Instrumentation & Measurement Magazine 24-9 - 24

offers directivity features, e.g., to separate signals from
counter-propagating beams. Complex, costly mechanics
that requires more real-estate compared to button
BPMs. Used for special applications, e.g., beam feedback
systems (dampers), tune monitors, accelerators with
long beam bunches, in ring colliders near the interaction
point, downstream shorted stripline BPMs are popular
in pulsed hadron linacs operating at the bunch repetition
frequency, can also be used as broadband beam kicker [5].
◗ Inductive BPM: Utilizes broadband RF transformercouplers
sensing the image current on the beam-pipe.
Typical frequency range 5-500 MHz. Compact and robust
mechanics. No commercially available sub-components,
requires RF design expertise and dedicated materials,
e.g., ferrite or nanocrystalline toroid cores. Used in some
accelerator test facilities [6] and was used at a hadron
booster synchrotron.
◗ Split-plane ( " shoe-box " ) BPM: Electrostatic BPMs with
a full 360° coverage angle, but with a linear, diagonal
cut between the electrodes to provide a linear position
characteristic, however, with a lower position sensitivity
compared to button or stripline BPMs. Mechanical
complexity, real-estate demand, and costs range somewhere
in-between that of button and stripline BPMs.
Develops substantial eigen-resonances already at
moderate frequencies, therefore only usable at lower
frequencies, typically < 300 MHz. Was very popular in
the early times of pure analog BPM signal processing electronics
due to the almost perfect linearity of the position
signal, i.e., no non-linear correction required. Today still
used in some hadron and heavy-ion machines [7].
◗ Cavity BPM: Passive, beam driven dipole-mode resonator
operating at a specific eigenmode frequency, typically
at some GHz. Has a very high position resolution potential,
down to a few 10 nano-meter for single bunches,
due to the high dipole-mode shunt (≡ transfer) impedance,
typically in the kΩ regime compared to a few Ω
transfer impedance of the above listed broadband BPMs.
Can deliver a common-mode free difference Δ-signal.
Requires a monopole-mode reference cavity to retrieve
the beam position from the dipole-mode cavity BPM. Has
many unwanted, higher-order eigen-resonances, therefore
a high beam-coupling impedance which cannot be
tolerated in ring accelerators. Complex RF design, costly
precision manufacturing, complex RF front-end electronics.
Used in FEL driver linacs [8] and electron linac test
facilities.
All BPM pickups are a passive, linear electromagnetic couplers,
and fall in two categories:
◗ Broadband BPM pickup, using symmetrically arrange
electrodes.
◗ Resonant BPM pickup, the mentioned cavity BPM.
While the resonant cavity BPMs have a very high-resolution
potential, their application is limited to linear accelerators
as of their high beam coupling impedance, an unwanted characteristic
of any device installed in the beam line vacuum system
24
[9]. Broadband BPM pickups, such as the button-BPM, have
two or four symmetrically arrange electrodes (see Fig. 2a), and
each electrode delivers an electrical signal (here defined in the
frequency-domain):
V xy s xy Z I b
elec
,
 ,,      
(3)
with Z(ω) being the transfer impedance of a BPM pickup electrode
(unit: Ω), Ib
(ω) being the beam or bunch current, i.e., the
frequency-domain equivalent of the time-domain envelope
function of the longitudinal particle distribution, and ω = 2f
being the angular frequency. Often the bunch current Ibunch
(ω)
is assumed as a Gaussian envelope function, which is a good
approximation for the distribution of electrons in a RF bucket,
but may have some limitations in case of bunched proton or
ion beams. s (x, y) is a sensitivity function reflecting the crosssection
geometry of the BPM pickup, to describe the transverse
position characteristic, i.e., the strength of the coupling between
beam and BPM electrode as function of the transverse
beam position (x, y). Eq. (3) is basically Ohm's law, applied and
expanded to a broadband BPM electrode, separating the frequency
independent, often non-linear, position sensitivity
s (x, y) from the frequency depending, linear transfer characteristic
Z(ω). (For non-relativistic beam velocities β << 1, the
position sensitivity function s becomes frequency dependent
because the beam develops longitudinal electromagnetic field
components [10], [11].)
Hidden in (3) is an electromagnetic coverage factor:

∮
∮
S elec
J dS
J dS
w

SBPM
w
which expresses the ratio of the wall currents Jw
of the pickup electrode Selec
on the surface
and the surface of the BPM beampipe
SBPM [5], for a centered beam, thus s(x = y = 0) = ϕ. For BPM
pickup electrodes mounted flush with the beam-pipe, without
large discontinuities, ϕ in good approximation resembles the
ratio of those surface areas [12].
Position Characteristic of a Line Charge
in a Circular Vacuum Chamber
The distribution of the image (or wall) charges, compensating
the charges of the particle beam, depends on the beam velocity
v = β / c, and is given by the distribution of the beam's electromagnetic
field with the conductive beam pipe as boundary.
Fig. 3a, Fig. 3b and Fig. 3c illustrate the electric field lines for
a point charge travelling off-center in a cylindrical beam pipe
at different velocities β, in a longitudinal section view. The
related longitudinal image charge distribution is indicated
above the electric field line plot and has a standard deviation
of
d
s
2 , with d being the distance between point charge
and beam-pipe wall, and γ the Lorentz factor mentioned in the
Introduction. At
 s
1
     , the longitudi0
nal
components of the EM-field of the point charge collapse
as it reaches a relativistic velocity. Fig. 3d illustrates the purely
IEEE Instrumentation & Measurement Magazine
December 2021
S
S
elec
BPM
(4)
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