A draft of the longitudinal beam injection method is shown in
High Power Laser and Particle Beams, Volume. 31, Issue 12, 125101(2019)
Nonlinear optimization for longitudinal beam injection in diffraction-limited synchrotron light sources
Storage rings of the next generation synchrotron light sources have quite small dynamic apertures with which transverse beam injection can hardly be efficient. The longitudinal beam injection may be a solution to this problem. To apply a longer kicker pulse, it is necessary to increase time offset of the injected beam to the stored one by reducing RF frequency. The beam with a longer time offset will have a higher momentum deviation due to synchrotron motion, thus full injection of this method requires the storage ring to provide large enough energy acceptance and off-momentum dynamic aperture. A candidate lattice of the upgraded Shanghai Synchrotron Radiation Facility (SSRF-U) was used to nonlinearly optimize the longitudinal beam injection. With the optimal results of a series of RF frequencies, it is found that there is a critical RF frequency below which lowering frequency could not help to lengthen the kicker pulse in a given lattice. The beam injection into the SSRF-U storage ring was simulated and reached high efficiency with its critical RF frequency and optimal sextupole gradients.
Figure 1.Longitudinal beam injection shown in the synchrotron phase space
A draft of the longitudinal beam injection method is shown in
Synchrotron radiation light sources, serving a large number of users for scientific experiments, have been developed with three generations over the past fifty years. Electron beam emittance was reduced from hundreds of nanometer-radians to sub nanometer-radians, which was pushed by users’ increasing requirements of photon brightness and coherence. With the progress of high-gradient magnet and high-precise alignment, a medium-sized light source employing the Multi-Bend Achromatic (MBA) lattice can obtain very low beam emittance down to the X-ray diffraction limit [
The Shanghai Synchrotron Radiation Facility (SSRF)[
where c is the speed of light, τ is the time offset, fRF is the frequency of the Radio Frequency (RF) system, and
1 Lattice of the SSRF-U storage ring
As new light sources with lower beam emittance and higher photon brightness have been successfully operated and many more light sources are conducting their research and development to the diffraction limit[
Ref.[
|
Figure 2.The beam optics in one fold of the SSRF-U storage ring
The RF voltage will modify the synchrotron phase space, and thus influence the longitudinal beam injection. A given momentum deviation usually corresponds longer time offset if the RF voltage is reduced. So it is better to use low voltage for lengthening the kicker pulse. However, the voltage cannot be too low to compensate the radiation loss from the dipoles and all the possible insertion devices. It was fixed by a tradeoff to be 1.5 MV in the SSRF-U storage ring.
2 Nonlinear optimization
The SSRF-U storage ring has six harmonic sextupoles in each cell to cancel high order aberrations. All the harmonic sextupoles were classified into nine families based on the periodicity of beam optics. Nonlinearity of the storage ring was optimized by these harmonic sextupoles, meanwhile the chromaticities in both transverse planes were always corrected to two by the sextupoles in the arc sections. Many nonlinear optimization methods are effective in lattice design and machine operation[
The energy acceptance or the full height of the bucket that always obtained a finite DA was the first penalty function. The size of the finite DA was determined to be around five times of the injected beam size, which is expected to capture most of all the particles in the injected beam. In this way, the energy acceptance is the one available for full beam injection. The following expression Eq. (2) with the beam parameter of the booster and the beam optics at the injection point in the storage ring can calculate the injected beam size, when the electron transport line matches well with the storage ring.
where
The horizontal and vertical tunes as a function of the momentum were then calculated within this energy acceptance. The larger tune variation amplitude in the two transverse planes was the second penalty function, which was desired to be small enough to prevent crossing the linear resonances. However, the smallest one is not the best. It is more suitable to find a distribution of the maximum energy acceptance within a range of the tune variation amplitude. Because of a large amount of the population there were too many different tune variation amplitudes in every generation, which created redundancy in parent choices in the optimization process. So within every slice of the tune variation amplitude with the width of 0.01, the sextupole solution with the maximum energy acceptance was selected as the parent solution for the next generation. To produce new population, the nine harmonic sextupole gradients of the parent solutions varied randomly within a range that decreased generation by generation. The particle tracking and the solution selection were then done again. When there was not any better solution, the optimization process ended.
Figure 3.The available energy acceptance as a function of the amplitude of the tune variation with momentum
The energy acceptance significantly increases generation by generation, and eventually reaches a maximum value of about 13% (from −7% to 6%) within the tune variation amplitude from 0.2 to 0.5. Unfortunately, it is impossible to get an energy acceptance of about 20%, with which the lattice provides the longest time offset by taking full advantage of the RF frequency of 100 MHz. The same optimization processes were repeated with different RF frequencies. The results are very similar with
3 RF frequency choice
The lattices with 12 different RF frequencies (from 100 MHz to 500 MHz) were optimized by the method described above. Their optimal harmonic sextupole gradients were obtained as well as the maximum energy acceptances. The maximum time offsets (distances) of the fully injected beam to the stored beam were determined by a scanning process in the synchrotron phase space (an example can be found in the next section). For every RF frequency, the time offset and the energy deviation of the center of the injected beam were scanned within an appropriate range. The injection efficiency on every scanning grid was obtained by 3000-turn 6-D tracking of 2000 particles normally distributed in the injected beam. The maximum time offset (distance) with the injection efficiency higher than 90% was found out for every RF frequency as shown in
Figure 4.The maximum distance of the fully injected beam to the stored beam as a function of the RF frequency
The distance increases when RF frequency is reduced from 500 MHz to 250 MHz, because the energy acceptance is dominated by the synchrotron motion and less than the effect of the nonlinear optimization. The distance doesn’t increase any more with RF frequency less than 250 MHz, as the energy acceptance has been limited by insufficient off-momentum DA. The synchrotron phase space will be distorted by radiation damping to show a golf-club shape (the black line in
4 An optimal solution for SSRF-U
The SSRF-U lattice performances with its critical RF frequency and optimal harmonic sextupole gradients were simulated.
Figure 5.Fractional tunes as functions of momentum deviation (top) and on- and off-momentum DAs (bottom)
Figure 6.Injection efficiency as a function of the center position of the injected beam in the synchrotron phase space (shown as contour maps)
5 Conclusion
A candidate lattice of the SSRF-U storage ring with the beam emittance of 97 pm·rad has approached the soft X-ray diffraction limit. A nonlinear optimization method was applied in this lattice. The optimal results showed that this method can significantly increase the energy acceptance available for full longitudinal beam injection. However, the energy acceptance of about 13% cannot permit lengthening the kicker pulse to 10 ns with the RF frequency of 100 MHz. The lattices with a series of RF frequencies were individually optimized and could not get any higher energy acceptance. The time offset of the fully injected beam to the stored beam was determined by beam injection simulation for every tested RF frequency. With these results, the critical RF frequency reaching the maximum time offset was found to be 250 MHz in SSRF-U. The lattice with the critical RF frequency and the corresponding harmonic sextupole gradients takes full advantage of the reduced RF frequency and makes the kicker pulse lengthen doubly. Full injection can be realized in the SSRF-U storage ring when the injected beam is in a large region of the synchrotron phase space. It is necessary to note that the performances of the lattice and the volume of the injected beam dominate the critical RF frequency and the maximum time offset. Their specific values may be variable in different light sources.
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Siqi Shen, Shunqiang Tian, Qinglei Zhang, Xu Wu, Zhentang Zhao. Nonlinear optimization for longitudinal beam injection in diffraction-limited synchrotron light sources[J]. High Power Laser and Particle Beams, 2019, 31(12): 125101
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Received: Jun. 3, 2019
Accepted: --
Published Online: Mar. 30, 2020
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