dip gives a quantitative measure of the mode-matching between the cavity mode and the Gaussian laser beam input field. In this way, we demonstrate improved coupling efficiency for our far-field optimized devices.

![Reflection dips for cavities with different outward shift parameters, including $s/a =$ 0.100, 0.115, 0.131, and 0.146. Data points are in blue, and the red curves are Fano lineshapes as predicted by our model, with $r_{DBR}$ = 0.95, and the mode-matching being the only free parameter. Larger dips are seen for cavities that were optimized for Gaussian-like far-field profiles. The measured far-field profiles (both H and V) for each of the cavities are shown below the reflection curves.[]{data-label="figfano"}](fig6.eps){width="100.00000%"}

The reflection dips typically exhibit a Fano lineshape, which have been observed previously in PhC cavities[@galli2009; @mccutcheon2011] and can be explained using the temporal coupled-mode theory developed by S. Fan et al.[@joannopoulos2003] Let us describe the photonic crystal membrane cavity as a resonator with central frequency $\omega_0$ and lifetime $\tau$, coupled to two ports, as illustrated in Fig. \[figmodel\](a). In structures with mirror symmetry, the relation between the field in port-2 and port-1 is given by: $$\label{dip1}
\left[
\begin{array}{c}
  E'_1 \\
  E'_2
\end{array}
\right]
=
\left[
\begin{array}{cc}
  \Sigma_{11} & \Sigma_{12} 