A conceptual design study of generating locally-round beam in a diffraction-limited storage ring using skew quadrupole triplets

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Abstract

Diffraction-limited storage rings (DLSRs), with sub-100pmrad electron emittances approaching the diffraction limit for hard X-rays, would provide unprecedented transverse coherence and much higher brightness than currently operational light sources to the synchrotron radiation scientific community. Instead of typical flat beam, some individual synchrotron light users prefer round beam, a beam with equivalent emittances between the horizontal and vertical planes, for the reasons of fully transverse coherent X-ray radiation and better photon–electron phase space match, and so on. To meet the requirements of different experimenters, obtaining a locally-round beam will be a significant subject in DLSR studies. We investigate the approaches of realizing locally-round beam in a storage ring, by means of a local emittance equalization of the transverse planes. In this paper, a novel method is proposed to achieve locally-round beam, performed with a combination of skew quadrupoles in insertion section of the storage ring. Theoretical analysis and application of this method to achieve a locally-round beam at DLSRs, particularly to High Energy Photon Source (HEPS) storage ring, are presented.

Introduction

Over the last few decades, accelerator technology and source development has continued in three generation sources, which provide synchrotron radiation photons in a spectrum spanning from infrared to hard X-rays for scientific applications. As newer light sources aim to further increase brightness, the diffraction-limited storage rings (DLSRs) based on fourth-generation ring light sources has been proposed [1]. It would provide beyond the brightness and coherence reached by existing light sources and strongly extend the capabilities of X-ray imaging with sub-100pmrad diffraction-limited emittances for multi-keV X-ray beams. With the diversification of experimental requirements, some synchrotron radiation users prefer round beam with equal transverse emittances, instead of horizontally elongated flat beam which has been used by general users for decades. Compared with the traditional flat beam, the round beam has two obvious advantages: (1) more conducive to photon–electron phase space matching, which results in brighter photon beam [2]; (2) meeting the requirements of some beamline users who prefer diffractive imaging utilize round pinholes [3]. To fulfil the requirements of different synchrotron radiation users, worldwide efforts of locally-round beam have been made to design and study based on DLSRs.

In principle, there are two approaches to generate globally-round beam, a beam with equivalent emittances between the horizontal and vertical planes along the whole storage ring. One method is introducing Mobius accelerator scheme [4]. At each turn of beam motion, this scheme cause an interchange of the betatron oscillation in transverse planes due to the coupling introduced by strong skew quadruples in the lattice. Meanwhile, optical functions of transverse planes will be coupled as well, resulting in difficulty for orbit correction, optics measurement and correction, and the re-analysis of beam dynamics is inevitably required. The other is to utilize linear coupling difference resonance to obtain globally-round beam [5]. This method is to move the tunes in horizontal and vertical planes close to each other (equal fractional betatron tunes in transverse planes) and introduce the coupling driving term by installing the weak skew quadrupoles in a storage ring, resulting in an interchange of the oscillation energy between the transverse planes due to the resonance effect. This method only requires weak skew elements installed in the ring, thus the horizontal and vertical optical functions have a weak coupling and still be close to a normal decoupled regime of operation. Realization of globally-round beam has been demonstrated by linear coupling difference resonance in APS [6] and NSLS-II [7]. To mitigate intra beam scattering effect and increase beam lifetime, and provide round beam to all beamline users, this method is being considered at several future storage rings such as APS-U [8], ALS-U [9] and HEPS [10].

There are two ways to achieve locally-round beam, a beam with equal emittances by means of a local exchange between the horizontal and vertical planes. One way is to introduce solenoid-insertion device–anti-solenoid (S-ID–AS) section [11]. A locally-round beam can be obtained when the parameters of solenoid satisfy the condition: BzL(2Bρ)=π4, with Bρ being the magnetic rigidity, Bz being the solenoid field, and L being the effective solenoid length. For a DLSR with a beam energy of 6 GeV, the solenoid must satisfy the condition: BzL=10π. Bz is inversely proportional to L, and it is difficult to set a S-ID–AS section with feasible solenoid field in a straight section of strict space limitation. Another way is to introduce a solenoid and two triplets of skew quadrupoles in a straight section, and generate a locally-round beam with emittances εx=εyεx0εy0 in undulator inside the solenoid, where εx0,εy0 are the transverse emittances of weak coupling [12]. In order not to spoil the potential gain in brightness, one need the undulator to be as long as possible (L > 5–10 m) and the solenoid as strong as possible (Bz>12T for 6 GeV). However, the requirements for the undulator and solenoid are very challenging in a straight section with strict space limitation, and a severe limitation is missing a way to match the electron optics to the photon optics of the insertion device. In principle, this scheme is appropriately applied to conventional third generation storage rings to provide diffraction limited X-rays even if ignoring these difficult requirements.

Due to unfeasible solenoid field and strict space limitation in the above two methods, we propose a novel method to achieve locally-round beam, performed with a combination of a skew triplet of quadrupoles (ST) and an anti-skew-triplet of quadrupoles (AST) in two different insertion sections across the standard cell periods (CELL). Theoretical analysis of the beam transport in a ST–CELL–AST section is presented in Section 2. In Section 3, taking a DLSR, particularly to the HEPS storage ring as an example, we show a detailed design to realize a locally-round beam by introducing a ST–CELL–AST section in two insertion sections, and calculate the equilibrium transverse emittances with the Accelerator Toolbox (AT) code [13]. In Section 4, numerical simulations with the ELEGANT code [14] are performed to demonstrate the agreement between the results of simulations and theoretical analysis. The conclusion is given in Section 5.

Section snippets

Beam transport through a ST-CELL-AST section

Taking analogy to the derivation used in the Ref. [11], we study the beam transport through a ST–CELL–AST section in this section. Presuming that a particle coordinate of flat beam in transverse planes is X0=x0,x0,y0,y0. x0=U1cosθx+V1sinθxx0=M1cosθx+N1sinθxy0=U2cosθy+V2sinθyy0=M2cosθy+N2sinθy U1,V1,M1,N1,U2,V2,M2,N2 are arbitrarily real numbers for representing general solution of the transverse betatron motion [15], xs=Axβxscosψxs+ψx0xs=Axαxscosψxs+ψx0+sinψxs+ψx0βxsys=Ayβyscosψys+ψy0ys

ST-CELL-AST section in HEPS

Until now, all light sources in China are in the medium and low energy regions. For national security and industrial innovation, a high-performance light source with high energy is urgently needed in China. The High Energy Photon Source (HEPS), with a beam energy of 6 GeV, a storage ring circumference of 1360.4 m and a natural emittance of 34.2 pmrad, will follow this intention and is constructed as a DLSR light source in Beijing [16]. 48 7BA cells form the HEPS storage ring and were divided

Tracking with elegant

As discussed above, a locally-round beam can be obtained with the ST–CELL–AST​ section at the HEPS storage ring. To verify the calculation results of transverse emittances in Section 3, a simulation of multi-particle tracking is performed with the ELEGANT code for the HEPS storage ring with the ST–CELL–AST section. To visually describe the simulation results, we put three points in the storage ring to record the particle coordinates as follows:

Point A: the entrance of the ST–CELL–AST section

Conclusion

In this paper, we propose and demonstrate that a ST–CELL–AST section can produce a locally-round beam, which provides photons with higher brightness and transverse coherence, and approaches the diffraction limit for hard X-rays in a DLSR. This method is practical and feasible in a storage ring, because the skew quadrupole fields can be realized with current technologies and the designs are straightforward without technical challenges. The main issue is inevitably reducing the dynamic aperture

CRediT authorship contribution statement

Chongchong Du: Conceptualization, Methodology, Software, Investigation, Writing - original draft, Data curation. Jiuqing Wang: Validation, Formal analysis, Writing - review & editing, Supervision. Daheng Ji: Visualization, Resources, Software. Saike Tian: Visualization, Resources, Software.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We would like to thank Peter Kuske and A. W. Chao for helpful discussion with the theory of round beam, M. Borland for helpful discussion with ELEGANT, and Y. Jiao for providing the designed HEPS lattice.

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