Resonant structures are the source of rich physical phenomena. Having a high quality factor Q and the ability for dynamic modulation are two crucial aspects in designing functional photonic devices based on resonance effects. Fano resonance and bound states in continuous spectra are currently two typical resonance phenomena that draw extensive attention from researchers. Fano resonance stems from the special interference effect between local (discrete) and continuous states in quantum or classical systems. Owing to its sharp asymmetric profile (linear shape), high spectral resolution, and extreme sensitivity to changes in structure and surrounding dielectric environment, Fano resonance can be applied to high-sensitivity sensing. Asymmetric linear Fano resonance can also be utilized in the design of optical switches. The bound state in the continuum (BIC) is a special resonant state, and its physical mechanism offers a superior solution for controlling and increasing the mode Q factor. A BIC with an infinitely large Q factor is an ideal model. Physically, optical quasi-BICs (QBICs) with extremely large Q factors can be used in various practical applications, such as lasers, sensors, light absorption, and harmonic generation enhancement. Up to now, diverse structures and mechanisms for realizing BIC have been proposed, among which the symmetry of the structure has become an important factor. Recently, researchers have proposed a grating waveguide structure based on guided mode resonance (GMR). The QBIC based on GMR is manifested through the Fano resonance peak. For any formation mechanism of BIC, there are strict requirements for structural parameters. Processing errors or defects in the structure can lead to a sudden reduction in the Q value. In addition, the adjustability of the mode, especially the dynamic continuous tuning, is essential in practical applications. However, most mode modulation relies on the tuning of structural parameters and cannot be achieved dynamically. Therefore, it is highly significant in optical research to discover a BIC structure model that is easy to implement and dynamically tunable. To fulfill the above objective, in our research, we design a composite structure model consisting of a grating and a waveguide, which generates a pair of BIC and resonance state. Through simple translation mismatching between two gratings, the BIC and Fano resonance states can be achieved and tuned dynamically, and some new photonic states have been discovered.

To overcome the difficulty in achieving dynamical tuning of BIC, a three-layer compound structure model composed of gratings and waveguide is designed. The structure has three independent tuning degrees of freedom: the width and height of the air groove, and the relative translation between the two compound grating layers. Based on the Comsol eigenfrequency solver, we construct a unit cell with Floquet periodicity boundary condition in the x direction. To ensure that the far-field polarization field is accurate enough, two thick enough air layers with two perfectly matched layers (PMLs) on each top are placed on the slab. For the TM modes (with field components E_z, H_x, and H_y), the frequency bands are obtained by scanning the wave number k_x in the Brillouin zone (BZ). Through the calculation of bands and mode analysis, as well as the calculation of transmission spectra, the mode characteristics and transmission properties of the structures are acquired.

The designed structure simultaneously generates a pair of BIC and Fano resonance at point Γ, the center of BZ. Their relative frequency position can be tuned through three independent degrees of freedom: the width and height of the air groove, and the relative translation between the two grating layers. The relative translation of the two grating layers enables continuous and dynamic tuning of the frequency positions and spectral line shapes of the BIC and Fano resonance. The Fano resonance has a peak in the transport spectrum, but the BIC cannot be detected from the transport spectrum due to its infinite Q value. With the symmetry broken by the translation mismatching between the two grating layers, BIC becomes QBIC. QBIC has a narrow peak in the transport spectrum. Through tuning of the relative translation, the QBIC resonance peaks and Fano resonance peaks experience an interesting merging. The merging of QBIC resonance peaks and Fano resonance peaks generates a special two-fold photonic state.

In this study, we have designed a composite structure of double-layer grating and waveguide, which simultaneously generates QBIC resonance and Fano resonance. The frequency positions and spectral shapes of QBIC resonance and Fano resonance are independently adjusted by the width and height of the air groove, and the relative translation distance of the two gratings. Compared with similar methods in adjusting modes by changing structural parameters, the method proposed in this study for controlling BIC resonance and Fano resonance is simpler, more accurate, and more feasible. The theoretical results and tuning method from this study have important application value in the design of optical switches and optical sensing.