Preamble

Compensation of emittance and bunch length variations induced by insertion devices using damping wigglers in fourth-generation light sources Yanwei Yang, Yi Jiao, Weihang Liu, Xiaoyu Li, Yu Zhao, Changdong Deng, and Sheng Wang 1, 2, 1 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, 100049, China Spallation Neutron Source Science Center, Dongguan, 523803, China Recent investigations suggest that in a fourth-generation light source, where both fundamental-frequency and harmonic RF cavities are commonly used for bunch lengthening, variations in radiation energy loss per turn ( could cause a significant change in bunch length. It is necessary to compensate for the variations caused by gap changes of the insertion devices (IDs). In this paper, we investigate the approach of using two horizontal or two vertical damping wigglers to simultaneously compensate for horizontal emittance and variations induced by IDs. Theoretical analysis and a specific example of application are presented.

Keywords

Emittance, Radiation energy loss per turn, Vertical damping wiggler, Parameters compensation

INTRODUCTION

Currently, multiple laboratories around the world have

either constructed or are planning to construct fourth- 3

generation light sources (4GLSs) [ ]. A remarkable fea- ture of 4GLS is that the emittance of the storage ring is re- duced to near or even reach the X-ray diffraction limit by us- ing the multi-bend achromat (MBA) lattice [ ]. Thanks to the reduction in beam emittance and adoption of the state- of-the-art insertion devices (IDs) [ ], the brightness of

4GLS is about two orders of magnitude higher than available 10

with the third-generation light source. More significantly, co- 11

herence of synchrotron radiation is improved, laying the foun- dation for realization of higher spatial resolution.

In 4GLSs, IDs cause radiation energy loss that matches or surpasses that in bending magnets. Adjustments of the gaps (and thus magnetic fields) of IDs according to exper- imental requirements will lead to evident variations in ra- diation energy loss per turn ( ) and other beam parame- ters, for instance, the emittance, energy spread, and bunch length. Among these parameters, the variations in emittance would lead to fluctuations in photon brightness and distribu- tion, which impact specific beamline experiments, such as

those conducted in scanning transmission X-ray microscopy 23

beamlines [ ]. The issue can be resolved by adopting hori- zontal damping wiggler (HDW) to compensate for horizontal emittance variations [ ]. And two HDWs have been pro- posed to simultaneously compensate for the variations of both emittance and energy spread [ Recent investigations suggest that, in a 4GLS where both

fundamental-frequency and harmonic RF cavities are com- 30

monly used for bunch lengthening, variations in U 0 could 31

cause evident changes in bunch length. Maintaining bunch 32

length stability is of significant importance for mitigating the 33

intra-beam scattering (IBS) effect, enhancing beam stability, and prolonging beam lifetime.

To compensate for bunch length variations, one approach is to integrate an additional feedback system into the RF cav- ities to adjust the relevant RF parameters accordingly, so as to keep the bunch length unchanged. In this paper, we in- vestigate another approach, i.e., using two HDWs or vertical damping wigglers (VDWs) to simultaneously achieve com- pensation for horizontal emittance and without the need for additional RF feedback system. Compared to the dual HDWs scheme, the dual VDWs scheme can directly generate vertical emittance without requiring additional methods and can compensate for the vertical emittance variations induced by non-planar IDs.

This paper is organized as follows. In Section II , we ana- 48

lyzed the characteristics of HDW and VDW and theoretically explained how two HDWs or VDWs achieve compensation for horizontal emittance and variations. In Section provided a specific example to facilitate a more intuitive un- derstanding. Section is the conclusion.

THEORETICAL ANALYSIS In this section, we will elucidate the reason for persistence variations following HDW’s compensation for horizon- tal emittance. Additionally, we will clarify how two HDWs or VDWs address this issue.

Horizontal emittance and variations induced by IDs In an electron storage ring, emittance (without linear cou- pling) and can be expressed in terms of synchrotron radi- ation integrals:

ϵ x = C q γ 2 I 5 I 2 − I 4 (1) 63

U 0 = C γ 2 π E 4 I 2 (2) 65

where, C q = 3 . 84 × 10 − 13 m, γ is Lorentz relativistic factor, 66

C γ = 8 . 85 × 10 − 5 m GeV − 3 , E is electron energy. The 67

synchrotron radiation integrals are expressed by:

I 2 = � 1 ρ 2 ds (3) 69

ρ 2 + 2 k ) ds, k = 1 Bρ ∂B y ∂x (4) 71

I 4 = � η ρ ( 1

| ρ | 3 ds, H = γη 2 + 2 αηη ′ + βη ′ 2 (5) 73

I 5 = � H

where, is curvature of reference particle, is dispersion function, is magnet rigidity, and are Twiss pa- rameters.

Considering IDs (we first consider only planar IDs, as was done in previous research on horizontal emittance compensa- tion [ ]), horizontal emittance and

ϵ x = C q γ 2 I 5 b + � N i =1 I 5 IDi ( I 2 b + � N i =1 I 2 IDi ) − ( I 4 b + � N i =1 I 4 IDi ) (6) 80

where, is HDW magnetic field strength, is HDW length, is HDW period length, is average horizon- tal beta function in HDW. Accordingly, netic field strength, length, riod length, is average horizontal beta function in

Here, � N i =1 I 4 IDi and I 4 HDW are ignored because their val- 103

ues are significantly smaller compared to � N i =1 I 2 IDi and 104

. When ID magnetic fields change, we can adjust the magnetic field strength of the HDW in real time to maintain the stability of horizontal emittance.

The equilibrium emittance of particle beams in a storage ring arises from a dynamic balance between two fundamental physical processes [ ]. First, the quantum excitation caused by the quantized emission of photons contributes to an in- crease of the emittance. Second, a damping effect, result- ing from the loss of some transverse momentum of the parti- cles during photon emission, contributes to a decrease of the emittance. The quantum excitation effects caused by IDs and

HDW are characterized by � N i =1 I 5 IDi and I 5 HDW . Follow- 116

ing the compensation of horizontal emittance using a HDW,

U 0 is automatically compensated only when � N i =1 I 5 IDi and 118

are sufficiently small to be considered negligible. In this scenario, the compensation equation can be expressed as follows:

i =1 B 2 IDi L IDi 2( Bρ ) 2 + B 2 H L H 2( Bρ ) 2 (9) 122

I c 2 = � N

U 0 = C γ 2 π E 4 ( I 2 b + � N

i =1 I 2 IDi ) (7) 81

where, the subscripts b and ID represent synchrotron radia- tion integrals contributed by bare lattice and IDs. is num- ber of IDs. During machine operation, users adjust ID mag- netic fields due to experimental requirements. This will result in alterations to synchrotron radiation integrals contributed by IDs, leading to variations in horizontal emittance and Horizontal emittance and compensation by two HDWs Evident variations in may still persist after compensat- ing for horizontal emittance using a HDW. To simultaneously compensate for both horizontal emittance and , we intro- duce two HDWs. For all theoretical considerations presented hereafter, the storage ring is assumed to contain perfectly achromatic straight sections.

For horizontal emittance compensation by a HDW, the compensation equation is

U c 0 = C γ 2 π E 4 ( I 2 b + I c 2 ) (10) 123

ϵ c x = C q γ 2 I 5 b ( I 2 b + I c 2 ) − I 4 b (11) 124

When the storage ring’s IDs consist primarily of low-field

undulators, I 5 b ≫ � N i =1 I 5 IDi , indicating that after employ- 126

ing a HDW to compensate for horizontal emittance, the mag-

nitude of U 0 variations is contingent upon I 5 HDW . When 128

the compensation amount of horizontal emittance variations is substantial, the magnetic field strength required for the HDW also increases. Thus, the quantum excitation induced

by HDW will produce a significant effect, especially in low 132

emittance rings (see Appendix ). The final result is that af- ter compensating for horizontal emittance, still exhibits evident variations.

To address the issue of variations, we split HDW into two parts (HDW1 and HDW2) without increasing total length. The compensation equations are as follows:

ϵ c x = C q γ 2 I 5 b + � N i =1 I 5 IDi + I 5 HDW ( I 2 b + � N i =1 I 2 IDi + I 2 HDW ) − ( I 4 b + � N i =1 I 4 IDi + I 4 HDW )

= C q γ 2 I 5 b + � N i =1 λ 2 IDi ⟨ β x ⟩ i B 5 IDi L IDi 15 π 3 ( Bρ ) 5 + λ 2 H ⟨ β x ⟩ B 5 H L H 15 π 3 ( Bρ ) 5

( I 2 b + � N i =1 B 2 IDi L IDi 2( Bρ ) 2 + B 2 H L H 2( Bρ ) 2 ) − I 4 b

U c 0 = C γ 2 π E 4 ( I 2 b + � N

Compensation of horizontal emittance and can be achieved with varying ID magnetic fields by adjusting the magnetic field strengths ( ) of two HDW. For given lengths of HDW ( ), appropriate selection of emittance and compensation values enables numerical determination of through computational proce- dures. The compensation equations can be further simplified:

i =1 B 2 IDi L IDi 2( Bρ ) 2 + B 2 H 1 L H 1 2( Bρ ) 2 + B 2 H 2 L H 2 2( Bρ ) 2 (14) 148

I c 2 = � N

I c 5 = λ 2 H 1 ⟨ β x ⟩ B 5 H 1 L H 1 15 π 3 ( Bρ ) 5 + λ 2 H 2 ⟨ β x ⟩ B 5 H 2 L H 2 15 π 3 ( Bρ ) 5 (15) 150

Horizontal emittance and compensation by two VDWs In this section, we propose an alternative method for si- multaneously compensating for horizontal emittance and Two VDWs are introduced. VDW was originally proposed to generate round beams instead of traditional linear cou- pling method [ ]. VDW is equivalent to rotating HDW around the longitudinal axis, thereby transforming the magnetic field from vertical direction to horizontal direction.

Consequently, the quantum excitation effect is transferred to the vertical direction.

After adding a VDW, emittances become:

ϵ x = C q γ 2 I 5 b + � N i =1 I 5 IDi ( I 2 b + � N i =1 I 2 IDi + I 2 V DW ) − I 4 b

= C q γ 2 I 5 b + � N i =1 λ 2 IDi ⟨ β x ⟩ i B 5 IDi L IDi 15 π 3 ( Bρ ) 5

( I 2 b + � N i =1 B 2 IDi L IDi 2( Bρ ) 2 + B 2 V L V 2( Bρ ) 2 ) − I 4 b

ϵ y = C q γ 2 I 5 V DW I 2 b + � N i =1 I 2 IDi + I 2 V DW

= C q γ 2 λ 2 V ⟨ β y ⟩ B 5 V L V 15 π 3 ( Bρ ) 5

I 2 b + � N i =1 B 2 IDi L IDi 2( Bρ ) 2 + B 2 V L V 2( Bρ ) 2

where, is VDW magnetic field strength, is VDW length, is VDW period length, is average vertical beta function in VDW. Here, vertical emittance is generated as a result of vertical dispersion induced by the horizontal field of VDW.

In the case where , the contribution of can be ignored. If a VDW is used to com- pensate for horizontal emittance variations, then is auto- matically compensated.

ϵ c x = C q γ 2 I 5 b

( I 2 b + � N i =1 B 2 IDi L IDi 2( Bρ ) 2 + B 2 V L V 2( Bρ ) 2 ) − I 4 b (18) 176

i =1 B 2 IDi L IDi 2( Bρ ) 2 + B 2 V L V 2( Bρ ) 2 )) (19) 177

( U c 0 = C γ 2 π E 4 ( I 2 b + � N

Since VDWs contribute radiation damping in both trans- verse planes but induce quantum excitation only in the ver- tical plane, the horizontal emittance benefits from increased damping without added excitation. This enables simultane- ous stabilization of both horizontal emittance and , while generating a small, controlled vertical emittance. We pro-

pose dividing the VDW into two units, designated VDW1 184

and VDW2, to maintain the stability of both horizontal and vertical emittance.

ϵ c x = C q γ 2 I 5 b + λ 2 H 1 ⟨ β x ⟩ B 5 H 1 L H 1 15 π 3 ( Bρ ) 5 + λ 2 H 2 ⟨ β x ⟩ B 5 H 2 L H 2 15 π 3 ( Bρ ) 5

ϵ c x = C q γ 2 I 5 b

ϵ c y = C q γ 2 λ 2 V 1 ⟨ β y ⟩ B 5 V 1 L V 1 15 π 3 ( Bρ ) 5 + λ 2 V 2 ⟨ β y ⟩ B 5 V 2 L V 2 15 π 3 ( Bρ ) 5

Stability of horizontal and vertical emittance can be achieved with varying ID magnetic fields by adjusting the magnetic field strengths ( ) of two VDW. Simul- taneously, is automatically compensated. One can gen-

erate a round beam by making ϵ c x = ϵ c y . The compensation 193

equations can be further simplified:

i =1 B 2 IDi L IDi 2( Bρ ) 2 + B 2 V 1 L V 1 2( Bρ ) 2 + B 2 V 2 L V 2 2( Bρ ) 2 (22) 195

I c 2 = � N

I c 5 = λ 2 V 1 ⟨ β y ⟩ B 5 V 1 L V 1 15 π 3 ( Bρ ) 5 + λ 2 V 2 ⟨ β y ⟩ B 5 V 2 L V 2 15 π 3 ( Bρ ) 5 (23) 196

APPLICATION TO SAPS LATTICE To intuitively understand the compensation of horizontal emittance and , we will take the Southern Advanced Pho- ton Source (SAPS) [ ] as an example.

SAPS is a mid-energy ultra-low emittance light source pro- posed to be built adjacent to the China Spallation Neutron Source (CSNS) [ ]. The main parameters of SAPS are listed in Table . The circumference of SAPS is 810 m and encom- passes 32 periods. The component layout and beam optics of one period are illustrated in Fig. . Nine IDs are scheduled for construction in first phase of SAPS. The IDs of SAPS consist of three types: in-air undulator (IAU), in-vacuum undulator

(IVU), and cryogenic permanent magnet undulator (CPMU), 209

and the parameters are presented in Table Parameter Value beam energy beam current circumference natural emittance pm rad tune (H,V) 78.21, 44.16 momentum compaction factor radiation energy loss per turn without IDs 0.768 uated using synchrotron radiation integrals formulas. The re- sults, presented in Fig. , were obtained by sequentially in- corporating each ID into the calculations (we assume that all

IDs are at minimum gap). The leftmost point represents the 215

scenario where no IDs are included, while the rightmost point

λ ω , B ωmax , L ω represent period length, effective magnetic field strength at ID minimum gap and length of ID, respectively).

CPMU16-1 CPMU16-2 IAU20 IVU16 IAU32.7 IVU18.5-1 IAU28 IAU50 IVU18.5-2 represents the scenario where all IDs are included. Horizontal by 28.34% (0.22 MeV) due to the influence of IDs.

In practice, horizontal emittance variations induced by IDs are less than 3.9 pm rad, as it is improbable that all IDs will

simultaneously operate at either the minimum or maximum 222

gap. The SAPS has not yet been constructed. Therefore, no operational data is available. We use a computer program and 1 for 9 IDs to simulate user’s behavior. Then the value of magnetic field strength for i-th ID used in calculation is bers (each consisting of 9 digits) to simulate IDs state of op- erating at 200 different times. The results are presented in pm rad, while varies between 0.78 and 0.92 MeV, with corresponding variations of 10.81% (2.6 pm rad) and 17.57% (0.14 MeV), respectively.

First, we employ a HDW to compensate for horizontal emittance. The capacity of a HDW to compensate for hori- zontal emittance depends on its capacity to reduce horizon- tal emittance (see Appendix ). We choose a HDW with a

Section , even after compensating for horizontal emittance, still exhibits evident variations, as shown in Fig.

Compared to the state before a HDW compensation, although variations of have been reduced, it still exhibits variations of 10.12%. length of 5 m and a periodic length of 42 mm. The hori- zontal emittance compensation capacity of the HDW is 2.7 pm rad, slightly higher than horizontal emittance variations induced by IDs (2.6 pm rad). The HDW can be used to com-

pensate for horizontal emittance variations, maintaining it at 243

23.4 pm rad. The details of parameter selection and relevant calculations about the HDW can be found in Appendix

For simplicity, the IBS effect [ 38 ] is neglected in this initial 246

study. Future work may include a more complete treatment, and we assume that the influences of IDs and HDWs (VDWs) on beam optics can be effectively corrected [ The simulation results of horizontal emittance compen- sation are shown in Fig. . The maximum magnetic field strength required by HDW is 2.48 T, corresponding to the mo- ment when horizontal emittance reaches its maximum value of 26.1 pm rad before compensation. The quantum excita-

tion effect induced by the HDW is significant. As analyzed in 255

variations will lead to evident changes in bunch

length. In SAPS, in order to reduce beam density, harmonic 262

cavity is used to stretch the bunch length. In steady state, the bunch length can be calculated through bunch density

σ 2 z = � z 2 ρ ( z ) dz − [ � zρ ( z ) dz ] 2 (24) 266

where, is RF cavity fundamental frequency, is revolu- tion time, is momentum compaction factor, is speed of light. For a given , RF voltage ( ) and phase ( ) satisfy

the optimal conditions for bunch lengthening: 273

sin ϕ 1 =

V 2 = V 1

tan ϕ 2 = − nU 0

For SAPS, after a HDW compensation, fluctuates between 0.92 and 1.02 MeV, varies between 1.05‰ and 1.16‰.

We take the average value of U 0 ( U 0 ave = 0 . 97 MeV) to set 279

the RF voltage and phase ( n = 3 , eV 1 = 1 . 4 U 0 ave ). When 280

the parameters of the RF cavity remain unchanged, the bunch length change induced by variations can be negligible.

However, the bunch length change resulting from varia-

tions is highly significant, as illustrated in Fig. 5 [FIGURE:5] . Variations in 284

will lead to the failure of bunch length stretching. It should be noted that our analysis has only focused on the simplified

case of an idealised harmonic cavity under flat-potential con- 287

ditions, whereas more complex scenarios—including beam loading in the cavities, variations in the filling pattern, distinc- tions between passive and active cavities, and the influence of machine impedance—warrant further investigation.

To address the issue of variations, we replace the 5 m HDW with two 2.5 m VDWs. The quantum excitation ef- fect of VDW has been transferred to the vertical direction, as described by Equations ( ) and ( ). Thus, the capacity of VDW to compensate for horizontal emittance is directly

ρ ( z ) = ρ 0 exp( − Φ( z ) ω rf T 0 Eα c σ 2 E ) (25) 267

The potential proportional to the square of magnetic field strength and is no longer constrained by its quantum excitation effect. How- ever, if the magnetic field is excessively strong, it will lead to a rapid increase in vertical emittance. We select the com- pensation value for horizontal emittance as 19.6 pm rad and variations. for vertical emittance as 5 pm rad, the results are displayed in also presented. After two VDWs compensate for horizontal emittance, is almost automatically compensated, as shown in Fig. c. The exhibits variations of only 0.1% (negligi-

ble variations) resulting from a small change of � N i =1 I 5 IDi 308

value. We also simulated the compensation using two HDWs. The magnetic field required by HDW during the compensation process is identical to that of VDW (see Appendix ). There- fore, both achieve the same and energy spread ( ) state after compensation. The difference is that (25.2 pm rad) is slightly greater than xV DW yV DW (24.6 pm rad), as shown in Fig. a and Table (see Appendix for explana- tions). (cos( (cos( Finally, we calculate the change in energy spread after adding VDWs (HDWs), the result is presented in Fig.

The fundamental (2.87 keV) brightness of CPMU16 (pm rad) (pm rad) (photons/s/mm /mrad /0.1%B.W) 17.57% 5.61% 0.125%) but remains acceptable, with variations decreasing from 5.61% before compensation to 4.33% after compensa- tion. If necessary, more complex scenarios involving energy ergy spread growth may influence the bunch length and pho-

of the CPMU16 and the bunch length after harmonic cavity 327

stretching under four different scenarios. In the calculations, ues of each case. For Case 1, the emittance was taken as the average value, while for Cases 2 and 3, a linear coupling coefficient of 0.25 was applied to the emittance (to maintain consistency with Case 3). Compared to the uncompensated case, the two VDWs (or two HDWs) compensation scheme results in an approximately 12% reduction in brightness (as ligible effect on the bunch length. We consider the trade-off of an acceptable brightness reduction for improved stability in photon distribution, brightness, and bunch length to be sci-

CONCLUSION

In this paper, we have demonstrated the approach of com- pensating for horizontal emittance and variations with two HDWs or two VDWs. Taking the SAPS as an example, we show that compensation can be achieved by adopting two HDWs or VDWs with a total length of 5 m. After compen- sation, the variations in emittance decreased from 10.81% to 0%, and the variations in decreased from 17.57% to 0.1%.

It is noted that this approach does not aim to compensate for energy spread variations. In the presented example, the energy spread variations remain at the same level with two HDWs or two VDWs. We consider that such a level of vari- ation could be acceptable for most user beamlines. If neces- sary, more complex scenarios involving energy spread com- pensation can be further explored. In addition, we also show that if two VDWs, rather than two HDWs, are used, the vari- ations in vertical emittance can be compensated. For light sources with non-planar IDs (such as the vertical or elliptical polarization modes of elliptically polarized undulators), this will be particularly beneficial. One another benefit of using VDW rather than HDW is that nonzero vertical emittance is required anyway for lifetime and intra-beam scattering rea- sons, and using the VDW to generate the vertical emittance allows a smaller horizontal emittance to be achieved. How- ever, if VDWs are to be employed, this may potentially affect the horizontal aperture of the storage ring and consequently affect off-axis injection, adoption of this approach necessi- tates thorough investigation to assess its feasibility and impli- Appendix A In this section, we will demonstrate that a lower natural emittance of the ring results in a more rapid increase in the quantum excitation effect generated by a HDW as magnetic field intensifies. It means that in rings with lower natural emittance, HDW has a lower capacity to compensate for hori- zontal emittance variations because its compensation capacity is limited by the quantum excitation effect. It also means that in rings with lower emittance, after compensating for horizon- tal emittance by a HDW, the variations in become more pronounced.

The HDW can utilize its low magnetic field region ( B H < 381

, see following text) to compensate for horizontal emit-

tance, as well as its high magnetic field region ( B H > 383

). The distinction is that the high magnetic field main- tains horizontal emittance at a higher value and induces a

more significant quantum excitation effect, thus exacerbating 386

variations in . Thus, this section focuses solely on the low magnetic field region of HDW.

In the low magnetic field region, the capacity of HDW to compensate for horizontal emittance variations induced by IDs depends on its capacity to reduce horizontal emittance.

We assume that the capacity of HDW to reduce horizontal emittance is denoted as xHDW max and the correspond- ing magnetic field strength of HDW is denoted as When HDW is not included, the range of horizontal emit- tance variations induced by IDs is denoted as When ID gaps are configured to achieve an emittance of , the magnetic field strength of HDW is set to resulting in a horizontal emittance of xHDW max When ID gaps are set such that horizontal emittance is between , the magnetic field strength of HDW should be reduced in real time to an appropriate value to maintain a stable horizontal emittance of xHDW max . Therefore, the prerequisite for compensating horizontal emittance variations induced by IDs with a HDW

is that ∆ ϵ xHDW max ≥ ϵ xmax − ϵ xmin . 411

Strictly speaking, xHDW max should be calculated un- der the state that horizontal emittance equals (26.1 pm rad). For simplicity, we calculate xHDW max under the bare lattice state (without IDs), where horizontal emittance

is ϵ xb = 26 . 3 pm rad. This has a negligible impact on the 416

calculation results. The capacity of a HDW to reduce horizontal emittance can be characterized by the emittance ratio of bare lattice with and without HDW. For a given length, the capacity of a HDW to reduce horizontal emittance depends on its magnetic field strength and period length.

ϵ xb = C q γ 2 I 5 b + I 5 HDW I 2 b + I 2 HDW − I 4 b C q γ 2 I 5 b I 2 b − I 4 b

= 1 + I 5 HDW

= 1 + λ 2 H ⟨ β x ⟩ B 5 H L H I 5 b 15 π 3 ( Bρ ) 5

∆ ϵ xHDW = ϵ xmax − ϵ xHDW ≈ ϵ xb − ϵ xHDW (A3) 425

∆ ϵ xHDW max = ϵ xb − ϵ xHDW ( B = B Hopt ) (A4) 427

The length of a straight section of SAPS is 6 m. We con- sider using only one straight section to achieve compensation for horizontal emittance because the straight section is very valuable for light source. We assume that the length of the HDW is 5 m. The emittance ratio varies under different val- ues of , as illustrated in Fig. . For a fixed

there exists an optimal B H ( B Hopt ) that minimizes the hor- 434

izontal emittance, corresponding to the maximum horizontal emittance reduction capacity of HDW ( xHDW max ). The

(For SAPS, ϵ xmin = 23 . 5 pm rad, ϵ xmax = 26 . 1 pm rad). By 397

employing a HDW, horizontal emittance can be consistently maintained at the value of xHDW max (For SAPS,

ϵ xmax − ∆ ϵ xHDW max = 23 . 4 pm rad ). 400

red dashed line represents the curve of for a given From the figure, it is evident that the HDW with a short pe- riod and high field strength has a greater capacity to reduce horizontal emittance. Taking into account the manufacturing technology of IDs, a superconducting wiggler with a period length of 48 mm and a peak field strength of 4.2 T is se- lected, as it has been demonstrated to be constructible [ The capacity of the HDW to reduce horizontal emittance first increases and then decreases with the increase in magnetic field strength. It reaches a maximum of 2.7 pm rad when

B H = B Hopt and decreases to zero when B H = B Hcri . 447

Magnet rigidity

Bρ = m 0 γv e (A5) 449

where, represent mass, velocity, and charge of elec- tron, respectively. Let the emittance ratio be equal to 1

ϵ xb = 1 + λ 2 H ⟨ β x ⟩ B 5 H L H I 5 b 15 π 3 ( Bρ ) 5

1 + B 2 H L H 2( I 2 b − I 4 b )( Bρ ) 2 = 1 (A6) 452

can be determined.

B Hcri = 3 �

The quantum excitation effect induced by HDW is propor- tional to fifth power of its magnetic field strength, while the radiation damping effect is proportional to square of its mag-

netic field strength. When 0 < B H < B Hcri , the radiation 460

damping effect induced by HDW exceeds the quantum exci- tation effect, resulting in a decrease in horizontal emittance.

When B H = B Hcri , the radiation damping effect cancels 463

out the quantum excitation effect, and horizontal emittance

remains unchanged. When B H > B Hcri , the quantum ex- 465

citation effect exceeds the radiation damping effect, resulting in an increase in horizontal emittance.

According to Equation ( is proportional to one- third power of ring natural emittance ( ); the lower natural emittance, the smaller . It implies that the quantum ex- citation effect induced by HDW increases more rapidly for low emittance rings. The lower natural emittance, the weaker horizontal emittance compensation capacity of HDW. When xHDW max

L H =5 m, the horizontal emittance ratio varies under different values of B H and λ H . The red dashed line represents the curve of B Hopt for a given λ H . The blue dashed line represents λ H =48 mm.

horizontal emittance variations induced by IDs are signifi- 474

cant, it is necessary to choose a sufficiently long HDW to ensure that its horizontal emittance compensation capacity is

adequate when B H = B Hopt . The lower natural emittance, 477

the longer required length of HDW under the same HDW pa- rameters, and the more pronounced the quantum excitation effect produced by HDW. Consequently, the variations of after a HDW horizontal emittance compensation remain evi- dent due to the strong quantum excitation effect of HDW.

Appendix B The compensation equations for HDW and VDW can be

expressed in a unified form. 485

I c 2 = � N

i =1 I 2 IDi + I 2 DW 1 + I 2 DW 2 (B1) 486

I c 5 = I 5 DW 1 + I 5 DW 2 (B2) 488

where, DW represents HDW or VDW. When ⟨ β x ⟩ = ⟨ β y ⟩ , 489

HDW compensation equations have same solutions as VDW

compensation equations. When ⟨ β x ⟩̸ = ⟨ β y ⟩ , we can make 491

VDW compensation equations be

I c 2 = � N

i =1 I 2 IDi + I 2 V DW 1 + I 2 V DW 2 (B3) 493

⟨ β y ⟩ ⟨ β x ⟩ I c 5 = I 5 V DW 1 + I 5 V DW 2 (B4) 495

In this way, the HDW and VDW compensation equations have the same magnetic field solutions.

B H 1 = B V 1 (B5) 498

B H 2 = B V 2 (B6) 499

The difference is that after compensation, emittance has dif- ferent values.

ϵ xV DW + ϵ yV DW = C q γ 2 I 5 b + � N i =1 I 5 IDi ( I 2 b + � N i =1 I 2 IDi + I 2 V DW 1 + I 2 V DW 2 ) − I 4 b

• C q γ 2 I 5 V DW 1 + I 5 V DW 2 I 2 b + � N i =1 I 2 IDi + I 2 V DW 1 + I 2 V DW 2

= C q γ 2 I 5 b + � N i =1 I 5 IDi ( I 2 b + � N i =1 I 2 IDi + I 2 HDW 1 + I 2 HDW 2 ) − I 4 b

I 2 b + � N i =1 I 2 IDi + I 2 HDW 1 + I 2 HDW 2

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