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Physical Review A

Cerenkov radiation at the resonance frequency of an amplifying medium

Demosthenes Dialetis

Published Nov 1, 1978• 1 citation
Publisher

Abstract

The propagation of C\ifmmode \breve{}\else \u{}\fi{}erenkov radiation through a medium in which the population of two energy levels is inverted has been studied. The population inversion is represented by an imaginary part of the index of refraction of the medium at the resonance frequency. The resonant atoms are embedded in a host medium which has the shape of a slab of finite width $L$ and dielectric constant ${\ensuremath{\epsilon}}_{h}$, which is real and varies slowly for frequencies close to the resonance frequency of the embedded atoms. It is assumed that the host medium and the embedded atoms inside the slab behave as an isotropic noncrystalline medium. The slab is covered by a mirror on each of its surfaces and the medium outside the slab and the mirrors has dielectric constant ${\ensuremath{\epsilon}}_{1}$ (${\ensuremath{\epsilon}}_{1}=1$ for vacuum). While each individual charged particle can stimulate coherent emission by different resonant atoms, the phase of radiation which is due to different particles is uncorrelated. It is shown, however, that radiation of high intensity in the direction of the C\ifmmode \breve{}\else \u{}\fi{}erenkov cone can be produced by highcurrent particle beams. Exact expressions, in integral form, are obtained for the electromagnetic field everywhere in space. Using the method of stationary phase approximate expressions are derived for the C\ifmmode \breve{}\else \u{}\fi{}erenkov fields as well as the radiated energy in the foward and backward direction, far away from the slab.

DOI
10.1103/physreva.18.2115

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The propagation of C\ifmmode \breve{}\else \u{}\fi{}erenkov radiation through a medium in which the population of two energy levels is inverted has been studied. The population inversion is represented by an imaginary part of the index of refraction of the medium at the resonance frequency. The resonant atoms are embedded in a host medium which has the shape of a slab of finite width LLL and dielectric constant \ensuremathϵh{\ensuremath{\epsilon}}_{h}\ensuremathϵh​, which is real and varies slowly for frequencies close to the resonance frequency of the embedded atoms. It is assumed that the host medium and the embedded atoms inside the slab behave as an isotropic noncrystalline medium. The slab is covered by a mirror on each of its surfaces and the medium outside the slab and the mirrors has dielectric constant \ensuremathϵ1{\ensuremath{\epsilon}}_{1}\ensuremathϵ1​ (\ensuremathϵ1=1{\ensuremath{\epsilon}}_{1}=1\ensuremathϵ1​=1 for vacuum). While each individual charged particle can stimulate coherent emission by different resonant atoms, the phase of radiation which is due to different particles is uncorrelated. It is shown, however, that radiation of high intensity in the direction of the C\ifmmode \breve{}\else \u{}\fi{}erenkov cone can be produced by highcurrent particle beams. Exact expressions, in integral form, are obtained for the electromagnetic field everywhere in space. Using the method of stationary phase approximate expressions are derived for the C\ifmmode \breve{}\else \u{}\fi{}erenkov fields as well as the radiated energy in the foward and backward direction, far away from the slab.
In this work we investigate the Cerenkov radiation in the case of motion of a charged particle on a background of the squeezed electromagnetic vacuum. Using the quantum approach (time dependent perturbation theory) we obtain the explicit expression for the Cerenkov radiated power. The analysis of the obtained expression shows that in contrast to the \classical case (where the Cerenkov radiation is a spontaneous process) the Cerenkov radiation in the squeezed vacuum has an induced character. The expression for radiated power contains two terms, namely the flrst term corresponds to radiation power, while the second one corresponds to absorption power. 1. INTRODUCTION Cerenkov efiect (1) have a long history but it remains relevant in relation to the difierent areas of application of such a phenomena (the wide list of publication on this subject takes place, see, e.g., (2{4) and related references). The expression for the radiated power was obtained by solving the equations of classical electrodynamics in the medium. A large number of studies have been devoted to other aspects of the Cerenkov efiect. They concern the extension to the case of magnetic media, a detailed analysis of radiation in crystals, the role of boundaries, etc. The classical theory of spontaneous Cerenkov efiect is fairly accurate, however the quantum analysis gives a more complete understanding of the phenomenon. Moreover, the quantum approach is most useful when considering the motion of a charged particle in a dielectric medium in the external electromagnetic wave. The radiation in this case has an induced nature and quantum approach gives a better understanding of the physical nature of phenomena. Creation of powerful sources of electromagnetic radiation initiated extensive research in the area of particle-fleld interaction. The squeezed states of light (5) play an important role in such researches. These states generalize the well-known class of coherent states and are characterized by the fact that the uncertainty of one of the canonical variables of the electromagnetic fleld is less than in a conventional coherent state. The squeezed states are important in such areas as optical communication, interferometry, nonlinear optics, the detection of gravitational waves, etc. It should also be noted that the squeezed states are objects of purely quantum nature. So the description of the motion of a charged particle on a background of the squeezed vacuum can be performed using the quantum fleld theory only. This problem is of great practical interest (new sources of electromagnetic radiation such as free-electron lasers and Cerenkov lasers). In this work we investigate the Cerenkov radiation in the case of motion of a charged particle on a background of the squeezed electromagnetic vacuum. Due to \purely quantum nature of the squeezed vacuum we use the quantum approach, namely the time dependent perturbation theory.
The coherent radiation of N atoms that are excited by an external field is studied close to a resonant transition. The resonant atoms are uniformly distributed over a sphere and the external field is a pulse that propagates in a fixed direction. The linear classical continuum theory is used for the above configuration. Exact expressions are obtained for the polarization, for the fields everywhere in space, and for the energy radiated and absorbed by the resonant atoms. Two equivalent methods for the derivation of the exact expression of the polarization are given, i.e., either by solving Maxwell’s equations and boundary conditions, or by solving an integral equation inside the sample. The equivalence of the two methods, in the general case of an arbitrary external field, has been proved elsewhere. Approximate expressions are also obtained for the physical quantities mentioned above in the extreme cases of a dilute and a dense resonant sample with radius much smaller than the radiation wavelength. Inside the dense small sample the polarization depends on position, and this space dependence is expressed in terms of spherical Bessel functions and vector spherical harmonics. It is shown that outside the dense small sample, a broadening of the original resonant line and a shift of the resonance frequency occur in the radiation spectrum of the atoms.
The propagation of a strong coherent "pump" wave traveling through a resonant medium is discussed. The medium is assumed to consist of identical stationary atoms, and the pump-wave frequency is assumed to coincide exactly with the resonance frequency for transitions between a particular pair of atomic states. A previously developed theory of the response of a strongly driven atom to a weak nearly resonant "signal" field is used to deduce the existence of a coupling between any two waves travelling parallel to the pump wave whose (nearly equal) frequencies sum to twice the pump-wave frequency. In the limit of very intense (highly saturating) pump fields, the coupling between the two waves of nearby frequencies leads to amplification of both waves if their frequencies lie within an interval about the pump-wave frequency equal to the Rabi frequency of population inversion. The basic process, which is roughly described by travelling-wave parametric-amplifier equations of motion, consists of the absorption of two pump-field photons followed by their emission at different nearby frequencies, and thus implies a frequency instability of the initially coherent pump wave.
The propagation of electromagnetic radiation through a resonant medium is studied. The slowly varying envelope of the electromagnetic field is assumed to have a slowly varying phase in addition to a slowly varying amplitude. The nonlinear coupled equations of motion are solved exactly under rather general assumptions. Three classes of solutions are obtained; two of them reduce to the solutions of Crisp and of McCall and Hahn when the slowly varying phase is zero. Because of the gauge invariance of the equations of motion, there is a term proportional to the vector potential in addition to the terms of its space and time derivatives in the equation describing the evolution in time of the electromagnetic field. Hence, a characteristic frequency exists in the infrared region below which the wave number becomes imaginary.
The coherent, radiative excitation modes of a resonant medium lend a resonant character to the amplitudes for reflection and transmission of light. For a plane wave incident upon a slab-shaped medium of two-level atoms, the reflection and transmission coefficients exhibit sharp peaks and dips as a function of the detuning of the wave from the central atomic resonance frequency. The locations and widths of these resonant features for a normally incident plane wave correspond to the imaginary and real parts, respectively, of the radiative decay constants of the coherent excitation modes of the medium discussed in the preceding paper [Phys. Rev. A 61, 063814 (2000)]. For the case of oblique incidence, generalizations of the modes discussed in the preceding paper are required. The richly detailed, resonant frequency dependence of the reflected and transmitted intensities is shown to depend significantly on the angle of incidence and the polarization of the incident light.
It is shown that coherent processes of elastic scattering of resonant radiation form a “buffer” electromagnetic field near the boundary of excited media. This part of radiation is not governed by the standard refractive index, but precisely this part of radiation forms the beams reflected and refracted by the excited medium. The presence of the “buffer” field causes the suppression of stimulated emission near the boundary and leads to the appearance of a frequency angular broadening of the beam transmitted through a thin film of excited atoms at an oblique angle.