Octupole based current horn suppresion in multi-stage bunch compression with emittance growth correction
Nicholas Sudar, Yuantao Ding, Yuri Nosochkov, Karl Bane, Zhen Zhang
OOctupole based current horn suppression in multi-stage bunch compression withemittance growth correction
N. Sudar, Y. Ding, Y. Nosochkov, K. Bane and Z.Zhang
SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA (Dated: September 7, 2020)High brightness linear accelerators typically produce electron beams with peaks in the head and/ortail of the current profile. These current horns are formed after bunch compression due to non-linearcorrelations in the longitudinal phase space and the higher order optics of the compressor. It hasbeen suggested that this higher order compression can be corrected by inserting an octupole magnetnear the center of a bunch compressor. However, this scheme provides a correlated transverse kickleading to growth of the projected emittance. We present here a method whereby octupole magnetsare inserted into two sequential bunch compressors. By tuning a π betatron phase advance betweenthe two octupoles, the correlated transverse kick from the first octupole can be corrected by thesecond, while providing a cummulative adjustment of the higher order compression. I. INTRODUCTION
Advancements in the production of high peak bright-ness electron beams have revolutionized the field of ultra-fast science through the advent of the X-Ray Free Elec-tron Laser (FEL). The ability to reach femtosecond levelX-Ray pulse durations at a growing number of FEL fa-cilities allows for the study of molecular and atomic scalestructures as well as femtosecond scale dynamical pro-cesses [1]. Meeting the growing demands of the scientificcommunity requires continued improvement in electronbeam quality and repetition rate at such facilities.In order to achieve the electron beam peak current re-quired to drive the FEL interaction, high brightness lin-ear accelerators typically employ multiple 4-dipole chi-cane bunch compressors. The total compression is lim-ited by non-linear correlations in the electron beam lon-gitudinal phase space. These correlations stem from RFcurvature[2], longitudinal space charge (LSC)[3], coher-ent synchrotron radiation (CSR) [4, 5], and resistive wallwakefields [6, 7]. Non-linear compression from the secondorder energy chirp is typically adjusted with a harmoniccavity [8]. Additional methods can be employed to reducethese correlations, [9–15]. However, if left unchecked,higher order compression can lead to the production ofhorns in the current profile as the head and/or tail ofthe electron beam are over-compressed compared to thelinear compression in the core of the beam [16, 17].These current horns can produce significant CSR in thebunch compressor causing further distortions in the lon-gitudinal phase space. This correlated longitudinal en-ergy variation can result in a correlated transverse kickleading to growth in the projected emittance, reducingFEL performance as portions of the beam will not bematched ideally to the transverse focusing lattice. Fur-thermore these current horns can produce significant en-ergy modulation from longitudinal space charge and re-sistive wall wakefields downstream [18]. In the Linac Co-herent Light Source-II (LCLS-II) linac the electron beammust be transported from the linac exit through a 2 kmbypass line, requiring control of these collective effects, [19]. Although in a normal conducting linac with a typ-ical beam rate of 100 Hz or lower, these current hornscould be removed with collimation in a dispersive section[20], this is not a viable option for high repetition ratesuperconducting linacs due to the significant increase inradiated power from beam losses.It was shown in [21–23], that the growth of currenthorns can be suppressed by adjusting the higher ordercompression with an octupole magnet inserted in a bunchcompressor. This scheme relies on placing an octupole ata point of significant dispersion, providing a transversekick correlated with the longitudinal beam coordinate.The octupole kick is then translated into a path lengthdifference through the remainder of the chicane, provid-ing an adjustment of the bunch compressor’s third orderlongitudinal dispersion, U . However, the correlatedtransverse kick from the octupole will generally remainimprinted on the beam after the bunch compressor caus-ing significant growth of the projected emittance.We present here a scheme whereby inserting octupolesin two successive bunch compressors, the projected emit-tance growth can be corrected. This scheme relies on thecancellation of the octupole kick from the first bunchcompressor (BC1) by the octupole in the second one(BC2). Provided that the beam undergoes a π beta-tron phase advance between octupoles, the second oc-tupole kick can be made equal and opposite to the firstone while providing an additive contribution to the total U of the system. This allows for effective suppressionof current horns without significant degradation of thetransverse beam quality.In this paper, we present a study of this scheme usingcurrent profile shaping in the LCLS-II superconductinglinac as a possible application. Section II gives an analyt-ical description of the longitudinal phase space transfor-mation from an octupole inserted in a bunch compressor.In section III we provide an analytical description of theemittance correction scheme. Section IV shows ELEGANT [24] simulations of a potential configuration for LCLS-II.Section V gives further simulation studies for optimiza-tion of the scheme. Section VI provides a discussion of a r X i v : . [ phy s i c s . acc - ph ] S e p FIG. 1. Illustration of the proposed scheme showing the firstlinac section (L1), first bunch compressor (BC1) with em-bedded octupole (O1), second linac section (L2) and secondbunch compressor (BC2) with embedded octupole (O2) alignment tolerances.
II. U FROM AN OCTUPOLE
An electron passing through an octupole magnet withnegligible vertical offset relative to the magnetic centerwill receive a horizontal kick depending on its horizontaloffset given by [25]: x (cid:48) = − B (cid:48)(cid:48)(cid:48) Bρ Lx ≡ − K Lx (1)Here L is the octupole length and K is the octupole’sgeometric strength. Placing an octupole at a point ofhigh dispersion we assume an electron’s transverse off-set is dominated by its energy deviation from the centralenergy, δ ≡ ( γ − (cid:104) γ (cid:105) ) / (cid:104) γ (cid:105) . Placing the octupole in thecenter of a chicane bunch compressor, the transverse off-set at the octupole entrance can be expressed by the R from a simple dogleg: x oct = R δ ≈ − θ ( l b + l d ) δ (2)Here l b is the chicane magnet length and l d is the driftlength between the 1st and 2nd, and 3rd and 4th chi-cane magnets, and θ is the bending angle of the chicanedipoles. The octupole kick then depends on the initialenergy offset as: x (cid:48) oct = 16 K L θ ( l b + l d ) δ (3)The path length difference induced by this kick at thechicane exit is given by the R from a simple dogleg.The transformation of an electron’s longitudinal position, s , due to the octupole kick is given by:∆ s foct = R x (cid:48) oct = − K L θ ( l b + l d ) δ (4)Here note we adopt the convention that the head of thebeam points to more negative s . The dispersive terms ofthe chicane including the octupole are given by: R ≈ − θ ( l d + 23 l b ) T ≈ − R U ≈ − K Lθ ( l b + l d ) + 2 R (5) We can approximate the transformation of the initialcurrent profile first considering the evolution of the lon-gitudinal phase space coordinates through the chicane,( s i , δ i ) → ( s f , δ f ). For an electron beam with a non-linear correlated energy chirp described by components, h i , this is given by: s i = s δ i = δ + h s + h s + h s + ...s f = s i + R δ i + T δ i + U δ i + ...δ f = δ i (6)In the limit of negligible initial energy spread the trans-formation of the current profile, I i , can be approximatedby I f ≈ (cid:0) ∂s f ∂s i (cid:1) − I i . This can be expressed in terms of thelinear compression factor, C and non linear compressionfactors, c and c : I f ( s f ) ≈ C χ ( s f ) I i [ s i ( s f )] (7) C ≡ (cid:18) ∂s f ∂s i (cid:12)(cid:12)(cid:12)(cid:12) s i =0 (cid:19) − c ≡ C (cid:18) ∂ s f ∂s i (cid:12)(cid:12)(cid:12)(cid:12) s i =0 (cid:19) c ≡ C (cid:18) ∂ s f ∂s i (cid:12)(cid:12)(cid:12)(cid:12) s i =0 (cid:19) χ ( s f ) = 1 + c s i ( s f ) + c s i ( s f ) + ... (8)Current horns will exist in the final current profile ap-proximately where the contribution from non-linear com-pression, given by χ ( s ), goes to zero. The adjustment of χ ( s ) from the octupole allows for a positive non-linearcompression factor along the bunch.Figure 2 shows an ELEGANT simulation of the nomi-nal compression scheme for the LCLS-II beamline witha strong current horn at the beam head (I) and a hypo-thetical configuration with an octupole magnet insertedat the center of BC2 again simulated in
ELEGANT (II).In both cases the incoming beam is generated from
IM-PACT [26, 27] simulations of the LCLS-II injector. Forthe single octupole case, K L = −
775 m − giving a to-tal U = 7 .
44 m − , with remaining electron beam andchicane parameters given in Table 1. For both cases, χ ( s )is shown both as functions of the initial and final beamcurrent.Here we see that in the nominal case, χ ( s i ) goes tozero within the initial current profile, leading to horns inthe final phase space. The octupole effectively suppressesthe current horns and can be used to produce a flat cur-rent profile with a factor of two increase in the core peakcurrent.Figure 2 also shows the s vs x (cid:48) phase space at the exitof the second bunch compressor. From this we see thatthe correlated octupole kick is preserved at the chicaneexit. The normalized projected emittance after BC2 is (cid:15) xn = 8 . µ m compared with (cid:15) xn = 0 . µ m for thenominal LCLS-II case.Assuming a gaussian energy distribution with RMS en-ergy spread, σ δ , the growth of the projected emittancefrom the octupole kick can be approximated by: (cid:15) x (cid:15) x ≈ (cid:115) β x (cid:15) x (cid:18) K L ( l b + l d ) θ σ δ (cid:19) (9)Here (cid:15) x is the projected geometric emittance at the chi-cane entrance and β x is the electron beam beta functionat the octupole.Figure 3 shows the emittance growth versus octupolestrength from ELEGANT simulations using an idealizedgaussian beam with purely linear correlated chirp. This isdone for both BC1 and BC2 using electron beam and chi-cane parameters given in Table 1. Comparison with theanalytical estimate from Eq. 9 shows qualitative agree-ment.
III. EMITTANCE CORRECTION
The projected emittance growth problem from the sin-gle octupole scheme can be mitigated by splitting the U needed to achieve the desired longitudinal shapingbetween BC1 and BC2.In this double octupole configuration, the correlatedkick induced by the first octupole is transported to thesecond octupole. Tuning the lattice between the two oc-tupoles to provide a π betatron phase advance in thebend plane of the chicanes, the x (cid:48) kick from the firstoctupole is inverted. The second octupole strength canthen be set to cancel the first octupole kick while provid-ing additional U .We can consider this transformation, writing the x (cid:48) kick at the first octupole in terms of the linear compres-sion factor of the first chicane, C , first octupole strengthand length, K (1)3 L , chicane magnet and drift lengths andbend angle. Writing this kick in terms of the compressedbeam coordinate, s , after propagation through half ofBC1 gives: x (cid:48) ( s ) = 16 K (1)3 L (cid:18) l d + l b l d + l b (cid:19) θ (cid:18) C − C + 1 (cid:19) s (10)Considering a more general case of nπ betatron phaseadvance between the two octupoles, the x (cid:48) kick from thefirst octupole at the entrance of the second octupole isgiven by: x (cid:48) ( s ) = ( − n +1 K (1)3 L × (cid:18) l d + l b l d + l b (cid:19) θ (cid:18) C ( C − C + C (cid:19) (cid:115) β E β E s (11)Here C is the linear compression factor of the secondchicane, β and β are the values of the beta function FIG. 2. (I) nominal LCLS-II beam (II) beam with currenthorns suppressed by single octupole showing: (a) Currentprofile (blue) and χ ( s i ) (red) at linac entrance. (b) Currentprofile (blue) and χ [ s i ( s f )] (red) at BC2 exit. (c) Longitu-dinal phase space after BC2 with current profile (blue) forreference. (d) s vs x (cid:48) phase space at BC2 exit. The bunchhead is on the left in all plots. in the bend plane and E and E are the values of thecentral beam energy at the first and second octupolesrespectively.The x (cid:48) kick provided by the second octupole is givenby: x (cid:48) ( s ) = 16 K (2)3 L (cid:18) l d + l b l d + l b (cid:19) θ (cid:18) C − C C + C (cid:19) s (12)The net x (cid:48) kick is cancelled when the second octupolekick is equal in magnitude and opposite in sign to thetransported kick from the first octupole. In reality, forLCLS-II the sign of the BC2 chicane bend angle is op-posite that of BC1, requiring an n π betatron phase ad-vance for the same cancellation effect. For illustrativepurposes we assume throughout that the BC1 and BC2chicanes have the same sign of the bend angle. In theprovided example, the phase advance between the two FIG. 3. Change in projected emittance varying octupolestrength from
ELEGANT simulations considering an idealbeam with a linear chirp and gaussian energy distribution(points) and the analytical expression from Eq. 9 (line) forthe BC1 octupole (top) and BC2 octupole (bottom). octupoles is 3 π .The cancellation requirement gives a condition for theratio between the two octupole strengths, α K : α K ≡ K (2)3 L K (1)3 L = (cid:18) ( l d + l b )( l d + l b )( l d + l b )( l d + l b ) (cid:19) (cid:18) θ θ (cid:19) × (cid:18) C ( C − C − C (cid:19) (cid:115) β E β E (13)For a given total U ≡ U tot we can split the octupolestrengths according to Eq. 13 and 5. K (1)3 L = 6 U tot + 24 θ ( l d + l b ) + 24 θ ( l d + l b ) θ ( l b + l d ) + α K θ ( l b + l d ) (14)Figure 4, shows the octupole kick from the first (a) andsecond (b) octupoles, the first octupole kick transportedto the entrance of the second octupole (c) and the kickcancellation at the second octupole exit (d) from EL-EGANT simulations of the LCLS-II linac configurationdescribed in Section IV. The analytical estimates fromequations 10-12 are shown for comparison. The discrep-ancy between the analytical expressions and simulationscan be attributed to second order chromatic focusing ef-fects in the transport between octupoles. This is high-lighted in particular by the comparison between Eq. 11and the transported first octupole kick from simulation.Additional discussion of the emittance correction schemecan be found in [28].
FIG. 4. From
ELEGANT simulations: (a) Correlated kickfrom the first octupole ( s vs x (cid:48) ) with the analytical estimatefrom Eq. 10 (white dashed), (b) Kick from the second oc-tupole with first octupole off with the analytical estimate fromEq. 12 (white dashed), (c) Kick from the first octupole trans-ported to the second octupole entrance with the analyticalestimate from Eq. 11, (d) Combined kick from the first andsecond octupoles with octupole strengths chosen to minimizeprojected emittance of the core of the beam. The bunch headis on the left in all plots. IV. LCLS-II DOUBLE OCTUPOLECONFIGURATION
From the single octupole case shown in section II, aninitial choice for the BC1 and BC2 octupole strengths isfound by splitting the total U between the two chi-canes according to Eq. 14. Some adjustment of the total U must be made to obtain an identical current profiledue to the change in higher order compression at BC1.Figure 5, shows the longitudinal phase space at the exitof BC2 and undulator entrance with the double octupoleconfiguration. The s vs x (cid:48) phase space at the BC2 exitshows the correction of the octupole kick. Parametersare given in Table 1.We define the core of the beam as the region within >
10% of the peak current, with this region shown bythe white dashed lines in Figure 5. The ratio between oc-tupole strengths for emittance growth cancellation can beadjusted in simulations to minimize the projected emit-tance over the core of the beam.Figure 6, shows the optimal octupole strength ratiofrom
ELEGANT simulations maintaining the total U .This gives an optimal ratio α K = 0 . α K = 0 . (cid:15) xn = 0 . µ m compared with (cid:15) xn = 0 . µ m at the BC1 entrance and (cid:15) xn = 3 . µ m TABLE I. Parameters for LCLS-II double octupole schemeParameter ValueBC1 e-beam energy (MeV) 250e-beam charge (pC) 100e-beam chirp @ BC1 entrance ( m ) 12.97emittance @ BC1 entrance (cid:15) x,y ( µm ) 0.3, 0.3BC1 R (mm), bend angle (mrad) -47.45, 95.66BC1 compression factors C , c , c K (1)3 L ( m − ) -4652.69BC1 U (m) 3.254Beta function @ BC1 octupole β x (m) 11.05BC2 e-beam energy (MeV) 1500e-beam chirp @ BC2 entrance ( m ) 8.34BC2 R (mm), bend angle (mrad) -44.93, 46.81BC2 total compression factors C , c , c K (2)3 L ( m − ) -511.11BC2 U (m) 4.811Beta function @ BC2 octupole β x (m) 55.1core emittance @ BC2 exit (cid:15) xn,yn ( µm ) 0.43, 0.41 from the single octupole case. The remnant increase inprojected emittance can be again attributed to secondorder focusing between octupoles. Varying α K by ± V. LINAC CONFIGURATIONCONSIDERATIONS
In choosing a linac configuration for the double oc-tupole scheme it is advantageous to minimize the emit-tance growth after the first bunch compressor. This canreduce the remnant emittance growth. In the case of theLCLS-II beamline, this is also necessary to avoid lossesin a collimator downstream of BC1.We can gain some insight towards an optimal linacconfiguration using Eq. 9 to estimate the BC1 emit-tance growth for varying linac and chicane parametersthat give approximately the same final current profile.This is done by constraining the total linear compression,BC1 and BC2 non-linear compression factors (Eq. 7), theelectron beam energy at BC2, and the emittance cancel-lation condition (Eq. 13). With current horn suppressionfrom the double octupole scheme, the energy modulationfrom linac wakefields dominates over LSC and CSR. Forchanges in the BC1 compression, the linac wakefield be-tween BC1 and BC2 can be approximately scaled by theBC1 linear compression factor.Figure 7 shows the estimated projected emittancegrowth after BC1 varying the BC1 R , electron beamlinear chirp at the entrance of BC1, and the electronbeam energy at BC1. Here we only consider linac con-figurations within the specifications of the LCLS-II linac.From this we see a general trend that the BC1 emittancegrowth decreases with increasing chirp and decreasing R . Furthermore, for the same chirp and R the emit-tance growth decreases with increasing electron beam en- FIG. 5. (Top) The longitudinal phase space at the exit ofBC2, with current profile (blue), the white dashed lines definethe core of the beam. (Middle) s vs x (cid:48) at the exit of BC2.(Bottom) Longitudinal phase space at the undulator entrancewith current profile (blue) and energy lineout (yellow). Thebunch head is on the left in all plots. ergy at BC1. The parameters for the LCLS-II examplecase are chosen to optimize the final phase space at theundulator entrance. Notably, Figure 7 illustrates thatthe desired longitudinal shaping can be achieved over awide range of linac parameters. A discussion of otherpossible LCLS-II linac configurations is given in [29]. VI. ALIGNMENT TOLERANCE
Misalignment of the octupole will result in a normaland skew sextupole like kicks. To see this we can write K ( ) / K ( ) c o r e ϵ x ( mm - m r ad ) FIG. 6. The normalized core projected emittance varyingthe octupole strength ratios maintaining the total U from ELEGANT simulations (points) with quadratic fit (line) the octupole field shifted by ∆ x and ∆ y : B y = B (cid:48)(cid:48)(cid:48) x − ∆ x ) − x − ∆ x )( y − ∆ y ) ] B x = B (cid:48)(cid:48)(cid:48) x − ∆ x ) ( y − ∆ y ) − ( y − ∆ y ) ] (15)Assuming this misalignment is small compared to thedispersion and the electron beam’s vertical offset relativeto the magnetic center of the octupole is dominated bythe vertical shift, we can write the field keeping the lowestorder terms in ∆ x and ∆ y : B y = B (cid:48)(cid:48)(cid:48) x − x ∆ x ) B x = − B (cid:48)(cid:48)(cid:48) x ∆ y ) (16)For the octupole inserted in the chicane, the horizontaloffset at the octupole is dominated by dispersion. Thehorizontal and vertical kick induced by the shifted oc-tupole is then given by: x (cid:48) = 16 K Lθ ( l b + l d ) δ − K Lθ ( l b + l d ) δ ∆ xy (cid:48) = − K Lθ ( l b + l d ) δ ∆ y (17)The additional sextupole kick will remain imprinted onthe transverse phase space causing additional emittancegrowth. Provided the octupole kicks are cancelled, theemittance growth at the exit of BC2 from a misalignedoctupole is approximately given by:∆ (cid:15) x (cid:15) x ≈ β x (cid:15) x (cid:18) K Lθ ( l b + l d ) σ δ ∆ x (cid:19) ∆ (cid:15) y (cid:15) y ≈ β y (cid:15) y (cid:18) K Lθ ( l b + l d ) σ δ ∆ y (cid:19) (18)Here l b , l d , θ , K and L refer to the BC1 or BC2 chicaneand octupole parameters, β x,y and (cid:15) x,y are the beta func-tion and geometric emittance at the octupole, and σ δ isthe RMS energy spread at the chicane entrance. This FIG. 7. The estimated emittance growth after BC1, varyingBC1 R and linear chirp at the BC1 entrance for BC1 energy230 MeV (top), 250 MeV (middle) and 270 MeV (bottom). expression gives 10% x emittance growth for ± µ moffset of the BC1 and BC2 octupoles and 10% y emit-tance growth for ± µ m offset of the BC1 octupole and ± µ m of the BC2 octupole.These alignment tolerances are relaxed when we con-sider emittance growth in the core of the beam includingthe additional emittance growth observed in ELEGANT simulations. Figure 8, shows the emittance growth vary-ing the x and y offset of the BC1 and BC2 octupoles in
ELEGANT . From this we see 10% y-emittance growth for ± µ m offset of the BC1 octupole and ± µ m for theBC2 octupole and 10% x-emittance growth for ± µ moffset of the BC1 and BC2 octupoles. We also observethat the remnant emittance growth caused by 2nd orderchromatic focusing is reduced by the sextupole kick fromx offset of the octupoles. Furthermore, the x-emittancegrowth due to BC1 sextupole kick is corrected by an op-posite x offset of the BC2 octupole. This effect can pos-sibly be utilized to reduce the final emittance growth. VII. CONCLUSION
The double octupole scheme demonstrates effectivesuppression of current horns by adjusting the higher or-der compression of a linac with multi-stage bunch com-pression. This allows for the generation of a flat currentprofile with an increase in the achievable peak currentand preservation of the transverse beam quality. Thismethod can be incorporated in most existing high bright-ness linacs, and could be improved upon if consideredin the initial design of such a facility. In the providedLCLS-II example, for an octupole length, L = 0 . ACKNOWLEDGMENTS
The authors would like to thank Agostino Marinelli,Tessa Charles and Paul Emma for useful discussion. Thiswork was supported by U.S. Department of Energy Con-tract No. DE-AC02-76SF00515.
FIG. 8. a) The x normalized emittance at the BC2 exit of thecore of the beam varying the BC1 octupole x offset (blue) andBC2 octupole x-offset (yellow) b) The y normalized emittanceat the BC2 exit of the core of the beam varying the BC1octupole y offset (blue) and BC2 octupole y-offset (yellow).c) The change in x normalized emittance at the BC2 exit ofthe core of the beam varying both BC1 and BC2 x-offset. d)The change in y normalized emittance at the BC2 exit of thecore of the beam varying both BC1 and BC2 y-offset. [1] P. Neyman, W. B. Colson, S. C. Gottshalk, A. M. M.Todd, J. Blau, and K. Cohn, Free electron lasers in 2017,in
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