200 unifying physics of accelerators, lasers and plasma
tion, which, if properly adjusted, produces repeating vertical
lines in the z − E plane (see Fig. 10.18), which correspond to
spatial density modulation of the beam at high harmonics of
the lasers.
Spatial harmonics that can be created with this method
are defined by k = nk 1 + mk 2 , where the integers n and m
can be large, granting the potential for seeding at very short
wavelengths.
10.3 Stability of beams
Stability of particle beams or laser pulses is usually one of
the most important requirements of any design. Any practical
realization of a beamline or laser system has imperfections:
static (caused by misalignments) or dynamic (caused, e.g., by
vibrations) positioning errors. In this section, we will look at
just a few selected examples from this wide topic.
10.3.1 Stability of relativistic beams
Head–tail effects are often the cause of instabilities and deterioration of bunch properties, and occur when an initial offset of the bunch head creates much larger oscillations in the
bunch tail.
7DLO
+HDG
FIGURE 10.19
Fields of the bunch and
head–tail effects.
However, we need to take into account that the fields of
a relativistic bunch are mostly transverse, as illustrated in
Fig. 10.19. Therefore, in the free space, the motion of the tail
of the bunch would be independent of the motion of the head,
and in particular as to whether the head has offset and/or oscillations.
For the head–tail instability to develop, it is necessary to
have an “agent” to carry the information about the offset from
the bunch head to the bunch tail.
This so-called agent can be, for example, the fields induced by the bunch in the surrounding accelerating structures. The field of the opposite colliding bunch can also play
the role of this agent. We will study our examples starting
with the latter.
10.3.2 Beam–beam effects
Modern electron–positron colliders aiming at the highest luminosity 4 require tiny beam sizes at their interaction region
−
(IR). An example of the beam sizes for a 500 GeV CM e + e
ILC collider project is shown in Fig. 10.20. Note that these
beams are extremely flat, to ensure that the energy losses of
the beams caused by synchrotron radiation during beam collision are significantly suppressed.
4 International Linear Collider Technical Design Report, 2013.
tion, which, if properly adjusted, produces repeating vertical
lines in the z − E plane (see Fig. 10.18), which correspond to
spatial density modulation of the beam at high harmonics of
the lasers.
Spatial harmonics that can be created with this method
are defined by k = nk 1 + mk 2 , where the integers n and m
can be large, granting the potential for seeding at very short
wavelengths.
10.3 Stability of beams
Stability of particle beams or laser pulses is usually one of
the most important requirements of any design. Any practical
realization of a beamline or laser system has imperfections:
static (caused by misalignments) or dynamic (caused, e.g., by
vibrations) positioning errors. In this section, we will look at
just a few selected examples from this wide topic.
10.3.1 Stability of relativistic beams
Head–tail effects are often the cause of instabilities and deterioration of bunch properties, and occur when an initial offset of the bunch head creates much larger oscillations in the
bunch tail.
7DLO
+HDG
FIGURE 10.19
Fields of the bunch and
head–tail effects.
However, we need to take into account that the fields of
a relativistic bunch are mostly transverse, as illustrated in
Fig. 10.19. Therefore, in the free space, the motion of the tail
of the bunch would be independent of the motion of the head,
and in particular as to whether the head has offset and/or oscillations.
For the head–tail instability to develop, it is necessary to
have an “agent” to carry the information about the offset from
the bunch head to the bunch tail.
This so-called agent can be, for example, the fields induced by the bunch in the surrounding accelerating structures. The field of the opposite colliding bunch can also play
the role of this agent. We will study our examples starting
with the latter.
10.3.2 Beam–beam effects
Modern electron–positron colliders aiming at the highest luminosity 4 require tiny beam sizes at their interaction region
−
(IR). An example of the beam sizes for a 500 GeV CM e + e
ILC collider project is shown in Fig. 10.20. Note that these
beams are extremely flat, to ensure that the energy losses of
the beams caused by synchrotron radiation during beam collision are significantly suppressed.
4 International Linear Collider Technical Design Report, 2013.