Design of a quasi-2D photonic crystal optomechanical cavity with tunable, large x²-coupling

We present the optical and mechanical design of a mechanically compliant quasi-two-dimensional photonic crystal cavity formed from thin-film silicon in which a pair of linear nanoscale slots are used to create two coupled high-Q optical resonances. The optical cavity supermodes, whose frequencies are designed to lie in the 1500 nm wavelength band, are shown to interact strongly with mechanical resonances of the structure whose frequencies range from a few MHz to a few GHz. Depending upon the symmetry of the mechanical modes and the symmetry of the slot sizes, we show that the optomechanical coupling between the optical supermodes can be either linear or quadratic in the mechanical displacement amplitude. Tuning of the nanoscale slot size is also shown to adjust the magnitude and sign of the cavity supermode splitting 2J, enabling near-resonant motional scattering between the two optical supermodes and greatly enhancing the x²-coupling strength. Specifically, for the fundamental flexural mode of the central nanobeam of the structure at 10 MHz the per-phonon linear cross-mode coupling rate is calculated to be g˜_+-/2π=1MHz, corresponding to a per-phonon x²-coupling rate of g˜^'/2π=1kHz for a mode splitting 2J/2π = 1 GHz which is greater than the radiation-limited supermode linewidths.