Mie Plasmons: Modes Volumes, Quality Factors, and Coupling Strengths (Purcell Factor) to a Dipolar Emitter

Abstract

Using either quasistatic approximation or exact Mie expansion, we characterize the localized surface plasmons supported by a metallic spherical nanoparticle. We estimate the quality factorQnand define the effective volumeVnof thenth mode in such a way that coupling strength with a neighbouring dipolar emitter is proportional to the ratioQn/Vn(Purcell factor). The role of Joule losses, far-field scattering, and mode confinement in the coupling mechanism is introduced and discussed with simple physical understanding, with particular attention paid to energy conservation.

Purcell factor of spherical Mie resonators

“…The normal modes of the system and their eigenfrequencies are computed using the formulas given by eqs. (17)(18)(19)(20)(21)(22)(23)(24)(25)(26). It will be shown that even in the more symmetric situations, the Purcell factor has contributions from more than one normal mode.…”

“…The reason stems from the definition of the normal modes of the systems given by eqs. (19)(20)(21)(22)(23)(24)(25)(26). In particular, the only dependence of E (x) j,mz,q in m z comes via Y jmz and χ mz , and it can be seen that an integral over 2π equalizes the behavior of the modes with m z and −m z .…”

“…(6) using the normal modes defined in eqs. (19)(20)(21)(22)(23)(24)(25)(26) and the eigenfrequencies given by eqs. (17)(18):…”

“…These changes can be summarized in two: extension of the Purcell factor to a scenario where the dipolar emitter couples to a superposition of modes E µ of the cavity; and redefinition of the effective volume V µ of each of the E µ normal modes of the resonator. One of the first attempts to redefine the effective volume for spherical cavities was proposed by Colas des Francs et al in 2012 [21,22]. The proposed method used the equality between the classical normalized decay rates [23][24][25] and the Purcell factor given by eq.(1).…”

Optical amplification of surface plasmon polaritons: review

“…Experimentally, spectrally narrow emission at 530 nm from gold nanoparticles coated with dye-doped glass (OG-488 in silica) has been observed 35 but no independent determination of the material gain of the doped glass was provided. In general, the interaction between realistic gain materials and bright and dark plasmon modes of spherical metal nanoparticles 36 and more complex nanostructures 37,38 will certainly be an active topic of theoretical and experimental research in the coming years.…”