Study of Fringe Field Transformations of Short Permanent Magnet Quadrupole Focusing Lens

Quadrupole magnet is an important element in the beam transport and manipulation. Permanent magnet quadrupole (PMQ) lenses appear frequently in various beamline attributed to their compact structure. In this paper, a method applicable to the fringing field transformation of a short permanent magnet quadrupole lens is proposed. The large electron beam envelope error, which occurs during the focusing process of short PMQ lenses, has been addressed. This achieved by replacing sharp cut-off fringing field with multi-rectangle cut-off fringing field. The causes of fringe field transformation errors are investigated by theoretical calculations of beam dynamics. The law of magnetic field gradient curve change of a short PMQ lens is analyzed. The scale parameters in the multi-rectangle approximation method are optimized by multi-particle tracking and finite element simulation. Applying the method to PMQ lenses of varying mechanical lengths confirms the feasibility and universality of the multi-rectangle approximation method for PMQ fringe field transformation.