10th Int. Partile Accelerator Conf. IPAC2019, Melbourne, Australia JACoW Publishing ISBN: 978-3-95450-208-0 doi:10.18429/JACoW-IPAC2019-WEPTS090 SUPPRESSION OF MICROBUNCHING INSTABILITY THROUGH DISPERSIVE LATTICE

∗ Biaobin Li1,2, Ji Qiang2, 1 University of Science and Technology of China, 230029, Hefei, China 2 Lawrence Berkeley National Laboratory, 94720, Berkeley, USA

Abstract is maintained after Linac-3, which calls for an isochronous dogleg design to avoid further bunch length compression. The microbunching instability from the initial small mod- ulation such as shot-noise can be amplified by longitudi- The longitudinal phase space smearing brought up by nal space-charge force and causes significant electron beam the coupling terms in BC1 can significantly smooth the up- quality degradation at the exit of accelerator for the next gen- stream density and energy modulation, thus suppresses fur- eration x-ray free electron laser. In the paper, we present an- ther instability amplifications in the following section of the alytical and numerical simulation studies of a novel method accelerator. Meanwhile, due to the existence of energy chirp using dispersion leakage from a quadrupole inside a bunch right before BC1, additional slice energy spread will also compressor chicane. be introduced by the coupling terms, which can effectively suppress the possible growth of the intrinsic and residual INTRODUCTION shot-noise modulation between the BC1 and the dogleg. Fol- lowing the equations in [12,15], the slice energy spread after The microbunching instability in linacs can limit the per- the BC1 is given as: formance of single-pass x-ray FELs by significantly de- s C h RA s grading the electron beam quality [1–5]. The conventional σδ ( 3) ≈ 1 1 51σx ( 2), (1) method to control the instability is to use a laser heater to increase the beam uncorrelated energy spread before bunch where C1 is the compression factor from s1 to s3, h1 and compressors [2, 6], which is typically tolerable for opera- σx (s2) are the energy chirp and the horizontal rms beam tion of self-amplified spontaneous emission (SASE) FELs. RA size before the BC1, 51 is the coupling term from the BC1 However, for seeded FELs it could limit the FEL gain [7]. section. Finally, the extra slice energy spread and the trans- Recently alternative techniques [8–13] based on transverse- verse emittance can be restored by adjusting the quadrupoles to-longitudinal coupling terms (R51, R52) or (R53, R54) are inside the dogleg, which cancels the induced couplings and proposed to suppress the microbunching instability. These makes the whole system an achromat. schemes are quite attractive because of the "reversible fea-

ture", which means the transverse emittance and slice energy 2019). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

spread can be recovered meanwhile effectively suppressing © the microbunching instability [14]. Among these methods, the scheme based on two bending magnets [9] is quite sim- ple without expensive hardware, however the use of bending magnets will change the beam line direction, which is not convenient and difficult to apply to the existing FEL facili- ties. Figure 1: (color online). Scheme Layout. In this paper, we propose a simple scheme to suppress the microbunching instability by inserting a quadrupole into a B A Here the linear transfer matrices R (R = R Tr R ) from four dipoles bunch compressor to introduce the longitudinal ′ s2 to s7 in (x, x , z, δ) coordinates are given in the above mixing terms to suppress the instability. Theoretical anal- A scheme, where R is the transfer matrix of s2 → s3, ysis and numerical simulations including longitudinal and transverse space charge are given to show the feasibility of RA RA RA * 11 12 0 16 + this scheme. .RA RA 0 RA / RA = . 21 22 26 /, (2) .RA RA R / 51 52 1 56,1 METHODS , 0 0 0 1 - We consider the machine layout shown in Fig. 1. One B quadrupole is placed in the middle of BC1 to leak out the R is the transfer matrix through the dogleg section s6 → s7, longitudinal mixing terms R51 and R52 for instability sup- RB RB RB pression. The energy chirp induced in Linac-2 is canceled * 11 12 0 16+ .RB RB 0 RB / out in Linac-3. For the energy chirp generated in Linac-1, it RB = . 21 22 26/, (3) .RB RB / 51 52 1 0 ∗ [email protected] , 0 0 0 1 - MC2: Photon Sources and Electron Accelerators WEPTS090 Content from this work may be used under the terms of the CC BY 3.0 licence (

This is a preprintT12 — the final version is published with IOP Beam Injection/Extraction and Transport 3325 10th Int. Partile Accelerator Conf. IPAC2019, Melbourne, Australia JACoW Publishing ISBN: 978-3-95450-208-0 doi:10.18429/JACoW-IPAC2019-WEPTS090

Tr is the middle accelerator section from s3 → s6

r11 r12 0 0 .* /+ .r21 r22 0 0 / Tr = . /, (4) . 0 0 (1 + ζ)/C2 R56,2E3/E5 / , 0 0 0 (C2E3/E7)/(1 + ζ)- where ζ = (C1 − 1)C2(R56,2E3)/(R56,1E5), Ej is the beam energy at position s . By using the symplectic condition of j Figure 2: (color online). Evolution of the beam transverse the transfer matrices (RT · S · R = S · E /E ) when f inal initial rms radius (left plot) and normalized transverse emittance taking acceleration effects into account and designing the (right plot) along the longitudinal position. dogleg section so that

A A A A A A A A [r21(R R − R R ) + r22(R R − R R )] RB = 52 11 51 12 52 21 51 22 , before the BC1 will still be amplified after electron beam 51 C ( + E E ) 2/ 1 ζ 7/ 3 passing through the dogleg because of the reversible feature. [−r (RA RA − RA RA ) − r (RA RA − RA RA )] RB = 11 52 11 51 12 12 52 21 51 22 By leaking some dispersion out from the reversible system, 52 , C2/[(1 + ζ)E7/E3] this part of amplification may also be suppressed [12]. We (5) here focus on the instability suppression in s2 → s7 sec- tion, neglecting the collective effects in Linac-1. Solving the entire transport system can be made an achromat with the microbunching integral equation provided in [17, 18] the linear transfer matrix from s2 to s7 and following the equations in [9], as the R56(s5→7) is zero, which drops off the amplification term due to the energy R11 R12 0 0 .* /+ modulation induced by collective effects inside Linac-3 and .R21 R22 0 0 / Rs = . R56,1 C1 R56,2 E3 / . also the coupled collective effects amplification term be- 2→7 . 0 0 (1 + ζ)/C2 + / . C2 E5 / tween Linac-2 and Linac-3, finally only two terms are left in 0 0 0 C2 E3 , (1+ζ)E7 - Eq. (7). Assuming an electron beam with zero energy chirp (6) and an initial current modulation factor b0 at the entrance to OPTICS DESIGN Linac-1, the final density modulation factor is given as

The optics design based on the first-order transfer map b[k(s7); s7] = b1[k(s7); s7] + b2[k(s7); s7], (7) without collective effects are given in this section. The 2 = normalized quadrupole strength (K1) in BC1 is 0.83/m where k(s) C(s)k0, C(s) is the compression factor and RA = k is the initial modulation wavenumber. Here b [k(s ; s )] with 0.2 m length in this example. This leaks out 51 0 1 7 7 2019).0 Any distribution of. this05 work must maintain attribution, to theR author(s), title ofA the work, publisher, and DOI = 0.18m for microbunching instability suppres- describes the evolution of modulation factor in the absence

© 52 sion. Generally, a stronger quadrupole in the BC1 will gener- of all collective effects, and is given as ate stronger instability suppression. However, the dispersion 2 2 2 b1[k(s7); s7] = b0 exp[−k (s7)Rˆ (s1→7)σ /2]; (8) leakage and beam tilt after the BC1 will make the beam 56 δ0 size in horizontal direction more difficult to control. The the second term b2 describes the amplification due to collec- maximum rms beam size after the BC1 is about 3 mm in tive effects between Linac-2 section, and is given as this example with the initial beam bunch length of 3 mm as shown in the left plot of Fig. 2. The initial beam trans- I(s3) b2[k(s7); s7] = ib0k(s7)Rˆ56(s3→7) Zˆ (s3→4) verse distribution is a uniform round cross section with 0.4 γ0 mm radius and 0.3 mm·mrad normalized transverse emit- k2 2 s 2 − D ( 3→7)σδ tance with a flattop current of 20 A. The initial uncorrelated × exp ( 0 0 ) (9) energy spread is 2 keV with zero energy chirp. The beam 2 2 is accelerated from 100 MeV to finally 5 GeV with a total −k H (s3→7)ǫ x,n × exp ( 0 ). compression factor of 72. The beam transverse size at the 2γ0 βx0 entrance of BC1 is matched back to the initial beam radius 0.4 mm. Evolution of the transverse normalized emittance The impedance term above is defined as is shown in the right plot of Fig. 2, horizontal emittance is sk 4πZ[k(τ); τ] Zˆ (s ) = dτ, restored after tuning the quadrupoles in the dogleg section j→k Z I Z according to Eq. (5) with the help of optimization tools of sj A 0 the Elegant [16]. where Z[k(τ); τ] is the impedance per unit length of col- THEORETICAL ANALYSIS lective effects and Z0 is the vacuum impedance, IA is the Alfvén current, σδ0 is the initial rms relative energy spread, As the whole section s2 → s7 is an achromat but with γ0 is the initial electron beam relativistic factor, ǫ x,n is the non-zero R56, the energy or density modulation induced normalized horizontal emittance, I(sj ) = C(sj )I0, I0 is the

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3326 T12 Beam Injection/Extraction and Transport This is a preprint — the final version is published with IOP 10th Int. Partile Accelerator Conf. IPAC2019, Melbourne, Australia JACoW Publishing ISBN: 978-3-95450-208-0 doi:10.18429/JACoW-IPAC2019-WEPTS090

initial current. The exponential damping to the modulation amplification due to initial energy spread and transverse-to- longitudinal coupling is given by:

2 2 2 D (s3→7) = U (s7, s3) + U (s3, s1), (10) H (s3→7) = T (s7, s3) + T (s3, s1), where T (s, τ) = ( β V − α W)2 + W 2, and x0 x0 Figure 3: (color online). Final longitudinal phase space

U(s, τ) = C(s)Rˆ56(s) − C(τ)Rˆ56(τ), when quadrupole in BC1 is turned off (left plot) and on (right plot). V (s, τ) = C(s)Rˆ51(s) − C(τ)Rˆ51(τ), (11)

W (s, τ) = C(s)Rˆ52(s) − C(τ)Rˆ52(τ),

where βx0 and αx0 are the initial horizontal Twiss param- eters. Here, we used the shorthand notation Rˆmn (s) = Rˆmn (s1 → s), where Rˆmn is defined as Rˆmn (sj→k ) = Rmn (sj→k )γ0/γj, in order to include acceleration effects [4]. SIMULATION

To verify the above scheme, we simulated an electron Figure 4: (color online). Left plot, Final projected current beam with an initial 0.6% current modulation at 140 µm when quadrupole in BC1 is turned off and on. Right plot, wavelength and the initial beam parameters described in the final horizontal normalized slice emittance (dashed line) and optics design section, transporting through the accelerator slice energy spread (solid line) with the quadrupole in BC1 system shown in Fig. 1 with the quadrupole in BC1 turned turned on. off and on using IMPACT [19]. We modified the source code to transport particles using first-order transfer map, but taking into account both the longitudinal space charge (LSC) and the transverse space charge (TSC) effects, to benchmark our analytical calculations. Figures 3 and 4 show the final longitudinal phase space distributions, the projected current, the final horizontal nor- malized slice emittance, and the slice energy spread at the exit of dogleg with residual energy chirp removed manually. With the help of the coupling terms leaking out from BC1, 2019). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI © both energy and current modulations are greatly suppressed. The slice energy spread after BC1 is enlarged to about 156 Figure 5: (color online). Microbunching gain defined as keV in the simulation, which agrees well with Eq. (1). The |b[k(s7); s7]/b0| when quadrupole in BC1 is turned off and additional slice energy spread greatly suppressed the insta- on. bility amplification process in s3→6 section, which showed smooth density profile from the simulation. The final rms un- correlated energy spread of the core beam section is restored compressor BC1 is proposed. The increased slice energy back to about 150 keV, slightly larger than 144 keV which is spread and the horizontal-to-longitudinal smearing after the the value of the initial uncorrelated energy spread enlarged BC1 are efficiently applied to suppress the density and en- by the total compression factor. The slice emittance in the ergy modulations in this scheme. The final slice energy core beam section shows about 60% growth when LSC and spread is almost restored back to the nominal energy spread TSC are taken into account. after the dogleg section. The factor that causes the final Figure 5 shows the total gain after dogleg with quadrupole slice emittance growth is still under study. More detailed in BC1 turned off and on based on the linear theory along work including coherent synchrotron radiation (CSR), the with IMPACT simulation results. The analytical gain and accelerator transverse structure wakefield effects, and the the simulation results agree quite well. The slight discrepan- nonlinear transfer map will be carried out in our future work. cies in short wavelength could be due to the approximation involved in the space-charge impedance model used in the ACKNOWLEDGMENTS linear theory. CONCLUSIONS This work was supported by the U.S. Department of En- ergy under Contract No. DE-AC02-05CH11231. One of In this paper, a simple microbunching instability suppres- the authors, B. L., thanks China Scholarship Council for sion scheme using a quadrupole inserting into a buncher financial support (CSC File No. 201706340072).

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