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 = 2f 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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