Analytic second- and third-order achromat designs

An achromat is a transport system that carries a beam without distorting its transverse phase space distribution. In this study, we apply the Lie algebraic technique to a repetitive FODO array to make it either a second-order or a third-order achromat. (Achromats based on reflection symmetries are not studied here.) We consider third-order achromats whose unit FODO cell layout is shown. The second-order achromat layout is the same except the octupoles are absent. For the second-order achromats, correction terms (due to the finite bending of the dipoles) to the well-known formulae for the sextupole strengths are derived. For the third-order achromats, analytic expressions for the five octupole strengths are given. The quadrupole, sextupole and octupole magnets are assumed to be thin-lens elements. The dipoles are assumed to be sector magnets filling the drift spaces