Impact of Kerr nonlinearity on the whispering gallery modes of a microsphere

To describe the temporal evolution of the mode amplitude of a spherical microcavity, a nonlinear equation is developed by considering loss and Kerr nonlinear effect as perturbations. In order to study the impact of Kerr nonlinearity, the tensor components of χ((3)) in spherical coordinates are calculated. To describe the impact of Kerr nonlinearity, effective mode volume and effective nonlinear coefficient are defined. We found that the resonant modes undergo a negative frequency shift proportional to the injected energy, consistant with the reported experimental observations.