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

Journal Article · · Physical Review Special Topics - Accelerators and Beams
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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 Uð2Þ 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 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 designs in the larger parameter space of general focusing lattices.

Research Organization:
Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)
Sponsoring Organization:
USDOE Office of Science (SC), High Energy Physics (HEP)
DOE Contract Number:
AC02-07CH11359
OSTI ID:
1151738
Report Number(s):
FERMILAB-PUB-14-092-APC; PRABFM; ArticleNumber: 044001
Journal Information:
Physical Review Special Topics - Accelerators and Beams, Vol. 17, Issue 4; ISSN 1098-4402
Country of Publication:
United States
Language:
English

References (36)

Symmetries and invariants of the oscillator and envelope equations with time-dependent frequency journal May 2006
Kinematic moment invariants for linear Hamiltonian systems journal March 1990
Generalized Courant-Snyder Theory for Charged-Particle Dynamics in General Focusing Lattices journal September 2013
Transverse-to-longitudinal emittance exchange to improve performance of high-gain free-electron lasers journal October 2006
Theory of the alternating-gradient synchrotron journal January 1958
Class of Generalized Kapchinskij-Vladimirskij Solutions and Associated Envelope Equations for High-Intensity Charged-Particle Beams journal February 2013
On Some Types of Topological Groups journal July 1949
New aspects in the theory of stability of Hamiltonian systems journal February 1958
Twiss parameters and beam matrix formulation of generalized Courant–Snyder theory for coupled transverse beam dynamics journal August 2010
On geometric phases and dynamical invariants journal October 1994
A Proposed Möbius Accelerator journal February 1995
On the generalized Twiss parameters and Courant-Snyder invariant in classical and quantum optics journal March 1992
Beam dynamics of the Neutralized Drift Compression Experiment-II, a novel pulse-compressing ion accelerator journal May 2010
Analysis of continuously rotating quadrupole focusing channels using generalized Courant-Snyder theory journal August 2013
Stability properties of the transverse envelope equations describing intense ion beam transport journal February 2004
Transverse to longitudinal emittance exchange journal August 2002
Review of single bunch instabilities driven by an electron cloud journal December 2004
Classical adiabatic angles and quantal adiabatic phase journal January 1985
An Exact Quantum Theory of the Time‐Dependent Harmonic Oscillator and of a Charged Particle in a Time‐Dependent Electromagnetic Field journal August 1969
Classical Mechanics journal September 1952
Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams journal January 1998
A physical parametrization of coupled transverse dynamics based on generalized Courant–Snyder theory and its applications journal May 2009
Generalized Kapchinskij-Vladimirskij Distribution and Envelope Equation for High-Intensity Beams in a Coupled Transverse Focusing Lattice journal November 2009
Class of Exact Invariants for Classical and Quantum Time‐Dependent Harmonic Oscillators journal November 1968
Generalized Courant-Snyder theory for coupled transverse dynamics of charged particles in electromagnetic focusing lattices journal June 2009
Control of emittance growth due to mismatches in accelerators that use stellarator transport journal January 1995
On the theory of time-dependent linear canonical transformations as applied to Hamiltonians of the harmonic oscillator type journal January 1977
Betatron motion with coupling of horizontal and vertical degrees of freedom journal October 2010
The Numerical Determination of Characteristic Numbers journal April 1930
The nonlinear differential equation $y”+p(x)y+cy\sp {-3}=0$ journal May 1950
Correspondence between Berry's phase and Lewis's phase for quadratic Hamiltonians journal September 1988
Twiss parameters and evolution of quantum harmonic-oscillator states journal October 1992
Generalized Courant–Snyder theory and Kapchinskij–Vladimirskij distribution for high-intensity beams in a coupled transverse focusing lattice journal May 2011
Parametrization of Linear Coupled Motion in Periodic Systems journal January 1973
General moment invariants for linear Hamiltonian systems journal February 1992
A new derivation of the generalized Courant-Snyder theory with coupling
  • Dattoli, G.; Mari, C.; Mezi, L.
  • Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Vol. 321, Issue 3 https://doi.org/10.1016/0168-9002(92)90053-7
journal October 1992

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