An ideal cavity is an empty space enclosed by conducting walls that allow electromagnetic (EM) fields to be generated and confined when charged bunches pass longitudinally through it. A cavity designed to excite only transverse magnetic (TM) modes will have a vanishing longitudinal component of the magnetic field and an electric field aligned with the beam path. This alignment provides a strong coupling between the cavity and the beam. The electromagnetic field produced contains information on the beam, which can be measured.

This paper presents the use of a rectangular cavity for beam position measurement. A rectangular cavity is easier to fabricate compared to a cylinder. It provides sufficient beam position measurement accuracy and can measure horizontal or vertical positions separately. This cavity is designed with dimensions of 500×250×100 mm, and beam holes are incorporated into the walls to allow for coupling with the beams. In the case of a rectangular cavity, the monopole and dipole modes are used. Figure 2 shows the electromagnetic field modes. The monopole mode TM110 ( Figure 2a ) can be easily excited and has the strongest electric field in the center of the cavity. The excited electric field is directly proportional to the bunch charge and is concentrated almost exclusively around the center region, making it ideal for applications such as beam acceleration or intensity detection. In contrast, the dipole mode TM120 ( Figure 2b ) has a frequency that is close to that of the monopole mode. Its electromagnetic fields are mirror symmetric and increase linearly in the y direction near the center of the cavity. Since TM120 mode exhibits linear field growth along the y-axis with x-independence, this configuration inherently enables spatial decoupling of x-directional contributions from y-position monitoring. However, to simultaneously achieve x-directional beam position measurement, an orthogonally positioned identical structure is required to establish independent field monitoring channels.

The eigenfrequency of TM modes only takes discrete values determined exclusively by the dimensions of the cavity. The resonant frequency of the TMmnp mode in a rectangular cavity is given by the following Equation:

In this Equation, m, n, and p represent the mode indices, while a, b, and L are the cavity dimensions along the x, y, and z axes, respectively. Here, c denotes the speed of light in a vacuum.

The resonant frequency of the TMmnp decreases as the transverse dimensions increase. In periodic multi-bunch diagnostics, cavities are typically tuned to an integer multiple of the bunch microstructure frequency. In the HUST - PTF, the microstructure frequency is 73 MHz, which is determined by the operating frequency of the superconducting cyclotron. To downsize the cavity-type BPM, its operating frequency is set at 146 MHz, precisely twice the repetition frequency of the proton microstructure.

To estimate the resolution expected from the cavity BPMs, further analysis of the energy exchange between the beam and the cavity is essential. This interaction is influenced by both the cavity’s geometry and the proton microstructure properties, which are characterized by shunt impedance (a critical parameter). Higher shunt impedance results in a higher sensitivity to the beam position. The normalized shunt impedance for each operational mode serves as a critical parameter for this assessment, which is defined as the shunt impedance divided by the unloaded quality factor (R_sh /Q ₀) (12, 15, 18), expressed mathematically as Equation 2:

Where V _ext represents the excited voltage across the cavity, ω denotes the resonance frequency, and U is the storage energy within the cavity.

For example, if a rectangle cavity works in the TM110 mode, the normalized shunt impedance in the center region (y=0) is defined as Equation 3:

Here, a, b, and L denote the cavity dimensions along the x, y, and z axes, respectively; T represents the transient factor; and ϵ₀ is the permittivity of free space. Increasing the normalized shunt impendence increases the beam position monitor signal.

To address these challenges mentioned above, the proposed solution is a novel rectangular cavity BPM with an off-center beam pipe working in monopole mode with dielectric loading, as shown in Figure 3 . Cavity BPMs traditionally use secondary operation modes (dipole modes). At the same operating frequency, their transverse dimensions are larger than those of cavities operating with fundamental modes. Equation 1 shows that when the operating frequency is 146 MHz, the transverse dimensions of a rectangular resonant cavity (with an aspect ratio of 2:1) working in the TM120 mode are 4236 mm × 2118 mm, which is too large for practical applications. Furthermore, the longitudinal electric field excited in the dipole modes is weaker than that of fundamental modes. This results in a lower signal amplitude being extracted from the TM120 mode, which, in turn, limits sensitivity. In contrast, operating in the TM110 mode reduces the transverse dimensions of the cavity to 2298 mm × 1149 mm, a 71% reduction in the transverse area.

GSI proposed using the fundamental mode TM110 in a rectangular cavity to measure the beam’s transverse position to improve the cavity BPM’s sensitivity and simultaneously reduce its transverse dimensions (19, 20). For a rectangular resonant cavity with dimensions a, b, and L along the transverse length, width, and longitudinal height, respectively, the longitudinal normalized shunt impedance expression for the TM110 mode is given by Equation 4:

In the region of the central axis of the rectangular cavity, the normalized shunt impedance for the TM110 mode remains essentially unchanged. However, if the center position of the beam pipe is shifted to the middle of the short semi-axis of the rectangular cavity (see Figure 3 ), then near the center of the beam pipe where x=x’ and y=1/4b+y’ (where x’ and y’ represent the displacement of the beam from the center of the beam pipe), substituting the expressions for x and y into the normalized shunt impedance equation yields Equation 5:

for y’<<b,

Equation 6 clearly shows that modifying the beam pipe’s position makes R_sh dependent on y’. This ensures accurate beam position measurements while maintaining the enhanced output signal characteristics of the TM110 mode.

According to Equation 6, when the center of the beam pipe is aligned with the midpoint of the short semi-axis of the rectangular cavity, the normalized shunt impedance exhibits minimal sensitivity to small displacements of the beam in the x-direction. In contrast, displacements in y primarily affect the normalized shunt impedance.

For a cavity length of 100 mm, we have theoretically calculated and simulated the normalized shunt impedance values for various transverse positions near the center of the beam pipe using CST Microwave Studio. The results are presented in Tables 2 , 3 , together with the results from Equation (6). Notably, the Normalized Shunt Impedance (NSI) values derived from Equation (6) agree within 0.04% with CST simulations, with discrepancies remaining within 0.1%. Figure 4a shows the calculation results.

The CST simulation shows that R_sh is independent of x for up to ±10 mm changes in x direction ( Figure 4b ). Consequently, employing two orthogonally placed rectangular cavities can be used to measure the beam’s transverse position.

While the off-center rectangular cavity operating in the TM110 mode provides smaller transverse dimensions compared to traditional designs, its dimensions still remain large. Leemann (21) proposed that the resonant frequency can be lowered by placing two metal rods along the axis, with gaps between the rods and the cavity wall on one side. The fundamental principle of this methodology lies in its concentration of the resonant electric field within the interstitial regions between the rod elements, thereby inducing a dual effect of enhancing cavity capacitance while reducing resonant frequency.

However, the aforementioned approach fails to achieve adequate field concentration within the gaps, thereby restricting the possibility of further dimensional reduction in the cavity structure. Given that the electric field of the TM110 mode is most intense along the central axis of the cavity, introducing dielectric material can further lower the resonant frequency (22, 23). As illustrated in Figure 5 , two pairs of metal rods are connected to metal trays designed to hold dielectric disks. Additionally, the resonant frequency can be finely tuned by adjusting the dimensions of the metal rods or trays.

Two crucial parameters significantly influence performance: 1) the gradient of the normalized shunt impedance along the beam pipe, which directly affects the BPM’s positional resolution, and 2) the frequency sensitivity, i.e. the change in frequency per change in the thickness of the dielectric disk, denoted as t. Following optimization, the cavity’s transverse dimensions are established at 500 mm×250 mm, with a longitudinal length of 85 mm. The chosen dielectric material is 99.5% pure alumina ceramic, which has a dielectric constant of ϵ= 9.8 and exhibits a low dielectric loss of less than 0.0001. Since the cavity does not include a tuner, it is critical to maintain the frequency sensitivity within 0.5 MHz per 0.1 mm to minimize frequency shifts. After optimization, the final parameters have been determined as follows: r ₁ = 15 mm, r ₂ = 28 mm, and t = 11.1 mm.

The electromagnetic field behavior excited by the passing beams within the cavity was analyzed using CST software, as depicted in Figure 6 . Figure 6a illustrates the electric field distribution in the y-z cross-section, while Figure 6b presents the magnetic field distribution in the x-y cross-section. Due to the presence of the dielectric disk, the electric field of the operating mode exhibits incomplete longitudinal concentration, with negligible components existing in the transverse direction, while the magnetic field is predominantly distributed transversely. Therefore, the operational mode of this dielectric-loaded off-axis rectangular cavity BPM can be categorized as a quasi-TM mode.

Figure 7 presents the map of the normalized shunt impedance (R/Q) near the center of the beam pipe. Within a ±15 mm range around the pipe’s center, the contour lines are approximately parallel to the x-axis. This observation indicates that when the beam moves laterally within this region, the amplitude of the BPM output signal is predominantly influenced by the beam’s displacement in the y direction, with minimal crosstalk between the x and y axes. The actual operation includes variations in pulse duration. However, the output signals of the BPMs primarily depend on the normalized shunt impedance. Consequently, these minor alterations in proton pulse structure only affect the average beam amplitude without compromising the positional information accuracy. Although the contours curve slightly at the edges of this region, potentially introducing some crosstalk, it is not anticipated to be a significant concern, as most protons are expected to traverse the central area where the contours are primarily straight. By normalizing for beam intensity, a pair of orthogonally arranged cavities can accurately determine the beam’s transverse position.

This paired BPM configuration allows for precise beam energy verification through phase analysis of the BPM signals. In the HUST-PTF facility, two critical proton energy parameters require continuous monitoring: the cyclotron’s nominal output energy (E₀) and the clinical treatment energy (Ec).

Theoretically, the cyclotron’s design output energy remains invariant, serving as the baseline for subsequent degrader adjustments to achieve the prescribed proton energy for therapeutic applications. However, variations in accelerator operating conditions, particularly phase fluctuations in the RF accelerating field, may induce deviations in the cyclotron’s output energy. Such energy deviations can be quantitatively characterized by examining the phase correlation between the BPM signal and the cyclotron’s RF accelerating field.

During clinical treatment, the degrader controls the therapeutic proton beam energy through precise mechanical adjustment. Nevertheless, intrinsic limitations in the degrader’s calibration accuracy may lead to discrepancies between the achieved proton energy and the prescribed clinical value. The paired BPM system offers comprehensive real-time monitoring capability for the delivered clinical beam energy. Considering that therapeutic proton energies operate in the non-relativistic domain (70–230 MeV), their velocity exhibits a well-defined energy dependence. Continuous real-time energy monitoring is accomplished by precisely measuring the time-of-flight (TOF) interval between successive BPM signals as the proton beam traverses the paired detector system. For instance, with a 1 m center-to-center separation between BPMs, the measured TOF is 9.1049 ns at 70 MeV, decreasing to 5.5946 ns at 230 MeV, thereby providing reliable energy verification during therapeutic beam delivery.

A prototype cavity was designed and fabricated for proof of principle, and stainless steel was chosen for the majority of its components to reduce cost. Using copper in constructing a real cavity BPM is expected to improve performance.

To enable precise tuning of the resonance frequency, the cavity was initially designed with a resonant frequency of 147.45 MHz, and its key components were intentionally manufactured slightly oversized to accommodate fine adjustments. Following the machining of the essential components of the rectangular cavity, a temporary assembly was constructed using fixtures for measurement purposes, as depicted in Figure 8 . A network analyzer was employed to measure critical parameters, including the resonant frequency. This arrangement facilitated the accurate assessment and optimization of the cavity’s performance before the final assembly was completed.

To tune the resonant frequency to 146 MHz, we adjusted the small metal rods within the rectangular cavity. According to the CST simulation results presented in Figure 9 , reducing the radius of these metal rods lowers the resonant frequency of the operating mode. Therefore, we decreased the rod radius by increments of 0.05 mm. Following each adjustment, we temporarily reassembled the cavity with fixtures to measure its resonant frequency. Once the frequency consistently stabilized at 146 MHz, we welded the cavity to ensure stability and performance.

Figure 10 depicts the welded rectangular cavity BPM, now operating at the frequency of 146.06 MHz. Figure 11 displays the results of the S-parameter measurements. The unloaded quality factor Q₀ of the cavity was measured at 780.2, exhibiting a 14.6% deviation from the designed specification of 913.4. Meanwhile, the external quality factor Q_ext demonstrated superior performance with a measured value of 1761, surpassing the design target by 17.4%. These measurement discrepancies primarily originate from machining tolerances exceeding acceptable limits, manifesting as microscopic surface irregularities on the internal wall surfaces. Notably, since this prototype cavity is primarily intended for proof-of-concept validation, the machining precision and material selection were not rigorously controlled to reduce development costs. Consequently, certain measurement parameters exceeded the initially specified tolerance limits but remained within permissible operational margins, without compromising the cavity’s functional integrity.

The output signal amplitude of the BPM cavity is primarily determined by the normalized shunt impedance along the beam direction. Therefore, accurate measurement of this impedance is critical for evaluating the cavity’s performance characteristics. However, the introduction of dielectric material at the cavity’s center introduces electromagnetic field components in both longitudinal and transverse directions, which significantly compromises the measurement accuracy. Traditional cylindrical and spherical perturbators exhibit inherent limitations in achieving directional electric field modulation along the axial axis. To address this issue, a birdcage perturbation device (24) has been engineered to induce controlled perturbations in the axial electric field. This device employs metal wires to specifically perturb the electric field along their alignment, enhancing the measurements’ resolution and sensitivity. Extensive details regarding the operational mechanism of the device are elaborated in the cited works (25). Figure 12a illustrates the birdcage perturbation device, consisting of a plastic disk with a 5.5 mm radius to hold wires (26, 27). Metal wires with a 0.5 mm diameter and 10 mm length are symmetrically positioned around a circumference that measures 5 mm in radius surrounding the disk. After calibrating, the number of wires was set to 12. This configuration induces controlled perturbations in the cavity’s electric field, enabling high-precision shunt impedance measurements.

An automated test bench has been built to measure the electric field within the cavity. This bench incorporates the birdcage perturbation device, which is driven by a stepper motor to traverse along the central axis of the beam pipe, as depicted in Figure 12b . The change in frequency directly depends on the distance of the birdcage along the beam pipe, and the resulting deviation frequency, Δf, is recorded at various positions. For enhanced accuracy, ten measurements are taken at each position, and the average deviation frequency at the reference position is calculated to minimize measurement uncertainty, as shown in Figure 13 . The calculated normalized shunt impedance at the beam pipe position was found to be 9.45 Ω.

Using the wire displacement method, beam simulation results for the cavity were conducted on the automated test bench, as illustrated in Figure 14 . Initially, a single thin metal wire was stretched through the center of the beam pipe. A 146 MHz sine-wave signal was then introduced into the cavity from an RF signal generator to simulate the transient response of the proton beam transition. The resulting induced voltage signal from the pickup within the cavity was subsequently extracted to an oscilloscope, allowing observation of the EM field distribution in both the time domain and frequency domain. The transverse position of the metal wire is scanned at constant RF signal generator output. The amplitude of the RF signal generator’s output is maintained constant while the transverse position of the metal wire, relative to the center of the beam pipe, is varied to simulate different transverse beam offsets. The wire is maneuvered over a range of ±15 mm with a displacement accuracy of less than 0.005 mm. For each position of the metal wire, the amplitude of the output signal is recorded. The measured results are depicted in Figure 15 , exhibiting excellent spatial correlation with the two-dimensional distribution of the cavity’s normalized shunt impedance obtained via computational simulations.

In practice, the dielectric-loaded off-center rectangular cavity BPM is strategically positioned immediately downstream of a reference cavity BCM, so the output of the BPM can be normalized by the BCM signal, assuming the same beam current traverses both devices. Our offline test data encompassed beam positions in the range of x= ± 10 mm and y= ± 10 mm, which is more than the maximum expected variation in the beam position.

The energy of the clinical proton beams typically spans from 70 MeV to 230 MeV, which impacts the output signal of the BPM used to track the beam’s position. Simulations show that a 70 MeV proton beam exhibits a transit time factor of 0.962, and the beam current is 0.35 nA in our machine. As it passes through the cavity BPM at different lateral positions, the output signal amplitude varies between 18.68 nV and 28.52 nV. The position sensitivity of the BPM at this energy is 0.49 nV/mm. At 230 MeV, the simulation shows a transit time factor of 0.985, and the beam current is 5 nA in our machine. This increased energy and current result in a larger output signal amplitude ranging from 650.79 nV to 993.13 nV at different lateral positions. The position sensitivity at this energy level is higher, at 17.12 nV/mm. These variations in output signal amplitude and position sensitivity at different proton beam energies need to be characterized for the proper operation of the BPM.

Figure 16 illustrates the simulated amplitude of the BPM output signal at various transverse positions for proton beams at both 70 MeV and 230 MeV. This data highlights the critical need for energy-specific calibration of the BPM.

The x-y directional coupling characteristics of the BPM output signal were experimentally studied. The experiments employed a stationary measurement approach: First, the beam position was fixed along the y-axis, and the output signal at x=0 was established as the reference baseline. We acquired normalized voltage response curves for x-directional displacements (Δx=−10~+10 mm) relative to the reference point. By calculating the normalized relative deviation δ(Δx)=|V(Δx)−V₀|/V₀×100% for each displacement, we quantified the coupling effect. The use of orthogonally aligned BPMs enables concurrent measurement of beam signals in the x- and y-directions, resulting in a consistent x-y coupling variation along the x-axis.

Figure 17 shows that as the beam moves along the y-axis, there is an increase in the coupling between the x and y directions. However, within ±10 mm of the center of the pipe—a typical range of operation—the output signal error due to this x-y coupling remains within 0.5%. This level of error is acceptable, especially when considering the BPM’s resolution requirement of 0.5 mm.

The position measurement accuracy of the cavity BPM is determined by two principal factors: the position signal sensitivity and x/y directional coupling effects. Experimental characterization of the prototype BPM confirm that, under clinical operating conditions, the output signal amplitudes attains the nanovolt scale (~nV). These nanoscale signals are vulnerable to be obscured by noise sources several orders of magnitude stronger, necessitating the use of lock-in amplification methodology for reliable signal retrieval. This technology with phase-sensitive detection to extract signal components at specific reference frequencies and phases while suppressing noise at other frequencies.

An SR844 model lock-in amplifier is used for evaluating the position measurement accuracy of this prototype BPM (for technical specifications and operating procedures, refer to the instrument manual). Performance evaluation was conducted at five characteristic energy points across the proton therapy energy range (70–230 MeV), with quantitative results shown in Table 4 . The measured position accuracy demonstrates energy dependence, ranging from 0.65 mm at 70 MeV to 0.02 mm at 230 MeV. Notably, the cavity BPM maintains sub-0.5 mm accuracy throughout most of the therapeutic energy spectrum.

Additional measurements reveal that when the beam deviates within ±10 mm from the central axis, x/y directional coupling induces output signal errors less than 0.5%. These results indicate that the fundamental limitation in positional measurement accuracy stems principally from the resonant cavity’s intrinsic sensitivity characteristics.

While the current prototype utilizes cost-effective stainless steel fabrication for proof-of-principle validation, this material selection inherently constrains its detection sensitivity. Substantial performance enhancement is theoretically achievable through precision-engineered copper cavity BPM systems, thereby satisfying the requirements (≤ 0.5 mm) for real-time, non-intercepting beam monitoring applications.