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Canadian Journal of Physics

Considerations of a classical model for spontaneous emission in a nonhomogeneous medium

André Reid, Michel Piché

Published Aug 1, 1992• 0 citations
Publisher

Abstract

The classical model for spontaneous emission from a source embedded in a nonhomogeneous medium is examined in a new light. It is shown that the general solution may be expressed in terms of an effective polarizability associated with the transition being modeled. The case of a transition occurring in the vicinity of a perfect mirror is then studied as an illustration. Analytical expressions for the frequency shift and changes in the transition rate as a function of distance are found.

DOI
10.1139/p92-101

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The classical model for spontaneous emission from a source embedded in a nonhomogeneous medium is examined in a new light. It is shown that the general solution may be expressed in terms of an effective polarizability associated with the transition being modeled. The case of a transition occurring in the vicinity of a perfect mirror is then studied as an illustration. Analytical expressions for the frequency shift and changes in the transition rate as a function of distance are found.
The definition of an effective polarizability associated with a dipole transition is shown to result from the use of a dyadic Green's function in solving the electron equation of motion obtained from a classical model of spontaneous emission that takes into account the presence of nonhomogeneities in the electric permittivity. This polarizability provides access to the line shape associated with the transition; it also provides a generalized procedure for studying the effects of changes in the electromagnetic environment on line shapes, frequency shifts, and transition rates.
Summary form only given. Over the last three decades of the 20 th century many experiments have demonstrated that the rate of spontaneous emission for an emitter is determined by its electromagnetic surrounding. This phenomenon has been usually studied for atoms or molecules placed in the vicinity of mirrors or in optical resonators. Furthermore, it is also well-known that the fluorescence lifetime of an emitter changes after being embedded in dielectric media where aside from a change in the refractive index, it experiences the local fields in the dielectric. For a spherical medium larger than the order of a wavelength it is known that the coupling to Mie modes of this dielectric resonator has a large influence on the emission rate of an emitter inside. However, when the medium becomes comparable or smaller than a wavelength one can no longer speak of resonator modes. Here we report on an experiment where we have studied the variation of the spontaneous emission rate as the dielectric medium surrounding an emitter is made smaller and smaller down to a few nanometers.
The spontaneous radiation of a classical oscillator and that of a two-level atom both in the vicinity of a phase-conjugate mirror are calculated and compared. In either case the phase-conjugate mirror may be replaced by a phase-conjugate "image" of the source. We study the dependence of the decay constants and the characteristic frequencies on the phase-conjugate reflectivity and the difference between the characteristic frequency of the radiator and the pump frequency of the phase-conjugate mirror.
Quantum electrodynamics formulas are applied to calculate the spontaneous emission rate of an excited atom in the vicinity of a perfect reflecting mirror. The result exhibits that damping oscillating patterns depend sensitively on the distance of the atom to the mirror. And then we use Fourier transform to extract the corresponding frequency of the osciations. In order to explain this phenomena, we are the first to utilize the closed-orbit theory to deal with the spontaneous emission rate of an emitted photon near a mirror. So this study provides a different perspective on the spontaneous emission of atoms in environments.
Starting from a quantum Maxwell equation for the vector potential operator, we study the spontaneous emission properties of a two-level atom interacting with the electromagnetic field in a lossless and inhomogeneous dielectric structure and express the spontaneous emission rate, the external quantum efficiency, and the spontaneous emission factor in terms of the classical Green function of the dielectric structure. Comparing these results with the corresponding classical results, we show that the above quantities can be calculated from the radiation field of a classical dipole. Based on this correspondence, a numerical algorithm is developed to calculate the modification of spontaneous emission in microcavities. The line shape of the emission spectrum of the light source and the decay of the cavity resonant modes are automatically taken into account in this algorithm. The approximate expressions for the spontaneous emission rate and the spontaneous emission factor are discussed.
We have calculated the effect of material absorption on the rate of spontaneous emission by an embedded atom using a macroscopic Green-function approach and a microscopic Hamiltonian method that includes reservoir damping. When local field effects are neglected, the free-space emission rate is modified by the real part of the refractive index at the transition frequency of the embedded atom. Local field effects are introduced via a local field correction factor. This modification of the spontaneous emission rate generalizes a well-known result for transparent media and is of crucial importance in assessing the degree of inhibition of spontaneous emission for frequencies close to the resonances of the dielectric.