Preamble

Injection upgrade using a nonlinear kicker for a storage ring based light source Zhe Wang, Kemin Chen, Tao He, Zhonghan Wang, Masahito Hosaka, Gongfa Liu, and Wei Xu 1 NSRL, University of Science and Technology of China, Hefei, Anhui, China A conventional local bump injection system with four pulsed dipole kicker magnets is currently adopted in the Hefei Light Source II storage ring to achieve top-off operation. Owing to the multipole magnets located inside the injection section, it is difficult to form a perfect closed bump during beam injection, which leads to large perturbations. In order to provide the near-transparent beam injection for the light source users, a new injection method using a nonlinear kicker (NLK) is proposed in this paper. The NLK generates magnetic fields with a nonlinear distribution, which provides an off-axis peak field for the injected beam while keeping a field-free region for the on-axis stored beam. To simplify the upgrade, the NLK is going to be installed in the arc section downstream of the current injection system and the original septum will remain. The physics design of the NLK is conducted by optimizing its field conditions, including the on-axis field gradient, the peak-field position and strength. The injection efficiency is maximized by tuning the NLK conductor current to match the injection acceptance with the injected beam. With reasonable error tolerance of the NLK fields, the injection perturbation on the stored beam is analyzed to be greatly reduced compared to the local-bump injection.

Keywords

Storage ring, Injection system, Local bump, Nonlinear kicker

INTRODUCTION

After a major upgrade in 2014, a full-energy linac as the injector and a local bump injection system were adopted to achieve top-off operation for HLS-II [ ]. Local bump in- jection is an off-axis injection method that is widely used for synchrotron light sources, e.g., Diamond [ ], SSRF [ ] and ESRF-EBS [ ]. Several (usually 3 or 4) kickers are used to form a closed local bump at the injection point in a straight section and capture the injected beam within the ring accep- tance. Ideally, a perfect local bump injection should be trans- parent to the stored beam outside the injection region. An im- perfect local bump will cause oscillations to the stored beam, leading to degradation of the synchrotron light source perfor- mance for user experiments. Owing to the multipole magnets located inside the local bump, the existing local bump injec- tion system in the HLS-II storage ring cannot form a perfect local bump during beam injection. This paper presents a new injection scheme using an NLK kicker in the HLS-II storage ring to mitigate the injection perturbation to the stored beam.

To simplify the injection system and achieve transparent in- jection, Photon Factory proposed an injection method with a pulsed quadrupole magnet (PQM) [ ]. Since the dipole mag- netic field is zero on-axis of a multipole magnet, in principle, the off-axis injected beam can be deflected without perturbing the stored beam. However, the PQM quadrupole component on-axis is estimated to disturb the stored beam by increasing its size up to 2.4 times in the PF-AR storage ring. To over- come this disadvantage, the injection with a pulsed sextupole magnet (PSM) was proposed and tested in the PF ring [ Due to the large inductance of the pulsed sextupole, it is difficult to create a fast-pulse power supply for the kicker, re- sulting in multi-turn injection and low injection efficiency for the PF ring. The multi-turn injection using a PSM has also Supported by the National Natural Science Foundation of China (No.11975227) been evaluated for MAX-IV [ ] and UVSOR-III [ ], and the evaluated injection efficiency is much lower than that of the single-turn injection. The HLS-II storage ring has a cir- cumference of only ]. To realized the single-turn injection with a PSM requires the pulse width to be shorter

than 220 ns , which is technically difficult. 39

To minimize the perturbation of the pulsed multipole mag- 40

net (PMM) on the stored beam, BESSY developed the NLK injection method on the basis of the PMM approach [ ]. Un- like the multipole magnet, the nonlinear field distribution of an NLK is achieved using four coils with a mirror symmet- ric geometry. It has a relatively low inductance and a short- pulse power supply can be realized for single-turn injection.

ALS optimized an NLK to inject the beam at the flat-top of the magnetic field and the injection efficiency is improved to nearly 100% while the perturbation on the stored beam is greatly reduced [ ]. This novel injection method has al- ready been successfully applied to several synchrotron radia- tion facilities including MAX-IV [ ] and Sirius [ Besides, more light sources, such as ESRF-EBS [ ], NSLS-

II [ 22 ], HALF [ 23 ] and TPS [ 24 ], are planning to adopt the 54

NLK injection scheme. In this paper, we propose a new injection scheme for the HLS-II storage ring to replace the current local bump injec- tion using four dipole kickers. The NLK is going to be in- stalled downstream of the last dipole kicker and the septum will be reused. The local bump injection system will remain working until the NLK injection is achieved after commis-

sioning in the storage ring, which helps reduce the influence 62

on the user operation time. In Sec. , we present an overview of the HLS-II storage ring and its current local bump injection system. In Sec. the NLK injection scheme for the HLS-II storage ring is de- scribed in detail and the simulation results of the NLK injec- tion are compared and discussed. Finally we conclude the paper in Sec.

LOCAL BUMP INJECTION SYSTEM OF THE HLS-II PARAMETERS OF THE HLS-II STORAGE RING HLS-II is a dedicated synchrotron light source consisting of a full-energy linac injector and a storage ring with Double- Bend Achromat (DBA) structure [ ]. With a recent dy- namic optimization, the main parameters of the storage ring is given in Table ]. The optical function of the storage ring is shown in Fig. and the dynamic aperture is shown in

Parameter Value Beam energy (MeV) 800 Circumference (m) 66.13 Natural emittance (nmrad) 33 Harmonic number 45 RF frequency (MHz) 204 Damping time [H,V,L] (ms) 19.9/21.1/10.8 Transverse tunes [H, V] (4.26, 2.22) Momentum compaction factor 0.0209

Dispersion (m) Using the linear beam dynamics, we can derive the equa- tions for the relationship between the kicker angle and the local bump height without considering the multipole magnet effects (magnetic field feed-down effects), which are listed below [

= 0 = 2% = -2% w/ septum

y (mm) x (mm)

θ 1 = b √ β 2 β 1 sin ∆Ψ 21 ,

cos ∆Ψ sin ∆Ψ sin ∆Ψ

θ 2 = − b β 2

cos ∆Ψ sin ∆Ψ sin ∆Ψ

θ 3 = − b β 3

θ 4 = b √ β 3 β 4 sin ∆Ψ 43 ,

where the subscript numbers indicate the positions of the kickers, is the phase advance between two kickers, and are the Courant-Snyder (C-S) parameters.

Currently, the orbit bump injection system of the HLS-II storage ring adopts four kickers and one septum to achieve top-off operation [ The eddy-current type septum has a septum sheet which is thick. Its maximum excita- tion current is with a peak magnetic field of

875 T

The pulsed bump kickers, which utilize the soft ferrite ma- terial, have a maximum excitation current of with a peak field of

1 T

. The injection system operates at a max- imum repetition rate of

2 Hz

. The HLS-II storage ring has four short straight sections of and four long straights . The septum is located at the end of a long straight section, as shown in Fig. . Considering the space limita- tion of the straight section, two kickers are placed in the same straight section, whereas the other two kickers are located at the nearby arcs. The magnet lattice of the injection system, the injected beam trajectory and the local orbit bump of the stored beam is shown in Fig.

The dynamic acceptance of the storage ring with and with- out orbit bump in the horizontal phase space is shown in wall and the injected beam is outside the acceptance. With the help of the orbit bump, the acceptance is shifted to cover the injection point while the stored beam is moved to the local bump with a height of 24 mm

Trajectory of injected beam Local bump orbit Undisturbed orbit Sextupole Quadrupole Kicker x (mm) s (m) at beam injection.

The first few turns of an injected bunch using the current local bump injection system in the horizontal phase space are shown in Fig. . The parameters of the injected bunch are SEPTUM listed in Table Acceptance at septum Injected beam Injected 2nd turn 3rd turn (mrad) 4th turn 1st turn injected beam x (mm) Parameter Value Horizontal emittance (nm·rad) Energy spread Bunch length ) (m) (-1,9) Injected beam position (mm) Injected beam angle (mrad) The misalignment of the dipole kickers and the power sup- ply jitters can lead to imperfection of the local bump. In the HLS-II storage ring, additional perturbations of the four sex- tupoles located inside the local bump should be considered.

When the stored beam is off-axis in the sextupoles, it sees ad- ditional dipole and quadrupole components of the magnetic

fields, which is called the feed-down effect. The magnetic field feed-down of a sextupole can be expressed as: ) + 2( ) + (

B y Bρ = 1

where is the normalized strength of the sextupole and is the off-axis distance of the particles. The first and sec- ond terms on the right-hand side of Eq. represents the ad- ditional dipole and quadrupole magnetic fields, respectively.

The dipole and quadrupole fields of the sextupole with the maximum bump height of 24 mm are estimated to be

23 T

Injection time 1st turn after injection 1st turn before injection Normalized excitation strength Closed orbit distortion (mm) 2nd turn after injection 2nd turn before injection Time (ns) (black).

Considering the feed-down effect, the formulae in Eq. no longer accurate to calculate the angles of the dipole kick- ers. However, the tracking method can be applied to match the angles. The first kicker is set to bring the ideal particle to the height we need at the second kicker, and the second kicker is to bring the particle’s to zero. The third and fourth kickers are set in the similar way as the first and second kick- ers. The matched kicker angles from simulation are (6.380, 3.698, 3.699, 6.383) with the maximum bump height of 24 mm . The injection pulse of the kickers is half-sine with a width of , which is approximately six times of the rev- olution time, as shown in Fig. . The excitation of the injec- tion kickers is optimized at the peak height of the local bump, where the feed-down effect is considered. According to Eq. the feed-down field of the dipole components is nonlinear to the bump height, which means perfect orbit bumps cannot be formed in the whole excitation process. Therefore, while the excitation is ramping up and down along the sinusoid, global orbit distortion outside the local orbit bump region is gen- erated, which causes oscillations to the stored beam during injection. The rms closed orbit distortion outside the local bump for different excitation strengths is plotted in Fig.

The closed orbits of the stored beam in phase with the in- jected beam are calculated and shown in Fig. . It is obvious that the orbit distortion is leaked to the outside of the injection bump.

Bump height: 1st turn 2nd turn 3rd turn Injection point 1st turn: 24.0 mm 2nd turn: 21.0 mm x (mm) 3rd turn: 12.6 mm s (m) STORED BEAM PERTURBATION To calculate the oscillation amplitude of the stored beam disturbed by the beam injection, six bunches with different timing to the injection pulse are tracked using the MATLAB Accelerator Toolbox (AT) [ ]. In order to accurately calcu- late the beam size, each bunch contains 1000 particles. The position of the particles are recorded turn by turn. The aver- aged beam centroid and beam size of the stored beam after injection is shown in Fig. . As shown in the figure, the max- imum oscillation amplitude of the stored beam centroid after injection is approximately , and the beam size is in- creased to (one sigma). The simulation does not take into account the field leakage of the septum. If considered, the stored beam would experience greater perturbations during

local bump injection. The initial increase in the beam size is 174

due to the quadrupole component of the sextupole feed-down fields. Further increase of the beam size for up to can be explained by the decoherence effect, i.e., the beam centroid

oscillation is transfered into the beam size through initial be- 178

tatron tune spread [ ]. The perturbation on the stored beam continues for a damping time with large amplitudes, which can interfere with the user’s experiments.

The injection perturbation on the stored beam degrades the performance of the synchrotron radiation light source with top-off operation, which should be mitigated to achieve trans-

parent injection. At ESRF, several techniques are applied to 185

reduce the injection perturbation on the stored beam, includ- ing introduction of nonlinear fields in the injection kickers, compensation of vertical and quadrupole perturbations using skew quadrupoles and octupoles, and compensation of kicker perturbations using shakers and a stripline, etc [ ]. At HLS- II, a nonlinear kickers is proposed to replace the current local bump injection system to mitigate the injection perturbation.

Beam centroid osc. (mm) Times (ms) Beam size osc. (mm) Times (ms) 20 ms INJECTION USING AN NLK FOR THE HLS-II STORAGE RING NONLINEAR KICKER INJECTION The injection process using a nonlinear kicker (NLK) is il- lustrated in a normalized horizontal phase space in Fig.

The beam is injected from outside of the septum and moves clockwise in the phase space. To achieve beam injection, the NLK should be located appropriately downstream of the in- jection point, and should provide a kick to bring the injected beam into the ring acceptance. To choose the location of the NLK, the phase advance between the injection point and the NLK should be considered.

HLS-II is a user facility which provides more than 5000 hours per year for user experiments. To reserve enough user operation time, the current injection system will remain until the new injection system is successfully commissioned. Con- sidering the space limitation of the storage ring and the phase advance, the NLK is planned to be placed in a downstream arc, as shown in Fig. . To simplify the upgrade, the original septum will be reused for the new injection system.

The preliminary matching of the injection could be treated

as a two-step process. The initial step is to make the injected 214

beam reach the NLK with an offset, and the subsequent step is to optimize the parameters of the NLK. To ensure that the injected beam has a appropriate offset at the NLK loca- tion, the deflection angle of the septum should be tuned. The magnetic field strength of the NLK can be estimated by the position of the injected beam in phase space. Fig. shows a preliminary match result of the NLK injection. is the phase space between the injection point and the NLK.

Sextupole Quadrupole Kicker Trajectory without NLK The injected beam Undisturbed orbit x (mm) s (m) HLS-II storage ring. The brown line represents the trajectory of the injected beam, and the red line represents the trajectory of the in- jected beam in the absence of the NLK kicker.

PHYSICS DESIGN OF AN NLK KICKER A nonlinear kicker that produces off-axis magnetic field peaks with near-zero center fields can be built using 8 current- driven conductors with mirrored horizontal and vertical sym- metry as shown in Fig. ]. Two conductors with opposite currents occupy each quadrant. On the other hand, the inner four conductors have the same current direction and the outer four reverse their polarity. All conductors share a common power supply.

The Biot-Savart law can be used to calculate the magnetic fields generated by a current , as described in Eq.

B = µ 0 4 π

The magnetic field in the middle horizontal plane with a distance to the axis (shown as point in Fig. ) can then be calculated by

Outer conductors Inner conductors Magnetic field Magnetic field strength

B y =

where is the conductor current and positive current denotes outward flow from the plane. is the permeability of free space, and indicates the number of the conductors. The

infinite-length conductors are assumed in the calculation, and 241

the hard-edge field is adopted in the following analysis.

The dipole kick of the NLK makes the ring acceptance move to cover the injected beam while the stored beam is not affected. Due to the length of the kicker, there is a horizontal drift of the injected beam in the NLK.

To achieve high injection efficiency and eliminate the ef- fect on the stored beam, the NLK magnetic field strength is expected to have a peak value at the appointed position and a field-free region on the axis. The magnetic fields of the NLK

can be optimized by tuning the positions and currents of the 247

conductors [ ]. And the intelligent optimization algorithms

and machine learning methods are widely used in the design 249

and optimization of the particle accelerators [ ]. Here the intelligent optimization algorithm of Multi-Objective Par- ticle Swarm Optimization (MOPSO) is adopted in the physics design of the NLK [ A preliminary injection matching result in the horizontal phase space with the ring acceptance using a dipole kick are shown in Fig. . According to the matching result, the NLK should have a peak magnetic field of at least

220 Gauss

Since the NLK has a length of in the -direction and the injected beam has a drift in the NLK, the location of

the NLK peak field can be set near x = − 11 mm , which 260

is outside the ring acceptance. The position with the peak NLK magnetic field is set as an objective in the optimization.

The other objective is the field flatness in the central region, which determines the influence on the stored beam. Owing to the symmetric configuration of the NLK, the on-axis field strength is zero. Due to the size of the stored beam, the gra- dient of the magnetic field can cause the quadrupole kick on it, which leads to increasement of the beam size. Considering the beam size at the NLK location, the field gradient is calcu- lated within the range of around the axis. The follow- ing simulation uses the real magnetic field strengths for dif- ferent particles according to the field distribution of the NLK.

The conductor current is fixed at

1000 A

and the peak field strength is set as a constrain which filters out the optimization results when it is smaller than

220 Gauss

. To ensure enough vertical space of the kicker vacuum chamber, the inner con- ductor is set to have at least off-axis distance in the -direction. Therefore, the gap of the NLK vacuum cham- ber can reach 12 mm , which is on the same level of the ID chambers at HLS-II.

Final iteration Pareto front Selected point Peak field position (mm) The Pareto front of the optimization of the the peak posi- tion and the field gradient is plotted in Fig. . One solution is selected as the optimization result with the peak-field loca- tion of 11 mm and the field gradient of

5 Gauss

. The optimized NLK and its magnetic field distribution is shown

Outer conductors Inner conductors Magnetic field Magnetic field B (Gauss) Vacuum chamber wall y (mm) Injected beam Stored beam x (mm) in the horizontal plane. The magnetic field is a calculated with the conductor current of

1000 A

, which can be adjusted to modify the field strength and the acceptance area. in Fig. . The main design parameters of the NLK are listed in the Table.

Parameter Value Outer conductor position 4.8, 18.5 Inner conductor position 5.9, 7.2 Peak magnetic field at

1000 A

Gauss Peak field position at The beam is injected on the central horizontal plane where tal one ( ), and the vertical size of the injected beam is jected beam (see table ), the quadrupole component (trans- fect of the horizontal magnetic fields of the NLK on the beam injection can then be ignored.

To determine the optimal NLK strength (or conductor cur- rent) for beam injection, the NLK magnetic field strength is varied to calculate the acceptance of the ring. Here we define two parameters, the acceptance angle and the acceptance area, as shown in Fig. , which represents the effective acceptance for the injected beam. The simulation results of the accep- tance angle and the acceptance area versus the peak magnetic of the NLK increases, the acceptance area also increases, but becomes slender in phase space, which may reduce the injec- tion efficiency. Therefore, the acceptance angle parameter is added to help select the NLK magnetic strength. Considering these two parameters, the magnetic field strength of the NLK is finally chosen to be

250 Gauss

Acceptance area (mm·mrad) Acceptance angle (mrad) Magnetic field (Gauss) As previously described, the pulse duration of the current power supply of the kickers is approximately six times of the revolution time.

We plan a new power supply with a pulse base width shorter than 440 ns , which is used to achieve single-turn injection with an NLK for the HLS-II storage ring.

The linac injector of HLS-II provides injected bunches for the storage ring with single-bunch mode. The bunch length is ), which is very short comparing to the pulse base width of the injection system. Due to the low inductance of the NLK, this power supply is easy to be realized with mod- ern technologies. The ring acceptance calculated by particle tracking before and after the NLK is shown in Fig. tracking the acceptance back to the location of the septum, we obtain the ring acceptance with the NLK at the septum, which is shown in Fig. . The position and phase space of the injected beam is then matched to the ring acceptance with the parameters presented in Table Parameter Value Horizontal emittance (nm·rad) Energy spread Bunch length ) (m) (0,7) Injected beam position (mm) Injected beam angle (mrad) To accurately calculate the injection efficiency of the new injection system, the injected beam errors should be included.

According to the performance of the HLS-II injector, the error of the NLK field strength also affects the injection efficiency.

The jitter of the NLK power supply is required to be less than

0.1%, which is technically achievable. 335

Acceptance before NLK Acceptance after NLK NLK magnetic field Magnetic field B (Gauss) (mrad) Injected beam before/after NLK Stored beam x (mm) NLK and the blue one after the NLK. The field of the NLK is shown as the green line. The NLK peak field strength is

250 Gauss

with an excitation current of

868 A

To calculate the injection efficiency as a function of the er- ror level, an error scaling factor is introduced by multiplying the error from 0 to 2. The random errors are generated with the Gaussian truncation of 3 . For each error setting, 100 bunches with each 1000 particles are used in the simulation.

The simulation results of the injection efficiency as a func- tion of the error scaling factor are shown in Fig. . The injection efficiency is about 95% with the error factor of 1, when the error factor increases to 2, the injection efficiency remain 85%.

Parameter Error ( Injected beam position (mm) Injected beam angle (mrad) Injected beam energy stability ( NLK field error ( Injection efficiency Beam centroid osc. (mm) Time (ms) Beam size osc. (mm) Time (ms) PERTURBATION ON THE STORED BEAM According to the physics design of the NLK, the central magnetic field is and the field gradient is optimized to be

less than 0 .

1 T

/ m . However, owing to the technical limita- 349

tion, the field leakage to the axis of a real NLK is usually larger considering the errors in conductor positions and the influence of the ceramic chamber. Referring to the previous

work reported by other facilities [ ], a loose error tolerance with a dipole field of

6 Gauss

and a field gradient

3 T

can be set to the NLK. With these field errors, the injection perturbation of the stored beam at the injection point is calculated and shown in Fig. . The global injec- tion perturbation along the whole storage ring including the maximum beam centroid and the change of the beam size is shown in Fig. . The typical beam orbit stability requirement for the light source users is 10% of the beam size, which can also be treated as the criterion for transparent injection. Since the changes of the beam centroid and size of the stored beam during beam injection are smaller than 10% of the beam size, transparent injection is realized using an NLK for the HLS-II storage ring. A more strict error requirement can be applied to the NLK to further reduce the injection perturbation on the stored beam.

Max. beam centroid change (mm) Beam size (mm) s (m) Max. beam size change (mm) Beam size (mm) s (m) The injection perturbations on the stored beam between the local bump injection and the NLK injection with an NLK are compared in Table . With the new injection scheme, the perturbation on the stored beam is less than 10% of the beam size, which means the transparent injection is achieved.

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