other unmarked lattice modes are not within the frequency range of our interest. Therefore, the others Fano resonances peaks in transmission spectra are generated by lattice modes, other than the coupling between sub-cells.
Although we find new phenomena and coupling mechanisms, there are still some shortcomings. For example, the Q factor of quasi-BIC cannot reach the same order of magnitude as the quasi-BIC generated by breaking the symmetry of the structure. The largest Q factor in the study is only 52.87. Firstly, due to the large intrinsic losses of metal, the Q factor of the quasi-BIC can generally only reach $10^{2}$ orders of magnitude with the loss [9,23]. Compared with the all-dielectric metasurface, Q factor of the quasi-BIC can exceed $10^{6}$ orders of magnitude [45,46]. Thus, if our structure is made of an all-dielectric metasurface, we believe that the Q factor of our quasi-BIC can much higher than our present work. Secondly, considering the limitation of the experimental system, the higher Q factor may not be observed in our current experiment. Thirdly, due to the limitation of our structures (for example, three identical CWs or SRRs), the resonant frequencies and phases of two sub-cells (one is the single CW or SRR and another one is two CWs or SRRs) can not be very close in every case. Therefore, it can not have a very high Q of quasi-BIC in our structure based on the theory. However, the idea of our paper is to demonstrate the quasi-BIC from the coupling of the sub-cells, which is not to obtain very high Q quasi-BIC for applications. We find a new physical mechanism of quasi-BIC based on the coupling of sub-cells in this paper.
In order to verify the theory that the differences between resonant frequencies negative correlation to the quality factor, we calculate the Q factor by varying the lengths of the CW in the middle (q) from 180$\mu m$ to 210$\mu m$ and maintaining the length of the CWs on two sides (L) as 230$\mu m$. The widths of the three CWs are 30$\mu m$, and the periods in the horizontal and vertical directions are both 280$\mu m$. The distance between CWs in one supercell is 20$\mu m$ while the distance between the closest CWs in adjacent supercells is 150$\mu m$. The CW with a fixed length on the left can be regarded as one sub-cell and the CW with variable length in the middle and fixed length on the right can be regarded as another sub-cell. According to the simulation, we find that when the resonant frequencies of two sub-cells split by a supercell are close, their phases are also very similar. Therefore, we only verify the relationship between the resonance frequency difference and the Q factor. We simulate the two sub-cells and the full structure separately to obtain the resonance frequencies of the sub-cells and the Q factor of the quasi-BIC peak in the transmission spectrum of the full structure. We define asymmetric parameters ($\Delta \omega$) by using the difference between the resonant frequencies of two sub-cells. By fitting the Q factor and the ($\Delta \omega$) between two sub-cells, we find that the Q factors are the inverse square dependence on frequency difference ($\Delta \omega$) as shown in Fig. 9. The theory mentioned in the paper that the smaller the difference in resonance frequency between the two sub-cells split by the supercell, the larger the Q factor of the quasi-BIC peak generated by the coupling between the sub-cells has been verified.
Symmetric periodic metasurface arrays can be regard as gratings [47]. Guided mode resonances (GMRs) arise from the coupling of diffracted waves in the grating to waveguide modes [48,49]. When the lattice of the grating is perturbed, the dispersion relation of the guided mode folds into a radiation continuum, which leads that the initial guided mode of the symmetrical array evolving to a leaky resonance. Thus, an asymmetrical Fano resonance peak occurs [50,51]. Therefore, the narrow resonance (Fano resonance peak) of standard CWs also can be explained by guided modes. In addition, this asymmetrical Fano resonance peak can be considered as the quasi-BIC [47,49].
However, the generation of this Fano peak can also be explained by the sub-cells coupling based on coupled-mode theory (CMT). The two sub-cells are excited by the external electric field with y-axis polarization, which can act as two bright modes. The radiation from two sub-cells is destructive interference in the far field. Therefore, the radiation losses of two sub-cells can largely decrease due to destructive interference, which brings the high Q Fano peak, so called quasi-BIC [25,27]. In this paper, we employ the CMT to explain this high Q Fano peak instead of guided modes. Furthermore, the most important is that when we slightly change the standard grating structure with maintaining the length of the CWs on both sides and varying the length of CW in the middle (we take the three CWs structures as examples, as shown in Fig. 9), this grating structure is no longer a standard grating structure. Based on the simulations, the non-standard grating structure also has the Fano peak. Thus, the standard guided mode theory does not work well in this case, while the coupling between sub-cells still works. This supercell can be separated into two sub-cells, as shown in the red box in Fig. 9. The coupling of two sub-cells can generate the quasi-BIC as well. Therefore, both CMT and guided mode can explain the quasi-BIC produced by periodically perturbed standard gratings, but the coupled-mode theory can also explain the quasi-BIC produced by non-standard gratings. Therefore, the CMT is a more universal theory to explain the generation of asymmetric Fano peaks.
In addition, we obtain the Q factors of the quasi-BIC of non-standard gratings structures with varying the difference resonant frequencies between two sub-cells ($\Delta \omega$), as shown in Fig. 9. The black balls represent the simulation results, and the red solid line represents the fitting result. It easily obtains that the Q factor is inversely proportional to the quadratic power of the asymmetric parameter $\Delta \omega$. This relationship between the asymmetric parameters ($\Delta \omega$) and Q factors is consistent with the relationship between the asymmetry parameter and the Q factor for symmetric-protected BICs [17,52], which further verifies that these asymmetric Fano peaks are quasi-BICs.
The lattice mode coupling to LC, dipole, and other resonance modes, can reduce the radiation loss of the resonance mode, thus narrowing the resonance linewidth and improving the Q factor [39]. The lattice mode can enhance the Q factor of the resonances by nearly an order of magnitude [53,54]. Based on the theory, the BIC comes from the radiation loss is completely coherent and destructive interference in the far field [1,52], which means the two coupled structures should radiate by driving the external field. Thus, two coupled structures should be bright modes at least. However, lattice modes are considered dark modes trapped in the surface of periodic structures and fail to radiate to the far field [39]. Therefore, the coupling of bright mode and lattice mode can not obtain the BIC. On the other hand, lattice modes can decrease the radiation loss of quasi-BIC by being well designed, we believe that lattice mode can enhance the Q factor of quasi-BIC. However, the resonance enhancement of the lattice mode cannot make the Q factor reach infinity [39]. To sum up, the lattice modes could not drive the structural metamaterial resonances towards infinite quality factor values, but could enhance them.
6. Conclusion
Compared to breaking the symmetry of the structure, the coupled mode theory is a more fundamental theory to explain the generation of quasi-BIC(s). Due to the coupling effect, quasi-bound states in the continuum phenomena acquire in identical structures without breaking the symmetry of the supercells. The phenomenon of quasi-BICs generated by symmetrical structures cannot be explained by breaking the symmetry of the structure. We have demonstrated the accuracy of our conclusion through theoretical, simulated, and experimental. This discovery provides a new avenue for future research, and we believe that it will contribute to the design of coupling devices and the study of coupling mechanisms between metasurface structures.
Funding
National Natural Science Foundation of China (12264010, 61965005, 62005059, 62205077); National Science and Technology Major Project (2017ZX02101007-003); Guangxi Key Laboratory of Optoelectronic Information Processing (GD20102).
Acknowledgments
W.H. acknowledges funding from Guangxi oversea 100 talent project; W.Z. acknowledges funding from Guangxi distinguished expert project.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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