  Using an evolutionary algorithm combined with a gradient descent method we
design optical cavities with significantly enhanced strong coupling rates
between cavity photons and a single quantum emitter. Our approach allows us to
find specially designed non-spherical mirrors which lead to high-finesse cavity
eigenmodes with large field enhancement at the center of the cavity. The method
is based on adding consecutive perturbations to an initial spherical mirror
shape using the gradient descent method for optimization. We present mirror
profiles suitable for fabrication which demonstrate higher cavity cooperativity
than any spherical cavity of the same size. Finally, we demonstrate numerically
how such a cavity enhances the operation frequency and purity of coupling a
Ca$^+$ ion to an optical fiber photon.
