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Physical Review A

Green function for multilayers: Light scattering in planar cavities

M. S. Tomaš

Published Mar 1, 1995• 247 citations
Publisher

Abstract

A compact and transparent plane-wave expansion of the complete Green function for an absorbing multilayered system is derived. The plane-wave solutions involved match, in the appropriate limit, the space-mode functions recently employed for quantization of the radiation field in a planar cavity and make this Green function particularly convenient for consideration of optical processes in realistic cavities and multilayers in general. This is illustrated by considering classically light scattering on a molecule in a planar cavity. It is demonstrated that the cross sections derived, besides generally describing light scattering on molecules embedded in a multilayer, correctly reproduce the effects of field confinement on light scattering in a planar cavity as predicted recently by the quantum-mechanical theory of Cairo et al. [Phys. Rev. Lett. 70, 1413 (1993)]. It is also pointed out that scattering into waves guided by the cavity lossy (metallic) mirrors may become an effective process in low-order microcavities. As a by product, a classical approach to spontaneous emission in absorbing planar cavities emerges from these considerations.

DOI
10.1103/physreva.51.2545

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A compact and transparent plane-wave expansion of the complete Green function for an absorbing multilayered system is derived. The plane-wave solutions involved match, in the appropriate limit, the space-mode functions recently employed for quantization of the radiation field in a planar cavity and make this Green function particularly convenient for consideration of optical processes in realistic cavities and multilayers in general. This is illustrated by considering classically light scattering on a molecule in a planar cavity. It is demonstrated that the cross sections derived, besides generally describing light scattering on molecules embedded in a multilayer, correctly reproduce the effects of field confinement on light scattering in a planar cavity as predicted recently by the quantum-mechanical theory of Cairo et al. [Phys. Rev. Lett. 70, 1413 (1993)]. It is also pointed out that scattering into waves guided by the cavity lossy (metallic) mirrors may become an effective process in low-order microcavities. As a by product, a classical approach to spontaneous emission in absorbing planar cavities emerges from these considerations.
The decay rate of an excited molecule (atom) embedded in a dispersive and absorbing planar cavity is derived by using a recently obtained compact form of the Green's function for a multilayer. As a by-product, a hint is provided for a straightforward extension of the results obtained for lossless cavities of other shapes to the corresponding absorbing cavities. The decay rate in an absorbing cavity consists of the spontaneous emission rate and of the nonradiative rates caused by the near-field interaction of the molecule with the cavity medium and, for nearby molecules, with the cavity mirrors. Only the spontaneous emission rate is satisfactorily described in the macroscopic approach adopted. The theory is applied to an analysis of the effects of the weak cavity absorption on the decay rate in a dielectric microcavity formed by two metallic mirrors. As expected, dissipation in the cavity medium spoils the conditions for controlled spontaneuos emission and strongly suppresses the intensity of spontaneous emission. However, its effect on the spontaneous emission rate is much less pronounced.
The method of thermodynamic Green functions is applied to the determination of the susceptibilities and absorption spectra of atoms in microcavities. The method is introduced by its application to an atom in free space, where the well known expression for the spectrum is very simply rederived. The calculations remain simple for more complicated systems, and results are obtained for atoms in single-mode cavities with lossy mirrors and with the additional damping by emission into free-space modes that occurs in open-sided cavities. The expressions for some limiting cases of the general theory are compared with the results of previous work and contact is made with the dressed-state theory. Special consideration is given to an atom placed in a planar microcavity; it is shown that non-Lorentzian spectra may in principle occur but that the parameter values of practical cavities produce Lorentzian spectra with linewidths given by Fermi's golden rule.
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.
The theory developed by Gruner and Welsch [Phys. Rev. A 54, 1661 (1996)] for calculating the 1D input-output relations of the quantized electromagnetic field at dispersing and absorbing dielectric (multilayer) plates is generalized to the three-dimensional case. First a general recipe for the derivation of the reflection and transmission coefficients at an arbitrary body that separates two half spaces from each other is presented. The general theory is then applied to the case of planar multilayer structures, for which the Green tensor is well-known.
A complete set of cavity modes in planar dielectric multilayer structures is presented which naturally includes the guided modes. We show that most of these orthonormal fields can be derived from a coherent superposition of plane waves incident on the stack from the air and from the substrate. Using this set of cavity modes, we present a quantization scheme in idealized lossless stratified dielectric structures. This enables us to discuss the vacuum quantum fluctuations of the fields within the multilayer stack. The spontaneous emission of photons by a two-level atom located at any position inside the stratified planar structure is derived using the quantum approach and standard perturbation theory. The spontaneous emission rate as well as the radiation pattern in the air, in the substrate and in the guided mode are presented. Finally, a numerical analysis of the radiative properties of an erbium atom located in an asymmetrical Fabry - Perot multilayer structure is investigated. We show that a large part of the emitted power is directed in one half-space although a significant amount of light remains trapped in the guided modes of the structure.
We present a one-dimensional quantization scheme for the electromagnetic field in arbitrary planar absorbing and dispersive dielectrics, taking into account explicitly the finite extent of media. The complete form of the electric field operator includes a part that corresponds to the free fields incident from the vacuum towards the medium and a particular solution which can be expressed by using the classical Green-function integral representation of the electromagnetic field. By expressing the classical Green function in terms of the light modes of the dielectric structure, we obtain a natural generalization to absorbing media of the familiar method of mode expansion which strictly applies to non-absorbing media. As an application of our mode expansion, we obtain general quantum-optical input-output relations relating the output photon operators in vacuum to the input photon operators and to the reservoir noise operators. The input-output relations obtained are uniquely determined by the knowledge of the classical light modes of the planar system. We finally show that our quantization scheme for arbitrary planar dielectrics satisfies the prescribed equal-time commutation relations of QED.