2 Beam Dynamics
27
2.1.10 Transport Matrices for Lattice Components
An empty space or drift length is the simplest of the lattice component matrices.
Figure 2.8(a) shows the analogy between a particle trajectory and a diverging ray in
optics. The angle of the ray and the divergence of the trajectory are related:
θ = tan
−1
x
.
(2.28)
The effect of a drift length in phase space is a simple horizontal translation from
(x, x ) to (x+ , x ) and can therefore be written as a matrix:
x 2
x
2
=
1
0 1
x 1
x
1
.
(2.29)
The next case is that of a thin quadrupole magnet of infinitely small length but
finite integrated gradient:
=
1
(Bρ)
∂B z
∂x
.
(2.30)
The optical analogy of a thin quadrupole with a converging lens is illustrated in
Fig. 2.8(b). A ray, diverging from the focal point arrives at the lens at a displacement,
x, and is turned parallel by a deflection:
θ ≈
1
f
x.
(2.31)
This deflection will be the same for any ray at displacement x irrespective of
its divergence. This behaviour can be expressed by a simple matrix, the thin lens
matrix:
x 2
x
2
=
1 0
− 1/f 1
x 1
x
1
.
(2.32)
a
b
Fig. 2.8 The effect of a drift—(a), left side—and a focusing quadrupole lens—(b) right side—on
a particle trajectory. The mathematical expressions are given in Eqs. (2.29) and (2.32)
27
2.1.10 Transport Matrices for Lattice Components
An empty space or drift length is the simplest of the lattice component matrices.
Figure 2.8(a) shows the analogy between a particle trajectory and a diverging ray in
optics. The angle of the ray and the divergence of the trajectory are related:
θ = tan
−1
x
.
(2.28)
The effect of a drift length in phase space is a simple horizontal translation from
(x, x ) to (x+ , x ) and can therefore be written as a matrix:
x 2
x
2
=
1
0 1
x 1
x
1
.
(2.29)
The next case is that of a thin quadrupole magnet of infinitely small length but
finite integrated gradient:
=
1
(Bρ)
∂B z
∂x
.
(2.30)
The optical analogy of a thin quadrupole with a converging lens is illustrated in
Fig. 2.8(b). A ray, diverging from the focal point arrives at the lens at a displacement,
x, and is turned parallel by a deflection:
θ ≈
1
f
x.
(2.31)
This deflection will be the same for any ray at displacement x irrespective of
its divergence. This behaviour can be expressed by a simple matrix, the thin lens
matrix:
x 2
x
2
=
1 0
− 1/f 1
x 1
x
1
.
(2.32)
a
b
Fig. 2.8 The effect of a drift—(a), left side—and a focusing quadrupole lens—(b) right side—on
a particle trajectory. The mathematical expressions are given in Eqs. (2.29) and (2.32)