Rotational symmetry of photonic bound states in the continuum

The bound states in the continuum (BICs) have been investigated by simulating the optical reflectivity of a tri-layer photonic crystal slab. We found that optical BICs can occur in a class of photonic crystal systems with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{1}^{z}$$\end{document}c1z, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{2}^{z}$$\end{document}c2z or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{4}^{z}$$\end{document}c4z rotational symmetries, which are constructed by three identical photonic crystal slabs. By applying the two mode coupled model, we obtain the reflectivity formula to fit the numerical data and evaluate the lifetime of radiation decay. In vicinity of BIC, the lifetime diverges as a power law form, when approaching the BIC point. The infinity life time of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{1}^{z} {\text{ } - \text{ BIC}}$$\end{document}c1z-BIC in the tri-layer structure indicate that it is a true BIC. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{1}^{z} {\text{ } - \text{ BIC}}$$\end{document}c1z-BIC occurs robustly in tri-layer structures, but the resonance frequency of the BICs is dependent on the permittivity of slab, air-hole size and hole shape.


Scientific Reports
| (2020) 10:18243 | https://doi.org/10.1038/s41598-020-75308-x www.nature.com/scientificreports/ behavior of modes indicates that the BIC at P 2 is a Friedrich-Wintgen type BIC 40 . Because Fano resonances originate from the interference between out-of-plane far-field radiation and in-plane waveguide modes, the resonance profiles can be changed by moving the middle layer along the y direction. In this tri-layer structure as mirrors to form a tunable optical cavity, the interference is highly sensitive to y 0 . When the displacement is suitable at the correct phase matching conditions, the blue and red branches should meet each other. Then, a BIC occurs. The y 0 -dependence interference indicates that the Fabry-Perot-type resonance plays an important role in the formation of BICs in the tri-layer structure. By continuously changing y 0 , we find three BICs guided by the arrows, as shown in Fig. 1b. BICs at P 1 , P 2 , and P 3 can also be observed for the p-polarized mode (the electric field along the y axis) by changing the incident angle ( θ ) shown in Fig. 1d-g. However, when s-polarized light normally incident on slabs, Fano resonances with finite resonant width are observed around the reduced frequency (0.086) and insensitive to the displacement as shown in Fig. 1f. By changing the incident angle, the tri-layer structures exhibit that, for the p-polarized mode, fano resonances still remain the finite resonant width shown in Fig. 1i-k.
In order to gain further physical insights of the BIC in the tri-layer structure, we plot the distributions of the electric field intensity in yz-plane and xy-plane at BICs in Fig. 2. We clearly see that the field distributions become localization inside the structure, while the outside field tends to vanish. Because the tri-layer structure always contains a mirror symmetry for different displacements in the z direction, the electric field intensities also have the mirror symmetry as shown in Fig. 2a-c. When y 0 = 0 , the structure has c z 4 rotational symmetry ε x, y, z = ε −y, x, z . The electric field intensity also has c z 4 rotational symmetry as shown in Fig. 2d. Then, we have ε(� r) = ε(R 4 � r) where the operator R n rotates vectors by an angle 2π/n about the z-axis. The appearance of a BIC may be understood as the geometric symmetry forbids coupling to any far-field radiation. The electric www.nature.com/scientificreports/ field of the resonance can be � E = e i � k·� r � u � r, � k by using Bloch's theorem, where � k = k x , k y , 0 and u is periodic function in (x, y) 41 . Then, we could use the vector field rotator O(R 4 ) to rotate the electric fields E as Because the structure is invariant under transformation R 4 , the electric fields R 4 � E R −1 4 � r are also the solutions of Maxwell's equations with the same resonance frequency of E , but they could differ by a phase factor � E=e iθ 4 R 4 � E R −1 4 � r . If applying the rotational transformation 4-times, we can return to the original electric fields.
When moving the middle layer, c z 4 symmetry of tri-layer structure is broken and c z 4 − BIC disappears. However, when y 0 /L = 2.131 , the accidental phase matching condition is satisfied to cancel the radiation mode and the c z 1 − BIC occurs. This BIC arises from the interaction of two modes 42 .This tri-layer structure only has c z 1 rotational symmetry shown in Fig. 2b, e, which means BICs can be found in the multi-layer structures without c z 2 rotational symmetry. Here, c z 1 means 360° rotation around z-axis. It is noted that the tri-layer structure always keeps the time reverse symmetry and the mirror symmetry for the any displacements of the middle layer.
When y 0 /L = 4 , the symmetry-protected BIC appears since the tri-layer structure recovers c z 2 rotational symmetry ε x, y, z = ε −x, −y, z as shown in Fig. 2c If we rotate the system twice, the electric fields add another phase factor � E=e i2θ 2 � E , so the phase θ 2 can only take on values of 0 or π . In fact, the displacements of middle layer, that break rotational symmetry, reduce the infinite lifetime of BICs but add a tuning parameter of resonant state. As a result, optical BICs can occur in the tri-layer photonic crystal systems with c z 1 ,c z 2 or c z 4 rotational symmetries. However, the distributions of the electric field intensity at BICs are sensitive to the rotational symmetry.
At BICs, the lifetime of radiation decay becomes infinity. Thus, we might apply the temporal coupled mode theory to estimate the lifetime from the numerical data. Here, we consider a system that possesses two modes coupling with each other, and the non-Hermitian Hamiltonian of the tri-layer structure is Here, f 1 and f 2 are the resonance frequencies of the resonators. The radiative-decay lifetime can be defined by τ i = 1/γ j . Due to the energy conservation and time-reversal symmetry [43][44][45] , the reflectivity can be easily obtained and given by Then, we could apply Eq. (2) to fit the numerical reflectivity as shown in Fig. 3a. From the fitting curves, the resonance frequencies and the lifetimes might be estimated for various y 0 values. The lifetime of the resonance goes to infinity at c z 1 − BIC shown in Fig. 3b. In vicinity of BIC, the lifetime can be written into a power law form as where y BIC 0 is the displacement for the BIC situation. α N/P is the exponent of power law and a positive real number. Naturally, τ become infinity, when y 0 =y BIC 0 , and a BIC occurs. Furthermore, the exponents of lifetime can be estimated by the power law relations shown in Fig. 4. Here, we define the dimensionless parameter t = y 0 − y BIC 0 /L . The Log-Log plots of τ versus t become straight lines with slopes α P and α N . For c z 4 − BIC , where the tri-layer structure has c z 4 rotational symmetry, two exponents of lifetime are identical with each other. When the structure is changed and has c z 2 rotational symmetry, the exponents are still same with each other, but the values become larger. For c z 1 − BIC , the estimated exponent is α N 1 = 1.7 in the negative region, and the exponent is α P 1 = 2.1 in the positive region.
Size and shape of holes. In order to examine the robustness of the relations between exponents, the size of square air hole in the slabs is enlarged, and the reflectivity spectra are plotted in Fig. 5. Then, the lifetime can be extracted by Eq. (4), and the exponents of lifetime are also estimated by the power law form. If the tri-layer structure has c z 4 rotational symmetry, two exponents of lifetime keep a constant value ( α N/P 4 = 2 ) for various air hole sizes. The integer form of exponents indicate that the c z 4 − BIC can be understood by a topological defect vortex 9,35 . Then, the tri-layer structure, for various air hole sizes, can be changed to produce c z 2 − BIC , it is found that two exponents ( α   , the reflectivity spectra of trilayer structure are shown in Fig. 6. We can observe three modes and two c z 1 − BICs guided by P ↑ and P ↓ . By applying the Eq. (2), the resonance frequencies of c z 1 − BICs could be estimated. When the permittivity of slab is increasing, the resonance frequencies of c z 1 − BICs are decreasing as shown in Fig. 7a, but the difference between resonance frequencies is increasing. However, the displacements y BIC 0 (P ↑ ) and y BIC 0 (P ↓ ) have a non-monotonic function of ε shown in Fig. 7b. These results illuminate that the waveguide modes in tri-layer structures can be adjusted by changing the permittivity of slab. Thus, the resonance frequencies of BIC can be manipulated by choosing various materials.

Conclusion
In conclusions, we have demonstrated the formation of BICs in tri-layer systems made of three identical PhC slabs with square or rectangle air holes. The c z 2 and c z 4 types of BICs are accessible due to the symmetry incompatibility with the radiation. A distinct characteristic of the c z 1 − BICs achieved in this study is that it originates from destructive interference of two resonances at the phase matching conditions. We introduce the two mode coupled model to obtain the formula to fit the reflectivity data and estimate the lifetime. The tri-layer structures, producing the various rotational symmetries, provide a new platform towards the BIC studies.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.