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Physical Review A

Saturated absorption line shape: Calculation of the transit-time broadening by a perturbation approach

Ch. J. Bordé, J. L. Hall, C. V. Kunasz, David G. Hummer

Published Jul 1, 1976• 202 citations
Publisher

Abstract

We present a third-order perturbation calculation of line shapes in laser spectroscopy based on the density-matrix formalism. The new feature of this theory is the inclusion of the Gaussian spatial structure of the laser beams. We study the linewidth as a function of relaxation and transit times. A shift is found when the wave fronts are not flat. General line-shape formulas are given as well as approximate formulas valid in various domains.

DOI
10.1103/physreva.14.236

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We present a third-order perturbation calculation of line shapes in laser spectroscopy based on the density-matrix formalism. The new feature of this theory is the inclusion of the Gaussian spatial structure of the laser beams. We study the linewidth as a function of relaxation and transit times. A shift is found when the wave fronts are not flat. General line-shape formulas are given as well as approximate formulas valid in various domains.
A density-matrix calculation of the line shape for a Doppler-broadened two-level system subjected to a plane-standing-wave laser field with a Gaussian intensity profile is presented to fifth order in perturbation theory. The primary new result is that, for fixed pressure and intensity, the power-broadening contribution to the linewidth is at a minimum when the beam radius is approximately equal to the mean free path. In addition, an analytic solution for the linewidth in the collision-dominated regime is presented for general collision parameters.
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The line shapes of saturation spectroscopy in optically thick two-level media are calculated for atomic and molecular transitions. Line narrowing and distortion effects due to propagation are presented. For Doppler-broadened media it is shown that, due to multiphoton processes, the natural linewidth limits the ultimate narrowing that can be reached. Thus, up to ${\ensuremath{\alpha}}_{0}L=20,$ only when the rate-equation approximation is valid subhomogeneous linewidths are expected.
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.
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This paper presents a calculation of the depth of the saturation dip observed when laser radiation of a Gaussian TEM00{\mathrm{TEM}}_{00}TEM00​ mode is absorbed in a thermal molecular beam or in a very-low-pressure gas. Transit-time broadening is assumed to be the only broadening mechanism, with neglect of collisions and other relaxations. There is no restriction to small saturation parameters. The absorption experienced by one molecule has been calculated via the optical Bloch equations. Averaging over all relevant parameters of the molecular trajectory and the laser field followed. To get a reference for the saturation, from which to measure the dip depth, simply twice the absorption from a running-wave field has been taken, which therefore also had to be calculated in addition to the calculation of the absorption in a standing wave. Preliminary experiments have been carried out for comparison, and the data agree very well with the calculated shape of the curve and the absolute values of the calculated dip depth. From the laser-power scaling factor, necessary to fit the experimental data to the universal theoretical curve, the dipole moment can be derived without having to know such hard-to-determine quantities as absorber density, partition function, collision relaxation rate, or even the diameter of the laser beam.