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Physical Review A

Quantization of evanescent electromagnetic waves: Momentum of the electromagnetic field very close to a dielectric medium

J. M. Vigoureux, L. D’Hooge, Daniel VAN LABEKE

Published Jan 1, 1980• 30 citations
Publisher

Abstract

The problem of the momentum of the electromagnetic field near a homogeneous dielectric plane surface is studied. In the particular case of evanescent waves, our calculations are consistent with the experimental result of S. Huard and C. Imbert which points out that, during an interaction between a moving atom and a surface wave of pulsation $\ensuremath{\omega}$, the exchanged momentum is greater than $\frac{\ensuremath{\hbar}\ensuremath{\omega}}{c}$. This work leads at first to a straightforward procedure for quantizing the momentum parallel to the diopter of the field. The authors proceed to investigate the form of the component perpendicular to the interface which is transmitted by the field to an atom during an interaction. (In the case of an evanescent wave, this problem involves the interpretation of the imaginary part of the wave vector.) A general theory shows that the presence of a medium widens and shifts the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ levels of the field. This result agrees with the uncertainty principle.

DOI
10.1103/physreva.21.347

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The problem of the momentum of the electromagnetic field near a homogeneous dielectric plane surface is studied. In the particular case of evanescent waves, our calculations are consistent with the experimental result of S. Huard and C. Imbert which points out that, during an interaction between a moving atom and a surface wave of pulsation \ensuremathω\ensuremath{\omega}\ensuremathω, the exchanged momentum is greater than \ensuremathℏ\ensuremathωc\frac{\ensuremath{\hbar}\ensuremath{\omega}}{c}c\ensuremathℏ\ensuremathω​. This work leads at first to a straightforward procedure for quantizing the momentum parallel to the diopter of the field. The authors proceed to investigate the form of the component perpendicular to the interface which is transmitted by the field to an atom during an interaction. (In the case of an evanescent wave, this problem involves the interpretation of the imaginary part of the wave vector.) A general theory shows that the presence of a medium widens and shifts the k\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}k\ensuremath→​ levels of the field. This result agrees with the uncertainty principle.
The problem of the quantization of evanescent waves, which appear in the angular spectrum representation of the electromagnetic field in a half-space, is discussed. Although evanescent waves are associated with material sources, scatterers, etc., we are able to treat the electromagnetic field, including the evanescent waves, effectively as a free field, by making use of the idea of the refractive index of a passive, macroscopically continuous medium. We consider a space which is filled with a homogeneous dielectric to the left of the plane z=0z=0z=0, and is empty to the right of the plane. Triplets of incident, reflected, and transmitted waves at the interface form the fundamental orthogonal modes of the space. By expanding the field in terms of these triplet modes, we show that the field Hamiltonian reduces to the sum of independent harmonic-oscillator Hamiltonians. The quantization is therefore straightforward. We introduce the creation and annihilation operators for the triplet wave modes, and encounter Fock states, coherent states, etc., for a field having evanescent wave components. The field commutator at two space-time points in the right half-space is shown to have an explicit contribution from evanescent waves, characterized by an exponential decay to the right and a propagation parallel to the interface. We also examine the problem of atomic excitation by quantized evanescent waves, and show that the results are of the form given by semiclassical treatments.
The problem of field quantization in a half space is discussed based on the detector modes, which provide a useful basis for the study of atomic or molecular interaction with radiation modes involving evanescent electromagnetic waves. The detector modes, each of which involves a single outgoing wave, are introduced in terms of the time-reversal and spatial-rotation transforms of the widely used triplet modes given by Carniglia and Mandel. The derivation of the orthogonal relation for the detector modes is therefore straightforward. We represent the creation and annihilation operators for the detector modes in terms of those for the triplet modes, and obtain the field operator, Hamiltonian, number operator, and pseudomomentum operator. Based on this formalism, we evaluate the differential radiation probability for atomic dipole radiation near a dielectric surface. The evaluation of the final-state mode density is straightforward for the detector modes involving single outgoing waves. With respect to the image dipole picture, the quantum-to-classical correspondence is clear in the detector-mode formalism.
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.
In this work the electromagnetic field in a dielectric medium is quantized in a simple and straightforward way based on the classical expression for the energy density for the case of weak absorption. The quantized field leads to expressions for spontaneous and stimulated emission rates, and gain and absorption coefficients, for the general case of a dispersive dielectric host medium. Local field corrections, when they apply, arise in the same manner as in classical electromagnetic theory. The theory also leads to a radiative level shift dependent on the dispersion of the dielectric, and this shift may be interpreted as a van der Waals interaction between the guest atom and the atoms of the host medium.>
The electromagnetic field in a dispersive dielectric medium is quantized in a straightforward manner based on the classical expression for the field energy away from absorption resonances. Results for spontaneous and stimulated radiative rates, and absorption and gain coefficients, are obtained and compared with other expressions in the literature. Local field corrections, when they exist, are included. A level shift for an atom in a host dielectric medium is obtained and interpreted as the van der Waals interaction between that atom and its neighbours in the host medium.
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.