Beam expansion with specified final distributions

The formation of nearly uniformly distributed beams has been accomplished by the use of multipole magnets. Multipole fields, however, are an inappropriate basis for creating precise distributions, particularly since substantial departures from uniformity are produced with a finite number of multipole elements. A more appropriate formalism that allows precise formation of a desired distribution is presented. Design of nonlinear magnets for uniform-beam production and the optics of an accompanying expansion system are presented.