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The Astrophysical Journal

The angular dependence of partially redistributed resonance radiation

R. J. Ballagh, J. Cooper

Published Apr 1, 1977• 32 citations
Publisher

Abstract

Explicit expressions for the angular distribution and frequency dependence of radiation scattered by a spatially degenerate atom undergoing collisions are derived using a previously developed formalism of the quantum theory of line broadening for an atom in its rest frame. Results valid in the impact regime for resonant scattering are presented for the cases of linear incident and scattered polarization circular incident and linear scattered polarization, and unpolarized incident radiation with no monitoring of the scattered polarization. These results relate to scattering involving a j - j + or - 1 - j atomic transition, with lower-state interactions neglected; the expressions obtained take a relatively simple form involving atomic multipolar relaxation rates of the order of k = 0 and k = 2. The angular distributions for each case are combined with the frequency dependences to yield a three-component frequency redistribution function for an s-p-s transition, the components being a coherent term, a frequency-redistributed term with the same angular dependence as the coherent term, and an isotropic frequency-redistributed term.

DOI
10.1086/155179

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Optics and Spectroscopy•Apr 1, 2002•2 citations
Explicit expressions for the angular distribution and frequency dependence of radiation scattered by a spatially degenerate atom undergoing collisions are derived using a previously developed formalism of the quantum theory of line broadening for an atom in its rest frame. Results valid in the impact regime for resonant scattering are presented for the cases of linear incident and scattered polarization circular incident and linear scattered polarization, and unpolarized incident radiation with no monitoring of the scattered polarization. These results relate to scattering involving a j - j + or - 1 - j atomic transition, with lower-state interactions neglected; the expressions obtained take a relatively simple form involving atomic multipolar relaxation rates of the order of k = 0 and k = 2. The angular distributions for each case are combined with the frequency dependences to yield a three-component frequency redistribution function for an s-p-s transition, the components being a coherent term, a frequency-redistributed term with the same angular dependence as the coherent term, and an isotropic frequency-redistributed term.
The techniques of modern line-broadening theory are used to investigate the scattering of polarized radiation in the rest frame of an atom undergoing collisions. The formulation explicitly includes both elastic and inelastic (quenching) collisions. When the lower state has zero width, a form for the redistribution function similar to that of Zanstra is obtained, but with the redistribution in the neighborhood of the resonance line being caused solely by elastic collisions. In the limit of no collisions, but with both levels of finite lifetime, the result of `veisskopf and Woolley is obtained. The effect of level-degeneracy is also explicitly included; in this case the results are a function of the polarization of the light and the different relaxation rates for the multipolar components of the atomic states.
We present a calculation of the scattering of monochromatic radiation by a degenerate atom in the binary-collision approximation. We limit ourselves to field strengths such that $\ensuremath{\Omega}{\ensuremath{\tau}}_{c}\ensuremath{\ll}1$, where $\ensuremath{\Omega}$ is the Rabi frequency for the atomic transition, and ${\ensuremath{\tau}}_{c}$ is the duration of a strong collision. Our calculation is not limited to the impact regime, $\ensuremath{\Delta}\ensuremath{\omega}\ensuremath{\ll}{\ensuremath{\tau}}_{c}^{\ensuremath{-}1}$, or to the region where thermal correlations may be neglected, i.e., $\ensuremath{\Delta}\ensuremath{\omega}\ensuremath{\ll}\frac{\mathrm{kT}}{\ensuremath{\hbar}}$. (Here $\ensuremath{\Delta}\ensuremath{\omega}$ is the difference between either the incoming or outgoing photon's frequency and the atom's natural frequency.) To do this we derive an equation of motion for the correlation function, valid outside the quantum-regression regime, whose solution for practical cases is straightforward. We present solutions for the weak-field (linear-response) regime in terms of generalized absorption and emission profiles that depend on the indices of the atomic multipoles created.
The close-coupled theory of atomic collisions in the presence of a radiation field may be used to calculate the distribution of final atomic states which results from absorption of polarized light during a collision. The theory applies equally well to optical collisions (line broadening) and to radiative collisions (laser-induced collisional energy transfer). For an optical collision the detuning $\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{\ensuremath{\infty}}$ is restricted to be larger than either the Rabi frequency or the widths due to natural, Doppler, or pressure broadening. The radiation field is assumed to be weak enough that the transition probabilities are linear in field intensity. The molecular picture is emphasized in which the wave function is expanded in a basis of field-free molecular states and the Hamiltonian is blocked in accordance with molecular quantum numbers. The quantities needed to predict experimental observables are reduced radiative scattering SSS-matrix elements ${s}_{\ensuremath{\omega}}$ in the asymptotic Hund's case-(e) representation. The theory of product orientation and alignment is equivalent to that which has been developed for molecular photodissociation. It requires coherent sums of ${s}_{\ensuremath{\omega}}$ matrix elements from different transition branches, that is, for PPP-, QQQ-, and RRR-type transitions corresponding to changes of -1, 0, and +1 in total molecular angular momentum. On the other hand, the total absorption coefficient and the branching ratio to different final fine-structure states irrespective of orientation or alignment depend on incoherent sums of the same matrix elements.
A formal investigation of the redistribution of weak near-resonance radiation is presented for the case that the radiative transitions of the scattering atom occur between two well-separated groups of states. The lower states are assumed stable while the upper states are allowed to undergo arbitrary interactions with an environment. If the lower states are also degenerate, a Rayleigh component which is infinitesimally narrow in the atom's rest frame is shown to exist under extremely wide conditions and independent of the approximations made in previous work on redistribution. Like the absorption profile, the intensity of this component as a function of the detuning (Rayleigh response function) can be expressed in terms of the mean atomic time evolution operator projected on the upper-state manifold. As compared to the absorption profile, however, the Rayleigh response function is different in shape and may provide complementary information on the scattering medium.
The scattering of radiation in the presence of collisions can be described quantum-mechanically in terms of essentially two processes. The first may be thought of as an absorption to the excited state followed subsequently (after propagating in the excited state) by emission. This gives rise to radiation redistributed about the transition frequency. The effects of m-degeneracy are particularly interesting for this first process. As an example a case is considered in which the incident frequency is in the quasi-static line wing, while the scattered frequency is close to the line center. It is found that under these circumstances the dominant contribution from this process in the scattered spectrum is obtained for an absorption of the incident frequency taking place during a strong (close) collision and reemission of the scattered frequency when the atom is essentially unperturbed
A compact expression for the cross section of scattering of an arbitrarily polarized light by aligned atomic systems is obtained, in which the dependence on the geometric parameters and the Stokes parameters specifying the state of partial polarization of the incident radiation is represented in explicit form. The effect of atomic alignment and the processes of dissipation of the light energy on the polarization specific features and the angular distribution of the scattered light is investigated. In particular, it is shown that, if a dissipative channel is accessible, the angular distribution and the degree of linear polarization of scattered light depend on the degree of circular polarization of the incident radiation η 2 . Dissipative processes also induce the circular polarization of the light scattered by aligned atoms when η 2 =0.