  Radial Bragg gratings are commonly used to enhance light extraction from
quantum emitters, but lack a well-suited, fast simulation method for
optimization beyond periodic designs. To overcome this limitation, we propose
and demonstrate an algorithm based on the transfer matrix model (TMM) to
calculate the free-space emission of such gratings. Using finite difference
time domain (FDTD) simulations, we characterize the free-space emission and
transfer matrices of single grating components. Our TMM then combines any
number of components to receive the total emission. Randomized benchmarks
verify that results from our method agree within 98 % with FDTD while reducing
simulation time by one to two orders of magnitude. The speed advantage of our
approach is shown by maximizing emission of a fifteen-trench circular grating
into a Gaussian mode. We expect that our novel algorithm will facilitate the
optimization of radial gratings, enabling quantum light sources with
unprecedented collection efficiencies.
