Third-order combined function bending magnets

It is pointed out that a complete third-order description of a charged particle optical system must include a combined function bending magnet. The magnetic field can have dipole, quadrupole, sextupole, and octupole terms. Midplane symmetry is assumed. Each additional order in the optical analysis requires inclusion of an additional multipole in the field expansion. First-, second-, and third-order expansions require quadrupole, sextupole, and octupole terms, respectively. Third-order matrix elements can be derived by an iterative Green´s function solution of the differential equations of motion. Third-order transfer matrix elements arise not only from third-order terms in the equations of motion, but also from the cascading effect of second-order terms. Solutions have been derived and incorporated into the computer program Transport and Turtle