Evaluation of second order optomechanical coupling strength in photonic crystal cavities including the case of degenerated modes

Abstract

This paper presents the potentialities of the perturbation method to provide physical interpretations of optomechanical coupling in a simultaneous photonic and phononic crystal cavity. The classical perturbation method as applied to photonic modes is recalled. The method is then extended to the degenerated modes including second order perturbation. The method ends with semi-analytical expressions for the first and second order correction terms. These semi-analytical expressions enable, on the basis of symmetry criteria, to analyze and/or predict the modulation behaviors and ultimately lead to design rules. Special attention is given to the case of degenerated optical modes and the event of degeneracy lifting is also analyzed. In order to illustrate this method, we use the simplest example of an L1 cavity: a missing hole in a square lattice. This structure displays a miscellaneous situation exhibiting interesting behaviors that may be exploited to clearly highlight the versatility of the perturbation method. We select two cases: one where the first order is dominant and practically hides the second order behavior whatever the modes are degenerated or not; the other where the first order correction vanishes in the case of non-degenerate modes, resulting in a second order behavior, but in the case of degenerate modes, both first and second orders express themselves.