to 610 nm, the air-mode band-edge falls from a normalized frequency of 0.441 to 0.409. This corresponds to increasing $n_{eff}$, which decreases $\omega_{o}$ and reduces coupling to radiation modes, and is reflected by $Q_o$ increasing towards $10^6$ as shown in Fig. \[figureOne\](c). A simulated optical mode electric field profile ($E_y$) for a $t$ = 500 nm device is shown in Fig. \[figureOne\](d), whose corresponding $Q_o$ is $\sim 10^5$. In fabricated Si$_{3}$N$_{4}$ PCC devices, $t$ generally ranges from 200-450 nm [@Khan:11; @Grutter:15IEEE]. In consideration of the single-mode condition and with a desire for a high-$Q_o$, we will use a $t \sim 500~\text{nm}$ for the remainder of this analysis.

To design a nanobeam PCC with elliptical holes that will form the basis of the optomechanical devices discussed below, we relax the constraint of requiring holes to be circular, leave the other nanobeam dimensions unchanged, and again apply the deterministic design procedure [@Hryciw:13]. There are a continuum of hole shapes with varying ellipticity that maintain the same $\omega_o$ as the round hole device, as discussed below. We begin by choosing the centre hole with maximum ellipticity as constrained by the waveguide width. The resulting device is shown in Fig. \[figureTwo\](a). The cavity centre 