Reciprocity approach for calculating the Purcell effect for emission into an open optical system

Based on the reciprocity theorem, we present a formalism to calculate the power emitted by a dipole source into a particular propagating mode leaving an open optical system. The open system is completely arbitrary and the approach can be used in analytical calculations but may also be combined with numerical electromagnetic solvers to describe the emission of light sources into complex systems. We exemplify the use of the formalism in numerical simulations by analyzing the emission of a dipole that is placed inside a cavity with connected single mode exit waveguide. Additionally, we show at the example of a practical ring resonator system how the approach can be applied to systems that offer multiple electromagnetic energy decay channels. As a consequence of its inherent simplicity and broad applicability, the approach may serve as a powerful and practical tool for engineering light-matter-interaction in a variety of active optical systems.