  Defect centers in diamond are promising building blocks for quantum networks
thanks to a long-lived spin state and bright spin-photon interface. However,
their low fraction of emission into a desired optical mode limits the
entangling success probability. The key to overcoming this is through Purcell
enhancement of the emission. Open Fabry-Perot cavities with an embedded diamond
membrane allow for such enhancement while retaining good emitter properties. To
guide the focus for design improvements it is essential to understand the
influence of different types of losses and geometry choices. In particular, in
the design of these cavities a high Purcell factor has to be weighed against
cavity stability and efficient outcoupling. To be able to make these trade-offs
we develop analytic descriptions of such hybrid diamond-and-air cavities as an
extension to previous numeric methods. The insights provided by this analysis
yield an effective tool to find the optimal design parameters for a diamond-air
cavity.
