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Physical Review

Nonlinear Effects in the Resonant Absorption of Several Oscillating Fields by a Gas

Carson L. Tang, H. Statz

Published Nov 1, 1962• 60 citations
Publisher

Abstract

The lowest order nonlinear effects due to saturation in the simultaneous resonant absorption of several oscillating fields by a low-pressure gas are considered. The analysis is based on a perturbation solution of the Boltzmann equation for the density matrix of the gas in a quantum statistical description. Within certain ranges of the parameters involved, it is shown that: (1) When the gas is irradiated by two oscillating linearly polarized fields of different frequencies well within the Doppler-broadened width of the same absorption line, maximum absorption of both fields occurs when the two linear polarizations are perpendicular to each other. (2) When the gas is in a static magnetic field and irradiated by two right-circularly polarized fields at ${\ensuremath{\omega}}_{0}$ and ${\ensuremath{\omega}}_{0}+\ensuremath{\Delta}\ensuremath{\omega}$, and two left-circularly polarized fields at ${\ensuremath{\omega}}_{0}$ and ${\ensuremath{\omega}}_{0}\ensuremath{-}\ensuremath{\Delta}\ensuremath{\omega}$, all well within the Doppler width of the respective Zeeman components ($\ensuremath{\Delta}m=+1 \mathrm{and} \ensuremath{-}1$) of the same absorption line, there exists a definite phase relationship among the fields whereby absorption of all four fields is simultaneously at a maximum. Although these effects are quite small in normal spectroscopic work, manifestation of both effects is readily observable in gaseous He-Ne optical masers.

DOI
10.1103/physrev.128.1013

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Physical Review•Oct 7, 1966•185 citations
The lowest order nonlinear effects due to saturation in the simultaneous resonant absorption of several oscillating fields by a low-pressure gas are considered. The analysis is based on a perturbation solution of the Boltzmann equation for the density matrix of the gas in a quantum statistical description. Within certain ranges of the parameters involved, it is shown that: (1) When the gas is irradiated by two oscillating linearly polarized fields of different frequencies well within the Doppler-broadened width of the same absorption line, maximum absorption of both fields occurs when the two linear polarizations are perpendicular to each other. (2) When the gas is in a static magnetic field and irradiated by two right-circularly polarized fields at \ensuremathω0{\ensuremath{\omega}}_{0}\ensuremathω0​ and \ensuremathω0+\ensuremathΔ\ensuremathω{\ensuremath{\omega}}_{0}+\ensuremath{\Delta}\ensuremath{\omega}\ensuremathω0​+\ensuremathΔ\ensuremathω, and two left-circularly polarized fields at \ensuremathω0{\ensuremath{\omega}}_{0}\ensuremathω0​ and \ensuremathω0\ensuremath−\ensuremathΔ\ensuremathω{\ensuremath{\omega}}_{0}\ensuremath{-}\ensuremath{\Delta}\ensuremath{\omega}\ensuremathω0​\ensuremath−\ensuremathΔ\ensuremathω, all well within the Doppler width of the respective Zeeman components (\ensuremathΔm=+1and\ensuremath−1\ensuremath{\Delta}m=+1 \mathrm{and} \ensuremath{-}1\ensuremathΔm=+1and\ensuremath−1) of the same absorption line, there exists a definite phase relationship among the fields whereby absorption of all four fields is simultaneously at a maximum. Although these effects are quite small in normal spectroscopic work, manifestation of both effects is readily observable in gaseous He-Ne optical masers.
Using classical fields and the density-matrix formalism, we analyze the nonlinear characteristics of a gas laser amplifier subject to an axial magnetic field, operating with two optical-frequency signals of arbitrary polarization. Perturbational solutions, valid for relatively weak intensities, are obtained which describe the saturation interaction of the signals and the generation of combination tones. An arbitrary amount of Doppler broadening is considered throughout. The simple $J=1\ensuremath{\rightarrow}J=0$ model is found convenient to describe the physical processes taking place. In general, however, the interactions are found to depend on the JJJ values of the levels. We find that only saturation and no combination-tone generation occurs for two opposite circularly polarized signals, while both effects are, in general, present for two linearly or elliptically polarized fields. The limiting case of zero magnetic field is examined separately. It is also found that in many cases the medium is made effectively anisotropic by the nonlinear interactions. Applications to measurements of atomic parameters such as JJJ values, ggg values, and lifetimes are discussed.
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.
The Lamb theory of the optical maser is applied to circularly polarized atomic transitions, and used to consider the beat frequencies and the coherence properties of such orthogonal fields when axial magnetic fields are applied to the gaseous laser. The beat frequency approaches zero in near-zero magnetic fields and synchronization can then occur between the right- and left-handed circularly polarized oscillations. For a resonator with no undue polarization constraint, such a strong coupling gives rise to a linearly polarized output, and to a rotation of the plane of polarization with increasing magnetic field. A self-consistent expression is derived for this rotation under steady-state conditions, and a maximum rotation of $\ifmmode\pm\else\textpm\fi{}\frac{1}{4}\ensuremath{\pi}$ with magnetic field is indicated before the synchronization breaks down and circularly polarized beat phenomena appear. The rotation with magnetic field depends on the laser intensity, on the anisotropy in the cavity losses, and on the position of the cavity resonance within the Doppler linewidth. Also, the angle of rotation is indeterminate unless such anisotropy is present. Other regions of such coherence can occur at higher magnetic fields, where the beat frequency again approaches zero. These depend on the detailed shape of the various dispersion curves of the laser medium. The results derived from the theory used are in general agreement with experimental observations on the 1.153-\ensuremath{\mu} He-Ne laser transition.
Using semiclassical radiation theory and a density matrix formalism we analyze the nonlinear characteristics of a gas laser amplifier operating with two optical frequency signals of arbitrary polarization and having an axial magnetic field. Both perturbational solutions, valid for relatively weak intensities and solutions valid for arbitrarily strong fields are obtained for two nonlinear effects: the saturation interaction of the electromagnetic waves, and the generation of combination tones. An arbitrary amount of Doppler broadening is considered throughout. The detailed treatment of J = 1 to J = 0 model yields the frequency, magnetic field and polarization dependence of the nonlinear effects. The results are presented analytically and graphically and are discussed using physical arguments. It is found that only saturation but no combination tone generation occurs for two opposite circularly polarized input signals while both are, in general, present for two arbitrary linearly or elliptically polarized fields. For two opposite circular waves the interaction is found to comprise three parts, each with a different behavior: self saturation, common level mutual saturation and a coherent double quantum interaction. The total interaction (coupling) between the two fields is always weak. The limiting case of a single linearly polarized field is considered separately, the zero magnetic field dip and the nonlinear behavior of the Faraday rotation is discussed. For two linearly (or elliptically) polarized waves the three nonlinear processes listed above take place between opposite circular components. In addition a modulation of the population inversion densities occurs due to the presence of two different frequencies with the same circular polarization. This results in the generation of new frequencies and also contributes to the coupling between the input fields. The coupling depends on the magnetic field, and on the frequency separation and the polarization states of the signals. The limiting case of zero magnetic field is examined. It is found that the medium is made effectively anisotropic by the nonlinear interactions. The polarization vectors of two linearly polarized fields rotate apart unless the angle between them is zero or 90 degrees. The results are extended to the general Ja to Jb transition. In zero magnetic field the nonlinear effects are found to depend on ΔJ, while for nonzero magnetic field resonances in the interaction occur whenever the frequency difference between two opposite circularly polarized transitions that have common level equals the frequency separation of the input fields. Combination tone generation takes place for all but two opposite circularly polarized signals.
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.