70 unifying physics of accelerators, lasers and plasma
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FIGURE 4.26
Lenses made from different
glasses compensate chromatic aberrations.
For light, one uses lenses made from different materials to compensate for chromatic aberrations, as illustrated in
Fig. 4.26. In this example, using a strong focusing lens made
from a low-dispersion crown glass coupled with a weak defocusing high-dispersion flint glass can correct the chromatic
aberrations.
For particle beams, chromatic aberrations are caused by
the dependence of the focusing strength on the beam energy.
There are no different materials and no specific Maxwell’s
equations in particle optics, therefore, other means have to
be used for chromatic compensation in accelerators.
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FIGURE 4.27
Compensation of chromatic aberration by inserting nonlinear
sextupole magnets in a dispersive region.
Let’s consider chicane arrangements of four bending
dipoles as shown in Fig. 4.27. Such an arrangement creates a
dispersive region, where orbit position depends on the energy.
Let’s insert a sextupole magnet with normalized strength S
(or K ' ) into the dispersive area
1 δ 2 B
S =
(4.8)
2!Bρ δx 2
The sextupole magnet produces the following effect (kick)
on the angle of the beam trajectory
x
' =
2
2
x
' + S x − y
and
y
' = y
'
− S 2xy
(4.9)
In the dispersive region, one needs to replace x with x +ηδ
where η is the dispersion, thus the sextupole kick will contain
the energy-dependent focusing since the terms in the equation below correspond to a quadrupole with an effective gradient equal to S · η:
x
'
⇒ S(x + ηδ)
2 ⇒ 2S η xδ + ..
(4.10)
y
'
⇒ −S 2 (x + ηδ) y ⇒ −2S η yδ + ..
Therefore, such an arrangement of nonlinear magnets placed
in a dispersive region can be used for chromatic correction in
the event of beam optics. One should also note that the terms
that are not shown in Eq. 4.3.3, containing x 2 and δ 2 , are unwanted additional aberrations — geometrical aberrations and
higher-order chromatic aberrations. They often need to be
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FIGURE 4.26
Lenses made from different
glasses compensate chromatic aberrations.
For light, one uses lenses made from different materials to compensate for chromatic aberrations, as illustrated in
Fig. 4.26. In this example, using a strong focusing lens made
from a low-dispersion crown glass coupled with a weak defocusing high-dispersion flint glass can correct the chromatic
aberrations.
For particle beams, chromatic aberrations are caused by
the dependence of the focusing strength on the beam energy.
There are no different materials and no specific Maxwell’s
equations in particle optics, therefore, other means have to
be used for chromatic compensation in accelerators.
'LSROH
(
%HDP
0DJQHWV
'LUHFWLRQ
( !
FIGURE 4.27
Compensation of chromatic aberration by inserting nonlinear
sextupole magnets in a dispersive region.
Let’s consider chicane arrangements of four bending
dipoles as shown in Fig. 4.27. Such an arrangement creates a
dispersive region, where orbit position depends on the energy.
Let’s insert a sextupole magnet with normalized strength S
(or K ' ) into the dispersive area
1 δ 2 B
S =
(4.8)
2!Bρ δx 2
The sextupole magnet produces the following effect (kick)
on the angle of the beam trajectory
x
' =
2
2
x
' + S x − y
and
y
' = y
'
− S 2xy
(4.9)
In the dispersive region, one needs to replace x with x +ηδ
where η is the dispersion, thus the sextupole kick will contain
the energy-dependent focusing since the terms in the equation below correspond to a quadrupole with an effective gradient equal to S · η:
x
'
⇒ S(x + ηδ)
2 ⇒ 2S η xδ + ..
(4.10)
y
'
⇒ −S 2 (x + ηδ) y ⇒ −2S η yδ + ..
Therefore, such an arrangement of nonlinear magnets placed
in a dispersive region can be used for chromatic correction in
the event of beam optics. One should also note that the terms
that are not shown in Eq. 4.3.3, containing x 2 and δ 2 , are unwanted additional aberrations — geometrical aberrations and
higher-order chromatic aberrations. They often need to be