The tilt sensitivity of a grazing incidence confocal unstable resonator with applications to free-electron lasers

The paraxial sensitivity to tilt misalignments of a four-element grazing incidence unstable resonator is calculated in terms of the beam displacement both at the center of a free-electron laser (FEL) wiggler and on the tilted component. A convenient representation of the resonator is made by splitting the diverging elements such that the resonator is modeled as three afocal telescopes. This formulation simplifies the round-trip system matrix and yields terms that are physically meaningful. The grazing incidence resonator is then compared with an equivalent nongrazing incidence two-element resonator. For a tilt misalignment of the small-angle-of-incidence components, the resulting beam displacement at the center of the FEL wiggler is calculated to be equivalent for both types of resonators and is found to be a function of the wavelength, the round-trip magnification, the fractional allowable beam displacement at the center of the FEL wigglers and the beam size On the small-angle-of-incidence components.