The wave annihilation technique and the design of nonreflective coatings

We develop theory and algorithms for the design of coatings which either el iminate or enhance reflection of waves from surfaces. For steady-state harm onic waves with continuous frequency spectrum that covers an arbitrarily pr escribed frequency band, coatings are designed that essentially eliminate r eflections of all frequencies within the band. Although we focus on acousti c waves in elastic media, the methods developed here can be adapted to elec tromagnetism or other phenomena governed by variants of the linear wave equ ation. To create a nonreflective coating which is to operate in a frequency band [ Omega(0),Omega(1)], we select n frequencies, uniformly distributed in the b and, and design an n-layer coating of given thickness that completely elimi nates reflections of waves at these frequencies. We show that if n is large , then the reflectivity of the coating designed by our method is small for all frequencies in the band. More precisely, the reflectivity at an arbitra ry frequency omega is an element of (Omega(0),Omega(1)) is O (1/n) if Omega (0) = 0, while it is O (alpha(n)) if Omega(0) > 0, where 0 < alpha < 1. Fur thermore, extensive numerical studies show that when this discrete n-layer coating is smoothed out by spline interpolation, the reflectivities remain small not only for frequencies in the original band but also for all larger frequencies. We also describe a procedure for designing coatings that maximizes reflecti vity (or, equivalently, minimizes transmissivity). We show that through a p roper layering technique, it is possible to obtain transmissivity of O(alph a(n)), 0 < alpha < 1, in an n-layer design.