by the red squares (labeled ‘with R’). (b) Simulated collection efficiency as a function of $s/a$ for NA $=$ 0.5 (in blue), 0.8 (in red), and 0.9 (in green). Simulations with (without) a bottom reflector are represented by squares (circles). For simulations without the bottom reflector, the collection efficiency maximum is 50$\%$; however, with the bottom reflector, the maximum collection efficiency is 100$\%$. The ideal compromise value, $s/a = 0.115$, is indicated by the cyan ellipse, where the Q is 15,000 and the coupling efficiency for an objective with NA of 0.8 is 80$\%$.[]{data-label="figQsh"}](fig2.eps){width="55.00000%"}

From our FDTD results, we calculate coupling efficiency to a microscope objective and to a single-mode fiber. For a given far-field pattern, the collection efficiency by a microscope objective of a given NA is found by dividing the power radiated within a certain angle by the total power radiated into the half-space above the membrane. For example, a microscope objective with an NA of 0.8 will collect light emitted within an angle of $53^\circ$ from the $\Gamma$ point in k-space. For simulations without the bottom reflector, the collection efficiency maximum is 50$\%$ because only the light emitted from the top of the membrane is collected, with 