Design of flat-to-round (Vortex) beam adapter with strong space charge

We describe the design of a symmetrical skew-quadrupole triplet and associated four-quadrupole matching section for a flat-to-vortex beam transformation in a low-energy, high current electron experiment at the University of Maryland. We review the basic principles involved, from the Courant-Snyder parameters, beam (sigma) matrix, conservation of canonical angular momentum and emittances, to the evolution of the beam envelopes, with emphasis on practical aspects of the design. The initial optimization involves the use of a standard sigma matrix code that includes direct space charge effects. Refinement of the calculations are made possible by a set of moment equations that can use rotated quadrupoles and an expanded set of parameters for optimization. Additional particle-in-cell (PIC) computer simulations and preliminary results from experiments are presented in an accompanying paper.