hole with semi-minor and semi-major axis dimensions ($R_{x_\textrm{c}}$, $R_{y_\textrm{c}}$) = (119.4, 350) nm has $a/\lambda~=~0.432$, and Fig. \[figureTwo\](b) shows how the bandstructure of this unit cell compares with that of the 150 nm radius round centre hole unit cell. If we quadratically taper to a mirror hole ($R_{x_\textrm{m}}$, $R_{y_\textrm{m}}$) = (210, 285) nm with maximum of the mirror strength following: $$R_{x_j,y_j}= R_{x_\textrm{c},y_\textrm{c}}+ (j/N_\textrm{c})^2(R_{x_\textrm{m},y_\textrm{m}}-R_{x_\textrm{c},y_\textrm{c}})$$ for integer $j\in[-N_\textrm{c},N_\textrm{c}]$, we find a simulated 3D $Q_o$ = $4.4~\times~10^4$.

For the purpose of comparison, we also simulated a device with a highly elliptical centre hole with the same fill factor as the round centre hole of the device in Fig. \[figureOne\](a). In Fig. \[figureTwo\](c), the fill factors of ellipses of dimensions ($R_x,R_y$) are calculated; the black line shows hole dimensions with the same fill factor as the round centre hole. The bandstructure for the hole with dimensions ($R_{x_c},R_{y_c}$) = (64.3, 350) nm is shown in Fig. \[figureTwo\](b) (red line) to have much lower bandedge frequencies; substituting this highly elliptical hole for the round cavity centre hole and tapering quadratically to the 215 nm round mirror holes results in the device shown in Fig. \[figureTwo\](d) with a 3D simulated $Q_o$ = $5.3~\times~10^5$. This is more than 