relevant parameter, $L/\lambda$ will be different for each device fabricated on a wafer; however, we chose to use $L=925$ nm for our fabricated samples.

![Top: FDTD simulated Q (blue solid curve) and fiber mode-matching (green dashed curve) of the H dipole mode as a function of air gap separation $L$. The best compromise, indicated by the red ellipse, occurs for $L=925$ nm. Bottom: Simulated far-field radiation patterns for devices of different $L$ values. Each profile is normalized by its own maximum intensity, with a scale bar shown at the far right. As the air gap separation increases, the mode profiles become more Gaussian, which show better coupling to a single-mode fiber. The white concentric circles correspond to NA = 0.2, 0.4, 0.6, 0.8, and 1.0.[]{data-label="figair1"}](fig3.eps){width="50.00000%"}

The effect of the bottom reflector on the radiation pattern can be understood by considering the interference of light that reflects multiple times between the PhC membrane and the bottom reflector. Following the treatment of Kim et al.[@kim2006], let the membrane be approximated as a uniform dielectric slab with an effective refractive index, $n_{eff}$, and $L$ and $d$ are the air gap thickness and the slab thickness, respectively. Waves initially propagate upward and downward from 