Phase conditions for free electron lasers

The phase conditions for free electron lasers are analyzed on the basis of energy equations and Lorentz equations for the interaction of relativistic electrons with magnetostatic field and laser field. It is shown that the energy of relativistic electrons is reduced when the initial phase of incident light lies in the first and the fourth quadrants with respect to the relativistic electron beam. When the initial phase lies in the second and the third quadrants, the relativistic electrons are further accelerated.