Analytical calculation of matched input in a helical undulator

The integrals of lateral components of magnetic field in a helical electromagnetic undulator of finite length have been calculated analytically, taking into account fringe fields. It is shown that an adiabatic input provides matched injection of beam only when the number of periods is sufficiently large (Na> 6−10) and requires additional correction when precision injection is necessary. On the basis of these analytical results, several methods of forming matched field by means of correcting elements are proposed. The minimal length of such elements is about 16 of an undulator period. The analytical results were confirmed by numerical calculations and experimental measurements. Depending on the specific requirements, one can choose one or another type of element. For more precise formation of fringe field, elements for independent correcton of only one of Bx and By field components are proposed.