Distorted orbit due to field errors and particle trajectories in combined undulator and axial magnetic field

Undulator and solenoid field errors cause electron trajectory deviation from the ideal orbit. Even small errors can result in a large lower frequency excursion from the undulator axis of a distorted orbit and of betatron oscillations performed now around it, especially near resonant conditions. Numerical calculation of a trajectory step by step requires large computing time and treats only particular cases, thus lacking generality. Theoretical treatment is traditionally based on random distribution of field errors, which allows a rather general approach, but is not convenient for practical purposes. In contrast, analytical treatment shows explicitly how distorted orbit and betatron oscillation amplitude depend on field parameters and errors and indicates how to eliminate these distortions. An analytical solution of the equations of motion can be found by expanding field errors and distorted orbit in Fourier series as was done earlier for the simplest case of a plane undulator without axial magnetic field. The same method is applied now to the more general case of combined generlized undulator and axial magnetic fields. The undulator field is a superposition of the fields of two plane undulators with mutually orthogonal fields and an arbitrary axial shift of the second undulator relative to the first. Beam space-charge forces and external linear focusing are taken into account. The particle trajectory is a superposition of ideal and distorted orbits with cyclotron gyration and slow drift gyration in the axial magnetic field caused by a balance of focusing and defocusing forces. The amplitudes of these gyrations depend on transverse coordinate and velocity at injection and can nearly double the total deviation of an electron from the undulator axis even after an adiabatic undulator entry. If the wavenumber of any Fourier harmonic is close to the wavenumbers of cyclotron or drift gyrations, a resonant increase of orbit distortion occurs