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Physical Review A

Theory of spectroscopy in an optically pumped effusive vapor

Thomas M. Stace, Andre Nicholas Luiten

Published Mar 29, 2010• 26 citations
Publisher

Abstract

We present a theoretical framework for studying spatially dependent absorption in a thermal vapor of multilevel atoms, of arbitrary optical thickness. The atomic state dynamics, governed by a standard atom-optical master equation, are self-consistently coupled to the axial evolution of the probe beam intensity and the effusive gas dynamics. We derive steady-state equations for the spatially varying distributions of atomic populations and the probe beam intensity. From the latter, absorption coefficients in both the saturated and unsaturated regimes can be calculated. We present solutions to the resulting equations at various levels of approximation, including an example of the full numerical solution of a saturated, optically thick vapor of three-level atoms, demonstrating a breakdown of Beer's law, among other measurable effects.

DOI
10.1103/physreva.81.033848

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We present a theoretical framework for studying spatially dependent absorption in a thermal vapor of multilevel atoms, of arbitrary optical thickness. The atomic state dynamics, governed by a standard atom-optical master equation, are self-consistently coupled to the axial evolution of the probe beam intensity and the effusive gas dynamics. We derive steady-state equations for the spatially varying distributions of atomic populations and the probe beam intensity. From the latter, absorption coefficients in both the saturated and unsaturated regimes can be calculated. We present solutions to the resulting equations at various levels of approximation, including an example of the full numerical solution of a saturated, optically thick vapor of three-level atoms, demonstrating a breakdown of Beer's law, among other measurable effects.
This work continues a theoretical investigation of the capabilities of the well-known method based on using a monochromatic probe light beam in combination with optical pumping of atoms (molecules) of a rarefied-gas medium by a broadband radiation in a thin cell the diameter of which is much larger than its internal thickness. In contrast to calculations carried out in the previous publications on this method of spectroscopy, here, we consider the case of arbitrary values of pump intensity and thickness of a cylindrical gas cell. Thus, all the possible mechanisms and specificities of velocity selection of atoms in optically excited levels caused by transit-time relaxation of such atoms in gas cells of this kind are analyzed. Within the framework of this approach, sub-Doppler absorption resonances of the probe light beam corresponding to quantum transitions from the upper level excited by optical pumping are investigated. The obtained results can be used in high-resolution spectroscopy of atoms (molecules), as well as for laser-frequency stabilization to established narrow spectral resonances.
This work examines absorption spectrometry by narrowband light in gaseous media with arbitrary optical thickness when the light induces optical saturation or optical pumping. Two quantities are defined: the observed absorbance, A(obs), and the true absorbance, A(true). The former is the absorbance that is measured under the existing conditions, whereas the latter represents the absorbance one would measure if the light acted solely as a probe of the populations of the various levels, and it is therefore directly proportional to the concentration or density of absorbers. A general integral equation for the propagation of light in media of arbitrary optical thickness in which the light influences the populations of the levels involved is derived. This expression is transcendental in the observed absorbance and cannot be solved analytically. It is shown that an analytical expression can be derived by investigating the inverse relationship, i.e., A(true) = f(A(obs)). Inasmuch as collision and Doppler broadened media react differently to optical saturation, they are considered separately. It is shown that a nonlinear response results if the medium is optically saturated (or pumped) and not optically thin. Expressions for the error introduced if the technique of standard additions is uncritically applied to such a system are derived.
Laser spectroscopy experiments are often conducted with a probe that does not significantly alter the medium’s properties. For a two-level atom a clear measure of the strength of a probe beam in terms of the saturation intensity is known. We show that for a multilevel atom the situation is very different, and the effects of optical pumping are crucial to understanding the opacity of the medium. We present a simple theoretical analysis for Doppler-broadened spectroscopy of alkali metals on the D2 line that emphasizes the importance of the transient nature of the population dynamics, and the crucial role of the distribution of the times of flight of atoms through the probe beam. Experimental results are obtained with room temperature rubidium vapor probed by an extended-cavity diode laser and confirm our theoretical prediction.
We present analytic solutions for saturated absorption spectra. The analytic forms of all the ground- and excited-state populations were obtained, and the absorption of a counterpropagating probe beam was calculated by solving the rate equations in the presence of a pump laser beam. The analytic solutions were compared with the numerical, experimental results, and Nakayama's model and good agreement was found between them. We found that the analytic theory could provide accurate spectra at an arbitrary pump beam intensity and diameter.
We have developed a model for calculating saturated absorption spectra for dipole transitions in multi-level atoms. Using a semiclassical density matrix formalism, we derive a set of coupled differential equations for the internal state of the atom in a standing wave light field. The equations are solved using standard integration techniques. The absorption at each laser detuning is found from an average of the absorption for a number of velocities along the laser field, thermally weighted. The method is relatively efficient computationally yet quantitatively predicts important details of saturated absorption spectra including saturation, crossover resonances, merging of absorption lines at high intensity and optical pumping between hyperfine levels. We have measured saturated absorption and fluorescence spectra of 85 Rb, and compare to our computational results for a 36-level model.
A theory of saturation spectroscopy in three-level gas-vapor systems, including collisional effects, is presented. Using a model of collisions in which they are phase interrupting in their effect on level coherences and velocity changing in their effect on level population densities, the authors calculate the absorption profile of a weak probe field on a given transition when an arbitrarily strong pump field acts on a coupled transition. Line shapes are derived for an arbitrary collision kernel describing the velocity-changing collisions, and these line shapes are evaluated for the collision kernel proposed by Keilson and Storer. Various limiting forms for the line shapes are derived and representative line shapes are displayed. Several new features of the line shapes, including collisionally induced increased probe absorption are discussed. The line shapes are seen to reflect the various collisional processes that may occur in atomic or molecular systems.