 Quarter-wave Bragg stack

At normal incidence, alternating layers with indices $n_H,n_L$ and optical thicknesses $n_H{d}_H=n_L{d}_L={\lambda }_0/4$ give reinforcing reflected waves near design wavelength ${\lambda }_0$. Their periodic transfer matrix has a stop band when its Bloch phase is complex. A finite stack has nonzero transmission and its exterior impedances matter. Introducing a suitably phased central defect can support a transmission resonance within the stop band; a one-dimensional stop band is not a complete three-dimensional photonic gap.