Analytical methods for describing charged particle dynamics in general focusing lattices using generalized Courant-Snyder theory

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 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 parameterization 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 β.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 design in the larger parameter space of general focusing lattices.

27 pages

References in corpus (1)

• Betatron motion with coupling of horizontal and vertical degrees of freedom

Cited by in corpus (12)

• Experimental Proof of Adjustable Single-Knob Ion Beam Emittance Partitioning
• Explicit High-Order Gauge-Independent Symplectic Algorithms for Relativistic Charged Particle Dynamics
• On the structure of the two-stream instability -- complex G-Hamiltonian structure and Krein collisions between positive- and negative-action modes
• Spontaneous and explicit parity-time-symmetry breaking in drift wave instabilities
• Generalized Kapchinskij-Vladimirskij Distribution and Beam Matrix for Phase-Space Manipulations of High-Intensity Beams
• Rotating system for four-dimensional transverse rms-emittance measurements
• Advanced fuel fusion, phase space engineering, and structure-preserving geometric algorithms
• Extension of Busch's Theorem to Particle Beams
• A necessary and sufficient condition for the stability of linear Hamiltonian systems with periodic coefficients
• Spectral and structural stability properties of charged particle dynamics in coupled lattices
• Linear Beam Stability in Periodic Focusing Systems:Krein Signature and Band Structure
• Envelope Hamiltonian for Charged-Particle Dynamics in General Linear Coupled Systems