Basics of beam dynamics 15
where MX i denotes the full map, not just the transfer matrix. The map can
be expanded into a Taylor series
X f = ∆X 0 + RX i + X
T
i TX i + · · · ,
(1.48)
where ∆X 0 is a constant coordinate shift that represents the accumulated
effects of dipole kicks throughout the section, R is the linear transfer matrix,
and T is the second order map. It is conventional to denote the Taylor map
up to the second order the TRANSPORT map [17].
The map of a composite section can be obtained by concatenating the map
of the individual elements sequentially.
1.2.3 Strong focusing principle and orbit stability
As pointed out previously, a quadrupole magnet always focuses in one transverse plane and defocuses in the other. This is unlike a convex lens for light
optics, which focuses simultaneously in both transverse planes. Therefore, according to the strong focusing principle, focusing and defocusing quadrupoles
are alternately placed along the beam path to keep the particle beam focused
in both planes.
Alternate focusing is often implemented with repetitive, identical magnet
lattice cells. The FODO cell is a basic cell structure, which consists of one focusing quadrupole and one defocusing quadrupole, separated by a drift space,
as illustrated in Figure 1.5. The transfer matrix for the cell, starting from the
center of the focusing quadrupole (QF) to the center of the next QF, can be
calculated using the transfer matrices of the individual components
M F = M 1
2 QF M D M QD M D M 1
2 QF ,
(1.49)
where subscript “D” stands for the drift space. Using the thin-lens approximation for the quadrupoles and assuming the defocusing quadrupole (QD)
and the QF have the same focal length, the horizontal transfer matrix for the
FODO cell at the QF center is found to be
M x,F =
1
0
−
1
2f
1
1 L
0 1
1 0
1
f
1
1 L
0 1
1
0
−
1
2f
1
=
1 −
L
2
2f 2
2L
1 +
L
2f
−
L
2f 2
1 −
L
2f
1 −
L
2
2f 2
,
(1.50)
where L is half the length of the cell and f is the focal length.
The transfer matrix for the vertical plane at the QF center can be obtained
from Eq. (1.50) by reversing the sign of f . It can be seen that if 2f is considerably larger than L, the FODO cell provides focusing in both transverse planes
since the (2, 1) elements of the horizontal and vertical transfer matrices will
both be negative, while the diagonal elements remain positive.
where MX i denotes the full map, not just the transfer matrix. The map can
be expanded into a Taylor series
X f = ∆X 0 + RX i + X
T
i TX i + · · · ,
(1.48)
where ∆X 0 is a constant coordinate shift that represents the accumulated
effects of dipole kicks throughout the section, R is the linear transfer matrix,
and T is the second order map. It is conventional to denote the Taylor map
up to the second order the TRANSPORT map [17].
The map of a composite section can be obtained by concatenating the map
of the individual elements sequentially.
1.2.3 Strong focusing principle and orbit stability
As pointed out previously, a quadrupole magnet always focuses in one transverse plane and defocuses in the other. This is unlike a convex lens for light
optics, which focuses simultaneously in both transverse planes. Therefore, according to the strong focusing principle, focusing and defocusing quadrupoles
are alternately placed along the beam path to keep the particle beam focused
in both planes.
Alternate focusing is often implemented with repetitive, identical magnet
lattice cells. The FODO cell is a basic cell structure, which consists of one focusing quadrupole and one defocusing quadrupole, separated by a drift space,
as illustrated in Figure 1.5. The transfer matrix for the cell, starting from the
center of the focusing quadrupole (QF) to the center of the next QF, can be
calculated using the transfer matrices of the individual components
M F = M 1
2 QF M D M QD M D M 1
2 QF ,
(1.49)
where subscript “D” stands for the drift space. Using the thin-lens approximation for the quadrupoles and assuming the defocusing quadrupole (QD)
and the QF have the same focal length, the horizontal transfer matrix for the
FODO cell at the QF center is found to be
M x,F =
1
0
−
1
2f
1
1 L
0 1
1 0
1
f
1
1 L
0 1
1
0
−
1
2f
1
=
1 −
L
2
2f 2
2L
1 +
L
2f
−
L
2f 2
1 −
L
2f
1 −
L
2
2f 2
,
(1.50)
where L is half the length of the cell and f is the focal length.
The transfer matrix for the vertical plane at the QF center can be obtained
from Eq. (1.50) by reversing the sign of f . It can be seen that if 2f is considerably larger than L, the FODO cell provides focusing in both transverse planes
since the (2, 1) elements of the horizontal and vertical transfer matrices will
both be negative, while the diagonal elements remain positive.