Betatron Motion with Coupling of Horizontal and Vertical Degrees of
Freedom§
S.A. Bogacz¨
Jefferson Lab, Newport News, VA 23606
V.A. Lebedev
Fermilab, Batavia, IL 60510
The Courant-Snyder parametrization of one-dimensional linear betatron motion is generalized to twodimensional coupled linear motion. To represent the 4´4 symplectic transfer matrix the following ten
parameters were chosen: four beta-functions, four alpha-functions and two betatron phase advances which
have a meaning similar to the Courant-Snyder parametrization. Such a parametrization works equally well for
weak and strong coupling and can be useful for analysis of coupled betatron motion in circular accelerators as
well as in transfer lines. Similarly, the transfer matrix, the bilinear form describing the phase space ellipsoid
and the second order moments are related to the eigen-vectors. Corresponding equations can be useful in
interpreting tracking results and experimental data.
1. Introduction
In many applications analysis of coupled betatron motion [1] [2] is an important part of the machine
design. The development of accelerator technology has stimulated additional interest in the subject in
recent years. Initially betatron coupling was an undesired effect and efforts were made to suppress it.
However, over recent years betatron coupling has become an intrinsic part of many accelerator proposals.
This paper introduces a new parametrizarion of coupled betatron motion, where in addition to four betafunctions and two betatron phases we introduce four alpha-functions. That yields a complete set of ten
independent functions to parametrize a 4´4 symplectic transfer matrix. The beta-functions have similar
meaning to the Courant-Snyder parametrization, and the definition of alpha-functions coincides with the
standard.
2. Equations of Motion and Condition of Symplecticity
The two-dimensional linear motion of a particle in a focusing lattice structure can be described by the
following matrix equations:
dxˆ
= UHxˆ ,
ds
(2.1)
where the Hamiltonian matrix H and the unit symplectic matrix U are introduced as follows,
é 2
R2
êK x + k +
4
ê
0
ê
H=
ê
N
ê
êë - R 2
N
0
R 2
1
2
R 2 Ky -k +
0
0
2
R
4
ù
- R 2ú
0 ú
ú
0 ú
ú
1 úû
,
é0
ê- 1
U=ê
ê0
ê
ë0
1
0
0ù
0 0 0úú .
0 0 1ú
ú
0 - 1 0û
(2.2)
Here x and y are the horizontal and vertical particle displacements from the ideal orbit; the derivatives are
calculated along the longitudinal coordinate s; K x , y = eB y , x / Pc ; k = eG / Pc ; N = eG s / Pc ; R = eBs / Pc ;
Bx, By and Bs are the corresponding components of the magnetic field; G is the normal component of the
magnetic field gradient; and G s is the skew component of the magnetic field gradient.
ˆ (0, s )xˆ , one can
Introducing transfer matrix from coordinate 0 to s, for the canonical variables, xˆ = M
0
expresses the symplecticity condition for particle motion in the following form
§
¨
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Work supported by the US DOE under contract #DE-AC05-84ER40150
e-mail: bogacz@jlab.org
ˆ (0, s) T UM
ˆ (0, s ) = U ,
M
(2.3)
Consider a circular accelerator with the total transfer matrix
li , and four corresponding eigen-vectors, v̂ i (i = 1, 2, 3, 4),
ˆ .
M
The transfer matrix has four eigen-values,
ˆ vˆ = l vˆ .
M
i
i i
(2.4)
We will consider the case of a stable betatron motion, meaning all four eigen-values are confined to a unit
circle and none of them is equal to ±1. For any two eigen-vectors the symplecticity condition of Eq. (2.3)
yields following set of orthogonality conditions:
+
vˆ 1 Uvˆ 1 = -2i ,
+
vˆ 2 Uvˆ 2 = -2i ,
T
vˆ 1 Uvˆ 1 = 0 ,
T
vˆ 2 Uvˆ 2 = 0 ,
T
vˆ 2 Uvˆ 1 = 0 ,
+
vˆ 2 Uvˆ 1 = 0 .
(2.5)
3. Relation between Eigen-vectors and Emittance Ellipsoid in 4D Phase Space
The turn-by-turn particle positions and angles (at the beginning of the lattice) can be represented as a
linear combination of four independent solutions,
(
xˆ = Re A1e -iy 1 vˆ 1 + A2 e -iy 2 vˆ 2
)
¢
²
¢
²
= A1 æç vˆ 1 cosy 1 + vˆ 1 sin y 1 ö÷ + A2 æç vˆ 2 cosy 2 + vˆ 1 siny 2 ö÷ ,
è
ø
è
ø
(3.1)
where four real parameters, A1, A2, y1 and y2 , represent the betatron amplitudes and phases. The
amplitudes remain constant in the course of betatron motion, while the phases change after each turn.
Introducing the following real symplectic matrix
ˆ = é vˆ ¢ ,- vˆ ² , vˆ ¢ ,- vˆ ² ù
V
1
2
2 ú
êë 1
û
ˆ -1 = -UV
ˆ TU ,
, where V
(3.2)
one can rewrite Eq. (2.1) in the compact form
ˆ Aξ ,
xˆ = V
A
(3.3)
where the amplitude matrix A is
é A1
ê0
A=ê
ê0
ê
ë0
0
0
A1
0
0
0
A2
0
0ù
0 úú
0ú
ú
A2 û
é cosy 1 ù
ê - siny ú
.
1ú
ξA = ê
ê cosy 2 ú
ê
ú
ë - siny 2 û
and
(3.4)
Let us consider an ensemble of particles, whose motion (at the beginning of lattice) is contained in a
4D ellipsoid. A 3D surface of this ellipsoid is determined by particles with extreme betatron amplitudes.
For any of these particles, Eq. (3.3) describes the 2D-subspace of single-particle motion, which is a
subspace of the 3D surface of the ellipsoid, described by the bilinear form
xˆ T Ξˆ xˆ = 1 .
(3.5)
ˆ , in terms of its diagonal form as follows :
Applying Eqs. (3.1) - (3.5), one can express the bilinear form, Ξ
ˆ = UV
ˆΞ
ˆ ¢V
ˆ T U T , where Ξˆ ¢ = A -1 A -1 is a diagonal matrix
Ξ
(3.6)
Therefore, a symplectic transform V̂ reduces matrix Ξˆ to its diagonal form. Then, in the new coordinate
frame the 3D ellipsoid enclosing the total 4D phase-space of the beam can be described by the following
equation:
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ˆ¢ = V
ˆ TΞ
ˆV
ˆ
Ξ
.
(3.7)
It is natural to define the beam emittance as a product of the ellipsoid axes:
e 4D =
1
ˆ ¢)
det(Ξ
=
( A1 A2 )2
= ( A1 A2 )
2
ˆ)
det(V
.
(3.8)
Thus, the squares of amplitudes A1 and A2 can be considered as 2D emittances e1 and e2 corresponding to
the eigen-vectors v̂ 1 and v̂ 2 . They coincide with the horizontal and vertical emittances of the uncoupled
motion so that e1 e2 = e4D.
ˆ , determines
Similarly to the one-dimensional case the particle ellipsoid shape, described by matrix Ξ
the beam emittances e1 and e2 , and the eigen-vectors v̂ 1 and v̂ 2 . In this case the beam emittances are
reciprocal to the roots of the following characteristic equation,
(
)
ˆ - il U = 0
det Ξ
.
(3.9)
ˆ ¢ , one can find the eigen-vectors
Furthermore, knowing the beam emittances and consequently Ξ
v̂1 and
v̂ 2 by solving the following equation
æˆ i ö
çç Ξ - U ÷÷ vˆ l = 0 ,
el ø
è
One can easily prove [3] that matrix
matrix .
where l = 1, 2
(3.10)
Ξ̂ is the inverse of the second moments of the beam distribution
4. Beta-functions for Coupled Motion
Employing previously introduced notation, one can parametrize the eigen-vectors
following form:
ù
é
b 1x
ê i(1 - u ) + a ú
1x ú
êú
ê
b 1x
vˆ 1 = ê
ú
in 1
b
e
1y
ú
ê
ê iu + a 1 y in ú
e 1ú
êb1 y
ûú
ëê
é
b 2 x e in 2 ù
ê iu + a
ú
2 x in 2 ú
êe
ê
ú
b 2x )
vˆ 2 = ê
ú
b2y
ê
ú
ê i(1 - u ) + a 2 y ú
êú
b 2y
ëê
ûú
.
v̂1 and v̂ 2 in the
(4.1)
Here b1x, b1y, b2x and b2y are the beta-functions; the alphas:a1x, a1y, a2x and a2y are negative half-derivatives
of the betas at regions with zero longitudinal magnetic field and two phases:n1, n2 constitute ten
independent parameters - the generalized Twiss functions.
The presented parametrization [3] has been proven useful for both analytic and numerical analysis of
coupled betatron motion in circular machines and transfer lines. Knowing the eigen-vectors, one can easily
obtain the generalized Twiss functions. The formalism also allows one to perform the inverse operation of
finding eigen-vectors from the generalized Twiss functions.
References
[1]
[2]
[3]
D.A. Edwards and L.C. Teng, IEEE Trans. Nucl. Sci. 20, 3, pp. 885-889 (1973)
F. Willeke and G. Ripken, Methods of Beam Optics, Proceedings of US Particle Accelerator School
(1987 and 1988), AIP Conf. Proc. 184, New York 1989
V.A. Lebedev and S.A. Bogacz, J-Lab internal publication, TN-00-022, May 2000
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