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Physical Review A

Collision broadening in Lamb-dip spectroscopy

Lewis S. Klein, Michel Giraud, A. Ben‐Reuven

Published Jul 1, 1977• 9 citations
Publisher

Abstract

A previously developed diagrammatic method for calculating the nonlinear response of a dilute gas at near resonance with strong classical radiation fields is extended to include velocity-dependent homogeneous and inhomogeneous broadening phenomena. The method is applied to two-level saturated-absorption (Lamb-dip) spectroscopy. Expressions are obtained for the collision-broadened Lamb dip probed by a weak field in the presence of an arbitrarily strong saturating field and are compared to the usual infinite-Doppler-width, weak-saturation expression. We study the effects of velocity and momentum transfer on the line-shape parameters and discuss the effect of phase-changing and strong inelastic collisions on the intensity.

DOI
10.1103/physreva.16.289

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Physical Review A•Jan 1, 1979•1 citation
A previously developed diagrammatic method for calculating the nonlinear response of a dilute gas at near resonance with strong classical radiation fields is extended to include velocity-dependent homogeneous and inhomogeneous broadening phenomena. The method is applied to two-level saturated-absorption (Lamb-dip) spectroscopy. Expressions are obtained for the collision-broadened Lamb dip probed by a weak field in the presence of an arbitrarily strong saturating field and are compared to the usual infinite-Doppler-width, weak-saturation expression. We study the effects of velocity and momentum transfer on the line-shape parameters and discuss the effect of phase-changing and strong inelastic collisions on the intensity.
We consider in this paper the Doppler-broadened absorption by two-level molecules of a gas from a weak traveling electromagnetic plane wave in the presence of an oppositely traveling strong plane wave at the same frequency. Here the absorption spectrum exhibits a dip that broadens and deepens as the gas is increasingly saturated by the strong wave. This saturation effect has been employed in the recent experiments on the fine structure of atomic hydrogen performed by H\"ansch, Shahin, and Schawlow (HSS). The form, linewidth, and strength of saturation dips in the fine-structure lines are studied for experimental parameters appropriate to the hydrogen experiment. In addition, saturation line-shape curves are calculated for experimental situations in which the strengths of the stronger field give rise to stronger or weaker saturation of the gas medium and varying degrees of Doppler broadening exist. The calculation of shape of the saturation dip and absorption curves for a Doppler-broadened gas of two-level atoms is made exactly; in effect, the strong field is treated as arbitrarily strong by avoiding a perturbation-theory treatment in favor of the exact solutions for a driven two-level system. The general behavior found is in accord with intuition and experimental observations by HSS. There is an optimum saturation power which gives a prominent dip and beyond which the dip suffers "power broadening."
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.
A three-dimensional theory for the resonant interaction of electromagnetic waves with a gas of two-level atoms is formulated in terms of macroscopic variables. The theory is utilized to find the steady-state attenuation of a plane wave in the presence of another plane wave running in the opposite direction with different amplitude. Contributions are included from the reflection of the oppositely running wave by an induced standing-wave inhomogeneity in the population inversion of the medium. The resulting attenuation and reflection coefficients are expressed as velocity integrals of continued fractions. Correspondence is made with existing gas-laser theories, yielding the formulation of a high-intensity ring-laser theory. Analytic approximations for the coefficients are presented for the Doppler-limit cases of both waves weak, one wave weak, and negligible reflection (rate-equation approximation). More-general cases have been calculated numerically. The attenuation coefficients exhibit a Lamb-dip feature. The relative depth of the dip increases rapidly with power at low saturation levels, slowly at high saturation, and is greater in the attenuation of the weaker wave. The width of the dip is nonlinearly power broadened. The shape of the dip is very nearly Lorentzian, except for one special case at high power in which the line splits. The propagation equations for the two waves are integrated over long absorption paths. A large resulting attenuation increases the relative size of the dip while decreasing the power broadening.
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.
We study inhomogeneously broadened three-level atoms ($\ensuremath{\Lambda}$ configuration). Two energy levels are strongly coupled by an intense standing wave. A weak laser field at an adjacent transition is used as a probe. The probe response is calculated by Green's functions involving continued fractions. The general solution is extendable to other three-level configurations as well to other types of multimode saturators. The connection of various approximate solutions to the exact one is discussed. We concentrate on the Doppler-free case where both transitions have nearly equal frequencies. For this we derive a reasonably accurate closed-form solution, the parameter dependencies of which are investigated in detail. In addition to its average changes the dynamics of the saturated transition plays an important role in the steady-state probe response. The spectra show sharp resonances which cannot be explained in terms of traveling wave superposition models. The width of the velocity distribution has a strong effect on the shape of the spectrum---some resonances are smeared away because of incomplete Doppler dephasing. The relaxation processes influence differently the resonances, which facilitates their interpretation.
Using the exact solution for a sine-driven two-level system we have calculated the nonlinear absorption coefficient of a gas of two-level molecules with homogeneous (hard collisions, spontaneous decay) and inhomogeneous (Doppler effect) broadening mechanisms. We have studied the dependence of the form of the absorption lines on relaxation parameters, on the strength of the electric field, and for various ratios between the homogeneous and inhomogeneous linewidths. We include considerations about saturation of the medium caused by the external field that gives rise to the so-called power broadening of the spectral lines. The dependence of the linewidth on the saturation parameters and on the relaxation parameters is determined. These parameters appear in the ratio of the purely homogeneous and purely inhomogeneous linewidths. Asymptotic behaviors of this linewidth for the homogeneous limit and for the inhomogeneous limit are also determined.