Betatron function parameterization of beam optics including acceleration

Betatron function parameterization of symplectic matrices is of recognized utility in beam optical computations. The traditional beta functions β, α, γ (= (1 + α2)/β) and φ (the betatron phase advance) provide an emittance-independent representation of the properties of a beam transport system. They thereby decouple the problem of matching injected beam envelope properties to the acceptance of a particular transport system from the details of producing a beam of a specific emittance. The definition and interpretation of these parameters becomes, however, more subtle when acceleration effects, especially adiabatic damping (with associated nonsymplecticity of the transfer matrix), are included. The authors present algorithms relating symplectic representations of beam optics to the more commonly encountered nonsymplectic (x, x', y, y') representation which exhibits adiabatic damping. Betatron function parameterizations are made in both representations. Self-consistent physical interpretations of the betatron functions are given and applications to a standard beam transport program are made. 5 refs., 3 figs