width $w$ = 760 nm and lattice constant $a$ = 670 nm such that the target PCC mode wavelength $\lambda_o$ $\sim$ 1550 nm ($\omega_o~\sim~200$ THz, $a/\lambda_o~\sim$ 0.432) can be achieved for the range of hole sizes considered here. Initially the nanobeam thickness was set to $t = 350~\text{nm}$, then varied to study its effect on the PCC modes. As part of this design recipe, the band structure and band-edge frequencies of unit cells of the waveguide for varying hole shape were calculated using a 3D frequency-domain eigensolver [@Johnson:01] with high spatial resolution. From these bandstructure calculations, we first find the central cavity hole dimensions that brings the unit-cell air-mode band-edge frequency close to the target $\omega_o$. We then find the outer mirror hole dimensions to maximize the mirror strength $\gamma$: $$\gamma=\left( \frac{(\omega_2-\omega_1)^2}{(\omega_2+\omega_1)^2}-\frac{(\omega_\textrm{o}-\omega_{mid})^2}{(\omega_{mid})^2} ) \right) ^{1/2}, 
% \gamma=[(\omega_2-\omega_1)^2/(\omega_2+\omega_1)^2-(\omega_\textrm{o}-\omega_{mid})^2/(\omega_{mid})^2]^{1/2},$$ experienced by the mode at $\omega_\textrm{o}$ that is within the mirror hole bandgap, defined by the dielectric and air band-edge frequencies $\omega_1$ and $\omega_2$, respectively, of the mirror hole unit cell [@Hryciw:13]. Here $\omega_{mid} = (\omega_2+\omega_1)/2$ is the mid bandgap frequency, and $\gamma$ is proportional to the inverse attenuation length.

![(a) The dielectric pattern and $E_y$ field profile of the elliptical hole air-mode 