Second approximation of the averaging method in electron maser theory

In the theory of electron masers, if the interaction length is much greater than the wavelength and Doppler corrections are applied to the resonator field frequency, use is made conventionally of an approximation that corresponds to the first approximation of the averaging method. In the present paper, the nonlinear differential equations with one fast phase, which describe the electron oscillations in a maser, are analyzed within the framework of the second approximation. Using a cyclotron resonance maser with a Fabry-Perot resonator whose axis is parallel to the static magnetic field, it is shown that a nonresonance wave should be taken into consideration, regardless of the value of the interaction length, and how the second approximation can be used to accomplish this.