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Japanese Journal of Applied Physics

Spontaneous Emission and the Concept of Effective Area in a Very Short Optical Cavity with Plane-Parallel Dielectric Mirrors

Kikuo Ujihara

Published May 1, 1991• 85 citations
Publisher

Abstract

Spontaneous emission in a model cavity with very short length and with plane-parallel dielectric mirrors is calculated quantum-mechanically under perturbation approximation. Outgoing modes as well as guided modes are considered. Results on the directivity and the rate of emission into outer space are compared with the estimate by Purcell's method. This leads to the concept of “effective area” of the “cavity mode”, which has a simpie physical interpretation.

DOI
10.1143/jjap.30.l901

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Spontaneous emission in a model cavity with very short length and with plane-parallel dielectric mirrors is calculated quantum-mechanically under perturbation approximation. Outgoing modes as well as guided modes are considered. Results on the directivity and the rate of emission into outer space are compared with the estimate by Purcell's method. This leads to the concept of “effective area” of the “cavity mode”, which has a simpie physical interpretation.
A detailed account of the calculation on the spontaneous emission rate in a very short optical cavity with plane-parallel dielectric mirrors is given. A cavity model consisting of a pair of plane dielectric slabs is used. Guided modes as well as outgoing modes are considered. The emission rate from a two-level atom is calculated under quantum-mechanical perturbation approximation. The enhancement factor for the rate is found to be of the order of unity except for extremely short cavities. The concept of quasimodes in a very short cavity and their effective area are discussed in relation to the enhancement factor. The coupling ratios into various quasimodes are derived.
The quantum theory of the spontaneous emission (SpE) in the optical microscopic cavity (microcavity) is reported we believe for the first time.1,2 The forms of the modes propagating orthogonally to the mirrors and previously studied by Ley and Loudon3 are here generalized to cover all spatial directions. Microcavity field quantization leads to the expressions of the SpE probabilities Γ||, Γ⊥ for dipoles parallel and orthogonal to the mirrors, respectively. These are computer calculated for some experimentally significant cases involving both metal and dielectric multilayer-mirror coatings.
In this chapter, the author calculates the modification of the spontaneous emission pattems and the corresponding rate change using classical physics. To derive the spontaneous emission pattems or rates the readers must average the emission rate at each frequency over this distribution function. It is important to understand that this assumption is only valid if the emission line is homogeneously broadened (albeit not necessarily at its natural linewidth limit due to the radiative decay). One more thing needs to be pointed out before the readers move on to a more exact calculation of the spontaneous emission pattern and the emission rate of a dielectric cavity. In the ideal mirror the reflection takes place at a well-defined plane. Integration of the emission over all solid angles and all wavelengths gives us information about the total spontaneous emission rate, or, equivalently, about the spontaneous emission lifetime of an emitter in a dielectric cavity.
A quantum theory of spontaneous emission from an initially excited two-level atom in a one-dimensional optical cavity with output coupling from both sides is developed. Orthonormal mode functions with a continuous spectrum are employed, which are derived by imposing a periodic boundary condition on the whole space with a period much larger than the cavity length. The delay differential equation of the atomic state of Cook and Milonni [Phys. Rev. A 35, 5081 (1987)] is re-derived in a strict manner, where the reflectivity of the cavity mirrors is included naturally in the mode functions. An approximate solution at a single-resonant-mode limit shows the results of ``vacuum'' Rabi oscillation in an underdamped cavity and enhanced spontaneous emission rate in an overdamped cavity. For the latter case, it is found that in the optical range the spontaneous emission rate is enhanced by a factor F (finesse of the cavity).
This chapter presents a tutorial review of spontaneous emission in optical resonators. Approximating correlation functions as delta functions means that the reservoir is assumed to lose its memory in a short time, compared to the time scale over which the dynamics of the small system occurs, the Markov approximation. In a generic cavity quantum electrodynamics experiment, an atom is coupled to one, or possibly a few, cavity mode(s). These modes are in turn coupled to the external world via mirror losses and diffraction. In addition, the atom might also be coupled to a continuum of vacuum modes, in the case where the cavity is open-sided. A simple physical interpretation of the spontaneous emission spectrum can be obtained if one adopts the point of view that the atom-cavity mode small system represents a single quantum system, and by analyzing its eigenenergy spectrum.
Enhancement or suppression of spontaneous emission by the presence of a cavity has been a subject of numerous papers [1]. The effective number of radiation modes was estimated in terms of the cavity Q. Sachdev [2] has improved on this by an exact quantum Langevin treatment of a lossy, single-mode cavity and predicted a vacuum Rabi oscillation for high-Q cavities. Cook et al. [3] included retardation effects by incorporating multiple Fabry-Perot modes and treated the coupling loss by a heuristic introduction of the mirror reflectivity. The present author [4] has considered the same problem in perturbation and Wigner-Weisskopf approximations by use of a set of continuously distributed radiation modes which describes a cavity having output coupling.