Correlated Energy-Spread Removal with Space Charge for High-Harmonic Generation

E. Hemsing, A. Marinelli, and G. MarcusSLAC National Accelerator Laboratory, Menlo Park, California 94025, USA

D. XiangDepartment of Physics and Astronomy, Key Laboratory for Laser Plasmas (Ministry of Education),

Shanghai Jiao Tong University, Shanghai 200240, China(Received 10 May 2014; published 22 September 2014)

We study the effect of longitudinal space charge on the correlated energy spread of a relativistic high-brightness electron beam that has been density modulated for the emission of coherent, high-harmonicradiation. We show that, in the case of electron bunching induced by a laser modulator followed by adispersive chicane, longitudinal space charge forces can act to strongly reduce the induced energymodulation of the beam without a significant reduction in the harmonic bunching content. This effect maybe optimized to enhance the output power and overall performance of free-electron lasers that producecoherent light through high-gain harmonic generation. It also increases the harmonic number achievable inthese devices, which are otherwise gain-limited by the induced energy modulation from the laser.

DOI: 10.1103/PhysRevLett.113.134802 PACS numbers: 41.60.Cr, 41.75.Ht, 41.85.Ct, 42.65.Ky

Free-electron lasers (FELs) use relativistic electronbeams to produce widely tunable light with exceptionalbrightness at wavelengths down to hard x rays for a broadrange of studies [1]. At suboptical wavelengths, FELstypically operate in self-amplified spontaneous emissionmode, or SASE [2–4], where the amplification process isinitiated by noise in the electron beam. This shot noise iscorrelated only over extremely short time scales (typically afew femtoseconds for x-ray FELs), so for long electronbeams the temporal and spectral SASE emission exhibits alarge number of uncorrelated spikes and large statisticalfluctuations.To improve the FEL performance, high-gain harmonic

generation (HGHG) is a technique used to stabilize thepower output and to generate fully coherent radiation [5–7].In HGHG, an external laser imprints a coherent modulationin the electron beam (e beam) that then seeds FELamplification of light at harmonics of the laser frequency.Because the modulation can be correlated over much longertime scales than SASE, the emitted light can have a muchnarrower bandwidth. In the standard HGHG setup, the laserfirst modulates the e-beam energy according to the trans-formation η ¼ η0 þ δη sinðks0Þ, where η0 ¼ Δγ0=γ ≪ 1 isthe relative energy deviation of an electron within the ebeam with average energy E ¼ γmc2, s0 is the electron’slongitudinal position within the beam, λ ¼ 2π=k is the laserwavelength, and δη is the laser modulation amplitude.The e beam then propagates through a dispersive sectioncharacterized by the transport matrix element R56, whichconverts the energy modulation into a periodic densitymodulation according to s ¼ s0 þ R56η. Electrons are piledinto sharp peaks longitudinally that are spaced at the laserwavelength, described by the distribution,

fsðsÞ ¼ 1þ 2X∞n¼1

bn cosðnksÞ; ð1Þ

where the density modulation is quantified by the e-beambunching factor bn. At a given harmonic h, the bunchingfactor for a beam with an initially uncorrelated Gaussianenergy spread σE is [5],

bh ¼ e−ðhBÞ2=2Jhð−hABÞ; ð2Þ

where A ¼ δη=ση0 is the laser energy modulation amplituderelative to the relative energy spread ση0 ¼ σE=E, B ¼kR56ση0 is the scaled dispersion, and Jh is the Besselfunction of order h.The harmonic number in HGHG FELs is typically

limited to h ∼ ρ=ση0 where ρ is the FEL frequencybandwidth at saturation [8]. From Eq. (2) the optimalenergy modulation to obtain significant bunching at thefrequency hkc is A ¼ 1=B≃ h. However, the FEL satu-rates when the e-beam energy spread approaches ρ, whichis typically around 10ση0 in modern devices. Thus, theHGHG harmonic number is limited to approximatelyh≲ 10–15 to obtain high FEL output power with goodtemporal coherence [9].Here, we propose and examine a scheme that enhances

the power output from HGHG FELs, and can also boost theaccessible harmonic number. Referred to as quieted highgain harmonic generation, or QHG, this technique exploitscollective longitudinal space charge (LSC) effects gener-ated by the sharp periodic density peaks to relinearizeportions of the modulated e-beam phase space whilepreserving the harmonic bunching. The LSC effect takesplace in a dedicated short drift or focusing section

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immediately downstream of the dispersive chicane whichmay already exist as part of an HGHG FEL setup to helpmatch the beam into the undulator (see Fig. 1). In thissection, the phase space at the chicane exit [Fig. 1(a)] ismodified by the LSC forces produced by the densitybunching. The result [Fig. 1(b)] is a beam with sharpdensity spikes that coherently seed the HGHG process, butwith a reduced projected energy spread between the spikesthat facilitates lasing up to full saturation power. Inprinciple, this approach may be used to increase theharmonic jumps in cascaded HGHG sections, or couldreplace “fresh bunch” seeding techniques [10] in certainregimes to enable use of the whole beam to further reducethe saturated spectral bandwidth.LSC effects in high-brightness beams have been exam-

ined recently in various different contexts, including themicrobunching instability (see, e.g., [11,12]), LSC ampli-fiers [13,14], shot noise suppression schemes [15–18],and as a method to generate a train of high peak-currentbunches [19,20]. Physically, LSC effects arise from theself-fields generated by longitudinal density perturbationsin the beam. Electrons near each density peak, which have awidth of Δs≃ λ=2A [21], receive an energy kick from therepulsive forces; those close in front of the peak experiencea positive energy kick, while those close behind have their

energy reduced [Fig. 1(c)]. In HGHG, the combination ofthe laser energy modulation and chicane dispersion used togenerate harmonic bunching generates a nearly ideal initiallongitudinal phase space distribution for the LSC forces toremove the positive energy chirp between density spikes.We refer to this portion as the working portion (WP) ofthe beam (i.e., the region everywhere outside the ∼3Δsregion at the spike), where the bulk of the harmonic FELamplification process takes place. In e beams with suffi-ciently small transverse and longitudinal emittances, theQHG effect works because the particle dynamics can bedominated by the motion in the energy space rather than inthe longitudinal space.We note that this HGHG enhancement scheme differs

from other techniques that propose secondary phase-shiftedlaser modulations to partially reduce the energy spread[22–24]. QHG exploits the fact that the LSC forces aregenerated by the bunching structure and thus naturallyphase locked to the correlated energy modulation. Thisavoids precise timing constraints between successive lasermodulation sections. Further, the shape of the harmonicLSC fields closely mirrors the phase space in the WP,allowing a nearly complete cancellation of the inducedenergy modulation in QHG [e.g., Fig. 1(d)].We describe the effect in a simplified model in which 3D

effects can be neglected in the high-frequency limit where λis small compared to the transverse beam size rb, namely,ξ ¼ krb=γ ≫ 1 [11,25]. Transverse motion of particles isalso neglected, assuming that the physical drift length isless than γhβi=hkϵn, where hβi is the average beta functionand ϵn is the normalized emittance. In this regime, thegeneral equations that describe evolution of the electronenergy and longitudinal position in the e-beam frame aregiven as [19],

dηdz

¼ qγmc2

Ez;dsdz

¼ η

γ2;

dEz

ds¼ qn0

ϵ0fsðsÞ; ð3Þ

where −q is the charge of an electron, n0 is the beamvolume density, and Ez is the longitudinal space chargefield. Inserting the longitudinal distribution of the bunchedbeam from Eq. (1), the space-charge fields due to theperiodic bunching structure are

Ez ¼2qn0ϵ0k

X∞n¼1

bnsinðnksÞ

n: ð4Þ

It is convenient to rescale the variables and definep ¼ η=ση0 as the scaled energy, τ ¼ kpz as the plasmaphase advance starting from the chicane exit,k2p ¼ q2n0=mϵ0c2γ3, and θ ¼ ks as the phase position ofa particle in the beam with respect to the laser. From theLSC fields in Eq. (4), we obtain two nonlinear equationsthat describe the evolution:

(a)

(c) (d)

(b)

FIG. 1 (color online). Top: Layout of QHG scheme. (a) Phasespace of the density modulated beam at the exit of the chicane(blue dots). The LSC fields give a kick to the nearby particlesduring a drift length τ (green arrows) that reverses the inducedenergy chirp, resulting in the distribution in (b). The initial sharpcurrent spike shown in (c) produces the force distribution (green)that cancels the modulation and is essentially unchanged duringthe QHG drift. In (d), the projected energy distribution of thebeam before the drift (solid line) shows the characteristic doublehorn shape, but returns to a narrow spike with a significantlyreduced energy spread afterward (dashed line).

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dpdτ

¼ 2

α

X∞n¼1

bnsin nθn

;dθdτ

¼ αp: ð5Þ

Note that bn ¼ bnðτÞ. We have also defined

α ¼ kση0kpγ2

; ð6Þ

which is the energy-spread parameter and sets the scale forthe overall dynamics. It can be interpreted as the ratio of thelongitudinal displacement due to thermal motion in aplasma period to the laser wavelength. The QHG regimerequires α ≪ 1 such that the thermal motion can beneglected on the time scale of the plasma period [25].The QHG scaling is obtained by linearizing the dynami-

cal equations for small changes in the phase space positionof a particle during the drift. Namely, we assumeθðτÞ≃ θð0Þ þ ΔθðτÞ, where ΔθðτÞ ≪ 1, and θð0Þ is theinitial position at τ ¼ 0. Similarly, the change in energy ispðτÞ≃ pð0Þ þ ΔpðτÞ where Δp < pð0Þ. The scaling of abeam optimized for density bunching at the harmonich ¼ A ≫ 1 is dominated by the n ¼ 1 space-charge termin Eq. (5). Accordingly, in this linear model, particles nearthe position (pð0Þ; θð0Þ) ¼ ðA; π=2Þ experience the largestchanges in energy and phase position. The optimal bunch-ing factor b1≃−AB=2¼−1=2 is constant in the ΔθðτÞ≪1limit, so both equations can be integrated directly over ashort drift Δτ to give Δp≃ −Δτ=α and Δθ≃ αAΔτ. Thedrift length over which the initial energy modulationinduced by the laser Δp≃ A is approximately canceledby space charge effects is then

Δτ≃ αh: ð7Þ

The corresponding change in phase is approximatelyΔθ≃ αhΔτ ≃ ðαhÞ2.The natural dispersion of particles with different energies

requires that

Δθ < 1=h; ð8Þ

such that the change in position is less than the desiredharmonic wavelength λh ¼ 2π=kh ¼ λ=h in order to pre-serve bunching at h. The approximate limit on the scaleddrift length is then Δτ <

ffiffiffiffiffiffiffiffi1=h

p. Together, Eqs. (7) and (8)

constrain the parameter α to

α <ffiffiffiffiffiffiffiffiffiffi1=h3

q: ð9Þ

If satisfied, Eq. (9) states that the bunching factor at theharmonic h remains essentially unchanged during the driftΔτ≃ αh, and that the correlated energy spread in the WPinduced by the laser is approximately minimized.

An example of the QHG process is shown Fig. 2, wherewe follow the evolution of the scaled energy spreadσp ¼

ffiffiffiffiffiffiffiffiffihp2i

pand corresponding phase space of a beam

along a drift. Results are obtained from particle simulationsgoverned by the 1D equations in Eqs. (5). We takeα ¼ 0.005, and the beam starts with an initial modulationof A ¼ 20 ¼ 1=B [Fig. 2(a)] that optimizes bunching ath ¼ 20. The beam is shown by the blue particles, some ofwhich are obscured by the overlaid red particles thatidentify only particles in the WP. During the drift theenergy modulation is steadily reduced by the LSC forces[Figs. 2(b)–2(e)]. The relative rms energy spread of all ofthe particles (blue line, upper plot) reaches a minimum nearτ≃ 0.8αA, [see Fig. 2(d)]. However, considering only theparticles in the WP, the energy spread continues to decrease(red line, upper plot) and is minimized at τ≃ αA [Fig. 2(e)],where the distribution of the red particles appears flat withan energy spread almost exactly equal to the initial value(σp ¼ 1). This is the end of the QHG process, at which pointthe transverse size is expanded to arrest the LSC effects.Otherwise the beam overshoots the minimum in energyspread [Fig. 2(f)]. Along the drift, the bunching factor atthe 20th harmonic (Fig. 2 top, inset) decreases slightlydue primarily to longitudinal dispersion, but remains

0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

0 2 4 6−40

−20

0

20

40

0 2 4 6−40

−20

0

20

40

0 2 4 6−40

−20

0

20

40

0 2 4 6−40

−20

0

20

40

0 2 4 6−40

−20

0

20

40

0 2 4 6−40

−20

0

20

40

(c)(a) (b)

(f)(e)(d)

0 0.5 18

9

10

11

FIG. 2 (color online). Top: Evolution of the relative energyspread for the whole beam (blue) and the WP just outside thedensity spike (red). Inset: Evolution of the bunching factor for the20th harmonic. Bottom: Phase space evolution along the drift.

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significantly larger than the shot noise such that the coherentharmonic signal dominates in the FEL.To evaluate the performance of the QHG beam in an

FEL, we performed time-dependent 1D FEL simulationsfor a simple case. Results are shown in Fig. 3 and arecompared with HGHG and SASE. The beam is assumed tohave a flat initial current profile (before modulation) andlases in an FEL tuned to the 20th harmonic of the lasermodulation, which covers the entire beam. The simulationcode is a variant of standard 1D FEL codes and calculatesthe performance using the high-gain FEL evolution equa-tions based on the exact input particles. However, it alsoallows for each of the slices that define the resonant FELwavelength λh to have a different number of electrons, aswould be the case in the physical density modulated beam.The simulations also permit electrons to pass betweendifferent slices each undulator period to account for theeffects of dispersion inside the FEL. This feature is neededto accurately model the modulated beams in QHG andHGHG where particles have large energy deviations andshift significantly in phase. The dynamics are calculated ingeneral form for an FEL specified by ρ according to thecommonly scaled variables for relative energy η=ρ, phaseθh ¼ khs, complex field amplitude a, and longitudinalcoordinate 2kuρz, where λu ¼ 2π=ku is the undulatorperiod [26].In this example we take ρ ¼ 2 × 10−3 and ση0 ¼ ρ=10,

similar to parameters in modern soft x-ray FELs. InFig. 3(a) the radiation power along the undulator is shownfor each scenario. The unmodulated SASE beam (blue line)reaches saturation (hjaj2i≃ 1) after ∼20 gain lengthsLg ¼ 1=2

ffiffiffi3

pkuρ as expected from high-gain FEL theory.

The HGHG beam fully saturates at a reduced power level(red line) and has a longer gain length due to theuncompensated energy structure from the laser. TheQHG beam reaches full saturation power (black line) atthe halfway mark thanks to the removed energy chirp inthe WP. The effect of reducing the energy modulation inQHG is also evident in the temporal and spectral structureof the output radiation [Figs. 3(b) and 3(c)]. The pulsefrom QHG has a roughly flat temporal profile at saturationand a narrow-band spectrum with rms bandwidthΔk=kh ¼ 0.03ρ. This is compared with the SASE and

HGHG cases at their respective saturation levels whichboth exhibit multiple temporal and spectral spikes and havea full bandwidth of ∼ρ. The premodulated beam in theHGHG case thus only reduces the output power withoutreducing the bandwidth. In the QHG case, however, theoutput pulse is temporally coherent and the bandwidth isdetermined simply by the beam length, which is 1000λh orapproximately five coherence lengths lc ≃ λh=

ffiffiffiffiffiffi2π

pρ. This

is consistent with the number of temporal spikes seen in theSASE case. The narrow QHG spectrum results because thecooperation length Lc ¼ λhLg=λu ≃ 23λh is slightly longerthan the distance between density spikes. This enablesphase information in the amplified radiation to be commu-nicated between density spikes over each gain length,which flattens out the temporal profile and narrows thespectrum. Narrowing the spectrum further is just a matter ofusing longer e beams.The differences between the QHG and HGHG beams

are illustrated by inspection of the phase space evolutionduring FEL amplification. Shown in Fig. 4(a), the HGHGbeam retains a large energy correlation in the WP. Asamplification of the 20th harmonic signal develops, only aportion of the particles near the center of the WP fullyparticipate in the interaction. Particles with the largestenergy deviation are outside the seeded FEL bandwidth andare thus detuned in frequency. Further, dispersion acts toslowly decompress the local chirp and shift the developingenergy modulation to longer wavelengths. This competeswith the resonant FEL wavelength, and beat waves in theenergy modulation emerge. The resulting frequency com-petition appears to suppress gain and leads to a multispikespectral structure. Conversely, the QHG beam in Fig. 4(b)has a flattened WP that permits monochromatic amplifi-cation. The particles in the WP contribute uniformly tothe FEL instability which generates more output power.

FIG. 4 (color online). Evolution of the e-beam phase space in(a) HGHG and (b) QHG during amplification of the 20th laserharmonic.

(a) (b) (c)

FIG. 3 (color online). FEL output power (a), temporal profile(b), and spectrum (c) for SASE, HGHG, and QHG scenarios.

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Dispersion also acts in this case to tilt the phase space of theparticles with the large residual energy deviation at eachspike. This leads to an observable dip in the harmonicbunching factor early on (not shown), but at that point thecoherent amplification process has been sufficiently devel-oped that the FEL output is essentially unaffected.We note that e beams characterized by non-Gaussian

energy spreads have been predicted to permit FEL satu-ration at h > 15 in HGHG [9]. The impact of the QHGscheme on such beams is worthy of additional investiga-tion, as this approach could further improve the harmonicrange and FEL performance through similar reductions inthe WP energy spread.Finally, 3D effects are likely to play an important role in

the implementation of this scheme in practice, particularlyfor beams where ξ ¼ rbk=γ ≲ 1 because the harmonicEz fields are each reduced and reshaped by a factorFnðr; ξÞ ≤ 1 which depends on the transverse distribution[11,25]. For a flattop distribution of radius rb, for example,this factor is Fnðr; ξÞ ¼ 1 − nξI0ðnξr=rbÞK1ðnξÞ, and 3Deffects can be included by inserting Fn into the sum inEq. (5). The drift length in (7) is then increased toΔτ≃ αh=F1ð0; ξÞ, and the energy-spread parameter con-straint is α <

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF1ð0; ξÞ=h3

p. For a flattop beam α can be

written in practical units as α ¼ ξση0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγIA=4I0

p, where

IA ¼ 17 kA and I0 is the peak current in the beam.From this scaling, a 1 GeV beam with I0 ¼ 5 kA andση0 ¼ 10−4, for example, requires a Δτ=kp ¼ 2.7 m drift ata spot size of rb ¼ 150 μm to reduce the WP energy spreadby a factor of 4 for the 20th harmonic of a 240 nm lasermodulation (ξ ¼ 2, α ¼ 0.008). In general, these effectsdepend sensitively on the beam parameters and are thesubject of future studies.

The authors thank Y. Ding for useful discussions. Thiswork was supported by the U.S. Department of Energyunder Contract No. DE-AC02-76SF00515 and the NationalNatural Science Foundation of China under GrantNo. 11327902.

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