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Physical Review A

Theory of laser oscillation on two or more coupled transitions

Farrokh Najmabadi, M. Sargent, F. A. Hopf

Published Oct 1, 1975• 22 citations
Publisher

Abstract

Aid or competition between two laser modes oscillating on different transitions of a three-level medium are studied following the method of Lamb. An arbitrary amount of Doppler broadening is allowed for. The "electric-quadrupole" term generated by back-to-back transitions of the two modes but having vanishing electric dipole moment is found to have little effect on the oscillation intensities in typically observed circumstances. This fact justifies, in part, the rate-equation approximation commonly used in multilevel analysis. For a bidirectional ring cascade configuration, however, this coherence term can be important because of constructive interference familiar from two-photon, Doppler-free spectroscopy. The ordinary (i.e., hole-burning) cross-saturation terms for cascade transitions produce mutual mode aid, rather than competition as in the competitive cases. A curious consequence of this difference is greater accuracy in the third-order theory of cascade transitions than in that of competitive transitions, as a result of partial cancellation of errors incurred through neglect of higher orders. The treatment is generalized to multilevel media (four or more levels) with numerical illustration and discussion of the method's relationship to laser oscillation involving diatomic molecules.

DOI
10.1103/physreva.12.1553

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Physical Review•Oct 18, 1965•10 citations
Aid or competition between two laser modes oscillating on different transitions of a three-level medium are studied following the method of Lamb. An arbitrary amount of Doppler broadening is allowed for. The "electric-quadrupole" term generated by back-to-back transitions of the two modes but having vanishing electric dipole moment is found to have little effect on the oscillation intensities in typically observed circumstances. This fact justifies, in part, the rate-equation approximation commonly used in multilevel analysis. For a bidirectional ring cascade configuration, however, this coherence term can be important because of constructive interference familiar from two-photon, Doppler-free spectroscopy. The ordinary (i.e., hole-burning) cross-saturation terms for cascade transitions produce mutual mode aid, rather than competition as in the competitive cases. A curious consequence of this difference is greater accuracy in the third-order theory of cascade transitions than in that of competitive transitions, as a result of partial cancellation of errors incurred through neglect of higher orders. The treatment is generalized to multilevel media (four or more levels) with numerical illustration and discussion of the method's relationship to laser oscillation involving diatomic molecules.
The semiclassical Lamb theory of gas lasers is extended to the case of lasers operating simultaneously on two coupled transitions. The theory is formulated in terms of the density-matrix equations for a three-level gaseous laser medium interacting with the field through electric-dipole transitions, so that there are two allowed transitions and one two-photon transition. It is assumed that there is a single resonator mode near each of the transition frequencies. Previous treatments of such three-level gas lasers either have used simple rate equations or have solved the density-matrix equations in an approximate fashion valid only for low intensities. Thus the numerical approach developed herein represents, to our knowledge, the first semiclassical theory of three-level gas lasers which is accurate for arbitrary detuning and intensity. Significant deviations from earlier, approximate treatments are found.
The multimode strong-signal theory of the preceding paper is applied to a number of two- and three-mode cases. A two-mode stability analysis yields strong-signal counterparts to the third-order coupling parameter. Two-mode homogeneous-broadened operation leads to the discussion of population inversion gratings and associated Bragg scattering. Symmetry properties of the equations for the bidirectional ring are examined. Results for Doppler-broadened two-mode special cases and the three-mode locked unidirectional-ring operation are presented. The strong-signal results are compared to those of the third-order and rate-equation-approximation theories. For the bidirectional ring the historical third-order domain of relative excitation from 1.0 to 1.2 is divided into a valid third-order region and a less accurate region. The condition of bistable operation vs monostable operation is seen to be dependent on the relative excitation and the amount of homogeneous broadening. Two types of bistable operation are seen to exist, one where one mode is completely suppressed and another where both modes may have nonzero intensities.
An analytical nonperturbative theory of a two-mode laser with coupled transitions is presented. The gain medium is modeled as a homogeneously broadened $\ensuremath{\Lambda}$-type three-level system. Stable stationary solutions of such a two-mode laser are studied with particular attention paid to the two-photon-induced atomic coherence effects. Our analysis shows that the nonradiative collisional decay of atomic coherence affects substantially the extent of mode competition. In particular, the coexistence domain of the two modes increases with a faster decay of the quadrupole atomic coherence that also leads to an increase in the power extraction efficiency.
We generalize Lamb's quantum theory of lasers to treat the off-resonance operation of two-transition lasers in inhomogeneously broadened media. The master equation is derived from the density-matrix equation of motion in the Doppler limit. We first find the stationary-state solution from which we then obtain such operation characteristics as the threshold condition, the mean photon number, and the half-width of photon distribution. Compared with the one-transition laser, it is found that in the two-transition laser it is much easier to start oscillation and to obtain higher gain. Furthermore, we find that the two-transition laser is less sensitive to the detuning effect and its deviation from the coherent state is smaller than that of the one-transition laser. The remarkably different hole-burning phenomenon is also discussed.
Two-photon transitions have provided the basis for numerous useful nonlinear interactions with radiation. In this paper we calculate the absorption and coupling coefficients for one-, two-, and three-wave mixing in two-photon homogeneously broadened media. The resulting field equations form the basis for understanding saturation spectroscopy, phase conjugation,1 and single-mode instabilities2 in lasers and optical bistability in two-photon media. We assume that the wave frequencies are not resonant with any intermediate state j and that the probe and conjugate waves cannot saturate the two-photon transition. This allows us to use the two-photon two-level model and to transfer much of our knowledge of the one-photon two-level systems to the two-photon case. Two major differences occur between the two models. First, dynamic Stark shifts of the active levels play an important role in generating asymmetric spectra and nonzero dispersion (≃ω s T1) at line center. Second, high pump intensities do not bleach the absorption coefficient at the Rabi sidebands.
Two- and three-step laser cascades have been recently detected experimentally in He-Ne mixtures and in pure neon. In the present paper a two-step cascade is treated in detail theoretically, starting with the density-matrix formulation. The corresponding equations can be simplified to rate equations under an assumption about the relative size of certain phase-memory times, which seems well justified in the present case. Because there are usually only minor differences in the results between a gaseous laser and a solid-state laser with an inhomogeneously broadened line, and because the mathematical treatment of the gas laser is more involved, the solid-state case will be treated here. The laser condition for both the lower and upper transitions is derived for the most general case, i.e., that all three levels are pumped. Finally, a special two-step cascade is treated in detail. The results are in qualitative agreement with the experimental findings, although a quantitative comparison is still impeded by the lack of an exact knowledge of some of the parameters, especially the radiative linewidth.