Normalized averaged equations of the helical motion of relativistic electrons in gyroresonance regimes. I. II Quasi-plane approximation

The present study derives a normalized system of differential equations of the averaged magnetically guided motion of relativistic electrons in the field of a resonant electromagnetic wave (possibly hybrid and nonsinusoidal) propagating with arbitrary phase velocity. The normalization is performed by a set of dimensionless parameters of transversely weakly relativistic theory and one new parameter, which is a specific property of the essentially relativistic theory. In addition, a determination is made of the approximate reducibility of the averaged equations of the gyroresonant motion of electrons in the field of a nonplane electromagnetic wave to a simplified form corresponding to the plane-wave case. It is shown that the quasi-plane approximation is valid only in a special narrow range of electron energy and wave phase velocity.