In this paper we use a recently suggested analytical solution of the pulse propagation equation in free-electron lasers (FEL) to get further information on the lethargy and on the velocity of propagation of the optical packet. We also introduce a refractive index for the FEL interaction. Finally, the supermodes relevant to the regime of long electron bunch are found analytically and a comparison with earlier numerical results is carried out.

The coherence properties of a $46.9\text{−}\mathrm{nm}$ soft-x-ray laser beam produced in a capillary discharge plasma have been investigated. The beam is produced in a regime that is free from the effect of refraction defocusing, with a divergence of only $0.5–0.6\mathrm{mrad}$. The discharge plasma is produced in capillary channels with length of $45\mathrm{cm}$, where the Fresnel number of the gain medium is of the order of unity. The full laser energy is about $300\mu \mathrm{J}$. The spatial coherence of the laser has been experimentally investigated using a Young’s double-slit interferometer and analyzed assuming a Gaussian Schell-model source and the generalized Van Cittert–Zernike theorem. We demonstrate the achievement of almost full spatial coherence and a near-Gaussian intensity distribution.

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/${\gamma }^3$ and the relevant physical consequences.