Fourier analysis for the rapid calculation of distorted undulator orbits and their correction

A simple method of electron trajectory calculation in undulators is proposed. Expanding the field errors in a Fourier series, one can calculate the distorted orbit and electron trajectory, which is a betatron oscillation around the distorted orbit with initial amplitude and velocity opposite in sign and equal in absolute value to those of the distorted orbit at the undulator input. The most dangerous components are “Resonant” harmonics of the field distortion with a period close to the betatron oscillation period. Correcting one, two or at most three dangerous harmonics is usually sufficient to obtain an electron orbit acceptably close to the ideal orbital. The above method of trajectory calculation is compared with some examples of numerical calculations published earlier in the literature.