to the first mirror holes with radius 215 nm. $N_m$ = 15 mirror holes were then added to both sides. The bandstructure for the centre hole unit cell plotted in Fig. \[figureOne\](b) shows the fundamental (TE-like, $y$-odd parity, $z$-even parity) air and dielectric mode frequencies in normalized units ($\omega a/2\pi c~=~a/\lambda$) as a function of $k_x$ for varying $t$. From the bandstructure, we see that the air-mode band-edge frequency for the $t=350~\text{nm}$ device is relatively close to the light-line. When we compute the fundamental optical mode of the full $t=350~\text{nm}$ PCC using a 3D finite-difference time-domain (FDTD) simulation [@ref:oskooi2010mff] we find a $Q_o \sim 10^4$. A similar $Q_o$ is observed if we keep the cavity hole radii constant and quadratically decrease the lattice constant from the centre of the cavity to the mirrors [@Grutter:15IEEE].

To increase $Q_o$ we then studied the effect of changing $t$. In nanobeam PCC’s, light is index-guided in the $z$ direction, and typically $t \le \lambda / 2n$ is chosen so that the underlying waveguide remains single-mode over the frequency range of interest [@Zhang:09]. Figure \[figureOne\](b) shows that increasing the thickness pushes the band structure lower in frequency, away from the light-line: as $t$ is increased 