research papers
Investigation of the damping wiggler effect and application on the PAL fourthgeneration storage ring
^{a}Department of Physics, POSTECH, Pohang, Gyungbuk 37673, South Korea, ^{b}Pohang Accelerator Laboratory, POSTECH, Pohang, Gyungbuk 37673, South Korea, and ^{c}SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
^{*}Correspondence email: tlssh@postech.ac.kr
An investigation of the damping wiggler effect to reduce the emittance in the Pohang Accelerator Laboratory (PAL), a fourthgeneration storage ring (4GSR) that uses a multibend achromat, is presented. A 4GSR lattice which has reduced emittance and increased dynamic aperture to amplify the synergy with two existing light sources (PLSII and PALXFEL) at PAL is described.
1. Introduction
A fourthgeneration storage ring (4GSR) that uses the multibend achromat (MBA) lattice concept may be able to surpass the ) was the first MBA machine; the next, Sirius (Liu et al., 2013), is currently under construction. Other projects are being conducted to convert existing 3GSRs such as ESRF, APS, SPring8 and ALS (Revol, 2017; APS, 2020; SPring8, 2014; Steier et al., 2016) to 4GSRs. These modifications will reduce beam emittance to a few hundred picometres, or even to ≤100 pm, but many efforts have been made to reduce the natural emittance further. The natural emittance in an electron storage ring is given by
and coherence that are attained using present thirdgeneration storage rings (3GSRs). 4GSRs are part of a worldwide effort to exceed the and coherence reached by present 3GSRs. MAX IV (MAX IV, 2020where C_{q} is a constant (for electrons ), is the relativistic factor for a particle, ≃ 1/ρ^{2} (where ρ is the unitcell bending angle) and j_{x} is the horizontal damping partition number (Borland et al., 2014). The natural emittance scales with the inverse cube of ρ, so MBA lattices enable emittance reduction by one to two orders of magnitude compared with thirdgeneration light sources.
The advent of 4GSRs enables a ring freeelectron laser (FEL) to perform highaveragepower extreme ultraviolet lithography (Luong, 2018) due to its ability to generate extremely low electron beam emittance. However, 4GSRs still have limitations such as relatively large energy spread and low repetition rate in FELs in the bypass line from the storage ring. A method to compensate for the effects of beam energy spread by introducing a transverse variation of the undulator magnetic field has been proposed (Huang et al., 2012), but the average power of the photon beam is insufficient because of the low repetition rate in a ring FEL that uses a bypass line (Cai et al., 2013). To further reduce the emittance, the concepts of the longitudinal gradient bend and reverse bends have been considered for a lattice design to achieve subtheoretical minimum emittance (Riemann & Streun, 2019). Use of a damping wiggler is an option to reduce emittance further, and its effect should be explored. The damping wiggler increases the repetition rate of the FEL beam by reducing the damping time; this effect must be quantified.
Recently a 4GSR has been designed at the Pohang Accelerator Laboratory (PAL4GSR) using concepts from ESRFEBS and APSU. As a result, a lattice with 3 GeV energy and 90 pm emittance with a 570 m ring was proposed. We investigate the effect of a damping wiggler on PAL4GSR, particularly focusing on emittance reduction and repetition rate increase. In this paper, we describe the PAL4GSR lattice, and the overall dampingwiggler effect on PAL4GSR. Section 2 describes the lattice design and beam dynamics studies for PAL4GSR. Section 3 presents derivation of analytic formulas for the damping wiggler effect on beam parameters. Section 4 applies the analytic formulas to PAL4GSR and compares them with simulations conducted using Elegant software (Borland, 2000). Section 5 presents a summary and concluding remarks.
2. PAL4GSR
The PAL4GSR storage ring (Table 1) is a hybrid sevenbend achromat lattice with a horizontal emittance of 90 pm. The ring is 570 m in circumference and is composed of 20 symmetrical cells. The length of the straight section is set as 6.5 m to accommodate two superconducting RF modules, as a result of experience on PLSII (Shin et al., 2013). The PAL4GSR lattice contains a 2 T superbend in the central dipole to produce radiation with a of 12 keV (Fig. 1).

The concepts of the ESRFEBS and APSU lattices were adopted in the PAL4GSR lattice. The dispersion was deliberately enlarged between the first and second dipoles and between the sixth and seventh dipoles, and chromatic sextupoles have been placed in this dispersionbump region to reduce the strength required to control the chromaticity. The betatron phase advances between the two dispersion bumps were set to ≃ 3π in the horizontal plane and ≃ π in the vertical plane. As a result, nonchromatic effects of the sextupoles are canceled naturally to first order. To minimize natural emittance, fivestep longitudinal gradient dipoles and reverse bending magnets were considered.
The sextupoles are twocell periodic in order to increase the number of control knobs. Therefore, a total of ten sextupoles in two cells, except two sextupoles for correcting the linear chromaticity (Fig. 2), are used in the optimization with a multiobjective (MOGA) (Deb, 2001). The objectives are an onmomentum dynamic aperture for offaxis beam injection and an offmomentum dynamic aperture (4%) to extend the Touschek lifetime. The result (Fig. 3) of MOGA optimization seems to have converged after 150 generations, at which the strengths of ten harmonic sextupoles reach a compromise between onmomentum dynamic aperture and 4% offmomentum dynamic aperture. The candidate point for the studies (marked by the star in Fig. 3) has a maximum onmomentum dynamic aperture area among the flat region for the offmomentum dynamics aperture area. As a result, large dynamic apertures were attained, enough for the offaxis injection with random errors (Fig. 4). Here, 30 µm r.m.s. offset, 200–500 µrad r.m.s. rotation and 0.1–0.2% r.m.s. field errors are assumed to generate 100 random errors. Despite assuming this harmful error scenario, a horizontal dynamic aperture from −9 mm to 12 mm is promising for 6 mm offaxis beam injection.
To obtain maximum Touschek lifetime, the RF voltage is scanned in the particle tracking with the optimized sextupole configuration (Fig. 5). At 3.1 MV RF voltage, which corresponds to 6% RF acceptance, Touschek lifetimes are 6 h for 1% coupling flat beam and 58 h for coupling round beam. Local momentum aperture tracking including error distributions is also compared with ideal local momentum aperture and RF momentum aperture (Fig. 6). Evaluated Touschek lifetimes in realistic conditions are 1.3 h for 1% coupling flat beam and 9.7 h for round beam at 1.9 MV optimum RF voltage, which corresponds to 4.3% RF acceptance. These lifetime reductions invoke the need of a harmonic cavity. Note that bunch lengthening from 3.45 mm r.m.s. to 20 mm r.m.s. with a harmonic cavity enables Touschek lifetimes to be increased up to 7 h for 1% coupling flat beam and 51 h for round beam. Here the factor 5.8 of bunch lengthening from the higher harmonic cavity is assumed to obtain sufficient lifetime in hybrid fill pattern operation with high single bunch current. In addition to the Touschek lifetime, the lifetime is 61 h, assuming a minimum half physical aperture of 2 mm, and the bremsstrahlung lifetime is 79 h.
The results of Elegant simulation of intrabeam scattering (Fig. 7) show a change in the zerocharge emittance along beam energy, which scales as ∼E^{2}. In the simulation, the algorithm uses the Bjorken & Mtingwa formula (Bjorken & Mtingwa, 1983) to simulate the intrabeam scattering (IBS) effect. For a 0.845 nC bunch, which is a single bunch current for a 400 mA multibunch, round beam operation must be used to avoid a large emittance increase due to the IBS effect. In addition, a harmonic cavity can be considered for bunch lengthening up to 20 mm. Here, the natural bunch length is 3.45 mm and the lengthening factor with harmonic cavity is 5.8.
3. Analytic formulas for the damping wiggler effect
In an electron storage ring, representative ring parameters can be expressed using synchrotron radiation integrals
where C_{q} ≃ 3.84 × 10^{−13} m, ≃ 8.846 × 10^{−5} m GeV^{−3}, is the natural horizontal emittance, E is the electron energy and τ is the damping time. The definition of the synchrotron radiation integral is given by
where k_{1} = , H = + + (Wiedemann, 1993), and ρ is the curvature of a reference particle. These integrals are changed by considering a damping wiggler that has a vertical magnetic field given by
where B_{w} = is the peak field, k_{w} = is the wavenumber, and is the wiggler period. For the 3 GeV electron storage ring, the magnet rigidity is ∼10 T m. The synchrotron radiation integrals for the wiggler in the straight section are (Appendix A)
where L_{w} is the length of the wiggler and is a horizontal betafunction. Equation (11) applies to a sinusoidal wiggler field versus s, but in reality the field of a longperiod insertion device (ID) includes high harmonics of k_{w}.
The emittance ratio between the storage ring emittances without and with wiggler is given by
where
which is very small compared with I_{2,wig}, so I_{4,wig} can be neglected for typical wiggler values in equation (12). The emittance ratio without I_{4,wig} becomes
where
and
The form of the function (1 + bL_{w})/(1 + aL_{w}) depends on the ratio of a and b, which are always positive. = 1 for a = b, = >1 for , and < 1 for . Here our main interest is the case of = <1. When a = 1.389B_{w}^{2} and b = 0.486B_{w}^{5} the emittance ratio (Fig. 8) along the length of a damping wiggler is limited on the asymptotic line given by b/a, despite the extremely long wiggler length. Therefore, short wiggler period, small horizontal betafunction, long wiggler length and low magnetic field are favorable conditions to reduce total emittance when the wiggler is infinitely long. Minimum emittance reduction ratio along natural storage ring emittance (Fig. 9) implies that the emittance reduction by a damping wiggler is not so effective when natural storagering emittance is extremely small.
In the same manner, the ratios of energy spread, damping time and momentum compaction factor are also formulated as (Helm et al., 1973)
where
where
The equation for energy spread ratio has a similar form to that of the emittance ratio. Energy spread reduction is also limited by an asymptotic line. Damping time is always reduced by the damping wiggler and the reduction of momentum compaction factor is negligible because f in equation (16) is small.
4. Application on PAL4GSR
The damping wiggler effects on the main beam parameters in the lattice of PAL4GSR were investigated using the analytic formulae derived in Section 3 and the Elegant simulation. Four straight sections among a total of 20 straight sections are considered for each 4.5 mlong damping wiggler. In this case, four ID straight sections should be sacrificed for insertion of damping wigglers. The synchrotron radiation integrals (Table 2) for PAL4GSR were obtained using simulation in Elegant. The theoretical results were also compared with the results of the Elegant simulation.

Assuming a damping wiggler that had a total length of 18 m, the emittancereduction ratios were investigated by scanning the wiggler period and the wiggler peak field (Fig. 10). The function plot from theoretical results agreed well with the simulation result obtained using Elegant. For a given wiggler period and wiggler length, an optimal wiggler peak field achieves the smallest emittancereduction ratio (Fig. 10, stars); this peak field increases as the wiggler period decreases.
The allowable peak field should be obtained for a given short wiggler period. The Halbach formula was used to determine the optimal wiggler field along the wiggler period and magnetic field versus wavelength (Fig. 11). For the total 18 m wiggler length in PAL4GSR, the minimum emittance reduction ratio is at most ∼0.83. As already shown in Fig. 9, emittance reduction by the damping wiggler is not so effective due to the small natural emittance.
For the FEL option through the bypass line, dampingtime reduction was also investigated to estimate the increase of repetition rate for high average FEL power. Horizontal, vertical and longitudinal damping times have the same form; the only difference is in the coefficients [equation (15)]. The coefficients are independent of the wiggler period. In PAL4GSR, they are
The dampingtime reduction (Fig. 12) by a wiggler was calculated. A halfdamping time (or double repetition rate) can be expected in a magnetic field >2.5 T. Similarly, the energy spread reduction can also be estimated (Fig. 13); an increase in energy spread is inevitable in a magnetic field >2 T. These results show that the damping wiggler does not have much effect on the increase of ring FEL power.
5. Conclusion
Analytic analysis and Elegant simulation were performed to quantify the damping wiggler effects on PAL4GSR which is a 3 GeV and 90 pm natural emittance storage ring with 570 m circumference. Analytic analysis agreed well with Elegant simulation, and described reductions in energy spread, damping time, and emittance. Emittance reduction by the damping wiggler was less effective than in thirdgeneration storage rings due to the small natural emittance in the fouthgeneration storage ring. Application of a damping wiggler on a ring FEL was also investigated in the aspect of increasing average Unfortunately the damping wiggler causes an increase in energy spread, and therefore cannot considerably increase the ringFEL power.
APPENDIX A
Synchrotron radiation integrals
Helm et al. (1973) evaluated the synchrotron radiation integrals. Synchrotron integrals in the damping wiggler have been investigated by many authors (Katoh & Kamiya, 1987; Wiedemann, 1988; Walker, 1995). This section presents specific details of the damping wiggler's contribution to synchrotron integrals.
Synchrotron integrals are described using η, k_{1} and . Therefore, to obtain synchrotron integrals in the damping wiggler, the form of η, k_{1} and in the damping wiggler should be realized. First, we obtained the dispersion function; assuming that the damping wiggler has a simple harmonic vertical field given by
where B_{w} is the peak wiggler field and k_{w} = , with being the wavelength of the damping wiggler, a Lorentz force is exerted on the electron in the damping wiggler as
where is the force, q is the charge, is the velocity vector and is the magnetic field. The time variable is usually replaced by the position variable z along the longitudinal coordinate. With the constantvelocity approximation, the lefthand side of the equation (19) becomes
where γ is the Lorentz factor, m is the mass, is a position vector, is dx/dz and v_{z} = dz/dt.
From now on, a prime denotes the derivative with respect to z. The righthand side of equation (19) becomes
where e is the charge on an electron.
Assuming that the longitudinal momentum is much larger than the transverse momentum, then the horizontal equation of motion is
where p is the magnitude of the momentum, p_{0} is the reference momentum, is the magnetic rigidity, and δ = ( pp_{0})/p_{0}.
Inserting equation (18) into equation (20) with the condition that δ is zero yields
When δ = 0, x is a reference orbit. By solving the differential equation, x can be obtained as
When δ ≠ 0, the orbit has an energydependent deviation from the reference orbit. The equation of motion is similar to equation (21),
Solving the differential equation (23) yields
The definition of the dispersion is given by
Comparing equations (24) and (25), the dispersion function can be expressed as
The normalized quadrupole strength k_{1} should be expressed with z. The definition of k_{1} is
The slope of ξ is perpendicular to the slope of the tangent line. The slope of the tangent line to the curve is dx/dz = . The product of and is −1 ( = −1), because they are perpendicular to each other. Then equation (27) becomes
Inserting equations (18) and (22) into (28) yields k_{1} as
Lastly, the bending radius is rewritten with the variable z. It is related to the magnetic field as
The damping wiggler contribution to a synchrotron integral will be denoted by the subindex wig. The integration is from the entrance of the wiggler to the exit of the wiggler, i.e. for wiggler L_{w}. By inserting equations (26), (29) and (30) into the synchrotron integrals, the contributions of the damping wiggler to the synchrotron integrals can be obtained as
where is the average value of the horizontal beta function from z = 0 to z = L_{w} and , and are the Twiss functions. The calculation of I_{5,wig} uses an approximation
Usually ≃ 1/k_{w} 1 m. For example, when the wiggler wavelength is 12.57 cm, 1/k_{w} = 5 ×10^{  3} m. An assumption is also used: that changes smoothly in the damping wiggler, so the integral of is approximately the same as the product of and the integrals of . Generally, the synchrotron radiation integrals are defined in a curvelinear coordinate. Here the synchrotron radiation integrals are integrated with z; they cannot represent the ring parameters related to path length. For example, I_{1,wig} does not represent the contribution of the damping wiggler to a momentum compaction factor. Then the new synchrotron radiation integrals
are defined to denote the ring parameters appropriately. With these new parameters, the damping wiggler contribution to the momentum compaction is expressed as = I_{1,wig}/C.
Acknowledgements
We thank H. Wiedemann for providing helpful information and many useful discussions.
Funding information
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF2019R1C1C1003412) and the Basic Science Research Program through the National Research Foundation of Korea (NRF2019R1A2C1004862).
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