Single-bunch operation mode

The loaded quality factor Q_L characterizes the total energy loss and is defined as $Q_{\text{L}}=\frac{\omega U}{P}\mathrm{,}$ (1) where ω is the resonant angular frequency, U is the energy stored in the cavity for a resonant mode, and P is the energy loss per angular oscillation of the mode, which consists of the power dissipation on the inner wall of the cavity caused by the ohmic loss p_wall and the loss caused by the coupling ports p_ext. The unloaded quality factor Q₀ and external factor Q_ext can be used to characterize p_wall and p_ext, respectively.

The normalized shunt impedance R/Q₀ describes the energy interaction intensity between the charged particle beam and the cavity [11]. If the phase shift between the electric field and the beam is neglected, the ideal normalized shunt impedance is defined as $\widehat{R}/Q_0=\frac{{\left|{\displaystyle \int }\text{E}\left(s\right)\text{d}s\right|}^2}{\omega U}=\frac{V^2}{\omega U}\mathrm{,}$ (2) where the numerator represents the integration of the electric field in the resonant mode along the beam passage. If a phase shift occurs during the beam transition, the normalized impedance can be represented as $R/Q_0=\frac{{\left|{\displaystyle \int }\text{E}\left(s\right)\cos \left(\frac{\omega z}{\beta c}\right)\text{d}\text{s}\right|}^2}{\omega U}=\widehat{R}/Q_0{T}^2\mathrm{,}$ (3) where β c represents the particle velocity, and T is the transit time factor, which is a function of the beam velocity. Considering the energy loss at the cavity wall and coupling ports, the decay time constant τ is defined as $\tau =\frac{Q_L}{\omega }\mathrm{.}$ (4) This represents the time required for the stored energy to reduce to 1/e of its original value. When the bunch interval is significantly longer than the decay time constant, there is no interference from the energy induced by successive bunches. In this case, the cavity monitor operates in single-bunch mode.

After a longitudinal Gaussian-distributed bunch whose root mean square (RMS) length is σz passes through the cavity, the output voltage at the port with characteristic impedance Z can be calculated as follows [12, 13]: $\begin{array}{c}V=\frac{q\omega }{2}\sqrt{\frac{Z}{Q_{\text{ext}}}\left(R/Q\right)}\exp \left(-\frac{{\omega }^2{\sigma }_z^2}{2c^2}\right)\exp \left(-\frac{t}{2\tau }\right)\exp \left(i{\omega }_{n}t\right)\\ =V_0\exp \left(-\frac{t}{2\tau }\right)\exp \left(i{\omega }_{n}t\right).\end{array}$ (5) In the TM010 (monopole) mode in a cylindrical cavity, the axial electric amplitude near the central axis remains relatively constant [14], making it suitable for measuring the beam current.

Multiple-bunch operation mode

When the bunch interval is less than or close to the decay time constant, the cavity monitor operates in multiple-bunch modes, and the output voltage is the superposition of each bunch signal. Assuming that the beam interval is T_b and the number of bunches is m, the output voltage of the cavity BCM is given by $\begin{array}{c}{V}_m=V_0{\displaystyle \sum _0^{m-1}\exp }\left(i{\omega }_{n}m{T}_{\text{b}}\right)\exp \left(-m\frac{T_{\text{b}}}{2\tau }\right)\\ =V_0\frac{1-\exp \left(i{\omega }_{n}m{T}_{\text{b}}\right)\exp \left(-m\frac{T_{\text{b}}}{2\tau }\right)}{1-\exp \left(i{\omega }_n{T}_{\text{b}}\right)\exp \left(-\frac{T_{\text{b}}}{2\tau }\right)}.\end{array}$ (6) To evaluate the effects of different working frequencies on the output signal of the cavity BCM, we define ${\omega }_n=N\frac{2\pi }{T_{\text{b}}}+\Delta \omega$, where N is an integer. Then, $V_m=V_0\frac{1-\exp \left(im\Delta \omega T_{\text{b}}\right)\exp \left(-m\frac{T_{\text{b}}}{2\tau }\right)}{1-\exp \left(i\Delta \omega T_{\text{b}}\right)\exp \left(-\frac{T_{\text{b}}}{2\tau }\right)}\mathrm{.}$ (7) As the number of bunches increases, the output voltage converges to a constant value. $V_m=V_0\frac{1}{1-\exp \left(i\Delta \omega T_{\text{b}}\right)\exp \left(-\frac{T_{\text{b}}}{2\tau }\right)}\mathrm{.}$ (8) When Δ ω =0, that is, the working frequency of the cavity BCM is an integer multiple of the bunch frequency, the output signal reaches its maximum [14, 15]. $V_m=V_0\frac{1}{1-\exp \left(-\frac{T_{\text{b}}}{2\tau }\right)}=V_0\frac{1}{1-\exp \left(-\frac{\omega T_{\text{b}}}{2Q_{\text{L}}}\right)}\mathrm{.}$ (9) Compared to the single-bunch operation mode, the multiple-bunch mode can amplify the output signal by the gain factor, which has an obvious advantage in the case of low-intensity measurements. In practice, we can adjust the external quality factor to ensure that the cavity monitor operates in the multiple-bunch mode.

Cavity geometry

In multiple-bunch mode, a cavity BCM prefers to operate in the frequency harmonics of the beam repetition rate to increase the output signal. The superconducting cyclotron produces Gaussian-shaped proton bunches at a repetition rate of 73 MHz. Gaussian-shaped bunches with a bunch charge q, bunch spacing T, and RMS length σz can be expanded using a cosine series with ${\omega }_0=\frac{2\pi }{T}$ [16]: $I_{\text{b}}\left(t\right)=\frac{q}{t}+{\displaystyle \sum _{m=1}^{\infty }{I}_m}\cos \left(m{\omega }_{0}t\right)\mathrm{,}$ (10) where $I_m=2\frac{q}{T}\exp \left(-\frac{m^2{\omega }_0^2{\sigma }_z^2}{2}\right)\mathrm{.}$ (11) It is evident that a lower harmonic contains more beam energy. However, if the cavity operates at the first harmonic frequency, it typically suffers from direct noise from the power generator. In addition, a low working frequency makes the cavity BCM considerably large. Therefore, the working frequency of the cavity BCM was matched to the second harmonic of the bunch repetition rate. Usually, a pillbox cavity serves as a BCM owing to its simple structure and precise machining [5]. However, when working at a low frequency of 146 MHz in TM010 mode, the pillbox cavity radius is 78.4 cm. Such a large size prevents precise manufacturing and installation of the cavity BCM.

A reentrant cavity was employed to minimize the cavity size, whose resonant characteristics can be analogized by two lumped elements [17, 18]: the capacitor plate (C_gap) and cavity wall inductor (L_wall). C_gap can be calculated as follows: $C_{\text{gap}}=\frac{{\epsilon }_r{\epsilon }_0\pi \left(r_{\text{max}}^2-r_{\text{min}}^2\right)}{d}\mathrm{,}$ (12) where ε _r is the relative permittivity of the material filling in the gap, ε ₀ is the permittivity of vacuum, d is the gap height, and r_min and r_max are the outer and inner radii of the capacitor plate, respectively. The cavity wall and inner metal cylinder can be regarded as a coaxial line with closed termination based on transmission line theory. The impedance Z, as seen at the input of the coaxial line, can be calculated as $Z=j{Z}_0\tan \left(\frac{2\pi h}{\lambda }\right)\mathrm{,}$ (13) where Z₀ is the characteristic impedance of the coaxial part, which is determined by the outer and inner radii of the coaxial line, h is the length of the coaxial line, and λ is the free-space wavelength. It is evident that the impedance Z is inductive when the length of the coaxial line is shorter than λ/4. The cavity resonates when the gap capacitance compensates for the cavity wall inductor, and the resonant frequency is given by ${\omega }_0=\frac{1}{\sqrt{Z_{\text{wall}}{C}_{\text{gap}}}}\mathrm{.}$ (14) At a working frequency of 146 MHz, λ/4 was 510 mm, rendering the coaxial length of the reentrant cavity BCM too large to be acceptable. Owing to the limited longitudinal installation space, the cavity length was set to within 150 mm, which led to a decrease in Z_wallC_gap must be increased to satisfy the resonance frequency, which can be achieved by decreasing the gap length or filling the gap with dielectric materials. For comparison, a common reentrant cavity and dielectric-filled reentrant cavity with the same working frequency of 146 MHz were designed, as shown in Fig. 2; the detailed dimensions are listed in Table 2.

Fig. 2Full-screen View

(Color online) Outline of a common reentrant cavity (a) and dielectric-filled reentrant cavity (b).

According to the comparison, inserting a dielectric material can effectively increase the gap length, leading to a longer interaction path between the charged particle and cavity, which amplifies the output signal. Increasing the gap length also reduces the sensitivity of the TM010 mode frequency. The simulation results show that if the gap height of the dielectric-filled reentrant cavity is 16.7 mm, the sensitivity of the working frequency to the gap height decreases from 23 to 3 MHz/mm, which makes fabrication easier. Dielectric materials use Macor ceramic because of its easy and precise machinability [19].

Output signal of BCM

Figure 3 shows the dielectric-filled reentrant cavity BCM modeled by the CST [20]. The simulation results show that the induced electric and magnetic fields are concentrated in the dielectric plate and coaxial area, validating the lumped-element circuit model, as shown in Fig. 4(a) and (b).

Fig. 3Full-screen View

(Color online) CST model of the dielectric-filled reentrant cavity.

Fig. 4Full-screen View

(Color online) Energy intensity distribution of the induced electric fields (a) and magnetic fields (b) inside the cavity.

A higher output signal requires more induced energy inside the BCM for a given external quality factor. Working in multiple-bunch operation modes, the BCM can accumulate more energy from each bunch, making it possible to determine the Q_ext flexibly to obtain a reasonable output signal. Equation 9 shows that the detectable signal of the multiple-bunch mode is the product of the gain factor and output signal of the single-bunch mode. An increase in the external quality factor can increase the gain factor. However, this reduces the output signal excited by a single bunch, as illustrated in Fig. 5. Considering that the dielectric loss accelerates the electromagnetic energy decay, the growth of the gain factor is limited. Eventually, the output signal in multiple-bunch mode decreases with an increase in the external quality factor (Fig. 5). Coupling loops were employed to extract the beam current signal, and the port impedance was tuned to 50 Ω. A smaller Q_ext requires larger coupling loops, which may severely disturb the electromagnetic field distribution inside BCM. Eventually, by adjusting the coupling loop size and orientation in the CST MWS, the external quality factor was optimized to 2000.

Fig. 5Full-screen View

Effects of the external quality factor on the BCM signal amplitude and gain factor.

In a practical cavity, the resonant frequency may deviate from the design owing to the mechanical tolerance and uncertainty of Macor permittivity. Thus, the output signal will change accordingly, as expressed by Eq. 12. As shown in Fig. 6, the signal amplitude decreased abruptly when the working frequency deviates from 146 MHz. To prevent frequency deviation from reducing the output signal, we installed four tuners to adjust the resonant frequency of the cavity. They were placed in the holes of the Macor plates to increase the tuning range.

Fig. 6Full-screen View

Effects of frequency offset on the BCM signal amplitude in multiple-bunch mode.

Manufacture of the prototype cavity

We manufactured the prototype cavity BCM shown in Fig. 7. It comprised three main mechanical components. Part 1 was a beam pipe, Part 2 was a concentric pipe that formed a coaxial line with the beam pipe with a closed termination, and Part 3 was the cavity gap filled with Macor ceramic. The cavity was stainless steel 316L. A 2-mm groove was arranged on a stainless-steel plate to fix the Macor ceramic. The coaxial probes were soldered directly onto the cavity wall. The coupling loops were detachable, which was convenient for tuning the external quality factor. Metal rods were employed as tuners to adjust the resonant frequency, and rulers were installed on the tuners to observe the penetration depth. We did not consider vacuum in the design of the prototype cavity. If the cavity BCM is installed in HUST-PTF, the corrugated pipes will replace the rulers to maintain the vacuum.

Fig. 7Full-screen View

(Color online) Mechanical design of the dielectric-filled cavity: (a) cross-section of mechanical design of the cavity BCM; (b) corresponding prototype cavity.

Working frequency and quality factors

Using the vector network analyzer KEYSIGHT E5071C, we obtained the resonant frequency and unloaded and loaded quality factors based on S-parameter measurements [21]. The normalized shunt impedance R/Q₀ was measured using the bead-pull method [22]. The measured data agreed well with the simulation results listed in Table 3. Figure 8 indicates that the tuners could easily adjust the resonant frequency and agree well with the simulation, which proves the correctness of the design. The penetration depth of the tuners could be changed by precession. The moving ranges of the tuners were set from 0 to 15 mm to guarantee sufficient frequency adjustment to satisfy the request. Because the tuners moving inside the cavity exhibited higher tuning sensitivity, the initial position of the tuners was inside the cavity at a depth of 12.5 mm to minimize the tuning sensitivity.

Fig. 8Full-screen View

Insertion depths of the tuners versus the working frequency of the cavity.

Beam analog measurement

Because HUST-PTF is currently being developed, we can only employ beam analog measurement to characterize the cavity BCM. Stretched-wire measurement is a typical method used to characterize beam monitors. The offline test platform is illustrated in Fig. 9. The prototype cavity was supported by a two-directional displacement platform. A vertical metal wire passing through the center of the cavity served as the longitudinal beam analog. A metal wire was fixed to both sides of the prototype cavity and stretched as tightly as possible. However, the cavity could move transversely with an accuracy of 0.001 mm, precisely simulating an off-axis beam. The 146-MHz sinusoidal signal from a signal generator was fed into the metal wire to excite the electromagnetic fields in the cavity and induce the corresponding voltage signals at the two feedthrough ports. Eventually, the feedthrough ports were connected to an oscilloscope with a fast Fourier transform (FFT) function for further analysis. The reliability of the beam analog was confirmed using the CST, in which the input signals of both the metal wire and bunch had the same longitudinal distribution and Gaussian distributions with RMS widths of 1 ns. The output signals in the frequency domain are shown in Fig. 10, which proves the equivalence between the metal wire and the actual bunch.

Fig. 9Full-screen View

(Color online) Layout of the offline test platform. On the left is a schematic of the stretched wire method, and on the right is a photograph of the beam analog measurement.

Fig. 10Full-screen View

Comparison of the output signals excited by a bunch and metal wire with the same electric signal.

When a metal wire is employed as a beam analog, the metal wire and beam pipes form a coaxial line, causing electromagnetic energy in the cavity to leak easily through the beam pipe [23]. Therefore, the output signal excited by the metal wire is weaker than that excited by the bunch. For each frequency component, the output amplitudes excited by the metal wire and bunch are linearly correlated, which can be written as $V_{\text{bunch}}=k{V}_{\text{wire}}\mathrm{,}$ (15) where k is a constant.

In practice, the proton bunch has a transverse size, and protons at different locations undergo different normalized shunt impedances. A weighted method based on the superposition principle is employed to simulate the bunch with a transverse size [24, 25]. A grid is defined in the cross-sectional plane of the beam, and each grid point represents the transverse position of the metal wire. According to the transverse distribution function of the bunch, the output signal is obtained by summing the signals of different grid points with weight factors. A smaller grid step size indicates a more accurate simulation, but a longer experimental time. In practice, we employed a 10 mm × 10 mm grid with a step size of 0.2 mm to approximate a Gaussian distribution with 2 σ of 5 mm [26]. The measurement results in Fig. 11 show satisfactory linearity between the output and excitation signals.

Fig. 11Full-screen View

Induced signal amplitude versus the excitation signal amplitude.

An ideal BCM should be independent of beam position. We evaluated the dependence of the output signal on the beam offset using an offline test platform and analogized the bunch at different transverse positions by shifting the platform. The experimental results showed that the transverse position of the beam had a negligible effect on the output signal (Fig. 12). The relative amplitude change was within 0.1% even with a 10-mm beam offset.

Fig. 12Full-screen View

Relative change in the induced signal amplitude versus beam offset.