Orbit and trajectory correction 71
The signal strength on each button when a beam with an offset from the
center passes can be calculated by integrating the surface charge density over
the extent of the button. The image charge density on a circular vacuum
chamber is given by
σ(φ) = −
λ
2πR
R
2 − r
2
R 2 + r 2 − 2rR cos(φ − θ)
,
(3.1)
where R is the radius of the vacuum chamber, λ is the line density of the beam
current, and (r, θ) is the beam center in polar coordinates. The sum signal,
defined as the sum of signals on all four buttons,
Σ = A + B + C + D,
(3.2)
is proportional to the beam intensity and the sum of the angles subtended by
the buttons. The horizontal and vertical difference signals are defined as
∆ x = A + D − B − C,
∆ y = A + B − C − D,
(3.3)
respectively. The difference signals depend on the position of the beam center
and the beam intensity. It can be shown that the ratio of the difference and
sum signals are related to the position of the beam center by
∆ x
Σ
=
√
2
sin ∆
R∆
x + O(r
3 ),
∆ y
Σ
=
√
2
sin ∆
R∆
y + O(r
3 ),
(3.4)
where ∆ is one half of the angle subtended by each button and O(r
3 ) represent
terms of r
3 or higher. Therefore, the beam position can be calculated from the
ratios with a linear conversion. This works well in the vicinity of the center
of the vacuum chamber. However, when the beam is far from the center, the
higher order terms in Eq. (3.4) will become important and hence the beam
position reading obtained with the linear conversion will deviate from the
actual beam position. The “measured” beam position vs. the actual beam
position for a round circular chamber is shown in the right plot of Figure 3.1.
In synchrotron light sources, the cross section of the vacuum chamber in
the arcs is typically not circular. In that case the signals on the BPM buttons
need to be calculated with the proper boundary conditions. The ratio of the
difference and sum signals will still be a good indication of the beam position
in the vicinity of the chamber center, although the conversion coefficient may
differ. The nonlinear response of the BPM reading to a large beam position
offset from the chamber center will also be different. Figure 3.2 shows the
configuration of the BPM buttons in the SPEAR3 vacuum chamber (left) and
the nonlinear response of the BPMs to the horizontal position offset [112].
The raw signals from the pick-up electrodes consist of a series of pulses
that correspond to the beam bunch passes. When processed through fast electronics, these signals could potentially yield the beam position of each bunch.
The signal strength on each button when a beam with an offset from the
center passes can be calculated by integrating the surface charge density over
the extent of the button. The image charge density on a circular vacuum
chamber is given by
σ(φ) = −
λ
2πR
R
2 − r
2
R 2 + r 2 − 2rR cos(φ − θ)
,
(3.1)
where R is the radius of the vacuum chamber, λ is the line density of the beam
current, and (r, θ) is the beam center in polar coordinates. The sum signal,
defined as the sum of signals on all four buttons,
Σ = A + B + C + D,
(3.2)
is proportional to the beam intensity and the sum of the angles subtended by
the buttons. The horizontal and vertical difference signals are defined as
∆ x = A + D − B − C,
∆ y = A + B − C − D,
(3.3)
respectively. The difference signals depend on the position of the beam center
and the beam intensity. It can be shown that the ratio of the difference and
sum signals are related to the position of the beam center by
∆ x
Σ
=
√
2
sin ∆
R∆
x + O(r
3 ),
∆ y
Σ
=
√
2
sin ∆
R∆
y + O(r
3 ),
(3.4)
where ∆ is one half of the angle subtended by each button and O(r
3 ) represent
terms of r
3 or higher. Therefore, the beam position can be calculated from the
ratios with a linear conversion. This works well in the vicinity of the center
of the vacuum chamber. However, when the beam is far from the center, the
higher order terms in Eq. (3.4) will become important and hence the beam
position reading obtained with the linear conversion will deviate from the
actual beam position. The “measured” beam position vs. the actual beam
position for a round circular chamber is shown in the right plot of Figure 3.1.
In synchrotron light sources, the cross section of the vacuum chamber in
the arcs is typically not circular. In that case the signals on the BPM buttons
need to be calculated with the proper boundary conditions. The ratio of the
difference and sum signals will still be a good indication of the beam position
in the vicinity of the chamber center, although the conversion coefficient may
differ. The nonlinear response of the BPM reading to a large beam position
offset from the chamber center will also be different. Figure 3.2 shows the
configuration of the BPM buttons in the SPEAR3 vacuum chamber (left) and
the nonlinear response of the BPMs to the horizontal position offset [112].
The raw signals from the pick-up electrodes consist of a series of pulses
that correspond to the beam bunch passes. When processed through fast electronics, these signals could potentially yield the beam position of each bunch.