The small-gain regime of free-electron lasers (FEL) without inversion is considered and seeming contradictions with the traditional theory of FEL are resolved. As a result, a generalized Madey's theorem is obtained for the case of a phase shift given to electrons between the two wigglers. It explicitly demonstrates the contribution of interference of radiation from the wigglers. It is shown, by considering the motion in the phase space, that Liouville's theorem applied to the motion of electrons in a two-dimensional (rather than one-dimensional) real space is consistent with a nonzero integral of the gain over electron energies. These conclusions are crucial for the achievement of orders-of-magnitude improvement in the FEL gain in the case of a large electron energy spread. [References: 30]
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