Quantum theory of magnetic bremsstrahlung gain

We present a quantum electrodynamic calculation of the gain associated with the stimulated emission and absorption of photons by high-energy electrons in the presence of a spatially periodic, static magnetic field. The calculation is relevant to the description of a high-energy electron laser in the long-interaction-length low-density (single-particle) regime. The different kinematic constraints imposed on the photon-emission and -absorption processes are shown to produce the effect of population inversion in an energetic electron beam which allows for a net gain in the number of photons produced per unit path length. The spectral dependence of the gain on the wave vector of the emitted photon has a width determined by the incident electron momentum distribution and a maximum in the vicinity of a k value [2(W₁₀mc²)²(2πλ₀)] determined by the central electron momentum W₁₀c and the spatial period of the magnetic field. Derived formulas for the gain and gain-minus-loss coefficients are compared with formulas obtained on the basis of the stimulated Comptonscattering model. While the separate gain and loss coefficients diverge in the limit →0, the gain-minus-loss coefficient is shown to have a finite value in this limit which agrees with the result of a classical derivation.