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Physical Review A

Electromagnetic field quantization in amplifying dielectrics

Reza Matloob, R. Loudon, M. Artoni, Stephen Mark Barnett, John Jeffers

Published Mar 1, 1997• 74 citations
Publisher

Abstract

The electromagnetic field is quantized for normal transmission of incident waves through a parallel-sided dielectric slab. The dielectric material is dispersive and it acts as a linear amplifier over limited ranges of the frequency and as a linear attenuator at the remaining frequencies. The field operators derived for the three spatial regions within and on either side of the slab are shown to satisfy the canonical commutation relations. The noise fluxes emitted by the slab are evaluated and shown to satisfy the general requirements for the minimum noise associated with linear amplifiers and attenuators. The behavior of the amplifier gain profile on the approach to the lasing threshold of the slab is determined, but the results are restricted to the below-threshold state of the system. The spectra of the electric-field fluctuations are evaluated for the three spatial regions and for amplifying and attenuating frequencies.

DOI
10.1103/physreva.55.1623

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The electromagnetic field is quantized for normal transmission of incident waves through a parallel-sided dielectric slab. The dielectric material is dispersive and it acts as a linear amplifier over limited ranges of the frequency and as a linear attenuator at the remaining frequencies. The field operators derived for the three spatial regions within and on either side of the slab are shown to satisfy the canonical commutation relations. The noise fluxes emitted by the slab are evaluated and shown to satisfy the general requirements for the minimum noise associated with linear amplifiers and attenuators. The behavior of the amplifier gain profile on the approach to the lasing threshold of the slab is determined, but the results are restricted to the below-threshold state of the system. The spectra of the electric-field fluctuations are evaluated for the three spatial regions and for amplifying and attenuating frequencies.
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 fully canonical quantization scheme for the electromagnetic field in dispersive and lossy linear dielectrics. This scheme is based on a microscopic model, in which the medium is represented by a collection of interacting matter fields. We calculate the exact eigenoperators for the coupled system and express the electromagnetic field operators in terms of them. The dielectric constant of the medium is explicitly derived and is shown to satisfy the Kramers-Kronig relations. We apply these results to treat the propagation of light in dielectrics and obtain simple expressions for the electromagnetic field in the medium in terms of space-dependent creation and annihilation operators. These operators satisfy a set of equal-space commutation relations and obey spatial Langevin equations of evolution. This justifies the use of such operators in phenomenological models in quantum optics. We also obtain two interesting relationships between the group and the phase velocity in dielectrics.
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 use of instruments in optical experiments needs careful examination with regard to their action on the quantum statistics of radiation. Instruments that respond linearly to radiation may be regarded as spatially structured dielectrics whose presence can be taken into consideration by quantizing the phenomenological Maxwell theory. Since radiation in dielectric matter undergoes dispersion and absorption, the losses must necessarily be included in a quantization scheme. This central problem can be solved by using a novel approach that is based on a Green’s function expansion of the radiation field in place of a mode expansion, which fails when the losses are taken into account. The theory is applied to calculate quantum-theoretically consistent inputoutput relations for dispersive and absorptive multi-slab dielectric plates. Assuming that the polarization field and the electric field are locally related to each other, the phenomenological Maxwell equations governing the propagation of radiation in linear dielectrics, without external sources, can be quantized by supplementing them by an appropriately chosen noise source associated with the losses in the dielectric matter. For the sake of transparency, let us consider a multi-slab dielectric plate with permittivity ε(x, ω) and restrict attention to radiation that propagates along the x axis and is polarized in z direction.
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.