  With a highly coherent, optically addressable electron spin, the nitrogen
vacancy (NV) centre in diamond is a promising candidate for a node in a quantum
network. However, the NV centre is a poor source of coherent single photons
owing to a long radiative lifetime, a small branching ratio into the
zero-phonon line (ZPL) and a poor extraction efficiency out of the high-index
host material. In principle, these three shortcomings can be addressed by
resonant coupling to a single mode of an optical cavity. Utilising the
weak-coupling regime of cavity electrodynamics, resonant coupling between the
ZPL and a single cavity-mode enhances the transition rate and branching ratio
into the ZPL. Furthermore, the cavity channels the light into a well-defined
mode thereby facilitating detection with external optics. Here, we present an
open Fabry-Perot microcavity geometry containing a single-crystal diamond
membrane, which operates in a regime where the vacuum electric field is
strongly confined to the diamond membrane. There is a field anti-node at the
diamond-air interface. Despite the presence of surface losses, quality factors
exceeding $120\,000$ and a finesse $\mathcal{F}=11\,500$ were observed. We
investigate the interplay between different loss mechanisms, and the impact
these loss channels have on the performance of the cavity. This analysis
suggests that the "waviness" (roughness with a spatial frequency comparable to
that of the microcavity mode) is the mechanism preventing the quality factors
from reaching even higher values. Finally, we apply the extracted cavity
parameters to the NV centre and calculate a predicted Purcell factor exceeding
150.
