Engineering complex dispersion relations

Motivated by recent theoretical and experimental interest in metamaterials comprising nonlocal coupling terms, we present an analytic framework to realize materials with arbitrary complex dispersion relations. Building on the inverse design method proposed by Kazemi [], which demonstrated arbitrary real dispersion relations with real nonlocal coupling terms, we show that using nonconservative, nonlocal couplings not only can band structures be drawn, but arbitrary attenuation of the supported modes can be achieved by tuning their trajectory in the complex frequency plane. This is demonstrated using a canonical mass-spring lattice with higher-order spatial connections; we consider the energy velocity in such systems and show that competing loss and gain of the complex interactions can produce exotic wave dispersion.