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We present the optical and mechanical design of a mechanically compliantquasi-two-dimensional photonic crystal cavity formed from thin-film silicon inwhich a pair of linear nanoscale slots are used to create two coupled high-$Q$optical resonances. The optical cavity supermodes, whose frequencies aredesigned to lie in the $1500$~nm wavelength band, are shown to interactstrongly with mechanical resonances of the structure whose frequencies rangefrom a few MHz to a few GHz. Depending upon the symmetry of the mechanicalmodes and the symmetry of the slot sizes, we show that the optomechanicalcoupling between the optical supermodes can be either linear or quadratic inthe mechanical displacement amplitude. Tuning of the nanoscale slot size isalso shown to adjust the magnitude and sign of the cavity supermode splitting$2J$, enabling near-resonant motional scattering between the two opticalsupermodes and greatly enhancing the $x^2$-coupling strength. Specifically, forthe 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$\tilde{g}_{+-}/2\pi = 1$~MHz, corresponding to a per-phonon $x^2$-couplingrate of $\tilde{g}'/2\pi=1$~kHz for a mode splitting $2J/2\pi = 1$~GHz which isgreater than the radiation-limited supermode linewidths.

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