A simple derivation of relativistic full-wave equations at electron cyclotron resonance

When a wave passes through an electron gyroresonance, in a plasma in the presence of a magnetic field gradient, there is a small spread in the resonance due to the electron's Larmor radius. Mathematically this is represented by the inclusion of the so called gyrokinetic term in the resonance condition, Lashmore-Davies and Dendy. The smallness of this term, compared with other effects such as relativistic broadening, suggests that it should be negligible. However, we shall show here, by extending the method of Cairns et al., into the relativistic regime, that its inclusion is vital for producing self consistent full-wave equations which describe electron gyroresonance. The method is considerably simpler than those used previously by Maroli et al., Petrillo et al., and Lampis et al., for obtaining similar equations. As an example we include a calculation for the O-Mode passing perpendicularly through the fundamental