Linear undulator brightness: Inclusion of sextupolar magnetic-field contributions and of higher-order energy corrections

Abstract

In this paper we discuss the theory of emission by relativistic electrons in linearly polarized undulators. We show that the analytical computations are greatly simplified by the introduction of the so-called generalized Bessel functions, originally introduced to treat problems for which the dipolar approximation does not hold. We also discuss the modification induced in the brightness by the inclusion of the sextupolar terms, which arise when electrons are injected off the undulator axis. We show that the phenomonology inherent to this problem is very rich, as additional harmonics come into play and may cause sizable effects. The analysis we develop may be useful to obtain further insight in the physics underlying the so-called inhomogeneously broadened regime. Finally, we also discuss the inclusion of energy corrections up to 1/${\ensuremath{\gamma}}^{3}$ and the relevant physical consequences.