line) to overlap very closely with that of the highly elliptical hole.

![(a) Schematic of the split-beam cavity with gap width $g$ and corresponding $E_y$ field profile. (b) Gap unit cell band edge frequencies (green) and $E_y$ field profiles for the optical modes at the points marked (and in blue at the same $w/t$ ratio as in Ref. [@Hryciw:13]) showing that the modes for the gap for each device do not cross. (c,d) Design and field profile of the paddle nanocavity with the axial motion of the paddle indicated (supports not shown in (d)).[]{data-label="figureThree"}](figure3v2.pdf){width="11.5cm"}

![(a) Schematic of the paddle nanocavity with $L$ = 50 $\mu$m supports. (b) Mechanical frequency $\omega_m$ (red) and effective mass $m_{eff}$ (blue) and (c) the single-photon to two-phonon coupling rate $\Delta\omega_{o}$ (red) given from the zero-point fluctuations $x_{zpf}$ (blue).[]{data-label="figureFour"}](figure4v7.pdf){width="11cm"}

A similar process can be followed to create a Si$_3$N$_4$ paddle nanocavity such as the device shown in Figs. \[figureThree\](c,d). In this device a Si$_3$N$_4$ paddle is inserted between the split-beam cavity nanocantilevers, with the same gap $g = 112\text{nm}$ between each nanocantilever and the paddle. The paddle length is chosen following the procedure used to design Si paddle nanocavities in Ref. [@Healey:15]. When simulated with a 