Modified quasi-phase-matching and spectral shaping in a nonlinear whispering gallery mode resonator
Résumé
We derive an analytic solution for the spectral and radial distribution of a sum-frequency generated wave, in a cylindrical whispering gallery mode (WGM) resonator with an arbitrarily modulated nonlinear coefficient along the angular direction. The cylindrical geometry dictates different wave vectors, spatial distribution, and spectrum of the generated wave with respect to a linear cavity with an identical round-trip length. The geometrical correction term to the phase-mismatch parameter is derived by exact analytic solution; it is found to be different by a factor of 2 with respect to the slowly varying envelope approximation result for WGMs, which has been previously suggested in the literature. This change occurs because the properties of the generated wave in the nonlinear process depend on the spatial distribution of the pump and signal, in addition to the geometrical boundary conditions. We also propose a scheme for the spectral shaping of optical pulses in a WGM resonator, based on a method taken from computer-generated holograms. This method, apart from having an advantage in the form of a considerable reduction of the necessary pump power due to the intracavity enhancement by the resonator, also should allow for arbitrarily shaping of the pulse envelope as well as the cancellation of group velocity dispersion per round trip.