Bound states in the continuum via singular transfer matrices

Bound states in the continuum (BICs) in photonic crystal slabs are Bloch eigenstates with zero linewidth and infinite lifetimes, though practical devices exhibit quasi-BICs due to geometric imperfections and material losses. Despite these limitations, quasi-BICs still maintain high $Q$ factors, enabling applications in lasing, high harmonic generation, and biosensing. The selection of specific geometrical or material parameters is essential to achieve devices that sustain BICs. Our work presents a framework for generating BICs by driving the transfer matrix of a device to a singular point, where it becomes noninvertible, thus defining the BIC states. This method applies to nonhomogeneous, tessellated structures and is supported by closed-form analytical expressions, facilitating efficient parametric exploration. The proposed framework aligns with the conjecture that the most robust BICs are associated with topological defects in specific parameter spaces. We identify regions in parameter space supporting BICs and validate our theoretical predictions through full-wave numerical simulations. The framework extends to a wide range of photonic structures, enabling robust design and optimization of devices for diverse applications.