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The theories of recoil-induced resonances (RIR) [J. Guo, P. R. Berman, B.Dubetsky and G. Grynberg, Phys. Rev. A {\bf 46}, 1426 (1992)] and thecollective atomic recoil laser (CARL) [ R. Bonifacio and L. De Salvo, Nucl.Instrum. Methods A {\bf 341}, 360 (1994)] are compared. Both theories can beused to derive expressions for the gain experienced by a probe fieldinteracting with an ensemble of two-level atoms that are simultaneously drivenby a pump field. It is shown that the RIR and CARL formalisms are equivalent.Differences between the RIR and CARL arise because the theories are typicallyapplied for different ranges of the parameters appearing in the theory. The RIRlimit considered in this paper is $qP_{0}/M\omega_{q}\gg 1$, while the CARLlimit is $qP_{0}/M\omega_{q}\lesssim 1$, where $% q $ is the magnitude of thedifference of the wave vectors of the pump and probe fields, $P_{0}$ is thewidth of the atomic momentum distribution and $% \omega_{q}$ is a recoilfrequency. The probe gain for a probe-pump detuning equal to zero is analyzedin some detail, in order to understand how the gain arises in a system which,at first glance, might appear to have vanishing gain. Moreover, it is shownthat the calculations, carried out in perturbation theory have a range ofapplicability beyond the recoil problem. Experimental possibilities forobserving CARL are discussed.Comment: 16 pages, 1 figure. Submitted to Physical Review

Comparison of recoil-induced resonances and the collective atomic recoil laser