PCC following the standard deterministic design procedure. (b) The band structures of a unit cell containing the three centre hole shapes and gap discussed: 150 nm round (purple), standard design ellipse (yellow), highly elliptical (red), 112 nm gap (blue). (c) Fill factors (FF) of ellipses with dimensions ($R_x,R_y$). The black line denotes elliptical hole dimensions with a fill factor similar to the round centre hole. Fill factors of the three centre hole shapes are noted, with the gap FF matching that of the highly elliptical hole. (d) Schematic of the PCC with a highly elliptical centre hole and hole axes quadratically tapering to round mirror holes, and the resultant $E_y$ field profile. (e) Mirror strength (MS) trajectories of the hole tapering of the devices in (a) (dotted) and (d) (dashed). The hole dimensions that have a mirror strength at the target frequency $\omega_o$ are shown with the solid white line.[]{data-label="figureTwo"}](figure2v6.pdf){width="12cm"}

Initially we restricted ourselves to circular holes. A cavity centre hole with radius 150 nm was found to have an air-mode band-edge frequency $\omega_2 \sim \omega_o$. To form an air-mode nanocavity, the surrounding hole radii were tapered quadratically over $N_c$ = 12 holes symmetrically on either side of the nanobeam 