Method of Computing First-, Second-, and Third-Order Transfer Coefficients for Arbitrary Fields

For given arbitrary electromagnetic fields, an analytical method to obtain transfer coefficients up to the third order in phase-space coordinates is described in this paper. The method is based on numerical tracking of a number of representative particles followed by the least-square minimization. The result is applied to an arbitrary beamline consisting of linear and nonlinear beam-optical elements and favorably compared with the numerical particle tracking. Transfer coefficients obtained with our method are also compared with those obtained from an existing computer code.