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Physical Review A

Solving the line-shape problem with speed-dependent broadening and shifting and with Dicke narrowing. I. Formalism

David A. Shapiro, R. Ciuryło, J. R. Drummond, A. D. May

Published Dec 5, 2001• 53 citations
Publisher

Abstract

In this paper, we present a formalism for the exact calculation of spectral profile of an isolated line undergoing speed-dependent collisional broadening and shifting and Dicke narrowing. It is couched in the language of the density matrix as in the closely related fields of transport phenomena, laser physics, and nonlinear optics. The formalism is ab intio in spirit, permitting the calculation of the line shape starting only from a given interaction potential. The line shape is obtained by solving a transport/relaxation equation for the off-diagonal element of the density matrix, a macroscopic quantity. The collision operator in the transport/relaxation equation, which is determined from the (microscopic) interaction between the active molecule and the perturber, is presumed known.

DOI
10.1103/physreva.65.012501

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In this paper, we present a formalism for the exact calculation of spectral profile of an isolated line undergoing speed-dependent collisional broadening and shifting and Dicke narrowing. It is couched in the language of the density matrix as in the closely related fields of transport phenomena, laser physics, and nonlinear optics. The formalism is ab intio in spirit, permitting the calculation of the line shape starting only from a given interaction potential. The line shape is obtained by solving a transport/relaxation equation for the off-diagonal element of the density matrix, a macroscopic quantity. The collision operator in the transport/relaxation equation, which is determined from the (microscopic) interaction between the active molecule and the perturber, is presumed known.
In the preceeding paper it was shown that the general problem of spectral line shapes in multiline, IR spectra may be determined by solving a transport relaxation equation for the off-diagonal elements of the density matrix. This is a semiclassical equation at the Wang-Chang--Uhlenbeck level, i.e., it treats the quantized internal states as nondegenerate. Here we apply the master equation to the case of Dicke narrowing, and by discretizing the velocity distribution show that Dicke narrowing of a single line may be treated in exactly the same manner as line mixing. Both effects lead to a narrowing of a spectral distribution. We indicate how the numerical technique can be extended and used to calculate profiles in the general case of spectra with speed-dependent broadening, shifting, and line mixing.
Borrowing heavily from the literature, we develop a transport relaxation equation for the off-diagonal elements of the density matrix. It is applicable to a large range of topics in laser theory, nonlinear optics and spectral line shapes, etc. Applying this single equation to the case of infrared electric dipole transitions permits us to recover, quickly and simply, all of the well-known spectral line shapes found in the literature. It also offers further insight into the process of line mixing, particularly in the weak mixing limit. An outline of a new treatment of Dicke narrowing and other speed-dependent contributions to spectral profiles is also given in terms of the transport relaxation equation. The treatment also provides a numerical method for including the effects of statistical correlation between the evolution of the internal and translational degrees of freedom.
The shape of an isolated spectral transition is analyzed in terms of an approximation to the Waldmann–Snider kinetic equation. This equation is written in the form of a drift and collision operator acting on a density matrix. With the use of the spherical approximation, the collision operator is subdivided into an elastic Boltzmann-like collision term, an inelastic loss term, and a dephasing term. The Boltzmann-like term is responsible for Dicke narrowing of spectral lines, the inelastic loss term leads to line broadening and the dephasing term may contribute both to line broadening and shifting. Simple approximations to these terms are powerful enough to account for some of the details of experimental line shapes such as asymmetrical deviations from a Lorentzian line shape. Model numerical calculations are carried out assuming classical scattering potentials of the form 1/rn in the Boltzmann-like term and single complex frequencies for the other two. It was found that as long as the density was scaled to give the same diffusion constant, the exact form of the scattering cross section had little effect on the final line shape.
The purpose of this work is to verify the possibility of the observation of the Dicke narrowing of atomic spectral lines in the optical domain. As an example we have chosen experimental results obtained by means of a laser-induced fluorescence method for the 114 Cd 326.1 nm line perturbed by xenon. Experimental results were carefully reanalysed using a line shape model which takes into account the speed dependence of collisional broadening and shifting, the velocity-changing collisions and the collision-time asymmetry. We have found no convincing evidence for the occurrence of Dicke narrowing of the 326.1 cadmium line perturbed by xenon. The Dicke narrowing (if it occurs at all) appears to be smaller than that resulting from the estimation based on the accessible data describing the diffusion of Cd in Xe. As a source of this behaviour the correlation between velocity-changing and dephasing collisions was indicated.
A general method for analysing line broadening caused by Doppler, phaseshift, inelastic and velocity-changing collisional effects is proposed for transitions at optical wavelengths. In a unified approach Poisson statistics are used to describe collision probabilities randomly distributed in a time interval which can terminate the optical coherence in a combination of the above effects; markoffian independence is assumed for each collision. Then the dipole moment correlation function is obtained in a form which is virtually exact in accounting for all effects of isotropic forces, and which is correct to second-order terms in the anisotropic interaction potentials. Phenomena as diverse as standard optical pressure broadening, broadening due to velocity-changing collisions in nonlinear spectroscopies, and Dicke narrowing of the Doppler contour are described. The implications of the theory are discussed in detail and applied to the case of the NaD lines perturbed by He buffer gas. Numerical calculations reveal that the substantially different lineshift and broadening parameters in nonlinear and in standard optical spectroscopies can be reasonably well understood within the limitations posed by present uncertainties in the Na 3 2 P-He interaction potentials. However, the large differences in shift and width between the D 2 and D 1 lines in saturation spectroscopy cannot be explained within the present theory.