Bifurcation of bound states in the continuum in periodic structures

In lossless dielectric structures with a single periodic direction, a bound state in the continuum (BIC) is a special resonant mode with an infinite quality factor ( Q factor). The Q factor of a resonant mode near a typical BIC satisfies Q ∼1/( β − β ∗ ) 2 , where β and β ∗ are the Bloch wavenumbers of the resonant mode and the BIC, respectively. However, for some special BICs with β ∗ =0 (referred to as super -BICs by some authors), the Q factor satisfies Q ∼ 1/ β 6 . Although super -BICs are usually obtained by merging a few BICs through tuning a structural parameter, they can be precisely characterized by a mathematical condition. In this Letter, we consider arbitrary perturbations to structures supporting a super -BIC. The perturbation is given by δF ( r ), where δ is the amplitude and F ( r ) is the perturbation profile. We show that for a typical F ( r ), the BICs in the perturbed structure exhibit a pitchfork bifurcation around the super -BIC. The number of BICs changes from one to three as δ passes through zero. However, for some special profiles F ( r ), there is no bifurcation, i.e., there is only a single BIC for δ around zero. In that case, the super -BIC is not associated with a merging process for which δ is the parameter.