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arXiv:0811.0373

Dipole emission and coherent transport in random media II. Density of states vs. index of refraction

M. Donaire

Published Nov 3, 2008• 0 citations
arXivHTMLPDF

Abstract

This is the second of a series of papers devoted to develop a microscopical approach to the dipole emission process and its relation to coherent transport in random media. In this Letter, we deduce a relation between the transverse decay rate of an emitter in a virtual cavity and the complex refraction index of the host medium. We argue on the possibility of a criterion for inhibition/enhancement of spontaneous emission in function of the transition frequency and the correlation length of the host scatterers. In addition, we study the radiative/non-radiative nature of the net power emission through a microscopical analysis of the scattering events involved. This study reveals essential discrepancies with previous interpretations.

DOI
10.48550/arxiv.0811.0373

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This is the second of a series of papers devoted to develop a microscopical approach to the dipole emission process and its relation to coherent transport in random media. In this Letter, we deduce a relation between the transverse decay rate of an emitter in a virtual cavity and the complex refraction index of the host medium. We argue on the possibility of a criterion for inhibition/enhancement of spontaneous emission in function of the transition frequency and the correlation length of the host scatterers. In addition, we study the radiative/non-radiative nature of the net power emission through a microscopical analysis of the scattering events involved. This study reveals essential discrepancies with previous interpretations.
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.
We present a simple theory on the effect of optical dephasing on the spontaneous emission (SE) rate in a microcavity containing (absorbing) dispersive dielectric media based on a photon Green's-function formalism in which the reservoir damping of medium polarizations is included in a phenomenological way. The theory yields the well-known result that the SE rate in free space is modified by the real part of the refractive index at the transition frequency of an atom in a bulk dielectric. On the other hand, it turns out that the degree of the alteration of the SE rate in a high-Q microcavity with dispersive media is affected not only by the real part of the refractive index but also by the imaginary part of the refractive index that is associated with the dephasing of the photon field due to the reservoir damping, i.e., the destruction of self-interference of the photon field.
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 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.
The effect of the dielectric medium upon the spontaneous emission rate was investigated using lead borate glass of various compositions lightly doped with the Eu3+{}^{3+}3+ lanthanide ion. A dual approach used the variation of (i) electric dipole/magnetic dipole (ED/MD) emission rates from spectral integrations and (ii) absolute ED emission rates from lifetime measurements for comparison with the variation of the refractive index of the glass medium. Contrary to a previous study [Phys. Rev. Lett. 91, 203903 (2003)], the results show the relevance of the virtual-cavity model in this case. The MD emission rates were taken to follow the n3{n}^{3}n3 increase in density of photon states in the medium. The justifications for employing Eu3+{}^{3+}3+ for identifying the appropriate model are given.