Theory of free space coupling to high-Q whispering gallery modes

A theoretical study of free space coupling to high-Q whispering gallery modes both in circular and deformed microcavities are presented. In the case of a circular cavity, both analytical solutions and asymptotic formulas are derived. The coupling efficiencies at different coupling regimes for cylinder incoming wave are discussed, and the maximum efficiency is estimated for the practical Gaussian beam excitation. In the case of a deformed cavity, the coupling efficiency can be higher if the excitation beam can match the intrinsic emission well and the radiation loss can be tuned by adjusting the degree of deformation. Employing an abstract model of slightly deformed cavity, we found that the asymmetric and peak like line shapes instead of the Lorentz-shape dip are universal in transmission spectra due to multi-mode interference, and the coupling efficiency can not be estimated from the absolute depth of the dip. Our results provide guidelines for free space coupling in experiments, suggesting that the high-Q ARCs can be efficiently excited through free space which will stimulate further experiments and applications of WGMs based on free space coupling.