# 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 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.more »

- Authors:

- Publication Date:

- Research Org.:
- Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1151738

- Report Number(s):
- FERMILAB-PUB-14-092-APC

Journal ID: ISSN 1098-4402; PRABFM; ArticleNumber: 044001

- DOE Contract Number:
- AC02-07CH11359

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review Special Topics - Accelerators and Beams

- Additional Journal Information:
- Journal Volume: 17; Journal Issue: 4; Journal ID: ISSN 1098-4402

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Qin, Hong, Davidson, Ronald C., Burby, Joshua W., and Chung, Moses.
```*Analytical methods for describing charged particle dynamics in general focusing lattices using generalized Courant-Snyder theory*. United States: N. p., 2014.
Web. doi:10.1103/PhysRevSTAB.17.044001.

```
Qin, Hong, Davidson, Ronald C., Burby, Joshua W., & Chung, Moses.
```*Analytical methods for describing charged particle dynamics in general focusing lattices using generalized Courant-Snyder theory*. United States. doi:10.1103/PhysRevSTAB.17.044001.

```
Qin, Hong, Davidson, Ronald C., Burby, Joshua W., and Chung, Moses. Tue .
"Analytical methods for describing charged particle dynamics in general focusing lattices using generalized Courant-Snyder theory". United States. doi:10.1103/PhysRevSTAB.17.044001. https://www.osti.gov/servlets/purl/1151738.
```

```
@article{osti_1151738,
```

title = {Analytical methods for describing charged particle dynamics in general focusing lattices using generalized Courant-Snyder theory},

author = {Qin, Hong and Davidson, Ronald C. and Burby, Joshua W. and Chung, Moses},

abstractNote = {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.},

doi = {10.1103/PhysRevSTAB.17.044001},

journal = {Physical Review Special Topics - Accelerators and Beams},

issn = {1098-4402},

number = 4,

volume = 17,

place = {United States},

year = {2014},

month = {4}

}

Works referenced in this record:

##
Symmetries and invariants of the oscillator and envelope equations with time-dependent frequency

journal, May 2006

- Qin, Hong; Davidson, Ronald C.
- Physical Review Special Topics - Accelerators and Beams, Vol. 9, Issue 5

##
Kinematic moment invariants for linear Hamiltonian systems

journal, March 1990

- Neri, Filippo; Rangarajan, Govindan
- Physical Review Letters, Vol. 64, Issue 10

##
Generalized Courant-Snyder Theory for Charged-Particle Dynamics in General Focusing Lattices

journal, September 2013

- Qin, Hong; Davidson, Ronald C.; Chung, Moses
- Physical Review Letters, Vol. 111, Issue 10

##
Transverse-to-longitudinal emittance exchange to improve performance of high-gain free-electron lasers

journal, October 2006

- Emma, P.; Huang, Z.; Kim, K. -J.
- Physical Review Special Topics - Accelerators and Beams, Vol. 9, Issue 10

##
Theory of the alternating-gradient synchrotron

journal, January 1958

- Courant, E. D.; Snyder, H. S.
- Annals of Physics, Vol. 3, Issue 1

##
Class of Generalized Kapchinskij-Vladimirskij Solutions and Associated Envelope Equations for High-Intensity Charged-Particle Beams

journal, February 2013

- Qin, Hong; Davidson, Ronald C.
- Physical Review Letters, Vol. 110, Issue 6

##
On Some Types of Topological Groups

journal, July 1949

- Iwasawa, Kenkichi
- The Annals of Mathematics, Vol. 50, Issue 3

##
New aspects in the theory of stability of Hamiltonian systems

journal, February 1958

- Moser, Jürgen
- Communications on Pure and Applied Mathematics, Vol. 11, Issue 1

##
Twiss parameters and beam matrix formulation of generalized Courant–Snyder theory for coupled transverse beam dynamics

journal, August 2010

- Chung, Moses; Qin, Hong; Davidson, Ronald C.
- Physics of Plasmas, Vol. 17, Issue 8

##
On geometric phases and dynamical invariants

journal, October 1994

- Monteoliva, D. B.; Korsch, H. J.; Nunez, J. A.
- Journal of Physics A: Mathematical and General, Vol. 27, Issue 20

##
A Proposed Möbius Accelerator

journal, February 1995

- Talman, Richard
- Physical Review Letters, Vol. 74, Issue 9

##
On the generalized Twiss parameters and Courant-Snyder invariant in classical and quantum optics

journal, March 1992

- Dattoli, G.; Mari, C.; Richetta, M.
- Il Nuovo Cimento B, Vol. 107, Issue 3

##
Beam dynamics of the Neutralized Drift Compression Experiment-II, a novel pulse-compressing ion accelerator

journal, May 2010

- Friedman, A.; Barnard, J. J.; Cohen, R. H.
- Physics of Plasmas, Vol. 17, Issue 5

##
Analysis of continuously rotating quadrupole focusing channels using generalized Courant-Snyder theory

journal, August 2013

- Chung, Moses; Qin, Hong; Gilson, Erik P.
- Physics of Plasmas, Vol. 20, Issue 8

##
Stability properties of the transverse envelope equations describing intense ion beam transport

journal, February 2004

- Lund, Steven M.; Bukh, Boris
- Physical Review Special Topics - Accelerators and Beams, Vol. 7, Issue 2

##
Transverse to longitudinal emittance exchange

journal, August 2002

- Cornacchia, M.; Emma, P.
- Physical Review Special Topics - Accelerators and Beams, Vol. 5, Issue 8

##
Review of single bunch instabilities driven by an electron cloud

journal, December 2004

- Zimmermann, F.
- Physical Review Special Topics - Accelerators and Beams, Vol. 7, Issue 12

##
Classical adiabatic angles and quantal adiabatic phase

journal, January 1985

- Berry, M. V.
- Journal of Physics A: Mathematical and General, Vol. 18, Issue 1

##
An Exact Quantum Theory of the Time‐Dependent Harmonic Oscillator and of a Charged Particle in a Time‐Dependent Electromagnetic Field

journal, August 1969

- Lewis, H. R.; Riesenfeld, W. B.
- Journal of Mathematical Physics, Vol. 10, Issue 8

##
*Classical Mechanics*

journal, September 1952

- Goldstein, Herbert; Twersky, Vic
- Physics Today, Vol. 5, Issue 9

##
Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams

journal, January 1998

- Simon, R.; Mukunda, N.
- Journal of the Optical Society of America A, Vol. 15, Issue 8

##
A physical parametrization of coupled transverse dynamics based on generalized Courant–Snyder theory and its applications

journal, May 2009

- Qin, Hong; Davidson, Ronald C.
- Physics of Plasmas, Vol. 16, Issue 5

##
Generalized Kapchinskij-Vladimirskij Distribution and Envelope Equation for High-Intensity Beams in a Coupled Transverse Focusing Lattice

journal, November 2009

- Qin, Hong; Chung, Moses; Davidson, Ronald C.
- Physical Review Letters, Vol. 103, Issue 22

##
Class of Exact Invariants for Classical and Quantum Time‐Dependent Harmonic Oscillators

journal, November 1968

- Lewis, H. R.
- Journal of Mathematical Physics, Vol. 9, Issue 11

##
Generalized Courant-Snyder theory for coupled transverse dynamics of charged particles in electromagnetic focusing lattices

journal, June 2009

- Qin, Hong; Davidson, Ronald C.
- Physical Review Special Topics - Accelerators and Beams, Vol. 12, Issue 6

##
Control of emittance growth due to mismatches in accelerators that use stellarator transport

journal, January 1995

- Krall, J.; Slinker, S.; Lampe, M.
- Journal of Applied Physics, Vol. 77, Issue 2

##
On the theory of time-dependent linear canonical transformations as applied to Hamiltonians of the harmonic oscillator type

journal, January 1977

- Leach, P. G. L.
- Journal of Mathematical Physics, Vol. 18, Issue 8

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

journal, October 2010

- Lebedev, V. A.; Bogacz, S. A.
- Journal of Instrumentation, Vol. 5, Issue 10

##
The Numerical Determination of Characteristic Numbers

journal, April 1930

- Milne, W. E.
- Physical Review, Vol. 35, Issue 7

##
The nonlinear differential equation $y”+p(x)y+cy\sp {-3}=0$

journal, May 1950

- Pinney, Edmund
- Proceedings of the American Mathematical Society, Vol. 1, Issue 5

##
Correspondence between Berry's phase and Lewis's phase for quadratic Hamiltonians

journal, September 1988

- Morales, D. A.
- Journal of Physics A: Mathematical and General, Vol. 21, Issue 18

##
Twiss parameters and evolution of quantum harmonic-oscillator states

journal, October 1992

- Dattoli, G.; Gallerano, G. P.; Mari, C.
- Il Nuovo Cimento B Series 11, Vol. 107, Issue 10

##
Generalized Courant–Snyder theory and Kapchinskij–Vladimirskij distribution for high-intensity beams in a coupled transverse focusing lattice

journal, May 2011

- Qin, Hong; Davidson, Ronald C.
- Physics of Plasmas, Vol. 18, Issue 5

##
Parametrization of Linear Coupled Motion in Periodic Systems

journal, January 1973

- Edwards, D. A.; Teng, L. C.
- IEEE Transactions on Nuclear Science, Vol. 20, Issue 3

##
General moment invariants for linear Hamiltonian systems

journal, February 1992

- Dragt, Alex J.; Neri, Filippo; Rangarajan, Govindan
- Physical Review A, Vol. 45, Issue 4

##
A new derivation of the generalized Courant-Snyder theory with coupling

journal, October 1992

- 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