On periodic structures, a bound state in the continuum (BIC) is a standing or propagating Bloch wave with a frequency in the radiation continuum. Some BICs (e.g., antisymmetric standing waves) are symmetry protected, since they have incompatible symmetry with outgoing waves in the radiation channels. The propagating BICs do not have this symmetry mismatch, but they still crucially depend on the symmetry of the structure. In this Letter, a perturbation theory is developed for propagating BICs on two-dimensional periodic structures. The Letter shows that these BICs are robust against structural perturbations that preserve the symmetry, indicating that these BICs, in fact, are implicitly protected by symmetry.

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Optical bound states in the radiation continuum (BICs) have interesting properties and potentially important applications. On periodic structures, the BICs are guided modes above the lightline, and they can be either standing waves or propagating Bloch modes. A one-dimensional (1D) array of circular dielectric cylinders is probably the simplest structure on which different types of BICs exist. Using a highly efficient numerical method, we perform an extensive numerical study for propagating BICs on 1D arrays of circular dielectric cylinders. In addition to the known Bloch BIC which is symmetric with respect to the axis of the array, we obtain a new BIC which is antisymmetric. The existence domains (in the plane of radius and dielectric constant of the cylinders) of both BICs are determined. The boundaries of these domains correspond to either standing waves which are not protected by symmetry or the opening of the second diffraction channel. Numerical results are also presented to illustrate the discontinuities of transmission and reflection coefficients at the BICs, and the resonant behavior near the BICs.

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A Fourier-matching pseudospectral modal method [PSMM(f)] is developed for analyzing lamellar diffraction gratings or grating stacks. A Chebyshev pseudospectral method is first used to accurately calculate the eigenmodes of the grating layers, and then the Fourier coefficients are matched at the interfaces between the layers. Compared with an existing pseudospectral modal method based on point matching, the PSMM(f) is more robust and accurate. The method performs better than the standard Fourier modal method for gratings involving metals.