### Focus

## BIT's research group has made important progress in the study of continuum bound states

News Resource: School of Physics

Editor: News Agency of BIT

Translator: Yuan Mengfan, News Agency of BIT

Recently, the research group of Professor Zhang Xiangdong from the School of Physics, Beijing Institute of Technology, has made important progress in the study of continuum bound states based on moiré photonic crystal plates. Relevant research results were published in the recent *Phy. Rev. Lett. *(128, 253902 (2022)). The research work was supported by the National Natural Science Foundation of China and the National Key R&D Program. Dr. Huang Lei (Class of 2021) and Dr. Zhang Weixuan (now postdoctoral fellow of the School of Integrated Circuits and Electronics) are the co-first authors of the paper.

In recent years, moiré superlattice systems in the field of condensed matter physics have attracted great research interest. Based on the lowenergy flat-band effect of moiré superlattices, various novel many-body correlated states are experimentally realized. Recently, the introduction of rotational degrees of freedom into the design of photonic structures to construct photonic moiré superlattices has received extensive attention. Based on this new control degree of freedom, the localization-delocalization transition of light can be realized on the moiré light lattice . In the twisted bilayer ÿ-MoO3 system, the researchers also observed a topological phase transition of the iso-frequency dispersion curve induced by the turning angle. In addition, numerous studies have demonstrated that the moiré flat-band effect corresponding to double-layer corner graphene can also be achieved by controlling the interlayer coupling of phonon/photonic crystal plates and double-layer circuit networks.

On the other hand, continuum bound states (BICs) are another important concept in the field of physics research. It corresponds to a wave state where the energy coexists with the radiation wave continuum and the wave function is zero at infinity. It is worth pointing out that optical systems with continuum bound states usually contain resonant modes with ultra-high quality factors and topological properties of momentum space, which make them very important for applications in ultra-low threshold lasers, ultra-sensitive sensors, and filters. In practical applications, researchers usually design quasi-BICs structures that are close to perfect BICs to achieve limited quality factor and resonance width, so that they can be effectively excited in practical applications. Recent studies have shown that quasi-BICs with high quality factors can be realized in different nanostructures. However, its significant dispersion effect also limits the use efficiency of quasi-BICs under wide-angle light sources. The Moiré optical structure provides an important platform for designing weakly dispersive photonic flat bands. An important question is therefore: Is it possible to combine moiré physics with BICs to construct quasiBIC photonic states with flat-band effects to make it have the dual characteristics of flat belt and quasi-BIC.

However, it is not an easy task to construct such novel photonic states. This is because the currently designed two-dimensional optical moiré flat bands are realized at the Dirac point of the Brillouin zone. Due to the complex optical mode coupling of high-frequency electromagnetic fields, it is difficult to find an ideal Dirac point above the light cone. Therefore, the current design of the moiré flat band of the photonic crystal plate is realized in the area below the light cone line. Paradoxically, the quasi-BICs in the photonic crystal plate must lie above the light cone. This makes it very difficult to combine BICs with moiré flat belts. Therefore, a new method had to be created to design the moiré photonic flat band above the light cone line

Research Highlight 1: the theoretical model of the moiré photon flat band on the light cone

The researchers first considered the collimated bilayer photonic crystal plate structure, as shown in the upper inset in Fig. 1(a). Figure 1(b) shows the band dispersion curve of the bilayer photonic crystal plate at very far distance. In this case, the interlayer coupling is negligible, which allows the energy band of the bilayer photonic crystal plate to be regarded as the dispersion curve of the doubly degenerate single layer photonic crystal plate. Notably, at the ÿ point of the collimated photonic crystal plate, there are high and low frequency photon modes of secondary dispersion, which are distinguished by blue and red lines, respectively. In subsequent calculations, the researchers used the low-frequency mode marked by the red line as a starting point to design the moiré photonic flat band.

Figure 1. Construction of the moiré photonic flat band above the light cone based on the photonic crystal plate structure.

By rotating the upper photonic crystal plate by an angle ÿ, the moiré photonic crystal plate structure can be realized, as shown in the lower inset of Fig. 1(a). The rotation changes the translational symmetry of the bilayer photonic crystal plate, resulting in a corresponding Brillouin zone change, as shown in Fig. 1(c). The red and blue large hexagons correspond to the first Brillouin zones of the top and bottom photonic crystal plates, respectively. Points K' and K'' are used to mark the equivalent valley points of the Brillouin zone of the upper and lower photonic crystal plates. The small black hexagons represent the Brillouin zone of the moiré photonic crystal plate. It should be emphasized that due to the band folding effect of the Brillouin zone, in addition to the ÿ point of the first moiré Brillouin zone, each ÿ point of the high-order moiré Brillouin zone also has a low frequency with secondary dispersion. photon energy band.

To describe the physics of the low-frequency moiré band, the researchers constructed an effective continuum model of the moiré photonic crystal plate in the first moiré Brillouin zone. Based on this effective model, the moiré energy bands corresponding to different interlayer couplings can be directly solved, as shown in Fig. 1(d-f). The band dispersion predicted by this efficient model agrees very well with the band structure solved by the finite element algorithm (Fig. 1(g-i)). Finally, through the analytical solution of the efficient model and rigorous first-principles numerical calculations, the researchers found that the moiré photonic flat band can be realized above the light cone by balancing the relationship between the turning angle and the interlayer coupling.

Research Highlight 2: Theoretical Design of Moiré Quasi-BICs

Based on the scheme of designing the moiré photonic flat band above the light cone, the researchers further combined the moiré flat band with BICs to realize quasi-BICs with flat band dispersion effect. At the center of the Brillouin zone, the monolayer photonic crystal plate has symmetryprotected BICs. Figure 2(a) shows the distribution of far-field polarization states and quality factors in the Brillouin zone of a single-layer photonic crystal plate. It can be clearly seen that the ÿ point corresponds to the singularity of the far-field polarization,and has a topological charge of size -2. Therefore, the low-frequency modes corresponding to the ÿ point of the monolayer photonic crystal plate are symmetry-protected BICs.

The period of the newly formed moiré primitive cells increased significantly after rotating the bilayer photonic crystal plate. In this case, besides the zero-order diffraction channel, there are other high-order diffraction channels in the moiré photonic state, as shown in Fig. 2(b) and Fig. 2(c). To analyze the far-field radiation properties of different diffraction channels, the researchers calculated the far-field polarization states of the zero- and first-order diffraction channels, as shown in Fig. 2(d) and Fig. 2(c). It can be seen from the figure that the zero-order far-field polarization state of the moiré photonic crystal changes from linear polarization to elliptical polarization due to the breaking of the upper and lower symmetry. In addition, the zero-order diffraction channel ÿ point still corresponds to the far-field polarization singularity. But for other first-order diffraction channels, a different form of far-field polarization exists at the ÿ point. The above results indicate that the eigenmodes in the center of the moiré Brillouin zone (ÿ point) can be coupled with the photon continuum to form leaky photon states with finite quality factors.

Further researchers found that by reducing the rotation angle of the double-layer photonic crystal plate system, the quality factor of the ÿ point photonic state corresponding to the moiré flat band will gradually increase and tend to infinity, as shown in Figure 2(f). This is because the interlayer coupling strength corresponding to the low-frequency moiré flat band decreases significantly when the rotation angle decreases. The weakened interlayer coupling can effectively reduce the radiation loss of high-order diffraction channels. Therefore, the eigenmodes at the ÿ point of the moiré flat band can be regarded as evolved from BICs of collimated bilayer photonic crystal plates, which can be called moiré quasi-BICs.

Figure 2. Quasi-BICs on the Moiré photonic flat band.

Research Highlight 3: nonlinear enhancement effect based on moiré quasi-BICs

Finally, the researchers proved through numerical calculations that moiré flat-band quasi-BICs with similar quality factors can achieve more efficient nonlinear enhancement than traditional quasi-BICs under the excitation of a wide-angle light source (shown in Figure 3(a)). effect. Figure 3(b-c) shows the second harmonic (SHG) of dispersive quasi-BICs and moiré flat-band quasi-BICs. Among them, the inset shows the enhanced efficiency of the second harmonic at different incidence angles. It can be seen that the moiré flat-band quasi-BICs all have a single resonance frequency under the excitation of different incident angles, so that the enhanced signals of different incident angles are superimposed at the target frequency position. However, the resonant frequencies of dispersive quasi-BICs shift significantly under excitation at different incident angles, resulting in a relatively small sum of the intensities of their nonlinear signals. By comparison, it can be seen that the second harmonic intensity of the moiré flat-band quasi-BICs is 10 times that of the dispersion.

Figure 3. Moiré quasi-BICs enhanced SHG scheme under wide-angle light source.

Paper link: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.128.253901.