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Journal of Physics B Atomic and Molecular Physics

Doppler-free saturated dispersion spectroscopy

J. I. Cotrim Vasconcellos, Antonio Guillermo Jose Balbin Villaverde, José Antonio Roversi

Published Mar 28, 1984• 4 citations
Publisher

Abstract

The authors consider the Doppler-broadened index of refraction of a two-level molecule of a gas for a weak travelling electromagnetic plane wave in the presence of a strong plane wave travelling in the opposite direction at the same frequency. The authors use the FVH vector model for a two-level system to calculate the real part of the non-linear susceptibility of the gas. The optical dispersion curve exhibits a wiggle at the centre line, that broadens and deepens as the gas is increasingly saturated by the strong incident wave.

DOI
10.1088/0022-3700/17/6/027

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Journal of Physics B Atomic and Molecular Physics•Apr 28, 1980•8 citations
The authors consider the Doppler-broadened index of refraction of a two-level molecule of a gas for a weak travelling electromagnetic plane wave in the presence of a strong plane wave travelling in the opposite direction at the same frequency. The authors use the FVH vector model for a two-level system to calculate the real part of the non-linear susceptibility of the gas. The optical dispersion curve exhibits a wiggle at the centre line, that broadens and deepens as the gas is increasingly saturated by the strong incident wave.
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."
It is shown that the dispersion dependence of the rotation of the plane of polarisation of weak radiation passing through a three-level atomic medium may be free of Doppler broadening, on the account of a velocity-dependent Stark-shift which is created by saturating the wave on an adjacent transition. In the case of two-photon resonance the use of opposing light beams with different frequencies gives the possibility for an essential increase of the signal on account of the resonance gain on an intermediate transition, without additional Doppler broadening.
The purpose of this work was to investigate coherence effects in the interaction of two counterpropagating electromagnetic waves with the atoms of a gas, in the case in which one of the waves is weak and frequency modulated, while the other is sufficiently intense to cause saturation. Solving the equations for the density matrix in first-order perturbation theory, the authors obtained in analytic form the real part of the susceptibility integrated over the atom's velocity, expressed in terms of the rate equation model (REM) predictions plus the coherence corrections. The relaxation constants of the levels involved in the transition were kept different in order to allow a wider range of applications of the results. Calculated line shapes related either to the frequency-modulated (FM) sideband or to the ordinary dispersion techniques are presented. The REM approximation and the coherence corrections are shown in detail for an optical line which is of interest for frequency standard investigations.
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.
Previously derived equations describing recoil effects in saturated absorption by a gas of two-level atoms or molecules are solved numerically for the case of standing plane-wave excitation and large Doppler broadening. The numerical techniques and accuracy estimates are discussed briefly. The lineshapes obtained are well approximated by the superposition of two Lorentzian functions whose width, heights, and separation parametrise the lineshape. The dependence of these parameters on saturation, recoil splitting, and relaxation rates is described and compared with corresponding results in the rate equation approximation and in the weak-probe-beam approximation. Saturation broadening depends strongly on these latter approximations, but very little on recoil splitting. When multiphoton processes are retained in the equations, increasing saturation causes a slight decrease in the fitted separation of the peaks. The ratio of the heights of the two Lorentzians is closely related to the ratio of relaxation rates for the level populations.