Fano resonance and wave transmission through a chain structure with an isolated ring composed of defects

We consider a discrete model that describes a linear chain of particles coupled to an isolated ring composed of N defects. This simple system can be regarded as a generalization of the familiar Fano-Anderson model. It can be used to model discrete networks of coupled defect modes in photonic crystals and simple waveguide arrays in two-dimensional lattices. The analytical result of the transmission coefficient is obtained, along with the conditions for perfect reflections and transmissions due to either destructive or constructive interferences. Using a simple example, we further investigate the relationship between the resonant frequencies and the number of defects N, and study how to affect the numbers of perfect reflections and transmissions. In addition, we demonstrate how these resonance transmissions and refections can be tuned by one nonlinear defect of the network that possesses a nonlinear Kerr-like response.