Determinants of bunch length

The bunch length in a storage ring can be determined using electron-optical machine parameters. If the beam energy spread follows a Gaussian distribution, the root mean square (RMS) bunch length can be given by ${\sigma }_{{\text{Δ}}_{\text{s}}}=\frac{c{\eta }_{\text{c}}}{\Omega }\frac{{\sigma }_{\text{p}}}{p_0}=\sqrt{\frac{c^3}{2\pi q}\frac{p_0{\beta }_0{\eta }_{\text{c}}}{h{f}_0^{2}V\cos \left({\varphi }_{\text{s}}\right)}}\frac{{\sigma }_{\text{p}}}{p_0}\mathrm{,}$ (1) where $\frac{{\sigma }_{\text{p}}}{p_0}$ is the relative momentum spread of the beam, p₀ is the equilibrium particle momentum, η_c is the momentum compaction factor, f₀ is the particle revolution frequency, V is the equivalent longitudinal electrical field (RF field, harmonic cavity field, and wakefield), and ϕ_s is the synchronous phase. The longitudinal distribution of the bunch is determined using the equivalent longitudinal electrical field, energy spread, acceleration phase, and momentum compaction factor. The bunch length is a crucial beam parameter that reflects whether the electron storage ring meets the design specifications or is in a good operational condition. Accurate measurement of the bunch length is essential for stable operation of the accelerator and acceptance testing of newly constructed facilities.

In the storage ring, variations in the bunch length arise primarily from changes in the effective longitudinal electrical field, V, and acceleration phase, ϕ_s. For example, longitudinal dipole oscillations can occur when longitudinal instabilities occur, resulting in changes in ϕ_s. When fresh bunches are injected, the initial phase of the injected bunches deviates from the equilibrium. Changes in V and ϕ_s can occur because of tuning of the RF system or transient effects from the beam wakefields. Precise quantitative analysis of these phenomena can be achieved by capturing and measuring the bunch-length evolution on a turn-by-turn basis. This provides a powerful tool for nonlinear beam dynamics research. Therefore, the accurate diagnosis and measurement of the bunch length on a bunch-by-bunch and turn-by-turn basis are crucial for storage rings.

Assuming that the bunch length is only related to V and ϕ_s, Eq. (1) can be written as ${\sigma }_{{\text{Δ}}_{\text{s}}}\left(V\mathrm{,}{\varphi }_{\text{s}}\right)=a/\sqrt{V\cos \left({\varphi }_{\text{s}}\right)}\mathrm{,}$ (2) where $a=\sqrt{\frac{c^3}{2\pi q}\frac{p_0{\beta }_0{\eta }_{\text{c}}}{h{f}_0^2}}\frac{{\sigma }_{\text{p}}}{p_0}$ denotes a constant for each machine. Taking the partial derivatives of Eq. (2), we obtain: $\frac{\partial {\sigma }_{{\text{Δ}}_{\text{s}}}\left(V\mathrm{,}{\varphi }_{\text{s}}\right)}{\partial V}=-\frac{{\sigma }_{{\text{Δ}}_{\text{s}}}}{2V}\mathrm{,}$ (3) $\frac{\partial {\sigma }_{{\text{Δ}}_{\text{s}}}\left(V\mathrm{,}{\varphi }_{\text{s}}\right)}{\partial {\varphi }_{\text{s}}}=\frac{{\sigma }_{{\text{Δ}}_{\text{s}}}}{2\cot \left({\varphi }_{\text{s}}\right)}\mathrm{.}$ (4) Thus, when the RF cavity voltage and bunch phase change within a small range, the change in bunch length Δ σ is linearly related to the change in the cavity voltage Δ V and phase Δ ϕ_s.

Bunch-length measurement principle

For an ideal Gaussian distribution, the beam signal in the time domain is given by $I\left(t\right)=\frac{Q}{\sqrt{2\pi }\sigma }\exp \left[-\frac{{\left(t-t_0\right)}^2}{2{\sigma }^2}\right]\mathrm{,}$ (5) where Q denotes the bunch charge, σ is the bunch length, and t₀ is the bunch phase. The principle of beam-signal acquisition in a storage ring is illustrated in Fig. 1a and 1b. When the beam passes through the button electrode, the detected signal can be expressed as $V_{\text{b}}\left(t\right)=\frac{\pi a^2}{2\pi b}\frac{1}{\beta c}Z\frac{t-t_0}{{\sigma }^2}I\left(t\right)F(\delta \mathrm{,}\theta )\mathrm{.}$ (6)

Fig. 1Full-screen View

(Color online) (a) Transverse cross-section of the storage ring beam pipe with four-channel button electrode. (b) Equivalent circuit for signal pickup in a BPM. (c) Ideal longitudinal beam charge distribution and waveform of the BPM acquisition signal. (d) Ideal frequency domain distribution of the BPM signal for bunch lengths ranging from 10 ps to 100 ps

In Eq. (6), a and b represent the physical dimensions of the button electrode, F(δ,θ) represents the transverse position of the bunch, and Z represents the transfer impedance of the button electrode. The beam distribution and the detected signal are shown in Fig. 1c, with the characteristics of a Gaussian distribution and bipolar pulse, respectively.

In the time domain, the detected signal (voltage signal) can be expressed as the convolution of the beam signal (current signal) and the detection system impulse response: $V_{\text{b}}\left(t\right)=I\left(t\right)*H_{\text{Button}}\mathrm{,}$ (7) where H_Button is the time-domain response function. In the frequency domain, we have $V\left(f\right)=I\left(f\right)\cdot R_{\text{Button}}\left(f\right)=Q\cdot \exp \left(-\frac{{\omega }^2}{2{\sigma }_{\text{f}}^2}\right)\cdot R_{\text{Button}}\left(f\right)\mathrm{,}$ (8) where $V\left(f\right)$ is the frequency-domain distribution of the detected signal, $I\left(f\right)$ is the frequency-domain distribution of the beam signal (Gaussian distribution), $R_{\text{Button}}\left(f\right)$ is the frequency-domain transfer impedance, and σ_f is the reciprocal of the time-domain bunch length. Therefore, by measuring σ_f, the bunch length can be determined.

Experimental platform

Beam experiments were conducted at HLS and SSRF. The key parameters of the two facilities are listed in Table 1.

Measurement system structure

The structure of the bunch-by-bunch measurement system is illustrated in Fig. 2. First, four-electrode signals from the BPM probes were acquired using a high-sampling-rate and high-bandwidth digital oscilloscope. The signals were imported into a Python script for further analysis. In the three-dimensional bunch position calculation module of the script, the four-channel data were concatenated to form a bunch-by-bunch beam signal. The response function was reconstructed, and the longitudinal center position (phase) information of the bunch was extracted in the time domain using the response function vector projection method. The reconstructed response function was resampled to obtain the signal amplitude for each bunch. The transverse position and charge information of the bunch were obtained by applying the difference and ratio algorithms.

Fig. 2Full-screen View

Overall schematic of the bunch-by-bunch measurement system in the storage ring

Bunch-length calculation process

The bunch-length calculation module calibrates the transfer impedance of the measurement system. Figure 3a-e shows the calibration process. The original continuous sampling data were sliced bunch-by-bunch and turn-by-turn. Single-bunch single-turn data were subjected to spectral analysis after baseline subtraction. A streak camera was used to measure the bunch length and calculate the expected signal spectrum distribution. The measured signal spectrum (voltage spectrum) was divided by the expected signal spectrum (current spectrum) to obtain the system transfer function (impedance). Assuming that the machine parameters of the storage ring remain constant, the system transfer impedance is a known quantity and does not require recalibration during subsequent data acquisition and processing. For the bunch-length calculation, all measurement data were sliced and harmonically analyzed, and the background was subtracted to obtain the voltage signal spectrum. Dividing the voltage signal spectrum by the system transfer function yields the current signal spectrum. Gaussian fitting was performed on this spectrum to determine the bunch-length measurement results for each bunch and turn. Figure 3f–i demonstrates the data processing with different charges of 0.18 nC, 1.16 nC, and 2.39 nC, showing distinct differences in spectrum distribution. Gaussian fitting of the spectra enables the determination of the bunch lengths as 65 ps, 83 ps, and 88 ps, respectively.

Fig. 3Full-screen View

(Color online) (a) Waveform of the raw signal from a single bunch in one turn. (b) Background noise in the absence of beam bunches. (c) Spectrum of the original signal, background noise, and the signal with the background noise removed. (d) Comparison of the frequency-domain distribution between the ideal signal and the measured data. (e) Frequency-domain transfer impedance calibrated using a streak camera. (f) Waveform of the raw signal from a single bunch with charges of 0.18 nC, 1.16 nC, and 2.39 nC. (g) Frequency-domain waveform with three different charges. (h) Frequency-domain distributions after calibration with three different charges. (i) Gaussian fitting on the three bunches with different charges, yielding their respective beam lengths ranging from 65 ps to 88 ps

Uncertainty evaluation

At HLS, where significant longitudinal instability occurs, the uncertainty of the bunch-length measurement system was evaluated in the single-bunch mode. By progressively increasing the charge of a single bunch and using principal component analysis (PCA) to separate the bunch oscillation modes during longitudinal instability, the residual measurement data were calculated as the measurement uncertainty of the system (Fig. 4a). In SSRF, where there is no apparent longitudinal instability in the normal operation mode, the random errors of the measurement system were evaluated in the multibunch operation mode. Assuming a constant bunch length for each bunch, the standard deviation of the bunch-length measurement for multiple turns of each bunch was calculated as the measurement uncertainty of the system. The dependence of the bunch-length measurement uncertainty on the bunch charges was analyzed for 500 bunches (Fig. 4b). Both experiments show that the bunch-length measurement uncertainty is inversely proportional to the bunch charge, aligning with expectations.

Fig. 4Full-screen View

(Color online) (a) Uncertainty assessment of bunch-length measurements for different charges in the single-bunch mode at HLS. (b) Multibunch operation mode at SSRF, the charge and bunch-length uncertainty for each individual bunch

Experiment at HLS

In the single-bunch experiment at HLS, the bunch charge varied from 0.2 nC to 2.6 nC. The bunch length was synchronously measured using a streak camera and high-speed oscilloscope in the time–frequency analysis system. The average bunch lengths obtained using both methods are in excellent agreement, as shown in Fig. 5a. Bunch length variation with different bunch currents is because of the longitudinal broad impedance, formulated by [35] ${\left(\frac{{\sigma }_{\text{t}}}{{\sigma }_{\text{t0}}}\right)}^3-\frac{{\sigma }_{\text{t}}}{{\sigma }_{\text{t0}}}=\frac{I_{\text{b}}\alpha }{\sqrt{2\pi }{v}_{\text{s}}^2{\omega }_0^3{\sigma }_{\text{t0}}^{3}E/e}\text{Im}{\left(\frac{Z_{\parallel }}{n}\right)}_{\text{eff}}\mathrm{,}$ (9) where α is the momentum compaction factor, ν_s=ω_s/ω₀ is the synchrotron tune, and σ_t0 is the bunch length at zero intensity. We calculated the longitudinal broadband impedance of HLS to be 7.1 Ω.

Fig. 5Full-screen View

(Color online) (a) HLS single-bunch experiment bunch-length distribution with respect to bunch charge. Time–frequency analysis system results show excellent agreement with streak camera results. (b) Under different charges, the bunch length exhibits noticeable oscillations when the charge exceeds a certain threshold. (c) The spectrum of bunch-length oscillations varies with different charges; the peak of the spectrum shifts to the left as the charge increases. (d) The spectrum of phase oscillations varies with different charges, exhibiting the same oscillation mode as the bunch length and a common oscillation peak

In addition to providing the average bunch length, the time–frequency analysis system captured the evolution of the bunch length on a turn-by-turn basis. As shown in Fig. 5b, the bunch length remained constant when the bunch charge was relatively low (0.18 nC). However, significant longitudinal instability occurred for bunch charges above a certain threshold, resulting in periodic oscillations in the bunch length. A harmonic analysis of the turn-by-turn bunch length variations (Fig. 5c) revealed that the oscillation frequency decreased as the bunch charge increased. Similarly, the harmonic analysis of the turn-by-turn phase variations (Fig. 5d) shows oscillation modes in complete synchronization with bunch-length oscillations.

Owing to the use of a room-temperature copper cavity as the main RF cavity at HLS, the higher-order mode (HOM) issues are quite severe. Additionally, there are numerous discontinuous beamline components in the storage ring, such as stripline beam position monitors (SBPM), beam scrubbing electrodes, and vacuum insertions, which result in a relatively high impedance and make the occurrence of longitudinal instabilities more likely. To investigate this, a typical dataset was recorded using the measurement system in normal operation mode at HLS (total current of 400 mA with 35 continuously filled bunches). Figure 6a–c, show the longitudinal center position and bunch length of the head, middle, and tail bunches, respectively, as a function of time. Figure 6d–f presents the corresponding reconstructed distributions. It is evident that both oscillations in the bunch center positions and bunch lengths are significant, and their frequencies are in agreement, as expected. Additionally, the amplitudes of the oscillations were larger at the head and tail of the bunch train than in the middle (Fig. 6).

Fig. 6Full-screen View

(Color online) (a)(b)(c) Oscillations of the longitudinal center position (phase) and bunch length for the head, middle, and tail bunches, respectively. (d)(e)(f) Reconstructed longitudinal distributions (including phase and bunch length) for the head, middle, and tail bunches, respectively

Experiment at SSRF

At SSRF, the charge of a single bunch was gradually increased from 2 nC to 9.5 nC. The variation in the bunch length with respect to the bunch charge was measured using the time–frequency analysis system with an oscilloscope, as shown in Fig. 7a. The bunch length exhibited localized nearly linear increases with the bunch charge. We also calculated the longitudinal broadband impedance as 0.3 Ω, which is in good agreement with the design value [36].

Fig. 7Full-screen View

(Color online) (a) SSRF single-bunch experiment relationship between beam length and uncertainty with respect to the beam charge. (b) Comparison of beam-length measurement results with and without harmonic cavity during the beam accumulation process in the multibunch at SSRF. (c) Injection mode for the multi-bunch at SSRF. (d) Distribution of the phase and beam length within each individual bunch in the bunch train

During harmonic cavity tuning, we measured the average bunch length as a function of the total beam current. The measurements were conducted in top-up mode with regularly spaced four-bunch trains, each containing 125 bunches. The total beam current varied from 80 mA to 180 mA. The results are shown in Fig. 7b (marked by red diamonds), with a comparison provided for the case of the harmonic cavity set to the non-working state, showing the variation in the average bunch length with the beam current (marked by blue circles). By comparing the measurement results, it is evident that the harmonic cavity exhibited a stretching effect on the bunches. However, the stretching factor was significantly smaller than the design target, indicating room for optimization.

This process was continued by decreasing the detuning frequency of the harmonic cavity and increasing the total beam current to 200 mA (Fig. 7c), owing to beam-loading effects, there were variations in both the bunch phase and bunch length at different positions within the bunch train. The results are presented in Fig. 7d, and are consistent with the simulation results of CEPC [37]. The difference in bunch length between the head and tail of the bunch train, which was less than 1 ps, was distinguishable. Additionally, a negative correlation was observed between the bunch length and phase within the bunch train. Assuming an uncertainty of 1 ps for the turn-by-turn measurements, the random error after averaging 7000 turns was determined to be less than 0.02 ps, which agreed with the measurement results.

During the process of optimizing the harmonic cavity, when there was a mismatch between the main RF cavity and the harmonic cavity, the stored bunches deviated significantly from their equilibrium positions owing to large longitudinal kick forces. This resulted in coherent longitudinal dipole oscillations with a substantial amplitude occurring at a frequency close to the stabilized working frequency (normalized frequency of 0.005) of the main RF system after closed-loop operation. When such instability occurs, the beam signal amplitude exceeds the normal operating state owing to coupling between the longitudinal and transverse motions. These random events can be captured by setting the appropriate signal trigger thresholds. To reduce random measurement errors, we averaged the phase and bunch-length data for all 500 bunches, as shown in Fig. 8a. Analysis of the time–frequency domain characteristics of the oscillation waveform revealed that the frequency of the longitudinal phase oscillation and the bunch length oscillation completely matched, but the oscillation phase was the opposite. The random error in the turn-by-turn bunch-length measurements, obtained through multibunch averaging, could be reduced to less than 0.05 ps, and the sinusoidal bunch-length oscillation waveform with a peak-to-peak amplitude of approximately 1.5 ps could be perfectly reproduced. Analyzing the changes in the bunch length and phase during the occurrence of instability, a strong negative linear relationship between the two was observed, which is consistent with previous theoretical analyses.

Fig. 8Full-screen View

(Color online) (a) Oscillations of beam length and phase when longitudinal instability occurs. (b) Spectrum of the beam length and phase oscillations when longitudinal instability occurs, with both normalized oscillation frequencies at approximately 0.005. (c) Negative correlation between the change in beam length and the change in phase