The Wave Annihilation Technique and the Design of Nonreflective Coatings

We develop theory and algorithms for the design of coatings which either eliminate or enhance reflection of waves from surfaces. For steady-state harmonic waves with continuous frequency spectrum that covers an arbitrarily prescribed frequency band, coatings are designed that essentially eliminate reflections of all frequencies within the band. Although we focus on acoustic waves in elastic media, the methods developed here can be adapted to electromagnetism or other phenomena governed by variants of the linear wave equation. To create a nonreflective coating which is to operate in a frequency band [Ω0,Ω1], we select n frequencies, uniformly distributed in the band, and design an n-layer coating of given thickness that completely eliminates 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 arbitrary frequency ω ∈ (Ω0,Ω1) is O(1/n) if Ω0 = 0, while it is O(αn) if Ω0 > 0, where 0 < α < 1. Furthermore, 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 reflectivity (or, equivalently, minimizes transmissivity). We show that through a proper layering technique, it is possible to obtain transmissivity of O(αn), 0 < α < 1, in an n-layer design.