Diffraction of plane waves by a periodic array of nonlinear circular cylinders

It is well known that standing waves, as special bound states in the radiation continuum, may exist on a periodic array of dielectric cylinders at a discrete set of frequencies if the medium is linear. Recent numerical studies indicate that nonlinear standing waves could exist continuously with respect to the frequency on a periodic array of cylinders with a Kerr nonlinearity. In this paper, we study the diffraction of a normal incident plane wave by a periodic array of circular cylinders with a Kerr nonlinearity. Using a perturbation method and a highly accurate numerical method, we show that a plane incident wave may couple to a nonlinear standing wave, and in general, there are four different couplings leading to four asymmetric solutions in two pairs. The existence of these asymmetric solutions provides another example for the symmetry-breaking phenomenon. Importantly, it seems that the asymmetric solutions (thus the symmetry-breaking phenomenon) appear for incident waves of arbitrarily low intensity.