Analytical Methods for Charged Particle Dynamics In General Focusing Lattices Using Generalized Courant-Snyder Theory
Authors: Hong Qin
Abstract:
The dynamics of charged particles in general linear focusing lattices with quadrupole, skew-
quadrupole, dipole, and solenoidal components, as well as torsion of the fiducial orbit and variation
of beam energy is parameterized using a generalized Courant-Snyder (CS) theory, which extends
the original CS theory for one degree of freedom to higher dimensions. The envelope function is
generalized into an envelope matrix, and the phase advance is generalized into a 4D symplectic
rotation, or an U(2) element. The 1D envelope equation, also known as the Ermakov-Milne-Pinney
equation in quantum mechanics, is generalized to an envelope matrix equation in higher dimen-
sions. Other components of the original CS theory, such as the transfer matrix, Twiss functions,
and CS invariant (also known as the Lewis invariant) all have their counterparts, with remarkably
similar expressions, in the generalized theory. The gauge group structure of the generalized theory
is analyzed. By fixing the gauge freedom with a desired symmetry, the generalized CS parame-
terization assumes the form of the modified Iwasawa decomposition, whose importance in phase
space optics and phase space quantum mechanics has been recently realized. This gauge fixing also
symmetrizes the generalized envelope equation and express the theory using only the generalized
Twiss function . The generalized phase advance completely determines the spectral and structural
stability properties of a general focusing lattice. For structural stability, the generalized CS theory
enables application of the Krein-Moser theory to greatly simplify the stability analysis.
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Submitted to: Physical Review Special Topics - Accelerator and Beams (December 2013)
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Download PPPL-4974 (pdf 502 KB 25 pp)
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