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 parametrized 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 a $U(2)$ 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 dimensions. 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 parametrization 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 expresses the theory using only the generalized Twiss function $\ensuremath{\beta}$. 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. The generalized CS theory provides an effective tool to study coupled dynamics and to discover more optimized lattice designs in the larger parameter space of general focusing lattices.
Highlights
In accelerators and storage rings, charged particles are confined transversely by electromagnetic focusing lattices
The envelope function is generalized into an envelope matrix, and the phase advance is generalized into a 4D symplectic rotation, or a Uð2Þ element
By fixing the gauge freedom with a desired symmetry, the generalized CS parametrization assumes the form of the modified Iwasawa decomposition, whose importance in phase space optics and phase space quantum mechanics has been recently realized
Summary
In accelerators and storage rings, charged particles are confined transversely by electromagnetic focusing lattices. Many different kinds of focusing lattices have been successfully designed and implemented. The fundamental theoretical tool in designing an uncoupled quadrupole lattice is the Courant-Snyder (CS) theory [1], which can be summarized as follows. For a given set of focusing lattices in the x- and y-directions κxðtÞ and κyðtÞ, a particle’s dynamics is governed by the oscillation equation qþ κqðtÞq 1⁄4 0; (1). Where q represents one of the transverse coordinates, either x or y. The solution of Eq (1) can be expressed as a symplectic linear map MðtÞ that advances the phase space coordinates
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