Nonlinear Eigenvalue Problem for Optimal Resonances in Optical Cavities

The paper is devoted to optimization of resonances in a 1-D open optical cavity. The cavity's structure is represented by its dielectric permittivity function epsilon(s). It is assumed that epsilon(s) takes values in the range 1 <= epsilon(1) <= epsilon(s) <= epsilon(2). The problem is to design, for a given (real) frequency alpha, a cavity having a resonance with the minimal possible decay rate. Restricting ourselves to resonances of a given frequency alpha, we define cavities and resonant :modes with locally extremal decay rate, and then study their properties. We show that such locally extremal cavities are 1-D photonic crystals consisting of alternating layers of two materials with extreme allowed dielectric permittivities epsilon(1) and epsilon(2). To find thicknesses of these layers, a nonlinear eigenvalue problem for locally extremal resonant modes is derived. It occurs that coordinates of interface planes between the layers can be expressed via arg-function of corresponding modes. As a result, the question of minimization of the decay rate is reduced to a four-dimensional problem of finding the zeroes of a function of two variables.