Introduction

The European Spallation Source (ESS) in Lund, Sweden, is constructing a high-intensity 2 GeV proton linear accelerator designed for 5 MW operation [1]. The accelerator comprises a normal-conducting (NC) front end operating at 352.21 MHz, followed by a superconducting (SC) linac at 704.42 MHz (excluding the spoke cavities). As shown in Fig. 1, the Drift Tube Linac (DTL) and SC sections are subdivided into individual tanks and cryomodules, with the gray-shaded cryomodules designated to remain unpowered during the initial 2 MW operation at 800 MeV [2-4]. The high-level parameters for the ESS are summarized in Table 1. Designed for a 62.5 mA peak beam current with a 14 Hz repetition rate, the ESS demands accurate and precise RF phase control in each cavity to maintain beam quality and minimize beam losses. By 2023, the initial commissioning had accelerated the beam through the early NC sections to approximately 74 MeV.

Fig. 1Full-screen View

(Color online) Schematic layout of the ESS linac operations at 800 MeV and 2 MW (upper plot) and a detailed view of the MEBT section (lower plot) [2]. In the upper plot, each segment in the Drift Tube Linac (DTL) and SC sections represents a DTL tank or a cryomodule, with the gray-shaded cryomodules indicating those that will remain unpowered during the initial 2 MW operation at 800 MeV. The lower plot provides an expanded view of the MEBT section, detailing its beam matching optics and diagnostic components essential for shaping and monitoring the beam. Three buncher cavities, labeled B1 to B3, are installed within the MEBT to maintain the longitudinal focus of the beam

Accurate measurement of the beam phase, defined as the phase experienced by the beam relative to the RF field, is critical for maintaining beam quality and minimizing losses, particularly in high-power accelerators such as the ESS. Various methods exist for calibrating beam phases. Direct approaches include phase-scan techniques (e.g., ΔT phase scan [4-7], phase-scan signature matching [7, 8]), which vary the RF phase and observe downstream beam changes. Although effective, these methods often need dedicated beam studies, requiring the sequential calibration of individual cavities with adjacent ones deactivated or significantly detuned, thereby disrupting normal operations. They are typically performed offline during commissioning or maintenance, making them labor-intensive. Furthermore, gradual ambient drifts, such as changes in temperature and humidity, within low-level RF (LLRF) control loops necessitate periodic recalibrations during routine operation.

Alternative strategies have been designed to minimize the disruption of the normal operation of the accelerator. The “drifting beam method,” pioneered at SNS [9], uses the LLRF system to measure transient RF signals excited by the beam in an unpowered SC cavity. The analysis of these transients yields the relative beam phase. This avoids extra beam diagnostics but requires temporarily unpowering the cavity. Researchers at DESY proposed another method based on the linear fitting of the trajectory of the RF transient induced by a beam pulse in the in-phase/quadrature (I/Q) plane. The beam pulse is typically short, on the order of tens of microseconds, and the measurement is performed within a powered SC cavity [10-14]. The DESY approach allows online measurement but usually requires operating the SC cavity in an open-loop configuration.

Adapting the DESY approach to the ESS NC buncher cavity poses a distinct challenge [15, 16]. The buncher cavity has a measured half-bandwidth (ω_0.5) of approximately 18 kHz, corresponding to a time constant () of about 9 μs. When using short diagnostic beam pulses (e.g., 5 μs) whose duration is comparable to τ, cavity detuning causes a significant non-linear rotation (deflection) of the beam-induced RF transient trajectory. Consequently, the trajectory deviates from the expected linear behavior, which substantially reduces the accuracy of the beam phase calibration based on linear fitting. This issue is less pronounced in SRF cavities, where the time constants are typically on the order of milliseconds owing to their much higher loaded quality factors (Q_L) [17-21].

To mitigate detuning-induced errors, we developed a detuning compensation algorithm by solving the cavity differential equations to predict the theoretical deflection angle () of the transient trajectory. Subtracting the calculated from the fitted angle from the RF transient trajectory enables accurate beam phase calibration.

However, operating RF cavities in the open-loop mode is neither practical nor safe for high-current proton linacs, such as the ESS. In open-loop operation, the RF field can deviate from its setpoint, and strong beam loading [22, 23] further increases the risk of beam losses. Therefore, it is preferable to maintain a closed-loop LLRF operation during measurements. To avoid interference from the feedback system during phase calibration, we deliberately selected a calibration window approximately equal to the LLRF loop delay ( μs) and restricted the analysis to the beam-induced RF transient before the feedback loop became active. This ensures that only the portion of the transient signal unaffected by the feedback is considered, allowing beam phase calibration under closed-loop conditions. It should be noted that the closed-loop measurement relies solely on the first 1.4 μs of transient data to determine the beam phase. Although the achievable phase precision is inherently limited compared to open-loop or BPM-based methods, this approach enables online phase monitoring under normal operating conditions of the device. The resulting accuracy is generally considered adequate for routine diagnostics, machine protection, and early fault detection [23].

In the following sections, we first revisit the cavity differential equation and analyze the impact of cavity detuning on the beam-induced RF transient. Based on this analysis, a detuning-aware beam phase calibration method is proposed, together with a practical implementation strategy under closed-loop conditions. Experimental validation was performed using the ESS MEBT buncher cavity, demonstrating the effectiveness of the proposed technique under realistic operating scenarios.

Principle of Beam Phase Calibration Using Beam-Induced RF Transients

The beam phase calibration method based on beam-induced RF transients in powered RF cavities was originally proposed by DESY [13, 14]. In the following subsection, we present the theoretical basis of DESY’s approach, including the derivation of the transient field trajectory and the conditions under which the linear approximation applies directly. Throughout this study, boldface symbols (e.g., V_c, I_b) denote complex-valued signals or quantities.

2.1

Transient RF Trajectory Calibration: Concept and Example

Figure 2 illustrates the basic principle of beam phase measurement using beam-induced RF transients. As shown in Fig. 2a, the cavity field is initially in a steady state with normalized amplitude and zero phase. When a short beam pulse passes through the cavity in the open-loop mode, it induces a transient perturbation, denoted as V_cb(t), shifting the cavity voltage from its nominal value to a new complex value . This transient response traces a trajectory in the I/Q plane (Fig. 2b), where the horizontal axis represents the in-phase (I, real) component, and the vertical axis represents the quadrature (Q, imaginary) component. The angle between the steady-state V_c vector and the direction of this trajectory reflects the beam phase [11, 13, 14, 24, 25].

Fig. 2Full-screen View

Principle of beam phase measurement using beam-induced RF transient response, as proposed by DESY. (a) Time-domain evolution of the cavity voltage amplitude and phase for the transient signal V_cb(t) when a short beam pulse traverses an RF cavity in the open-loop mode. (b) Corresponding trajectory of V_cb(t) in the I/Q plane, where the direction of the trajectory reflects the beam phase

Figure 3 shows the measured beam-induced RF transient in the ESS buncher cavity B2 (see Fig. 1) under open-loop conditions, generated by a 5 μs, 55 mA beam pulse with a beam phase of approximately -44°. The relevant RF parameters are listed in Table 2. As seen in Fig. 3a, the cavity remains in a steady state prior to beam arrival. At approximately t=177 μs, the cavity voltage V_c1(t) in the presence of the beam exhibits a pronounced drop in both the amplitude and phase relative to the no-beam case V_c0(t). A zoomed-in view of the shaded region in Fig. 3a, as shown in Fig. 3b reveals the detailed structure of the beam-induced RF transient. Figure 3c displays the corresponding transient trajectory of V_cb(t) in the I/Q plane, reconstructed from the sampled RF waveforms acquired by the digital LLRF system.

Fig. 3Full-screen View

(Color online) Measured beam-induced RF transient in the ESS buncher cavity B2 under open-loop conditions with a 5 μs and 55 mA beam pulse. (a) Time evolution of the forward voltage V_f, cavity voltage with beam V_c1, and cavity voltage without beam V_c0. (b) Zoomed-in view of the beam-induced RF transient in both the amplitude and phase. The red dots represent the sampled values of V_cb recorded by the LLRF DAQ system, with a sampling period of approximately 0.119 μs. (c) Trajectory of the beam-induced transient vector V_cb in I/Q plane. Due to a detuning angle of approximately -11° in the superconducting cavity, the tragectory is slightly deflected toward the negative direction of the X-axis (real axis)

Table 2Full-screen View

RF parameters for ESS buncher cavity B2 (see Fig. 1)

Parameter	Value
Beam pulse repetition rate (Hz)	14
Cavity voltage (kV)	59.7
RF frequency (MHz)	352.21
Cavity half bandwidth (ω0.5) (kHz)	~18
Normalized shunt impedance (r/Q) (Ω)	~75
Coupling factor (β)	~1

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In the following, we present the theoretical basis of this approach, including the derivation of the transient field expression and the conditions under which the linear approximation remains valid.

The time evolution of the cavity field, with and without beam loading, can be described by the following differential equations [24, 26-30]: (1)In these equations, t denotes the time. The variables V_f0 and V_f1 denote the cavity forward voltages without and with the beam, respectively. The detuning terms and account for the cavity detuning in the absence and presence of the beam, respectively. The parameter is the cavity half-bandwidth, assumed to be independent of beam loading, and β is the coupling coefficient of the input coupler. The beam-induced voltage V_b(t) is given by (2)where I_b(t) is the RF component of the bunched beam current. The commonly referenced beam current typically refers to its average (DC) value. For short bunches, the RF current amplitude satisfies , where is the DC beam current [26]. The loaded resistance is defined as (3)where is the loaded quality factor of the cavity and is the normalized shunt impedance.

If the cavity operates in an open-loop mode, the cavity forward signal remains approximately unchanged during beam injection, such that . We also assume that the cavity detuning is approximately constant throughout the beam pulse. Accordingly, we set . Under these assumptions, subtracting the first equation in Eq. (1) from the second yields: (4)The general solution of Eq. (4) for zero initial condition (i.e., the zero-state response) is given by: (5)where .

To simplify the analysis without loss of generality, we assume that the beam-induced voltage V_b(t) points along the X-axis (real axis) in the complex plane; that is, its phase angle is set to zero. Under this assumption, . For analytical convenience, we further model the beam input as a unit step function: , where for and for .

This approximation is justified by the fact that, as shown in Sect. 4, the beam current pulse typically resembles a quasi-rectangular waveform with a sub-microsecond rise time. Hence, in the early stage, when the beam first enters the cavity, its effect closely resembles that of a step excitation. With this simplification, the solution to Eq. (4) becomes [27]: (6)If the cavity is operated exactly on-resonance (i.e., Δω=0), Eq. (6) simplifies to (7)In this ideal case, the exponential term remains purely real, and both and V_b(t) lie along the X-axis in the I/Q plane. Consequently, the trajectory of remains aligned with V_b(t), leading to no phase distortion.

2.2

Detuning Impact Analysis

When the cavity operates off-resonance (), the solution in Eq. (6) becomes complex, containing both real and imaginary components. Consequently, its trajectory deviates from the direction of V_b, undergoing a rotation in the I/Q plane. The direction of this rotation, clockwise or counterclockwise, depends on the sign of the detuning.

To gain analytical insight into the early-time behavior, we consider the limit , where denotes the cavity time constant. We further assume that , which is typically satisfied by most RF cavities. Under this assumption, the product remains small, allowing the exponential term to be approximated using a second-order Taylor expansion as follows:Substituting this approximation into Eq. (6), we get:which can be separated into real and imaginary parts: (8)Accordingly, the ratio between the imaginary and real components is (9)

According to Eq. (9), as , the ratio between the imaginary and real components approaches zero, indicating that the imaginary part becomes negligible compared with the real part. Consequently, V_cb,step(t) remains nearly aligned with the direction of V_b, which is assumed to lie along the X-axis.

Figure 4 shows V_cb,step(t) under various detuning conditions. In Fig. 4a, both real and imaginary components evolve over time, with the response angle determined by the detuning angle . The early time behavior () is shown in Fig. 4b, where the imaginary component remains small. The I/Q trajectories in Fig. 4c illustrate that detuning causes the response to deviate from the direction of V_b, which is assumed to lie along the X-axis. For negative detuning, the real part remains unchanged, whereas the imaginary part reverses sign, resulting in symmetry about the X-axis (not shown). As shown in Fig. 4d, the very initial trajectory () remains nearly aligned with the X-axis even for detuning angles up to 40°, which is consistent with Eq. (9). According to this expression, as , the imaginary-to-real ratio approaches zero, indicating that V_cb,step(t) initially aligns with the direction of V_b. This confirms that for , the detuning-induced phase distortion is typically negligible.

Fig. 4Full-screen View

(Color online) Step response of Eq. (4) and the corresponding I/Q-plane trajectories under different detuning angles . (a) Simulated step response of V_cb,step(t) for various . The upper and lower subplots show the real and imaginary parts of the signals, respectively. (b) Zoom-in view of the gray-shaded region in sub-plot (a), highlighting the early time behavior of the transient response. (c) I/Q-plane trajectories of V_cb,step(t) under different . (d) Zoom-in view of the I/Q-plane during the very early stage (). In this regime, the trajectory of V_cb,step(t) closely follows a straight line along the real axis. This effect is particularly pronounced when is small, where the imaginary component remains negligible compared to the real part

Figure 4 and Eq. (8) provide insight into the applicability of the DESY method. When cavity detuning is present, the step response V_cb,step(t) includes an imaginary component, causing the corresponding transient vector to deviate from the direction of V_b in the complex I/Q plane. The angle between V_cb,step and V_b is determined by both the detuning magnitude and observation time t. It is important to note that t must remain less than the beam pulse width (see Fig. 2a), because the quasi-rectangular beam pulse can only be approximated as a step input within this interval.

For SC cavities, the loaded quality factor Q_L is typically high, resulting in a narrow half-bandwidth (tens to hundreds of Hz) and a large time constant on the order of milliseconds. Assuming ms (as in the KEK cERL main linac cavity [17, 18]) and a beam pulse width of T_ps=9 μs, with a detuning angle of , Eq. (9) yields , corresponding to a phase deviation of , which can be safely neglected.

In contrast, for the ESS NC buncher cavity B2 (see Fig. 1) with a time constant μs, the same 9 μs beam pulse leads to a significantly different behavior. Even with a small detuning angle of , the resulting deviation of V_cb can exceed 5°, which is no longer negligible. As shown in the experimental results of Fig. 3, the buncher cavity B2 exhibits a detuning angle of approximately -11°. With a 5 μs beam pulse, this results in a slight deflection of the transient response V_cb toward the negative direction of the X-axis in the I/Q plane.

It should be emphasized that Eq. (6) is valid only for the zero-state response of Eq. (4), where the beam-induced voltage V_b(t) is modeled as a unit step function aligned along the X-axis. In practical measurements, however, the observed beam-induced RF transient also includes the steady-state cavity field, and the beam-induced voltage vector V_b(t) may point in an arbitrary direction within the 360-degree I/Q plane. Therefore, the normalized transient signal V_cb(t) is expressed as: (10)where V_c,ss denotes the steady-state cavity voltage.

Substituting V_b in Eq. (10) with I_b using Eq. (2) yields: (11)where denotes the phase difference between the vectors I_b and V_c. The beam phase is related to by the expression , as shown in Fig. 2.

Figure 5 shows the simulation results based on Eq. (11), where different detuning angles produce distinct trajectories of V_cb(t) in the I/Q plane. The simulation assumed a peak beam current of I_b0=55 mA, pulse width of 5 μs, and beam phase of approximately -45°. Figures 5a and b illustrate the time evolution of the amplitude and phase and the corresponding I/Q-plane trajectories, respectively.

Fig. 5Full-screen View

(Color online) Simulation results of beam-induced RF transient under various detuning angles Δθ, assuming open-loop cavity operation ( mA, beam pulse width T_ps=5 μs, beam phase ≈-45°). These parameter values were specified as the input conditions in the simulation model. (a) Time-domain waveforms of amplitude and phase under different Δθ. (b) Corresponding trajectories of V_cb in the I/Q plane

In this section, the beam excitation is approximated as a quasi-rectangular pulse with a sub-microsecond rise time, as is typically observed in accelerator systems [22]. Based on this, the beam input was modeled as a unit step function to analyze the detuning effects on the beam phase measurement for both SC and NC cavities. Although this assumption simplifies the analysis, the qualitative conclusions remain valid for arbitrary beam pulse shapes. A general derivation is provided in Appendix A.

Beam Phase Calibration Algorithm with Detuning Compensation

To enable accurate phase calibration under detuning, this section presents a correction framework based on a theoretical deflection estimation. We first derive the deflection angle induced by detuning and then discuss its application under closed-loop operation.

3.1

Theoretical Estimation of Deflection Angle Caused by Detuning

According to Eq. (6) in Sect. II, the theoretical phase deflection angle induced by detuning Δω at time (T_w represents the calibration window for beam-phase, T_w<T_ps) can be expressed as: (12)As shown in Fig. 6, the step response V_cb(t) in Eq. (6) is discretized as , where is the sampling period of the digital LLRF data acquisition system (μs). Assuming that V_b points along the X-axis, the resulting discrete samples are plotted as blue dots in the I/Q plane. A linear regression of these points yields the green dashed line, with its angle relative to the X-axis denoted by . The theoretical deflection angle , calculated from Eq. (12), corresponds to the angle between the X-axis and the line connecting the first and last sampled points. Because reflects only the trajectory defined by the start and end points, whereas accounts for the overall sample distribution, the residual phase error is generally nonzero. In practical applications, is first extracted from the measured data and then corrected by subtracting (see Sect. 3.2), making it necessary to evaluate whether the resulting can be neglected in the analysis.

Fig. 6Full-screen View

Comparison of the fitted deflection angle and the theoretical deflection angle caused by cavity detuning. The blue dots represent the discretized I/Q-plane trajectory of the step response V_cb(t), calculated from Eq. (6)

Figure 7 presents 2D contour plots of the theoretical deflection angle (a) and the residual phase error (b) as functions of the detuning angle and the observation time , where and . Although and are derived differently, Fig. 7b shows that remains negligibly small across the entire scan range. This confirms that in practical applications, when the fitting interval is limited to , the phasor V_b (or I_b) can be accurately recovered by rotating the fitted trajectory by . Because can be directly calculated from Eq. (12), this method enables the effective compensation of detuning-induced phase errors in beam phase measurements.

Fig. 7Full-screen View

(Color online) Theoretical phase deflection angle and residual phase error versus detuning angle and calibration window . (a) ; (b) , representing the difference between the two angles, as illustrated in Fig. 6. The dashed line indicates

3.2

Feasibility of Closed-Loop Implementation

Having established the theoretical framework for detuning compensation, we now turn to its practical application under closed-loop conditions, which are essential for safe and stable beam delivery in high-current proton linacs, such as the ESS.

Accordingly, we exploit the LLRF loop delay window , during which the cavity field responds only to the beam and not to the feedback correction. As shown in Fig. 8a and Fig. 8b, the beam enters the cavity at approximately 151.6 μs, and the feedback system begins adjusting V_f at about 153 μs. Within this short interval ( μs), the system behaves equivalently to an open-loop operation, enabling an accurate measurement of the beam-induced transient. The close agreement between the open-loop and closed-loop trajectories in Fig. 8c demonstrates the feasibility of beam phase calibration in this mode.

Fig. 8Full-screen View

(Color online) Measured beam-induced RF transient in the ESS buncher cavity B2 under open- and closed-loop mode (T_ps=5 μs, I_b0=55 mA, ). (a) Time evolution of the cavity voltage V_c for both modes and forward voltage V_f in the closed-loop mode. (b) Zoom-in view of the beam-induced RF transient. The beam arrives at approximately 151.6 μs, and the closed-loop system starts responding at around 153 μs due to a loop delay of μs. During this delay window, the amplitude and phase of V_cb were nearly identical in both the open- and closed-loop cases. (c) Corresponding trajectories of V_cb in I/Q plane. The trajectories within the initial window show excellent agreement in the open- mode and closed-loop cases

Figure 9 illustrates the calibration process under closed-loop and detuned conditions. The beam phase is calculated as follows:

Fig. 9Full-screen View

Beam phase calibration procedure in the closed-loop mode with cavity detuning. Blue dots show the trajectory of V_cb in the I/Q plane within the loop delay ; the green dashed line is the linear fit, giving . After compensating for detuning-induced deflection , the beam phase is calculated via

Normalize the steady-state cavity voltage V_c prior to beam arrival to , which serves as the reference point in the I/Q plane.

Extract the transient response V_cb(t) within the loop delay window , and perform linear fitting to obtain the fitted deflection angle . Here, is extracted from the measured samples of V_cb at an arbitrary beam phase. For reference, Figs. 6 and 7 present obtained from the simulated samples for the special case in which the beam-induced vector is aligned with the X-axis. Because this case is physically equivalent to the general situation, we retain the same notation throughout this paper.

Compute the cavity detuning angle and corresponding frequency detuning at the beam arrival time (assuming detuning remains constant during the beam pulse): (13)Here, V_c and V_f are the calibrated cavity and forward voltages, respectively, obtained by applying complex scaling factors derived from a linear regression calibration [24, 25]. Then, the theoretical deflection angle at is calculated using Eq. (12):

Finally, compute the beam phase using: (14)

Experimental Validation

The experimental validation of the detuning-compensated beam phase calibration algorithm, including its closed-loop feasibility, was conducted using the ESS MEBT normal-conducting buncher cavity (B2, Fig. 1). The beam pulse width was fixed at T_ps=5 μs, with the beam current and phase adjusted as needed. BCTs and a BPM (in Q5, Fig. 1) independently measures the beam current and phase. RF pulse waveforms (Fig. 3, 8) and key RF parameters (Table 2) characterize the setup.

Figure 10 compares the measured beam-induced RF transients V_cb with the simulated results under various detuning angles in the open-loop mode. The measurements, shown as filled circles, aligned well with the simulated waveforms computed from Eq. (11), confirming the validity of the transient-response model. With validated predictions, we evaluated the performance of the detuning compensation algorithm.

Fig. 10Full-screen View

(Color online) Comparison of measured and simulated beam-induced RF transient V_cb in cavity B2 under open-loop operation for different detuning angles . (a) Time-domain evolution of the amplitude and phase. (b) Corresponding trajectories of V_cb in the I/Q plane. The simulation curves (solid lines) were calculated using Eq. (11) and the experimental results are plotted as discrete dot markers

Figure 11 shows the open-loop beam phase measurements (I_b0=55 mA) before and after the detuning correction. Subfigures (a) and (b) use μs () and 3μs. Individual (small markers) and average (larger filled circles/squares) measurements are shown. Applying the correction in Sect. III.B makes the measured beam phase insensitive to detuning, thereby confirming its effectiveness. Without correction, detuning angles of (corresponding to ) result in phase errors of approximately 9° at μs and 18° at μs (approximately ). These results are consistent with the theoretical predictions shown in Fig. 7a.

Fig. 11Full-screen View

(Color online) Comparison of beam phase measurements before and after detuning correction under open-loop operation (I_b0=55 mA). (a) and (b) correspond to the calibration window T_w of approximately 1.4 μs () and 3 μs, respectively. Small markers represent individual measurement results, whereas larger filled circles and squares denote the averaged values over multiple measurements

To validate the method under realistic accelerator conditions, closed-loop beam phase measurements were conducted. Figure 12 shows the BCT and BPM measurements of the beam current (top) and phase (bottom). Figures 12a and b show low- (I_b0=5 mA) and high-current (I_b0=60 mA) cases. Notably, at a higher current, intra-pulse phase fluctuations of approximately 2° are evident, and their impact on measurement precision is examined later.

Fig. 12Full-screen View

Measured beam current (top) and beam phase (bottom) using the BCM and BPM systems. Subfigures (a) and (b) correspond to beam currents I_b0=5 mA and 60 mA, respectively

Figure 13 compares the trajectories of the beam-induced transient signal V_cb in the I/Q plane under open-loop and closed-loop conditions. For the 5 mA case (Fig. 13a), the signal is relatively weak and more susceptible to noise, leading to reduced accuracy in phase determination. For the 60 mA case (Fig. 13b), although the feedback introduces a slight curvature to the trajectory, the open-loop and closed-loop traces remain nearly identical within the loop delay window (approximately 13 samples, including the initial sampling point), confirming the closed-loop feasibility.

Fig. 13Full-screen View

(Color online) Trajectories of the beam-induced RF transient V_cb in the I/Q plane under open-loop (red) and closed-loop (blue) operations. (a) I_b0=5 mA, (b) I_b0=60 mA. The four trajectories in each subplot correspond approximately to beam phases of 1°, -18°, -44°, and -89°

The impact of the calibration window length T_w was further evaluated for 5 mA and 60 mA beam currents in the open- and closed-loop modes. Figure 14 summarizes the results. Figures 14(a) and (b) show probability density histograms of the measured beam phase for various T_w, with upper plots for open-loop and lower for closed-loop. Different colors represent different T_w.

Fig. 14Full-screen View

(Color online) (a, b) Histograms of beam phase measurements obtained under different calibration window lengths T_w, for open-loop and closed-loop operations, respectively. The beam current and phase settings used in (a) and (b) correspond to those shown in Fig. 12a and Fig. 12b. (c, d) Mean values (top) and RMS errors (bottom) of beam phase measurements versus calibration window T_w. In (d), under open-loop operation with I_b0=60 mA, the mean value results suggest that the beam phase reaches steady state at approximately 3.5 μs. The vertical dashed lines indicate the results obtained at the loop delay μs

At 5 mA (Fig. 14a), under open-loop operation, increasing the calibration window T_w leads to a narrower beam phase distribution, corresponding to a lower root-mean-square (RMS) error. The histogram peak stabilizes at approximately -85.5°, which is consistent with the theoretical prediction shown in Fig. 12a. After switching to the closed-loop operation, the measurements remained in good agreement with the open-loop results when . However, deviations appear for because the feedback response begins to affect V_cb (see Fig. 8c).

At 60 mA (Fig. 14b), under open-loop operation, the phase distributions are more concentrated due to improved signal-to-noise ratio. Nevertheless, intrapulse beam phase fluctuations (as observed in Fig. 12b) cause the histogram peak to shift with increasing T_w, for example, from approximately -42.5°at T_w=1 μs to -45° at T_w = 4 μs. Under closed-loop operation, a similar trend is observed: the results match the open-loop reference when , but deviate for as the feedback response begins to affect V_cb.

Figures 14c and 14d present the average (top) and RMS error (bottom) of the measured beam phase versus T_w for the 5 mA and 60 mA cases. For 5 mA (Fig. 14c), the average phase remains stable across different under open-loop conditions. Under closed-loop operation, it matches the open-loop reference when , but deviates for larger T_w. The RMS error decreased monotonically with increasing T_w, indicating that a larger calibration window improved the robustness of the phase measurement. For 60 mA (Fig. 14d), although the overall measurement precision is higher, intra-pulse phase fluctuations still introduce deviations in both open- and closed-loop results, as previously discussed.

Figure 15 compares the calibrated beam phase (blue), obtained from V_cb fit under closed-loop operation, with BPM5 measurements (green) across different beam currents. The BPM data are obtained by averaging the phase signal over the 2.5–5 μs interval (see Fig. 12). BPM measurements revealed beam phase differences at various currents. This may be attributed to the defocusing effects of the buncher cavity, where beams of different intensities pass through the cavity along slightly different orbits, thereby causing variations in the BPM readings. As shown in Fig. 15, at low beam currents, BPM5 measurements match well with the calibrated beam phase from the V_cb fit under closed-loop operation. However, at higher currents, increasing deviations (approximately 1.5°) are observed owing to beam phase fluctuations during the pulse duration (see Fig. 12b). It should be noted that BPM-based measurements are generally considered more accurate because they average the beam phase over the entire pulse, particularly during the steady-state portion. In this study, a dedicated phase-scan calibration was carefully performed a few days prior to the experiment, which minimized the impact of possible ambient drifts on the BPM results.

Fig. 15Full-screen View

(Color online) Comparison of the calibrated beam phase (blue) and BPM measurements (green). Small dots represent individual measurements, and larger markers indicate the average over multiple shots. The BPM phase values are calculated by averaging the beam phase signal over the 2.5–5 μs interval (see Fig. 12)

Figure 16 presents the closed-loop beam phase measurement results with a calibration window of , at a beam current of I_b0=60 mA. Subfigures (a), (b), and (c) correspond to the different beam phase settings. Before calibration (blue), the measured beam phase exhibited a clear linear dependence on the cavity detuning. After applying the correction method described in Sect. III.B (red), this dependence was effectively eliminated, and the beam phase remained stable across the full detuning range. These results provide strong experimental validation for the proposed detuning compensation algorithm.

Fig. 16Full-screen View

(Color online) Comparison of beam phase measurements under closed-loop operation before and after detuning compensation at a beam current of I_b0=60 mA. Subfigures (a), (b), and (c) correspond to different beam phase settings. Small dots represent individual measurements, whereas larger solid markers (circles and squares) denote the averaged results before and after correction, respectively

In summary, the experimental data validate the effectiveness of the proposed method in compensating for detuning effects and achieving accurate beam phase measurements under realistic accelerator conditions. This approach enables non-invasive, online beam phase calibration under fully closed-loop operation, even in high-current normal-conducting cavities, such as those at the ESS. By avoiding disruption of routine operations, this method provides a practical and scalable solution for real-time diagnostics, routine calibration, and early fault detection in modern proton linacs [23].

Conclusions and Future Work

In this study, we systematically investigated a beam phase measurement method based on transient beam loading and quantitatively analyzed the influence of cavity detuning on the measurement results for both superconducting and normal-conducting RF cavities. A theoretical expression for the detuning-induced phase deflection angle was derived, and a compensation algorithm was proposed. The feasibility of implementing this method was experimentally validated using the buncher cavity (B2) in the MEBT section of the ESS accelerator. The effectiveness of the proposed algorithm was demonstrated by comparing the open-loop and closed-loop measurements under various detuning scenarios.

The results confirm that within the LLRF loop delay window (), the transient vector V_cb evolves similarly under both open- and closed-loop modes, thereby enabling a reliable beam phase determination. Nonetheless, the limited length of restricts the number of samples available for fitting, and low-current conditions reduce the phase resolution owing to the lower signal-to-noise ratios. Under higher beam currents, intrapulse phase fluctuations introduce systematic offsets compared with BPM-based measurements. Despite these limitations, closed-loop calibration offers enhanced operational stability and is better suited for continuous beam phase monitoring in linacs.

When both detuning and feedback are present, the trajectory of V_cb in the I/Q plane becomes more complex, but still follows identifiable patterns. Future research may leverage artificial intelligence (AI) techniques to overcome the current limitation, thereby enabling accurate phase recovery even beyond the initial transient regime. Preliminary investigations in this direction have shown promising results [16].