the $E_y$ component of the H mode. In addition, the $E_x$ far-field pattern is most intense at large angles and so would be virtually undetected by a microscope objective with an NA of 0.8 or smaller, as shown in Fig. \[fignfff\](e). Therefore, we can consider the H mode to be primarily $y$-polarized. For the V mode, the reverse is true, and it can therefore be considered to be primarily $x$-polarized.

We use finite-difference-time-domain (FDTD) software[@fdtd] to simulate the electromagnetic field in our PhC cavities, and we perform a near- to far-field transformation[@vuckovic2002] using the E-field components at a plane 10 nm above the membrane surface to calculate the far-field emission profiles of the cavity modes. Simulation parameters for the near-field and far-field patterns shown in Fig. \[fignfff\](d, e) are the following: lattice constant $a = 260$ nm, radius $r = 78$, and membrane thickness $d=130$ nm. In fabricated devices, the PhC membrane is separated from a Distributed-Bragg-Reflector (DBR) or the bottom substrate by an air gap. This bottom reflector, which significantly affects the far-field patterns[@kim2006] and improves collection efficiency of light emitted out of the bottom surface of the membrane, is modeled in our simulations as a dielectric substrate of 