Theoretical study of transverse-longitudinal emittance coupling Page: 3 of 5
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Theoretical study of transverse-longitudinal emittance coupling*
H. Qin"2, R. C. Davidson', M. Chung3 J. J. Barnard4, and T. F. Wang5
1. PPPL, Princeton, NJ 08543, USA. 2. USTC, Hefei 230026, China.
3. Handong Global Univ., Pohang 791-708, Korea
4. LLNL, Livermore, CA 94550, USA. 5. LANL, Los Alamos, NM 87545, USAAbstract
The effect of a weakly coupled periodic lattice in terms
of achieving emittance exchange between the transverse
and longitudinal directions is investigated using the gen-
eralized Courant-Snyder theory for coupled lattices.
INTRODUCTION
Recently, the concept and technique of transverse-
longitudinal emittance coupling have been proposed for ap-
plications in the Linac Coherent Light Source [1, 2] and
other free-electron lasers to reduce the transverse emit-
tance of the electron beam. Such techniques can also be
applied to the driver beams for the heavy ion fusion and
beam-driven high energy density physics, where the trans-
verse emittance budget is typically tighter than the longitu-
dinal emittance. The proposed methods consist of one or
several coupling components which completely swap the
emittances of one of the transverse directions and the lon-
gitudinal direction at the exit of the coupling components.
The complete emittance exchange is realized in one pass
through the coupling components. In the present study, we
investigate the effect of a weakly coupled periodic lattice in
terms of achieving emittance exchange between the trans-
verse and longitudinal directions. A weak coupling compo-
nent is introduced at every focusing lattice, and we would
like to determine if such a lattice can realize the function of
emittance exchange.
For simplicity, we will only study the coupling between
one of the transverse directions, the x-direction, and the
longitudinal direction, the z-direction. The focusing lat-
tice in the x-direction is a periodic FODO lattice speci-
fied by a focusing coefficient q(s), where s is the distance
along the longitudinal direction. The longitudinal dynam-
ics is modelled by a synchrotron oscillation with a constant
synchrotron focusing coefficient Kz. In every FODO lat-
tice, two small-amplitude transverse-longitudinal coupling
components are introduced. As discussed in Refs. [1, 2],
such components can be realized by a dipole mode cav-
ity, which generates a longitudinal acceleration force pro-
portional to the transverse displacement, and a transverse
acceleration force proportional to the longitudinal position
relative to the beam centroid. The coupling can be viewed
as a skew-quad between the x-direction and z-direction.
The coupling focusing strength will be represented by
Ks (S).* Research supported by U.S. Department of Energy.
We will study the emittance dynamics from the view-
point of the beam covariance matrixX 2)
_(Xz)
67= n(px)
Kpzx)(xz) (XPX) (xpz
(z2) (zpx) (zpz2
(pXZ) KpX2) (P2Kz
(pzz) (pxpz) pzN'
where () f fbdxdzdpxdpz represents average over the
particle distribution function. After the beam propagates
through the coupled lattice, the covariance matrix is trans-
formed to
a (s) M(s)f ,oM(s),
where M(s) is the transfer matrix from s 0 to s s,
go =a(0), and M(s)t is the transpose of M(s). The
emittance dynamics in the coupled lattice is therefore com-
pletely specified by the transfer matrix M (s) .
GENERALIZED COURANT-SNYDER
THEORY AND REPRESENTATION FOR
COUPLED DYNAMICS
We will use the recently developed generalized Courant-
Snyder (CS) theory and parameterization method for cou-
pled lattices to parameterize the transfer matrix [3, 4, 5, 6,
7]. The main result of the theory is summarized as fol-
lows. The Hamiltonian for the coupled transverse dynam-
ics is given byH =utAu, ti= (x,z,pX,pz)t,
A %q/2 Ks/2)
Of I s Kz/2(1)
(2)Here, the 2 x 2 matrix K(t) is time-dependent and sym-
metric, and I is the 2 x 2 unit matrix. The transverse and
longitudinal dynamics are coupled through the Ks (t) term.
The solution of the linear coupled system corresponding to
H is given by a transfer matrix M (t) , which is a time-
dependent 4 x 4 symplectic matrix [8]. Because there are
10 free parameters for a 4 x 4 symplectic matrix, many
different mathematical parameterization schemes for M (t)
exist. Teng and Edwards [9, 10, 11] first systematically
studied the transfer matrix and derived various parameteri-
zation schemes [9], among which the "symplectic rotation
form" [10] has been adopted in lattice design and particle
tracking codes. Other possible parameterizations have also
been considered [12, 13]. The generalized Courant-Snyder
theory adopted here gives a complete description of the
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Qin, H.; Davidson, R. C.; Chung, M.; Barnard, J. J. & Wang, T. F. Theoretical study of transverse-longitudinal emittance coupling, article, April 14, 2011; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc837409/m1/3/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.