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arXiv:1504.05787

The $E1$ & $M1$ Spontaneous Decay Rates for an Emitter Inside a Cavity Within a Medium

Jaideep Taggart Singh

Published Apr 21, 2015• 0 citations
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Abstract

We discuss the $E1$ and $M1$ spontaneous decay rates of the an emitter residing inside of a real cavity carved out of a vast, uniform, homogenous, isotropic, linear, lossless, dispersionless, and continuous medium. The ratio of the medium rate to vacuum rate is given by $Γ_m/Γ_0 = [G(u)]^2 n^3 / u$, where $G(u) = 3u/(2u+1)$ is the local field correction factor, $n = \sqrt{εμ/(ε_0μ_0)}$ is the index of refraction of the medium, $ε(ε_0)$ is the electric permitivity of the medium (vacuum), $μ(μ_0)$ is the magnetic permeability of the medium (vacuum), and $u = ε/ε_0$ for $E1$ transitions or $u = μ_0/μ$ for $M1$ transitions.

DOI
10.48550/arxiv.1504.05787

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The spontaneous-emission rates of emitters embedded in a thin layer 0 of a loss-free dielectric depend on their optical environment, i.e., on the optical properties of the two media 1 and 2 adjoining layer 0. For electric- and magnetic-dipole transitions the spontaneous-emission rates, normalized with respect to those in an infinite medium 0, are expressed in terms of the Fresnel reflection coefficients for plane and evanescent waves incident from medium 0 on the interfaces to the planar-stratified or homogeneous media 1 and 2, respectively. From this result which is valid for arbitrary layer thicknesses d0{d}_{0}d0​ we derive an approximation for extremely thin layers of optical thickness ${n}_{0}{d}_{0}\ensuremath{\ll}\frac{\ensuremath{\lambda}}{8}$ (where $\ensuremath{\lambda}$ is the emission wavelength) between two homogeneous loss-free dielectric media 1 and 2. For the normalized spontaneous-emission rates as functions of the refractive indices nj{n}_{j}nj​ of media j=0,1,and2j=0, 1, \mathrm{and} 2j=0,1,and2 analytical expressions are obtained. We have used these expressions previously without proof [Phys. Rev. B 21, 4814 (1980)] to analyze experimentally observed changes in the fluorescence lifetime of Eu3+{\mathrm{Eu}}^{3+}Eu3+ ions in varying optical environments, which yielded the quantum efficiency of the emitting state.
We consider the spontaneous emission in an absorbing medium by calculating the quantum flow of energy from an emitting atom. For this purpose, we extend the analysis by Power and Thirunamachandran of quantized Maxwell fields in the vicinity of an atom in the free space to a situation where the emitter is in a dielectric medium. We represent the quantum flow in terms of the non-equal time commutators involving electric and magnetic fields, as well as the local displacement field. The commutators are calculated applying a previously developed microscopic theory of quantization of radiation in dielectrics. Consequently, we derive from first principles the rate of spontaneous emission in an absorbing medium containing local field factors and other refractive contributions. The results support the previous introduction of the (virtual cavity) local field correction by Barnett et al on a phenomenological basis, as well as our previous approach in which the spontaneous emission is considered as a resonance energy transfer to the far-zone atoms. A distinctive feature of the present formalism is that it deals with the spontaneous emission directly by calculating the quantum flow from the emitting atom. Therefore, the method is free from divergences appearing in other approaches analysing the spontaneous decay in absorbing media.
In this paper, four different ways are described in which homogeneous, isotropic, nonferromagnetic, Maxwellian media at rest may exhibit absorption and stimulated emission. These four ways may be identified by a positive or negative conductivity, a sink or source-type electric field, a complex dielectric constant, and a complex permeability. The forms of the electromagnetic waves, the refractive index, and the positive or negative extinction coefficient are determined for each process for both absorbing and active media. Spontaneous emission is associated with source-type electric and magnetic fields which generate conduction or displacement currents in the medium with improper phases to contribute to the radiant flux in the incident electromagnetic wave. Stimulated emission is characterized by exponentially increasing incident electric and magnetic fields and an excess of the time average of the outflow of radiant flux over the surfaces of an arbitrary volume. Stimulated emission appears when the time average of the rate at which the source-type electric and magnetic fields do work in carrying certain conduction or displacement currents in directions opposite to the incident electric and magnetic fields is greater than the time average of the rate at which the incident electric and magnetic fields dissipate energy by doing work in carrying other conduction or displacement currents in directions parallel to the incident electric and magnetic fields.