  We realize mirror templates on the tips of optical fibers using a single-shot
CO$_2$ laser ablation procedure. We perform a systematic study of the influence
of the pulse power, pulse duration, and laser spot size on the radius of
curvature, depth, and diameter of the mirror templates. We find that these
geometrical characteristics can be tuned to a larger extent than has been
previously reported, and notably observe that compound convex-concave shapes
can be obtained. This detailed investigation should help further the
understanding of the physics of CO$_2$ laser ablation processes and help
improve current models. We additionally identify regimes of ablation parameters
that lead to mirror templates with favorable geometries for use in cavity
quantum electrodynamics and optomechanics.
