Using semiclassical radiation theory and a density matrixformalism we analyze the nonlinear characteristics of a gas laseramplifier operating with two optical frequency signals of arbitrarypolarization and having an axial magnetic field. Both perturbationalsolutions, valid for relatively weak intensities and solutions validfor arbitrarily strong fields are obtained for two nonlinear effects:the saturation interaction of the electromagnetic waves, and thegeneration of combination tones. An arbitrary amount of Dopplerbroadening is considered throughout.
The detailed treatment of J = 1 to J = 0 model yields thefrequency, magnetic field and polarization dependence of the nonlineareffects. The results are presented analytically and graphicallyand are discussed using physical arguments. It is found that onlysaturation but no combination tone generation occurs for two oppositecircularly polarized input signals while both are, in general, presentfor two arbitrary linearly or elliptically polarized fields. Fortwo opposite circular waves the interaction is found to comprise threeparts, each with a different behavior: self saturation, common levelmutual saturation and a coherent double quantum interaction. Thetotal interaction (coupling) between the two fields is always weak.The limiting case of a single linearly polarized field is consideredseparately, the zero magnetic field "dip" and the nonlinear behaviorof the Faraday rotation is discussed.
For two linearly (or elliptically) polarized waves the threenonlinear processes listed above take place between opposite circularcomponents. In addition a modulation of the population inversiondensities occurs due to the presence of two different frequencies withthe same circular polarization. This results in the generation of newfrequencies and also contributes to the coupling between the inputfields. The coupling depends on the magnetic field, and on the frequencyseparation and the polarization states of the signals. Thelimiting case of zero magnetic field is examined. It is found that themedium is made effectively anisotropic by the nonlinear interactions.The polarization vectors of two linearly polarized fields rotate apartunless 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 onΔJ, while for nonzero magnetic field resonances in the interactionoccur whenever the frequency difference between two opposite circularlypolarized transitions that have common level equals the frequencyseparation of the input fields. Combination tone generation takesplace for all but two opposite circularly polarized signals.
