FEL indulators with the hollow-ring electron beam

A conceptual design of undulators with a modulated longitudinal magnetic field is proposed. The magnetic field is created by use of a solenoid with axis coincident with the electron beam axis. In order to modulate the magnetic field we propose an insertion of a row of alternating ferromagnetic and superconducting diaphragms in line with electron beam. The simulation of two-dimensional distribution of the magnetic field in the plane containing undulator axis was made using the computer code open-quotes Mermaidclose quotes. The magnetic field was analysed as a function of the system geometry. The dependence on the spacing l between superconducting diaphragms, inner a and outer b radii of the last ones is investigated. Two versions of the device are considered: with ferromagnetic rings made of magnetically soft material placed between the superconducting diaphragms and without them. It is shown that the field modulation depth increases with ratio of b/l and can exceed 50% in case of the ferromagnetic insertions. An approximate analytical calculation of the magnetic field distribution is performed as follows. The axial-symmetrical magnetic field can be defined by the vector potential with only one nonzero component A(r,φ) where r and φ are the cylindrical coordinates. The solution of the Laplace's equation is found under the assumption that the magnetic field is infinitely extended and periodic along the z-axis. The boundary conditions are defined by the undulator design. The result is used for the calculation of the particle dynamics and for the investigations of the trajectory stability. The spectral and angular distribution of the radiation emitted from the described systems is found. The estimations show that the proposed design allows to create relatively high magnitude of the magnetic field (up to 1 T) with a short period about 1 cm or less