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Physical Review

Saturation Behavior of a Doppler-Broadened Transition Involving Levels with Closely Spaced Structure

H. Schlossberg, Ali Javan

Published Oct 7, 1966• 185 citations
Publisher

Abstract

The nonlinear absorption or gain characteristics of an optical-frequency, Doppler-broadened atomic resonance involving levels with closely spaced structure are analyzed. The level structures are assumed to be resolved with respect to their natural widths, but not necessarily with respect to the Doppler width at the optical transition. The radiation field consists of two closely spaced monochromatic frequencies lying within the Doppler width of the resonance. This type of radiation field may be obtained, for example, from a laser operating in two of its Fabry-Perot resonator modes. It is shown that, because of saturation of level populations and double-quantum Raman transitions between levels, appreciable nonlinear coupling takes place between the two fields. This coupling shows a resonance behavior when the frequency separation of the two applied fields becomes equal to the frequency splitting between two of the components which form either level structure. The width of this resonance is determined entirely by the natural widths of these two level components and not by the Doppler width of the optical transition or the natural width of other level components. When such a resonance occurs, the over-all gain or attenuation characteristics of the atomic resonance change drastically. In practice the frequency spacing of the fields may be kept constant while the level structures are tuned, e.g., magnetically. The effects are analyzed for cases of running-wave and standing-wave radiation fields. The use of this effect in precise determination of level structures as well as mode or transition coupling of a gas laser is discussed. Portions of the analysis are applicable to resonances with more general forms of inhomogeneous broadening.

DOI
10.1103/physrev.150.267

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The nonlinear absorption or gain characteristics of an optical-frequency, Doppler-broadened atomic resonance involving levels with closely spaced structure are analyzed. The level structures are assumed to be resolved with respect to their natural widths, but not necessarily with respect to the Doppler width at the optical transition. The radiation field consists of two closely spaced monochromatic frequencies lying within the Doppler width of the resonance. This type of radiation field may be obtained, for example, from a laser operating in two of its Fabry-Perot resonator modes. It is shown that, because of saturation of level populations and double-quantum Raman transitions between levels, appreciable nonlinear coupling takes place between the two fields. This coupling shows a resonance behavior when the frequency separation of the two applied fields becomes equal to the frequency splitting between two of the components which form either level structure. The width of this resonance is determined entirely by the natural widths of these two level components and not by the Doppler width of the optical transition or the natural width of other level components. When such a resonance occurs, the over-all gain or attenuation characteristics of the atomic resonance change drastically. In practice the frequency spacing of the fields may be kept constant while the level structures are tuned, e.g., magnetically. The effects are analyzed for cases of running-wave and standing-wave radiation fields. The use of this effect in precise determination of level structures as well as mode or transition coupling of a gas laser is discussed. Portions of the analysis are applicable to resonances with more general forms of inhomogeneous broadening.
The ground electronic state of even such simple atoms as those of alkali elements has the set of sublevels. These sublevels are hyperfine (HF) components and magnetic sublevels. The collisions with particles of nonmagnetic buffer gas or even with walls (when walls have proper coating) do not mix practically the populations of ground state HF sublevels, if the orbital moment of electrons in the ground state is equal to zero. Due to this reason these sublevels may be extremely long lived. Because of this, for instance, the so-called effects of optical orientation and optical pumping are apparent in atoms of alkali metals. The radiation can pump practically all the atoms (with which there is effective interaction) to a single HF component. In a situation in which optical pumping is a minor effect, the monochromatic absorption line consists of separate lines corresponding to intra-atomic transitions. The optical pumping is not significant, for example, if the radiation intensity is so low that now pumping between HF sublevels occurs in the time it takes the atom to traverse the light beam. Under the conditions of optical pumping and when other nonlinear effects are present, the absorption line profile can undergo considerable alteration. Many theoretical and experimental investigations have been done in the laser spectroscopy of resonant transitions of atoms (atoms of alkali metals) under the conditions of optical pumping within ground state HF structure. A characteristic feature of all such investigations is the velocity-selective excitation of absorbing atoms. Also, as a rule, the value of HF splitting is not very big in comparison to Doppler broadening in these experiments. Under such conditions, the absorption spectra consists of Doppler- broadened lines with nonlinear resonances, which emerge when two fields of different frequencies or different propagation directions are employed. When only one monochromatic traveling wave is employed, optical pumping leads to formation of a single smooth absorption line occupying a position between the resonant transitions frequencies associated with HF components. As HF splitting grows, the line profile remains smooth, its width is approximately equal to the size of the HF splitting. The strength of the optical pumping from one ground state HF component to another is greater for greater HF splitting in comparison to the Doppler width. In this regard, Cs and Rb are the most typical of the alkali atoms (the HF splitting exceeds the Doppler width by a factor close to 20). This is the qualitative picture of the absorption line profile (particularly for a monochromatic radiation) under the conditions of optical pumping between the HF sublevels of the electronic ground state, this qualitative picture had formed by now.
Previous theoretical results on the influence of a laser on the line shape of a coupled transition-laser-induced line narrowing-have been restricted to the case where the laser is detuned from the center of its atomic gain profile or is in the form of a traveling wave. This paper extends those results to include the case where the laser is an intense standing-wave field tunable to the center of its atomic gain profile (conditions for Lamb dip). The effect may be observed either in transmission, by probing the coupled transition with a weak traveling-wave field coaxial with the laser field, or else in spontaneous emission from the coupled transition viewed along the axis of the laser field. It is now well known that for a laser detuned from the center of its atomic gain profile, two narrow Lorentzian resonances of different widths appear superimposed upon the broad background signal at frequencies symmetrically located about the corresponding line center. When the laser is tuned to the center of its gain profile, however, additional fine structure develops. This structure, which is particularly significant when the laser field is intense, may have important applications in highresolution spectroscopy and laser-frequency stabilization. In this paper the laser frequency may be smaller or larger than the frequency of the coupled transition. In the latter case an intense laser introduces additional splitting effects, even when the laser is detuned. Splitting effects due to weakly saturating laser fields are also discussed. The problem is formulated by expanding elements of the ensemble-averaged density matrix in an infinite series of spatial Fourier components. A perturbation technique is employed, valid for a weak probe field and a standing-wave field of arbitrary intensity. One obtains an expression for emission induced by the probe field due to atoms moving with one velocity, written in terms of continued fractions in the general case and with Bessel functions in an important special case. This expression is integrated over the atomic velocity distribution by means of a computer to obtain the total emission due to atoms moving with all velocities. In some cases the integrated expressions may be written in closed algebraic form. A detailed discussions of line shapes and of the physical processes involved is included.
The review deals with resonance phenomena resulting from saturation of low-pressure-gas absorption lines by a coherent optical field, when the spectral lines are inhomogeneously broadened by the Doppler effect. Principal attention is paid to narrow resonances of width much smaller than the Doppler widths, which are produced upon saturation of the absorption by light waves under various experimental conditions (inside and outside the laser resonator, in traveling and standing waves, in spatially separated waves, etc.), owing to different effects (absorption saturation, mode interaction in the laser). Each of the phenomena is considered from both the theoretical and the experimental points of view. The phenomena considered serve as a basis of a new branch of optical spectroscopy, namely nonlinear laser spectroscopy of ultrahigh resolution.
We have studied theoretically nontrivial specific features of sub-Doppler resonances of saturated absorption in a multilayer gas cell with a rarefied gas medium, which is a compact analog of a large number of plane-parallel atomic beams. Spatially separated saturating and probing monochromatic laser beams that co- and counterpropagate (at the same frequency) in this cell have been considered. In this situation, the action of the light radiation of the saturating beam on the probing beam is determined by optically pumped atoms that fly between the beams under conditions that the longitudinal components of their velocities experience selection due to a specific geometry of the multilayer cell. Such a selection leads to a narrowing of the sub-Doppler resonance in the absorption of the probing beam and decreases the difference between the structures of this resonance for the cases of co- and counterpropagation of the saturating radiation. We have found that, in the considered multilayer cell, the effective width of the sub-Doppler resonance can be smaller (by a factor of about 1.5) than the extremely narrow characteristic width of the well-known Lamb dip in spectroscopy of saturated absorption in the standard gas cell. Results of this study can be used in atomic spectroscopy of ultrahigh resolution and for the laser-frequency stabilization.
The effect of collisions on the resonance fluorescence spectrum of three-level atoms with a V-configuration of levels when excited by two monochromatic resonance fields has been theoretically investigated. The analysis has been carried out for the systems with a small Doppler broadening compared to the collision frequency (high buffer gas pressures) and for the general case of an arbitrary change (from complete failure to complete preservation) of the phase memory at any of the three transitions in the V-system. It has been shown that in the three-level V-type atoms excited by two monochromatic fields, the resonant fluorescence spectrum (the number of spectral lines and their widths) at one of the transitions strongly depends on the presence of resonant radiation at the adjacent transition. Under certain conditions, the line widths will be completely determined by the relaxation characteristics of the adjacent transition and, in particular, can be repeatedly narrowed with an increase in the radiation intensity at the adjacent transition. It has been shown that despite the homogeneous nature of the broadening of the absorption lines, the resonance fluorescence spectrum has a pronounced anisotropy with respect to the mutual orientation of the wave vectors of spontaneous and exciting radiation. It has been established that the line widths in the resonance fluorescence spectrum are proportional to the diffusion coefficient of the atoms interacting with radiation under certain conditions. This fact can serve as the basis for a spectroscopic method for measuring the transport frequencies of the collisions of the radiation-absorbing particles with the buffer ones. It has been shown that the effects of phase memory in the collisions lead to a strong qualitative and quantitative modification of the resonance fluorescence spectrum.
The influence of nonlinear interference processes and of changes in populations in strong electromagnetic fields, and also of the Doppler broadening on resonant four-wave frequency mixing is investigated for a system of Raman-type transitions. The case in which a quantum system interacts with one weak field and two strong fields is considered. Each of the energy levels is assumed to be perturbed by just one strong field. It is shown that the Doppler broadening alters drastically the contributions of the populations of the various levels to the investigated process and that this can alter by several orders of magnitude and nonlinear susceptibilities averaged over the velocities and influence significantly the selection of optimal configurations of the transitions and of the experimental conditions for effective resonant four-wave conversion of the radiation frequencies. A numerical analysis of the derived expressions is used to account for the experimentally observed considerable differences between the spectral dependences of the output power and the conclusions drawn from perturbation theory.