Focusing properties of an undulator with anharmonic magnetic fields

The focusing properties of an undulator are considered for the common case that the undulator magnetic fields are not pure sinusoidal (anharmonic). The exact 3D solution of the Laplace equations for the magnetic field near the axis is found. Using equations of electron motion in the fields it is shown that an admixture of higher harmonics in the undulator fields can significantly increase the focusing strength of the undulator. It is demonstrated that this higher harmonic transverse focusing of the electron beam does not violate the phase synchronism condition between the electron and the field, what is similar to the natural undulator focusing of the basic harmonic field.