Adjacent Asymmetric Tilt Grating Structure With Strong Resonance Assisted by Quasi-Bound States in the Continuum

In the field of optics, bound state in the continuum (BIC) as a special state to be researched in many photonic crystals and periodic structures, which can produce strong resonance and an ultrahigh Q factor. Some designs of narrowband transmission filters, lasers, and sensors are proposed based on excellent optical properties of quasi-BIC. In this paper, we consider symmetrical rectangular grating structure firstly. Then cut off a corner of rectangular gratings, the Fano line of quasi-BIC can be observed in the spectrum. After that, we change the oblique parameter of the other rectangular grating, which can further decrease the Fano line width. In the momentum space, the change of structure means topological charges split from q = 1 into half charges q = 1/2. We analyze guided mode resonance (GMR) excitation of the grating structure, and discuss the dispersion relations in the waveguide layer with the position of BIC in energy bands. In addition, the spectrum exhibits asymmetric line-shapes with different values of the asymmetry parameters, M1 and M2. BIC is transformed into quasi-BIC as the symmetry of the structure broken. Thus, a large Goos-Hänchen shift can be achieved as a result of ultrahigh Q factor of quasi-BIC. This work demonstrates a double trapezoid structure with strong resonance properties, which has significant implications for exploring the phenomenon of BIC.

medium threshold. BIC was first proposed by von Neumann and Wigner in 1929 [1], but it's not demonstrated in quantum systems. It was not until 1992 that Capasso's group proposed a BIC experiment in quantum systems [2]. Then, many specific structures were used to support BIC, caused a leaky mode with a resonance state of zero width [3]. Until now, there are a great deal of researches on resonant states, which lie in the continuum with a finite lifetime. They are born from continuum states and decay eventually. Regulating a structure with specific parameters, resonance states interfered by some different channels generate zero-width resonance, corresponding to BIC without energy loss.
In this work, we propose a double trapezoid grating as an asymmetric structure to discuss the relationship between BIC and quasi-BIC. For a normal grating, it can store energy in the symmetrical structure without any loss. When we cut off a corner of rectangular gratings, symmetry of the structure will break and the energy in the structure leak into free space. For the BIC turn This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ into quasi-BIC, a Q factor will become finite from infinite. In the momentum space, topological charges split and couple with the change of grating. Based on guided mode resonance (GMR), the dispersion relations of grating can be theoretically derived, and we can find the position of BIC in energy bands. We focus on changing corners of the grating, and utilize finite element method (FEM) to calculate reflection spectrum, from which we find a Fano peak. Our structure can produce a large phase change at the wavelength corresponding to Fano peak, which can be used to achieve a large Goos-Hänchen (GH) shift. Our work provides an integrated platform for the BIC applications, such as optical switches, high-performance sensors and wavelength division multiplexers.

II. BIC IN THE TRAPEZOID GRATING AND THE ANALYSIS OF DISPERSION RELATIONS
In this letter, we focus on the transforming between BIC and quasi-BIC with double trapezoidal grating. Based on the original oblique structure [34], we further optimize structural parameters and achieve application of GH shift. We firstly discuss rectangular grating in a periodic photonic crystal slab as shown in Fig. 1(a), the structure maintains symmetry with BIC. When we cut off one corner of the rectangular grating, the symmetry of structure is broken in Fig. 1(b). For the another rectangle is also damaged in Fig. 1(c), a narrow Fano Peak can be observed with limitation of light field mode.
The unit cell of grating can be divided into three parts, grating layer, waveguide layer. The grating layer is chosen as Si and the waveguide layer is SiO 2 . The refractive index of Si and SiO 2 are 3.48 and 1.46 respectively [34], the period length of the whole unite grating is D = 1544 nm and the thickness of waveguide layer is h 1 = 2000 nm. For the grating layer, the air gap splits into two parts d c = d a = 0.23(1-x)D. Here, the rectangular grating is a fixed value d b = 0.27D and the height of grating layer is h 2 = 500 nm. For the part of trapezoidal grating, the schematic diagram of the structure can be seen in Fig. 1(b) and (c). We define parameters M 1 and M 2 to control trapezoid angle, the rest are controlled by N 1 and N 2 , which satisfy M 1 +N 1 = 1 (M 2 +N 2 = 1) Based on guided mode resonance (GMR) excitation of the grating waveguide layer [35], [36], [37], we can give a schematic representation of the dispersion relation. The propagation diagram is shown in Fig. 1(d). In the system, tangential component of wave vector is defined as k i , angular frequency is ω, c is light speed in the air. For the background of air, we have k i = k 0 sinθ with the wave vector in air k 0 = ω/c.
When the light beam spreads in the grating, a portion of the light passes through the grating and the other part is trapped in the waveguide layer. In the grating, we define where the 2π/D is the basic vector of the reciprocal lattice. The propagating constant is β, and the expression for GMR is expressed as Then the difference of the modal propagation coefficient β x can be written as where the θ is given by Substituting Eqs. (3) into (2), based on the Lorentzian model with the assumption, the resonance condition is explained as a pole in the complex domain [38], i.e., Δβ x = γ, where γ is the coupling loss coefficient. The Δλ represent linewidth at the peak position. Then we can derive We can initially calculate the half-height width of the resonance peak. For the slab waveguide theory, β is calculated based on the eigenvalue equation with TE where i is the order of waveguide mode. We can determine the location of the resonance peak based on the above equations. Considering the incident angle θ = 7.1°, ω 0 = 2πc/h 1 and TE 0 guided mode, the dispersion relation in the waveguide layer can be expressed as The dispersion relation of waveguide layer corresponds to the green dotted line in Fig. 2(a). The first and second order tangential wave vector components are blue and red solid lines. We set ω 0 = 2πc/h 1 , the intersection points in the figure we mark as X, angular frequency ω = 0.138ω 0 = 0.276πc/h 1 ≈129.96 THz, it means that GMR condition is caused by k i and β. Besides, we give the picture of energy bands of grating in Fig. 2(b) to match the GMR condition. The frequency of the BIC point is well matched to the angular frequency ω and finding the position of Fano peak in the next section based on energy bands.
This periodic grating structure is calculated by the COMSOL Multiphysics. We set up the characteristic frequency solver, two dimensional plane grating sets periodic boundary conditions on both sides, and the size of first Brillouin zone is 2π/D. In the next work, we calculate reflection and transmission spectra with finite element method (FEM). After the model analysis, we can obtain the enhanced electric field modes with the different oblique angle. Breaking the symmetric rectangular grating, there will produce some leakage of the light field. In the following, we will further discuss the change of the models and topological polarization by COMSOL Multiphysics.

III. SIMULATION RESULTS
In the last few years of reported works, discussing the symmetry of the structure as a way to study BIC. Grating as an optical device, it is favorable to study the BIC phenomenon. In our work, firstly, we consider a rectangular gratings structure, a BIC state with high symmetry. However, some optical phenomena cannot be observed due to symmetry-protected BIC. After that, we try to cut off a corner of the rectangular grating as a way to breaking symmetry. We keep one of the rectangle unchanged in a unit cell, and then define an asymmetric parameter M 1 to control the part of trapezoidal grating. When we scan the parameter M 1 from 0 to 1, the distribution of reflection as a function of the asymmetric parameter M 1 and wavelength can be seen in Fig. 3(a). We have circled the location of the quasi-BIC at the wavelength λ = 2387 nm in the Fig. 3(a) and the narrowest Fano peak found when M 1 = 0.3 with normal incidence. Therefore, we set the oblique angle in one of the rectangles to 0.3, and then control another parameter M 2 .
We adjust another trapezoid while keeping the parameter M 1 constant as shown in Fig. 3(b). Similarly, we define a new oblique parameter M 2 , and show the distribution of reflections in Fig. 3(b). We can see the color gradient map, where the red areas represent the location of the peak. In order to get narrow line widths while maintaining high reflectivity, we choose M 2 = 0.5 in Fig. 3(b). Whether it is greater than 0.5 or less than 0.5, none of Fano peaks can be better than the condition M 2 = 0.5. We refer to this situation as quasi-BIC, which is formed by the interference of two channels [39]. The two channels are discrete spectrum narrowband originated from GMR and the other is continuous spectrum broadband provided by resonance in the waveguide layer. In addition, resonance peak will produce a red shift due to the change with M 2 .
The topological natures of BIC is discussed with the change of grating. We show the distributions of the field polarization states in Fig. 4(b) and (d). In the surface momentum space of the grating, total winding of the polarization position remains 2π. For a symmetrical rectangular grating, topological charges locate at the point q = 1 as shown in Fig. 4(b), which corresponds to symmetry-protected BIC. The sum of topological charges can be calculated as [40]  Where the L is a closed loop that encloses one BIC, ψ is the orientation angle, and n w is defined as the winding number. When the planar symmetry of the structure is broken, the topology integer charge q = 1 will split into half topology in Fig. 4(d). Then they will move in the momentum space, with one being left-handed circularly polarized (blue) and the other being right-handed circularly polarized (red). As a result, the field of grating exhibits polarization diversity, where elliptical polarizations, linear and circular polarizations appear near the center of the first Brillouin Zone.
Here, we have calculated reflection and transmission spectra based on finite element method (FEM), the top of model is perfect match layer, and a incident port is set. For the asymmetric grating structure, the tilt parameter of one of the trapezoid is taken as 0.3, and the other parameter of trapezoid is taken as 0.5. An asymmetrical Fano line can be found in Fig. 5(b) and (c), where their peak reflection efficiency is extremely high and reach to 1, and the intensity distributions of the electric field along the y-axis is also calculated and shown in the insets. Firstly, we discuss the primitive case with M 1 = 0 and M 2 = 0, a symmetry-protected structure with BIC. Interestingly, when the structure maintains perfect symmetry, the reflection curve is a smooth line without any steep peak, and the resonance width vanishes completely in this case. In the case of BIC, discrete and continuous state will mutually couple in the structure, and its energy will be confined in the grating without any leakage. BIC has an infinite Q factor, and an infinite narrow line width is appeared in the spectrum, which agree with the phenomenon in spectrum of Fig. 5(a). The mode of electric field can be seen in the inset of Fig. 5(a), it shows the concentration of energy.
Considering quasi-BIC is an important way to achieve various good optical properties. Quasi-BIC is a state that infinitely close to the point of BIC, so that it always has an ultra-high Q factor and an ultra-narrow Fano peak in the spectrum. In the Fig. 5(b), we can find an asymmetrical Fano peak at the wavelength equal to 2387 nm with M 1 = 0.3, this state corresponds to the blue dotted line in Fig. 3(a). Compared to the initial structure, when the one of the rectangles is cut off a corner, the mode of electric fields will produce leakage due to the symmetry broken. Cutting off a corner of the grating leads to a change in the structure, topological charges start splitting from q = 1 into charges q = 1/2 in the momentum space.
Only changing the oblique angle of one grating in a unit cell, we can get an asymmetrical narrow Fano peak as shown in Fig. 5(b). In order to getting a more narrow line widths, this structure needs to be optimized further by controlling another oblique parameter in a unit cell. When the oblique parameters M 1 = 0.3 and M 2 = 0.5, the Fano line width will decrease to a minimum. In Fig. 5(c), an ultra-narrow Fano peak can be got, the line width is much narrower compared to that in Fig. 5(b). The result of peak is benefit from the optimization of the structure, this state corresponds to the white dotted line in Fig. 3(b). The obtained reflection peak is close to complete reflection at wavelength λ = 2410 nm. Theoretical calculated angular frequency ω = 129.96 THz in the Section II, and the corresponding wavelength λ≈2400 nm. The simulation results match the theoretical calculation results very well. Topological charges have been splitting into half charge totally in the momentum space, and it's electric field energy can be seen in the inset of Fig. 5(c) with a strong energy bondage. In this case, the mode represents leak and loss and the radiation will propagate into free space, which leads to a finite lifetime rather than infinite lifetime.
The trapezoidal grating has an ultra-narrow Fano peak, which can achieve a large GH shift. Next, we utilize the quasi-BIC to achieve a large GH shift based on a phase change. The asymmetry geometric parameter M 1 and M 2 determine the value of the GH shift, i.e., resonance will become stronger and result in a larger GH shift for specific values of the asymmetric parameter. Here, for the stationary phase method, the lateral GH shift for the reflected and transmitted beams can be calculate by using the expression [41], [42], [43], Where ϕ r is the reflection phase and the GH shift is proportional to the partial derivative of the reflection phase. Here, we discuss the structure which cut off a corner of a rectangular grating. The phase can change with the increased incident angle under λ = 2387 nm and the GH shift angular spectrum is shown in Fig. 6(a). The GH shift shows significantly different behavior around the resonance angle of θ = 7.12°. For the double trapezoid structure, the phase change with the incident angle is calculated in Fig. 6(b), which undergoes a abrupt phase at the incident angle θ = 7.15°. The GH shift in the Fig. 6(b) is far larger than the shift in the Fig. 6(a). In contrast with traditional GH shift enhanced by transmission-type resonances, a maximum GH shift assisted by the quasi-BIC locates at a reflection peak and it can be detected and utilized more easily.
Based on the large GH shift, we design an ultrasensitive sensor with the quasi-BIC, and define n 0 as refractive index of surrounding medium. For the surrounding environment of Fig. 5. (a) The spectra of symmetrical rectangular grating, reflection is a smooth curve without any peak protrusion, and the electric field pattern is symmetrically distributed. (b) When one of the rectangles is cut into a trapezoid, the original smooth curve appears a steep Fano peak at wavelength equal to 2387 nm, a little rotation has occurred in the electric field pattern. (c) Grating is optimized further by control another tilt parameter M 2 , where an ultranarrow Fano peak is found. the structure, we firstly calculate GH shift with air n 0 = 1 at the room temperature. Then we increase the refractive index of surrounding medium from 1, which can be seen in Fig. 6(c). With the index of refraction increases, the peak of GH shift will shift toward a larger incident angle. In addition, the sensitivity of sensor can be defined by the value change of GH shift and the refractive index of surrounding medium, i.e., S(n 0 ) = |d(S GH )/dn 0 |, and the dependence of the sensitivity on the refractive index can be seen in the Fig. 6(d). The blue solid line represents the refractive index of air n 0 = 1, when the medium changes, the GH shift as a red dotted line can be seen in Fig. 6(d). The sensitivity is 349.885 μm/RIU with the refractive change from 1 to 1.002. In order to better study the sensing performance, we discuss the relationship between the GH shift and refractive index n 0 in the Fig. 7(a), the relationship between sensitivity S and refractive index n 0 can be seen in the Fig. 7(b), as the refractive index changes, the value of the GH shift and sensitivity S are basically unchanged. The structure has a good robustness. In the Fig. 7, the green dashed line and the blue It indicates that the sensor we designed has a high sensitivity with the change of medium. So far, we have designed an ultrasensitive surrounding sensor based on large GH shift with quasi-BIC, this application provide a platform to discuss other applications with quasi-BIC.

IV. CONCLUSION
In conclusion, we have discussed a double trapezoid structure with a strong resonance and the split of topological charge. In addition, the point of the BIC in the energy bands can be found by finite element method. We use the GMR and dispersion relations to analyze the interaction between light and matter, which causes asymmetric Fano line shape. We break the symmetry by cutting off a corner of the rectangle, and control the degree of oblique angle by changing parameters M 1 and M 2 . With the optimization of parameters, a typical asymmetric Fano line width can reduce further, where the parameters are M 1 = 0.3 and M 2 = 0.5. In the momentum space, topological charge will split into half charge q = 1/2 with the destruction of symmetry. Assisted by strong resonance with two different channels, a saltatory phase can be obtained, then the maximum GH shift is achieved at the position of saltatory phase. Our work provides a new perspective to realize BIC, which can support many optical applications such as optical switches, high-performance sensors and wavelength division multiplexers.