Linear and nonlinear Fano resonance in the main chain-structure of additional defects with an isolated ring composed of defects

As is well known, Fano resonance originates from the interference between a continuum energy band and an embedded discrete energy level. We study transmission properties of the discrete chain-structure of additional defects with an isolated ring composed of N defect states, and obtain the analytical transmission coefficient of similar Fano formula. Using the formula, we reveal conditions for perfect reflections and transmissions due to either destructive or constructive interferences. It is found that a nonlinear Kerr-like response leads to bistable transmission, and for either linear cases or nonlinear ones, the defects in main arrays have a major impact on perfect reflections, but has no effect on perfect transmission.