RELATIONSHIP BETWEEN MEAN RADIATED ENERGY, MEAN SQUARED RADIATED ENERGY AND SPONTANEOUS POWER SPECTRUM IN A POWER-SERIES EXPANSION OF THE EQUATIONS OF MOTION IN A FREE-ELECTRON LASER

To lowest order in the electric field and the inverse electron energy, the classical mean energy radiated by electrons in a free-electron laser with a symmetric magnet is equal to one-half the derivative, with respect to energy, of the classical mean squared radiated energy. The integral for the mean squared energy is also shown to be identical to the integral for the classical spontaneous power spectrum. © 1979 Società Italiana di Fisica.