Bunch-by-bunch transverse position measurement during injection

Bunch-by-bunch systems have been developed at the Taiwan Light Source and the Taiwan Photon Source to monitor transverse position and filling patterns. The system comprises four channels with a 500 MHz sampling rate synchronized with the accelerator’s radio frequency. It is utilized to diagnose injection transients due to kick mismatch, beam oscillations resulting from damped betatron oscillations and wakefield, and monitor the injection current of each bunch.


Introduction
The Taiwan Photon Source (TPS), which has been in operation since 2016, is a 3-GeV light source with a storage ring circumference of 518.4 m at the NSRRC [1].The storage ring's harmonic number is 864.In contrast, the Taiwan Light Source (TLS) is a 1.5-GeV light source that has been in operation since 1993.Its circumference is 120 m, with a harmonic number of 200 [2].Four kickers are used in the injection scheme, making the beam perturbation unavoidable during injection [3,4].In a previous study [3], turn-by-turn (TByT) data were used to analyze the injection transient.However, the bunch oscillation is not in phase during injection, and TByT data cannot precisely explain the behavior of the beam.Thus, the bunch-by-bunch (BByB) system [5,6] is necessary to set up.
The numerical analysis of fundamental frequencies (NAFF) [7] is a method to obtain precise frequencies within a few samples and has been successfully applied in chaotic systems and accelerator physics [8,9].In this paper, we first use NAFF to calculate the beam motion tune from turn-by-turn (TByT) data, and study the tune evolution during the damping process of betatron oscillation.Second, we analyze the correlation between bunch oscillation and tune variation.Finally, we present the process to obtain the filling pattern and injection current of each bunch in storage, which can help understand the injection efficiencies of all or individual bunches.

Tune variation during injection
At the TLS, approximately 150 bunches are filled in the storage ring.The length of the electron beam emanating from the linear accelerator is around 50 nsec, and the injection step is 4 bunches.During identical injections, the measured tune by various beam position monitors (BPMs) is the same, while the amplitude and phase vary with betatron function and phase advance.However, when the beam is injected into different bucket addresses, the timing of the kickers and septum must be changed accordingly, which causes the beam in the storage ring to be perturbed and related to the injected bunch address.Hence, the injection horizontal and vertical tunes from TByT data vary with the injected bucket address, as shown in Fig. 1.For the tune variation during the damping oscillation, there are several frequencies for the horizontal oscillation and the frequency varies with the revolution turns after injection, as shown in Fig. 2. Because there is more than one frequency in horizontal beam motion during the betatron damping, the chromaticity decoherence [10] can be observed during the time evolution, shown in Fig. 3(a).For the vertical beam motion during injection, only one frequency mainly existed so the betatron beam motion damps exponentially in Fig. 3(b).
At the TPS, ~630 multi bunches and one isolated bunch are stored in the storage ring.Because the electron length from the linac can be more than 660 nsec, the stored beam is injected in three steps.One is from bunch #42 to 364, one is from #365 to 687 and the other is to the isolated bunch as shown in Fig. 7.The variation of horizontal tune is also related to injected bucket address.Because the coupling between the horizontal and vertical beam is much smaller and the vertical betatron oscillation is damped by the BByB feedback system, the vertical injection tune is hard to observe in routine operation.

Bunch-by-bunch position measuring system
To measure the transverse motion of each bunch during injection, a BByB system was installed at both TLS and TPS facilities, utilizing the high-speed data acquisition system Libera Digit 500.Due to the non-in-phase of beam motion in each bunch during injection, the system provides crucial information to analyze tune variation originating from the behavior of individual bunches.With a sampling rate of approximately 500 MHz, synchronized with the radio frequency (RF) clock, and a bandwidth of around 2 GHz, the system is capable of accurately capturing the beam motion.The sampling phase of each channel can be fine-tuned and is typically set to the peak of the positive phase to accurately measure the transverse motion of each bunch.The position (x, y) of the beam motion in each bunch is calculated by the counts of four channels (Sa, Sb, Sc, and Sd) using where kx and ky are constants decided by the structure of BPM.The results show in Fig. 4.  The amplitude and damping behavior of beam motion vary across different bunches.When considering the main tune (~0.325), the tune varies with the amplitude of beam motion due to amplitudedependent shift [11], as illustrated in Fig. 5.The maximum root-mean-square amplitude of each bunch corresponds to its injected bucket address, resulting in variation of the tune with respect to the injected bucket address.The tune variation can be grouped into three main categories, centered around 0.300, 0.325, and 0.305, as shown in Fig. 6.The chromaticity decoherence is likely attributed from the betatron beating of these three frequencies.

Filling pattern and injection current pattern
Additionally, this system has the capability to monitor the filling pattern of the storage and booster ring [12] by utilizing the sum (Sk) of the aforementioned four channels, as depicted in Fig. 7.The beam current of the isolated bunch can be calculated by where Ii and Id represent the beam currents of the single bunch and total current measured by a direct current current transformer, respectively; Si and Sk denote the signal values of the isolated bunch and each bunch.The filling pattern monitoring system can also be utilized to determine the injection current pattern by subtracting the beam current before and after injection, as depicted in Fig. 8.The obtained injection current pattern enables the calculation of injection efficiency and fine-tuning of the timing of the four kickers to achieve maximum injection efficiency.

Conclusions
The bunch-by-bunch monitoring system was developed at the Taiwan Light Source and the Taiwan Photon Source to investigate the transverse beam motion during injection.In addition to this, it is capable of monitoring the filling pattern of the storage ring and booster ring, as well as the injection current of individual bunches.
During the injection process at the Taiwan Light Source, three main horizontal tunes are excited, resulting in observable decoherence.As the amplitude of beam motion increases, the horizontal tune decreases due to the amplitude-dependent shift, leading to variations in the injection tune with the injected bucket address.In contrast, only one vertical tune is excited during injection, leading to exponential damping of the betatron beam motion.At the Taiwan Photon Source, the horizontal tune also varies with the injected bucket address, but the low coupling makes it difficult to observe the vertical tune during injection.

Figure 1 .
Figure 1.Relationship between the injected bucket address (black dashed line) and the variation of (a) horizontal tune (red line) and (b) vertical tune (blue line).

Figure 2 .Figure 3 .
Figure 2. The tune variation during damping as the beam injects to (a) bucket address #20 and (b) bucket address #160.

Figure 4 .
Figure 4. Turn-by-turn and bunch-by-bunch beam motion of bunch #50, #100, and #150 when the beam is injected into the 20th bucket address.To avoid overlapping, the mean values of the beam motion curves are shifted accordingly.

Figure 7 .
Figure 7.The filling pattern of the storage ring at the Taiwan Photon Source.

Figure 8 .
Figure 8.The bunch current in the booster ring after ramping and the injection bunch current to the storage ring.