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Physical Review A

Electromagnetic field quantization in absorbing dielectrics. II

Reza Matloob, R. Loudon

Published Jun 1, 1996• 112 citations
Publisher

Abstract

A quantization scheme for the electromagnetic field in absorbing dielectrics developed previously is extended to cover more complicated arrangements of dielectric media and to investigate various limiting cases of the general formalism. The limiting cases include media that have vanishing imaginary parts in their dielectric functions, because either the refractive index or the extinction coefficient vanishes. The further limit of a unit real dielectric function establishes the connection of the formalism with the well-known quantized field expressions in free space. Detailed calculations are presented for the quantization in the system of two different absorbing dielectrics in contact at a plane interface and for the cavity formed in the free space between two separated absorbing dielectrics. The forms of the field operators are determined for both systems, the canonical commutation relations are verified, and the spectra of the vacuum field fluctuations are calculated and illustrated. The calculations are restricted throughout to fields that propagate perpendicular to the dielectric interfaces. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.4567

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•Dec 1, 1996•0 citations
A quantization scheme for the electromagnetic field in absorbing dielectrics developed previously is extended to cover more complicated arrangements of dielectric media and to investigate various limiting cases of the general formalism. The limiting cases include media that have vanishing imaginary parts in their dielectric functions, because either the refractive index or the extinction coefficient vanishes. The further limit of a unit real dielectric function establishes the connection of the formalism with the well-known quantized field expressions in free space. Detailed calculations are presented for the quantization in the system of two different absorbing dielectrics in contact at a plane interface and for the cavity formed in the free space between two separated absorbing dielectrics. The forms of the field operators are determined for both systems, the canonical commutation relations are verified, and the spectra of the vacuum field fluctuations are calculated and illustrated. The calculations are restricted throughout to fields that propagate perpendicular to the dielectric interfaces. \textcopyright{} 1996 The American Physical Society.
The electromagnetic field is quantized in dielectric media that show both loss and dispersion. The complex dielectric function of the medium is assumed to be a known function and the loss is modeled by Langevin forces in the forms of noise current operators. The noise current correlation function is related to the assumed dielectric function by the fluctuation-dissipation theorem. Field quantization is carried out for the infinite homogeneous dielectric, the semi-infinite dielectric, and the dielectric slab, where the fields in the second and third cases are restricted to propagation perpendicular to the dielectric surfaces. The forms of the vector potential operator are obtained in the different spatial regions for all three geometries, and in each case the required canonical commutation relation for the vector potential and its conjugate generalized momentum operator is verified. The spatial dependence of the vacuum field fluctuations is calculated for the two dielectric geometries that have surfaces.
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.
We quantize the electromagnetic field in an absorbing resonant medium with translation symmetry broken along one direction. In this case the general polarization of the material system is driven by the field via a nonlocal complex susceptibility. We extend recently developed schemes for field quantization in absorbing dielectric media with local susceptibilities to dielectric systems described by a nonlocal susceptibility. Dissipation inside the medium is described by considering the coupling of polarization quanta of the system with the reservoir oscillators in the usual Langevin approach. The method is applied to the case of a semiconductor quantum well embedded in infinite barriers. Expressions for the electromagnetic-field operators are obtained in the different spatial regions and the spatial and frequency dependencies of the vacuum-field fluctuations are calculated.
The electromagnetic field is quantized in a material whose dielectric function varies with frequency and one spatial dimension. The dielectric function is assumed to have a known form and to be real over the range of frequencies important for a particular application, for example the propagation of an optical signal. General properties of the mode functions are derived and employed in the quantization procedure. Expressions are obtained for the energy and momentum density and current operators, and these are shown to satisfy the appropriate conservation and continuity relations. The general formalism is illustrated by application to the examples of a homogeneous dielectric and two different dielectrics with a sharp interface.
A three-dimensional electromagnetic field quantization scheme is developed using the Green's-function method in a homogeneous medium that shows loss and dispersion. Special attention is paid to the decomposition of the quantized fields into their longitudinal and transverse parts. The characteristic nature of these two parts is discussed. To demonstrate the generality of this formulation, the field quantization in lossless dispersive media and in free space is obtained by imposing appropriate limits. We also calculate the vacuum fluctuations of the field for the transverse and longitudinal parts separately.
The electromagnetic field is quantized in nonlinear media whose dielectric functions and nonlinear susceptibilities may vary with space. Given the dielectric function and nonlinear susceptibilities, the Hamiltonian of the electromagnetic field is determined completely by this quantization method. From Heisenberg equations we prove that Maxwell's equations are satisfied for the field operators. When the nonlinearity goes to zero, this quantization method returns to the generalized canonical quantization procedure developed in Ref.[1] for linear inhomogeneous media. The general formalism is illustrated by applications to quantization of the second-order and third-order nonlinear quasi-steady-state processes.