Capacitive BPM electromagnetic design optimisation

Capacitive beam position monitors (BPM) are widely used as diagnostics tools in particle accelerators. Typically due to a large number of BPM in an accelerator, their contribution to the beam coupling impedance cannot be neglected. In addition to the broadband part at low frequencies, the impedance exhibits resonant peaks at higher frequencies due to electromagnetic fields trapped around the BPM button and in the feedthrough assembly. Coupling of these peaks with beam spectrum lines can result in the BPM overheating. In this paper, we discuss the BPM design optimization aimed at the beam coupling impedance minimization while keeping the BPM signal transfer impedance.


Introduction
In the field of particle physics, button-type beam position monitors (BPMs) play an important role in measuring and controlling the position of charged particle beams within vacuum chambers.However, their use is not without challenges.One major issue is the contribution of BPMs to both broad-band and narrow-band impedances, which can lead to heating effects and coupled-bunch instabilities.Another concern is the heating that can occur in the BPM's ceramic and metallic walls due to the power carried by the trapped modes and signal.These issues make designing the BPM electrode a complex task.
To address these challenges, this study focuses on the optimization of the narrow-band beam coupling longitudinal impedance by using adapted feedthrough geometries.Specifically, the researchers examine how the button geometry can be modified to minimize the harmful effect of the resonance peak narrow-band impedance while keeping the transfer impedance, button capacitance, and gap between the electrode and vacuum chamber on the same level.

General considerations
The button electrode primarily responds to the electric field of the beam.An electrostatic monitor can typically be represented by an equivalent circuit that includes a current generator with the same value as the fraction of intercepted image current, which is then shunted by the electrode capacitance to ground.To connect the button to the detector circuit, a short length of modified coaxial cable with a characteristic impedance of R 0 is used, and an R 0 resistor terminates it.When the beam is centered, the transfer impedance, which is the complex ratio of the voltage induced by the beam at the external termination to the beam current, can be expressed as: here the frequencies ω 1 and ω 2 , are calculated based on a number of other variables [1].These include R 0 , C b , r, c, and b, where C b is the capacitance to ground, r is the radius of the button, c is the speed of light, and b is the radius of the beam pipe.The coverage factor, ϕ, is also defined in terms of r and b, as ϕ = r/4b.
The frequency response is classified as high-pass, and two separate regimes can be distinguished for low frequency (ω ≪ ω 1 ) and high frequency (ω ≫ ω 1 ) respectively: At low frequencies, the electrode behaves like a time differentiator.The transfer impedance mainly depends on the load resistance R 0 and geometrical factors b and r, and is largely unaffected by the capacitance C b .At high frequencies, the response is purely resistive, meaning that the electrode voltage is proportional to and in phase with the beam current.The transfer impedance scales inversely with the button capacitance and depends on the same geometrical factors, but not on R 0 .It is possible to shape the response for a given frequency range by adjusting the load resistor.For instance, increasing the value of the load resistor shifts the corner frequency ω 1 towards a lower value, thus extending the low-frequency response, without affecting the high-frequency response as long as the capacitance remains the same.To enhance the response at high frequencies, the capacitance can be reduced, but this causes the corner frequency ω 1 to shift towards a higher value, which is a drawback.
The absolute value of the transfer impedance increases quadratically with the button radius, making a larger radius preferable for higher sensitivity across all frequencies.Enhancing the electrode surface improves the overall response, but increasing capacitance reduces the response at high frequency, while decreasing capacitance has the opposite effect.To reduce the low-frequency corner of the differentiator response, the load resistor value must be increased.However, these parameters are not independent, and increasing the radius may result in a linear or quadratic increase of capacitance, depending on the design.It is not feasible to increase the button radius beyond a certain limit, as the longitudinal coupling impedance scales as r 4 .
The expression 5 underestimates the coupling impedance, as it only considers the fields that contribute to the output signal formation.Numerical simulations have revealed other modes that do not dissipate their power in the external termination, such as the TE 11 -like mode, which has a wavelength close to the mean perimeter of the annular cut created by the button and beam pipe walls.A smaller button radius is required to shift the frequency of the first parasitic mode towards higher frequencies, avoiding high power losses in the button BPM.The annular cut also contributes to the coupling impedance, which can be estimated analytically at low frequencies.
The coupling impedance is heavily dependent on the button radius, as evidenced by eq.6, where Z 0 represents the free space impedance, and by w size of the gap is denoted.Thus, in order to perform an impartial comparison of different shapes it is necessary to assume that the transfer impedance, button radius, capacitance, and gap are the same.

Modelling
The detailed layout of the simulated BPM electrodes attached to the vacuum chamber is shown in Fig. 1. References [2,3,4,5,6,7,8] are indicated as a source for approximated example models for simulated geometries.Each structure consists of the button geometry, the tapered part that adjusts from button sizes to the standard SMA connector and coaxial line.For the canonical geometries Fig. 1 a),b), and c) the button capacitance can be calculated analytically, while for the bell-shaped button, the same capacitance is adjusted numerically.Each button electrode has a 3 mm radius at the annular cut with the vacuum chamber and a 3 mm height.The bunch length used in the wake simulation is 8 mm traveling along the structure's central axis.The BPM electrode tapered and coaxial parts match the impedance of a 50 Ohm.A ceramic ring for vacuum insulation is located near the button region.It has a dielectric constant of about 5. The ceramic ring's inner radius must be altered for optimum impedance matching.Because of symmetry, only half of the structure is simulated.One button of the BPM is situated on the top of the vacuum chamber, and it tapers gradually to a coaxial line above.The simulations are done in the time and frequency domains, which consists of two kinds of calculations, namely wakefield and port transmission calculations respectively.
In wakefield calculation, a Gaussian beam is introduced in the z-direction, leading to the generation of electromagnetic fields at the BPM structure.These fields then influence the particle beam.To prevent the reflection of electromagnetic waves at the beam entrance and exit planes, waveguide boundary conditions are imposed.At the top boundary of the coaxial line, an outgoing waveguide port is considered to determine signal transmission.Initially, a twodimensional eigenvalue problem is solved to identify the propagating and evanescent modes of the coaxial line.Since the beam stimulates a wide range of frequencies, a broad-band boundary must be established at the waveguide port [9,10,11,12,13,14].
To evaluate the impedance of a button, one can either use the wakefield or its Fourier transform.The latter allows us to identify potential resonant modes that the particle bunch excites in the BPM.The resolution of narrow resonances in the impedance spectrum is directly proportional to the number of sampling points used in the wakefield calculation.Therefore, we compute the wakefield up to s = 10 m, where s represents the bunch coordinate.By computing the transfer impedance at the coaxial line port as a function of frequency, we can determine the signal strength picked up by the BPM as the beam traverses the corresponding section of the vacuum chamber.

Results
When dealing with high beam current, BPMs can generate significant broad-band and narrowband impedances that can cause single-bunch and coupled-bunch instabilities, respectively.The accepted limit of the total broad-band effective impedance for a given machine is determined beforehand to prevent single-bunch instabilities.It is desirable that BPMs contribute a small fraction to the total broad-band impedance budget.Narrow-band impedances can also be generated due to the excitation of trapped modes in the BPMs, and their values must be controlled below certain limits to avoid coupled-bunch instabilities.The most impactful higherorder mode excited by the beam is the TE 110 -like mode with respect to the button axis, and its frequency increases with the decrease in the diameter of the button.Figures 2 and 3 show the real and imaginary impedances at low frequencies, respectively.The graph of the imaginary part is intentionally displayed using the linear scale to confirm that it grows linearly with frequency, i.e.: it remains inductive.Figure 4 shows the result obtained in the 0−5 GHz frequency range for transfer impedance [15,16,17].As it can be seen in Figs. 2, 3, 4, for all considered button geometries, both the broadband coupling impedance and the transfer impedance remain essentially similar till rather high frequencies.On the contrary, the high frequency narrow-band impedance is substantially different, as demonstrated in Fig. 5 and Fig. 6.In Fig. 5 the wake potential for the different button geometries and the bunch length of 8 mm is displayed.The wake lasts longer after bunch passage indicating the presence of the narrow-band impedance, [18].
In Fig. 6, we show the real part of the impedance spectrum as a function of frequency.Sharp peaks for different buttons are seen at around 10.62 GHz, 12.93 GHz, 15.02 GHz, and 18.42 GHz that correspond to the trapped TE 110 mode for different types of buttons.The rather small peaks in the impedance spectrum, appearing around 11.98 GHz and 14.7 GHz correspond to the cut-off frequencies of the TM 01 and TE 11 modes for the vacuum chamber.
Among the considered button geometries the first strong resonance peak stays at the highest frequency for the bell-shaped button.Respectively, the amplitude of the wakefield oscillations is lower for such a geometry, Fig 5.

Conclusion
Among the studied geometries, the semispherical and bell-shaped buttons showed better handling of narrow-band beam coupling impedance resonance peaks.
However, it is essential to consider several factors when designing button electrodes for a specific machine, such as the machine's total impedance budget, transfer impedance, mechanical tolerances, and production costs.Based on the results presented in this study, it is evident that a choice of button geometry can significantly affect the narrow-band beam coupling longitudinal impedance of BPMs in vacuum chambers.
The correspondingly optimized design helps to shift the resonance peak narrow-band impedance to higher frequencies while maintaining the transfer impedance, button capacitance, and the gap between the electrode and vacuum chamber at the same level.
As a result, the issues of heating effects and coupled-bunch instabilities caused by BPMs' contribution to both broad-band and narrow-band impedances can be effectively addressed.

Figure 2 .
Figure 2. Real part of the BPM coupling impedance at low frequencies.

Figure 3 .
Figure 3. Imaginary part of the BPM coupling impedance at low frequencies.

Figure 6 .
Figure 6.Real part of the longitudinal impedance spectrum as a function of the frequency.