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The previous theory of high-gain free-electron laser (FEL) operating in the quantum regime is semiclassical because the electron dynamic is quantized but the radiation field is classically described. Here, we present for the first time a fully quantum mechanical theory where also the field is quantized. We shall restrict to the FEL operation in the steady state regime where the slippage length is much smaller than the bunch length. The results predicted by this theory are quite different from those of the semiclassical theory.

A unified analysis is presented to calculate the incoherent spontaneous power of cooperative radiations based on self-amplified spontaneous emission. Using quantum mechanical tools, we derive analytical expressions for the incoherent spontaneous power of undulator and Cherenkov free-electron lasers (FELs). The undulator and Cherenkov FELs are considered as two different examples for the radiation that accumulate cooperatively. In the case of the undulator FEL, we show an excellent agreement between an expression for the incoherent radiation power derived in the present work and that obtained using a completely different approach [Phys. Rev. E 65 (2002) 026501]. For the Cherenkov radiation, we demonstrate a satisfactory agreement between the incoherent power predicted in our analysis and previous experimental results.