Generalized Kapchinskij-Vladimirskij Distribution and Beam Matrix for Phase-Space Manipulations of High-Intensity Beams

Abstract

In an uncoupled linear lattice system, the Kapchinskij-Vladimirskij (KV) distribution, formulated on the basis of the single-particle Courant-Snyder (CS) invariants, has served as a fundamental theoretical basis for the analyses of the equilibrium, stability, and transport properties of high-intensity beams for the past several decades. Recent applications of high-intensity beams, however, require beam phase-space manipulations by intentionally introducing strong coupling. In this Letter, we report the full generalization of the KV model by including all of the linear (both external and space-charge) coupling forces, beam energy variations, and arbitrary emittance partition, which all form essential elements for phase-space manipulations. The new generalized KV model yields spatially uniform density profiles and corresponding linear self-field forces as desired.The corresponding matrix envelope equations and beam matrix for the generalized KV model provide important new theoretical tools for the detailed design and analysis of high-intensity beam manipulations, for which previous theoretical models are not easily applicable.

A necessary and sufficient condition for the stability of linear Hamiltonian systems with periodic coefficients

“…In this section, we present the method of time-dependent canonical transformation in preparation for the proof of Theorem 1 in the next section. It is necessary to emphasize again that the results and techniques leading to Theorem 1 have been reported previously in the context of charged particle dynamics in a general focusing lattice [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. These contents are included here for easy reference and self-consistency.…”

“…Techniques of normal forms for stable symplectic matrices and horizontal polar decomposition for symplectic matrices are developed to prove Theorem 2. The results and techniques leading to Theorem 1 have been reported previously in the context of charged particle dynamics in a general focusing lattice [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. These contents are included here for easy reference and self-consistency.…”

Linear beam stability in periodic focusing systems: Krein signature and band structure

“…When skew quadrupole components are present in addition to the standard quadrupoles, the transverse dynamics in the x− and y−dimensions are coupled. Such couplings between two dimensions can be introduced either intentionally [32], or as a result of misalignment of the quadrupole magnet [4,33]. The single-particle linear dynamics with a skew quadrupole is described by…”

Particle beam eigen-emittances, phase integral, vorticity, and rotations

“…Since this set may not fit requirements of certain beam applications, tailoring of eigen-emittances became a subject of extensive theoretical and experimental research. The first proposal of eigen-emittance modification was made in [2] followed by other fundamental investigations [3][4][5][6][7][8][9][10][11][12][13][14][15] and experimental applications in linear electron [16][17][18][19] and ion accelerators [20][21][22]. Measurements of eigen-emittances and beam coupling have been reported in [23][24][25][26][27][28] for instance.…”