Abstract
The Courant-Snyder theory gives a complete description of the uncoupled transverse dynamics of charged particles in electromagnetic focusing lattices. In this paper, the Courant-Snyder theory is generalized to the case of coupled transverse dynamics with two degrees of freedom. The generalized theory has the same structure as the original Courant-Snyder theory for one degree of freedom. The four basic components of the original Courant-Snyder theory, i.e., the envelope equation, phase advance, transfer matrix, and the Courant-Snyder invariant, all have their counterparts, with remarkably similar expressions, in the generalized theory presented here. In the generalized theory, the envelope function is generalized into an envelope matrix, and the envelope equation becomes a matrix envelope equation with matrix operations that are noncommutative. The generalized theory gives a new parametrization of the 4D symplectic transfer matrix that has the same structure as the parametrization of the 2D symplectic transfer matrix in the original Courant-Snyder theory. All of the parameters used in the generalized Courant-Snyder theory correspond to physical quantities of importance, and this parametrization can provide a valuable framework for accelerator design and particle simulation studies. A time-dependent canonical transformation is used to develop the generalized Courant-Snyder theory. Applications of the new theory to strongly and weakly coupled dynamics are given. It is shown that the stability of coupled dynamics can be determined by the generalized phase advance developed. Two stability criteria are given, which recover the known results about sum and difference resonances in the weakly coupled limit.
Highlights
AND THEORETICAL MODELThe transverse dynamics of a charged particle in a linear focusing lattice qðtÞ is described by an oscillator equation with time-dependent spring constant q€ þ qðtÞq 1⁄4 0; (1)where q represents one of the transverse coordinates, either x or y
Using a time-dependent canonical transformation technique, we have extended the Courant-Snyder theory for dynamics with one degree of freedom to the case of coupled transverse dynamics with 2 degrees of freedom
The envelope function is generalized to a 2 Â 2 envelope matrix, and the envelope equation is generalized to a matrix envelope equation
Summary
AND THEORETICAL MODELThe transverse dynamics of a charged particle in a linear focusing lattice qðtÞ is described by an oscillator equation with time-dependent spring constant q€ þ qðtÞq 1⁄4 0; (1)where q represents one of the transverse coordinates, either x or y. The transverse dynamics of a charged particle in a linear focusing lattice qðtÞ is described by an oscillator equation with time-dependent spring constant q€ þ qðtÞq 1⁄4 0; (1). The variable t appearing in Eq (1) is not really time, but rather the path length that plays the role of a timelike variable. The Courant-Snyder theory [1] gives a complete description of the solution to Eq (1), and serves as the fundamental theory that underlies the design of modern accelerators and storage rings. There are four main components of the Courant-Snyder theory: the envelope equation, the phase advance, the transfer matrix, and the Courant-Snyder invariant. The Courant-Snyder theory can be summarized as follows. Because Eq (1) is linear, the solution to Eq (1) can be expressed as a time-dependent linear map from the initial conditions, i.e.,
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