Real transmission and reflection zeros of periodic structures with a bound state in the continuum

For lossless periodic structures with proper symmetry, the transmission and reflection spectra often have peaks and dips that are truly $100%$ and $0%$, respectively. The full peaks and zero dips typically appear near resonant frequencies, and they are robust with respect to structural perturbations that preserve the required symmetry. However, current theories on the existence of full peaks and zero dips are incomplete and difficult to use. For periodic structures with a bound state in the continuum (BIC), we present a theory on the existence of real transmission and reflection zeros that correspond to the zero dips in the transmission and reflection spectra. Our theory is relatively simple, complete, and easy to use. Numerical examples are presented to validate the theory.