transverse dynamics 31
overall linear matrix of the beamline, computed as a step-bystep matrix multiplication of all individual elements, would
satisfactorily describe the propagation of particles through
the beamline.
As a practical example, let’s consider a pair of thin
quadrupoles of different polarity separated by a drift space
as shown in Fig. 2.12. The overall horizontal transfer matrix
of such a system is given by
(
)(
)(
) ⎛
⎞
1 0
1 L
1 0
⎜ ⎜ 1 −
L
L ⎟ ⎟
M
1→2
⎜
f 1
⎟
=
= ⎜
⎟
x
1
1
0 1
−
1
1
⎝ −
1
1 +
L ⎠
f 2
f 1
f ∗
f 2
1
1 1
L
where
= − +
(2.32)
f ∗ f 1 f 2 f 1 f 2
The quantity f ∗ can be considered as an effective focal distance of a system of two lenses — such a system is usually FIGURE 2.12
called a focusing doublet. The overall vertical transfer ma- For illustration of transfer
trix M 1→2 is obtained by reversing the signs of f 1 and f 2 . By matrix of two quadrupoles
y
referring to Eq. 2.32, we can see that there is a region of pa- separated by a drift.
rameters where the sign of f ∗ is the same and positive for
both the horizontal and vertical planes (for example, when
f 1 = f 2 ), which corresponds to focusing in both planes.
2.4.5 Analogy with geometric optics
As we can see, the particle trajectories can be described using matrix formalism in a very similar way to ray propagation in an optical system. The magnetic quadrupoles play the
role of focusing and defocusing lenses but, unlike an optical
lens, a magnetic quadrupole focuses in one plane and defocuses in the other plane due to the nature of Maxwell’s equations. However, as we have just shown in the previous section, a doublet of quadrupoles can focus simultaneously in
both planes, as illustrated in Fig. 2.13.
I
I
I
I
KRUL]RQWDO
YHUWLFDO
FIGURE 2.13
A doublet can focus simultaneously in both planes.
The similarity and difference with geometrical optics can
be highlighted by the following example. Let’s consider An optical telescope with two
an optical telescope consisting of two lenses as shown in lenses can provide arbitrary
Fig. 2.14. It is intuitively clear and in fact can be proven that demagnification.
overall linear matrix of the beamline, computed as a step-bystep matrix multiplication of all individual elements, would
satisfactorily describe the propagation of particles through
the beamline.
As a practical example, let’s consider a pair of thin
quadrupoles of different polarity separated by a drift space
as shown in Fig. 2.12. The overall horizontal transfer matrix
of such a system is given by
(
)(
)(
) ⎛
⎞
1 0
1 L
1 0
⎜ ⎜ 1 −
L
L ⎟ ⎟
M
1→2
⎜
f 1
⎟
=
= ⎜
⎟
x
1
1
0 1
−
1
1
⎝ −
1
1 +
L ⎠
f 2
f 1
f ∗
f 2
1
1 1
L
where
= − +
(2.32)
f ∗ f 1 f 2 f 1 f 2
The quantity f ∗ can be considered as an effective focal distance of a system of two lenses — such a system is usually FIGURE 2.12
called a focusing doublet. The overall vertical transfer ma- For illustration of transfer
trix M 1→2 is obtained by reversing the signs of f 1 and f 2 . By matrix of two quadrupoles
y
referring to Eq. 2.32, we can see that there is a region of pa- separated by a drift.
rameters where the sign of f ∗ is the same and positive for
both the horizontal and vertical planes (for example, when
f 1 = f 2 ), which corresponds to focusing in both planes.
2.4.5 Analogy with geometric optics
As we can see, the particle trajectories can be described using matrix formalism in a very similar way to ray propagation in an optical system. The magnetic quadrupoles play the
role of focusing and defocusing lenses but, unlike an optical
lens, a magnetic quadrupole focuses in one plane and defocuses in the other plane due to the nature of Maxwell’s equations. However, as we have just shown in the previous section, a doublet of quadrupoles can focus simultaneously in
both planes, as illustrated in Fig. 2.13.
I
I
I
I
KRUL]RQWDO
YHUWLFDO
FIGURE 2.13
A doublet can focus simultaneously in both planes.
The similarity and difference with geometrical optics can
be highlighted by the following example. Let’s consider An optical telescope with two
an optical telescope consisting of two lenses as shown in lenses can provide arbitrary
Fig. 2.14. It is intuitively clear and in fact can be proven that demagnification.