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How to "fix" Purcell's formula for leaky optical cavities and plasmonic nanoresonators


Two of the most common and useful metrics for characterizing the properties of optical cavities are the Q value (quality factor) and the effective mode volume V. The Purcell effect is a beautiful example of a situation in which a cavity with a large Q/V ratio enhances the spontaneous emission rate of an atom or quantum dot. In Purcell's original paper, a modest abstract published in the proceedings of the American Physical Society meeting at Cambridge in 1946, Purcell formulated the enhanced spontaneous emission factor in a very elegant way as scaling with Q/V. It is no exaggeration to say that Purcell's formula has been the workhorse for cavity physics for decades, but it turns out to be wrong! At least it turns out to be wrong in general with the way that the modes and effective mode volume are obtained for open and dissipative resonators. In this talk, I will argue that most, if not all, confusion about cavity modes can be removed by a proper treatment within the framework of quasinormal modes (QNMs), defined as the frequency domain solutions to the wave equation with open boundary conditions. Using these QNMs, I will describe a newly developed mode expansion technique that can be used to evaluate the electric field from a dipole emitter at arbitrary positions outside and within optical cavities and plasmonic resonators. I will then introduce a rigorous definition of the Purcell factor and enhanced spontaneous emission factor and point out why the usual expression for effective mode volume is wrong. Several applications of the theory for modelling hybrid plasmonic-coupled emitter systems will be exemplified, including metal-dimer single photon sources and plasmon-mediated entanglement between two quantum dots.