  Spin-bearing color centers in the solid state are promising candidates for
the realization of quantum networks and distributed quantum computing. A
remaining key challenge is their efficient and reliable interfacing to photons.
Incorporating minimally processed membranes into open-access microcavities
represents a promising route for Purcellenhanced spin-photon interfaces: it
enables significant emission enhancement and efficient photon collection,
minimizes deteriorating influence on the quantum emitter, and allows for full
spatial and spectral tunability, key for controllably addressing suitable
emitters with desired optical and spin properties. Here, we study the
properties of a high-finesse fiber Fabry-P\'erot microcavity with integrated
single-crystal diamond membranes by scanning cavity microscopy. We observe
spatially resolved the effects of the diamond-air interface on the cavity mode
structure: a strong correlation of the cavity finesse and mode structure with
the diamond thickness and surface topography, significant transverse-mode
mixing under diamond-like conditions, and mode-character-dependent
polarization-mode splitting. Our results reveal the influence of the diamond
surface on the achievable Purcell enhancement, which helps to clarify the route
towards optimized spin-photon interfaces.
