  We present a monolithic semiconductor microcavity design for enhanced
light-matter interaction and photon extraction efficiency of an embedded
quantum emitter such as a quantum dot or color center. The microcavity is a
hemispherical Fabry-Perot design consisting of a planar back mirror and a top
curved mirror. Higher order modes are suppressed in the structure by reducing
the height of the curved mirror, leading to efficient photon extraction into a
fundamental mode with a Gaussian far-field radiation pattern. The cavity
finesse can be varied easily by changing the reflectivity of the mirrors and we
consider two specific cases: a low-finesse structure for enhanced broad band
photon extraction from self-assembled quantum dots and a moderate-finesse
cavity for enhanced extraction of single photons from the zero-phonon line of
color centers in diamond. We also consider the impact of structural
imperfections on the cavity performance. Finally, we present the fabrication
and optical characterisation of monolithic GaAs hemispherical microcavities.
