Multimode Phonon Cooling via Three Wave Parametric Interactions with Optical Fields

We discuss the possible cooling of different phonon modes via three wave mixing interactions of vibrational and optical modes. Since phonon modes exhibit a variety of dispersion relations or frequency spectra with diverse spatial structures, depending on the shape and size of the sample, we formulate our theory in terms of relevant spatial mode functions for the interacting fields in any given geometry. We discuss the possibility of Dicke like collective effects in phonon cooling and present explicit results for simultaneous cooling of two phonon modes via the anti-Stokes up