Matrix formalism of synchrobetatron coupling

Abstract

In this paper we present a complete linear synchrobetatron coupling formalism by studying the transfer matrix which describes linear horizontal and longitudinal motions. With the technique established in the linear horizontal-vertical coupling study [D. Sagan and D. Rubin, Phys. Rev. ST Accel. Beams {\bf 2}, 074001 (1999)], we found a transformation to block diagonalize the transfer matrix and decouple the betatron motion and the synchrotron motion. By separating the usual dispersion term from the horizontal coordinate first, we were able to obtain analytic expressions of the transformation under reasonable approximations. We also obtained the perturbations to the betatron tune and the Courant-Snyder functions. The closed orbit changes due to finite energy gains at rf cavities and radiation energy losses were also studied by the 5$\times$5 extended transfer matrix with the fifth column describing kicks in the 4-dimension phase space.Comment: 25 pages, 6 figures; The subject-class should be soly "Accelerator Physics"; The comment (the last setence of the 3rd paragraph) in version 1 about the first appearance of the closed orbit formula was wrong. To my best knowledge PRE 72, 056501 was the first paper giving that formula, not SLAC technote PTM-5

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