  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 the
collective atomic recoil laser (CARL) [ R. Bonifacio and L. De Salvo, Nucl.
Instrum. Methods A {\bf 341}, 360 (1994)] are compared. Both theories can be
used to derive expressions for the gain experienced by a probe field
interacting with an ensemble of two-level atoms that are simultaneously driven
by 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 typically
applied for different ranges of the parameters appearing in the theory. The RIR
limit considered in this paper is $qP_{0}/M\omega_{q}\gg 1$, while the CARL
limit is $qP_{0}/M\omega_{q}\lesssim 1$, where $% q $ is the magnitude of the
difference of the wave vectors of the pump and probe fields, $P_{0}$ is the
width of the atomic momentum distribution and $% \omega_{q}$ is a recoil
frequency. The probe gain for a probe-pump detuning equal to zero is analyzed
in 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 shown
that the calculations, carried out in perturbation theory have a range of
applicability beyond the recoil problem. Experimental possibilities for
observing CARL are discussed.
