On the Robustness of Propagating Bound States in the Continuum

Bound states in the continuum (BICs) are localized eigenmodes with their frequencies in the radiation continuum of scattering states. The existence of a BIC implies the loss of uniqueness for scattering problems with given incident waves. Perturbed wave systems close to the ideal ones with a BIC exhibit strong resonance effects that are essential to numerous practical applications. A question of fundamental importance is whether a BIC is robust, i.e., whether it can continue its existence when the structure is slightly perturbed. In an earlier work [Yuan and Lu, Optics Letters, Vol.~42, pp.~4490-4493, 2017], for a class of BICs governed by the two-dimensional (2D) Helmholtz equation, which are not trivially protected by symmetry, we uncovered the conditions that ensure robustness and formally constructed the BIC in perturbed systems using a perturbation method. In this paper, we present a rigorous theory on the robustness of BICs in 2D dielectric structures with a single periodic direction. Specifically, we analyze the solvability and provide estimates for each order in the perturbation series, and prove the convergence of the series.