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

Chung, М.; Qin, Hong; Davidson, Ronald C.

doi:10.1063/1.3474930

Cited by 11 publications

(10 citation statements)

References 21 publications

(15 reference statements)

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“…It generalizes the 1D envelope equation (7), or the Ermakov-Milne-Pinney equation [28][29][30], as well as the previous matrix envelope equation for cases with only quadrupole and skew-quadrupole magnets, i.e., R ¼ 0 [21][22][23][24]. For n degrees of freedom, the envelope matrix w will be n × n, and the generalized envelope equation has the same form as Eq.…”

Section: Generalized Courant-snyder Theorymentioning

confidence: 99%

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

Qin

Davidson

Burby

et al. 2014

Phys. Rev. ST Accel. 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.

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Section: Generalized Courant-snyder Theorymentioning

confidence: 99%

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

Qin

Davidson

Burby

et al. 2014

Phys. Rev. ST Accel. Beams

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“…We generalize the classical KV solution and the associated nonlinear envelope equations for high intensity beams to the case of a coupled lattice [18]. To construct the generalized KV solution for high intensity beams in a coupled lattice, we need to first generalize the Courant-Snyder (CS) theory for a single charged particle to the case of a couple lattice [19][20][21]. In particular, it is necessary to find a generalized CS invariant.…”

Section: Introductionmentioning

confidence: 99%

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

Qin

Davidson

2011

Physics of Plasmas

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1The Courant-Snyder (CS) theory and the Kapchinskij-Vladimirskij (KV) distribution for high-intensity beams in a uncoupled focusing lattice are generalized to the case of coupled transverse dynamics. The envelope function is generalized to an envelope matrix, and the envelope equation becomes a matrix envelope equation with matrix operations that are non-commutative. In an uncoupled lattice, the KV distribution function, first analyzed in 1959, is the only known exact solution of the nonlinear Vlasov-Maxwell equations for high-intensity beams including self-fields in a self-consistent manner. The KV solution is generalized to high-intensity beams in a coupled transverse lattice using the generalized CS invariant. This solution projects to a rotating, pulsating elliptical beam in transverse configuration space. The fully self-consistent solution reduces the nonlinear Vlasov-Maxwell equations to a nonlinear matrix ordinary differential equation for the envelope matrix, which determines the geometry of the pulsating and rotating beam ellipse. These results provide us with a new theoretical tool to investigate the dynamics of high-intensity beams in a coupled transverse lattice. A strongly coupled lattice, a so-called N-rolling lattice, is studied as an example. It is found that strong coupling does not deteriorate the beam quality. Instead, the coupling induces beam rotation, and reduces beam pulsation.

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“…Recently we have generalized the CS theory for uncoupled dynamics to treat coupled dynamics induced by a skew-quadrupole lattice [17][18][19][20]. However, this generalization is valid only for cases without torsion and solenoidal components, i.e., cases with R ¼ 0 in Eq.…”

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confidence: 99%

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

et al. 2013

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The Courant-Snyder (CS) theory for one degree of freedom is generalized to the case of coupled transverse dynamics 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. The envelope function is generalized into an envelope matrix, and the phase advance is generalized into a 4D sympletic rotation. The envelope equation, the transfer matrix, and the CS invariant of the original CS theory all have their counterparts, with remarkably similar expressions, in the generalized theory.

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

Abstract: A physical parametrization of coupled transverse dynamics based on generalized Courant-Snyder theory and its applications Phys. Plasmas 16, 050705 (2009);

Cited by 11 publications

References 21 publications

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

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

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

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

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