ECHO ENABLED HIGH MODE GENERATION FOR X-RAY FELS

E. Hemsing∗, SLAC National Accelerator Laboratory, Menlo Park, CA, USAA. Marinelli, Particle Beam Physics Lab, UCLA, USA

AbstractWe describe a simple technique based on a modified

echo-enabled harmonic generation (EEHG) scheme to ma-nipulate the three-dimensional electron beam microbunch-ing distribution in order to generate higher-order opticalmodes in an FEL. As with EEHG, the concept uses twomodulators and two chicanes to produce microbunching.However, in one of the modulators, the resonant inter-action with the laser has a well-defined transverse struc-ture that becomes strongly correlated to the longitudinalmicrobunching distribution. Both high-harmonic frequen-cies and high transverse mode numbers can be generatedthrough a transversely-dependent echo effect.

INTRODUCTIONModern x-ray free-electron lasers (FELs) have become a

remarkable tool for probing matter at Ångstrom length andfemtosecond time scales. In most X-ray FELs, the ampli-fication process starts from noise in the electron beam (e-beam) emission spectrum, which leads to limited temporalcoherence. To achieve fully coherent radiation, the echo-enabled harmonic generation (EEHG) scheme has beenproposed to frequency up-convert long wavelengths withhigh efficiency [1, 2, 3].

The standard EEHG scheme uses two modulators andtwo chicanes to generate longitudinal microbunching in thee-beam at the frequency k = nk1 +mk2, where k1 and k2are the frequencies of the lasers in the modulating sections.High harmonics can be obtained for large values of n orm by precise tuning of the modulation amplitudes and dis-persions. Because the light emitted by the bunched beamwill have a phase structure determined by the microbunch-ing distribution, the transverse phase and intensity profileof the modulating lasers are typically assumed constantover e-beam profile so that the final harmonic microbunch-ing structure has no transverse spatial dependence. Thisensures maximal harmonic bunching and also facilitatesemission into the Gaussian-like fundamental FEL mode,which is peaked on-axis and has an axisymmetric trans-verse phase profile.

There are, however, numerous theoretical and experi-mental contexts in which higher-order light beams are de-sirable or required. Of particular recent interest are opticalvortices, or light beams that carry well-defined, discrete az-imuthal phase lφ about the axis of propagation, where l isan integer called the topological charge. Starting from thework of Allen [4], who showed that certain types of opti-cal vortex modes carry discrete values lh̄ of orbital angularmomentum (OAM) per photon, there has been considerable

∗ ehemsing@slac.stanford.edu

R56(1) R56(2)

k2, l2k1, l1

k, l

A1(r,φ) A2(r,φ)

Figure 1: General setup for echo-enabled harmonic modegeneration.

work exploring the interaction of optical vortices with mat-ter. OAM can be transferred to trapped particles [5], atoms[6], or Bose-Einstein condensates [7]. Optical vortices alsohave a corresponding ringlike intensity profile that can fa-cilitate subwavelength microscopy[8], sub-diffraction fluo-rescence microscopy [9], imaging [10], and optical pumpschemes [11]. At x-rays, optical vortices have been pro-posed as a method to separate quadrupolar from dipolarelectronic transitions at K-edges [12].

The generation of low-order l OAM modes in FELs hasbeen examined previously [13, 14]. Here, we examine anew variation on the EEHG scheme that enables the gen-eration of coherent, high-order optical vortices at x-rays byup-conversion of the frequency and/or helical mode num-ber l of a microbunched e-beam. In this scheme, the en-ergy modulation imposed on the e-beam has a well-definedhelical dependence. This would be the case if the com-plex laser mode field profile is an OAM mode (as in Fig-ure 1), or, more generally, if the resonant modulator phasebucket has a helical phase structure [15]. In contrast tostandard EEHG, recoherence of the imprinted helcial mod-ulations here occurs as a correlated function of transverseposition in the e-beam after passage through the beamline.As a result, both the frequency and helical mode number lcan be up-converted through a process we refer to as echo-enabled harmonic mode generation, or EEHMG. The final,highly-correlated helical microbunched distribution can betailored to preferentially emit optical vortices with varyingl values and harmonics in a downstream radiator throughproper adjustment of the laser modulation profile, ampli-tude, and beamline dispersion.

A striking feature of the spatial recoherence effect inEEHMG is that it allows large up-conversion of the fre-quency and lmode either simultaneously, or independently.By upconverting both together, one can generate large lmodes at harmonics. Alternately, large frequency harmon-ics can be generated without changing the magnitude of lso that the helical distribution generated at one frequencycan be passed to a vastly different frequency.

SLAC-PUB-15780

Work supported in part by US Department of Energy under contract DE-AC02-76SF00515.

Presented at the 3rd International Particle Accelerator Conference (IPAC 2012)New Orleans, Louisiana, USA, May 20 - 25, 2012

FORMULATIONThe modified echo scheme is shown Figure 1. The e-

beam distribution at the entrance to the first modulatoris given by fi(r, φ, p), where r and φ are the radial andazimuthal coordinates, p = (E − E0)/σE is the scaledelectron energy with respect to the central beam energyE0 = γmc

2, and σE is the rms energy spread. We considerthe complex modulation amplitude in the first modulator tobe of the form A1(r, φ) = A1(r)Exp(il1φ), where l1 is aninteger that describes the azimuthal phase variation acrossthe profile, and A1 is the radially dependent field ampli-tude. At the exit of the first modulator section, the beamenergy is modulated such that an electron acquires a newenergy p′ = p + A1 sin(k1z + l1φ) where k1 = 2π/λ1 isthe laser frequency and z is the initial longitudinal positionof the electron in the beam. Due to the transverse depen-dence, the energy kick depends on the electron’s azimuthalposition φ radial position r. Note that the mathematicalform of the modulating field is in fact quite general, in thatit describes a modulation generated either by an OAM laserseed, or by the interaction of the e-beam with a Gaussianlaser at harmonics of a helical undulator, as in HGHMGseeding [14]. The beam then passes through a longitu-dinally dispersive section characterized by the matrix el-ement R(1)56 , which shifts the electron to the position z

′ ac-cording to z′ = z + B1p′/k1, where B1 = R

(1)56 k1σE/E0.

The beam is then modulated by the second laser, with fre-quency k2 = 2π/λ2, in the second modulator according thethe field distributionA2(r, φ) = A2(r)Exp(il2φ). The newenergy is p′′ = p′+A2 sin(k2z′+l2φ). The second chicane,with dispersion R(2)56 , transforms the longitudinal positionas, z′′ = z′ + B2p′′/k1, where B2 = R

(2)56 k1σE/E0. In

terms of the final variables at the end of the last dispersiveelement, the e-beam bunching factor at the frequency k andat the azimuthal mode l is given by

b(l)(k) =1

N0

∣∣∣〈e−ikz′′−ilφff (r, φ, z′′, p′′)〉∣∣∣ (1)where ff (r, φ, z′′, p′′) is the final e-beam distribution,and brackets denote averaging over the final coordinates.Transforming back to the initial coordinates, the exponen-tial term is then,

Exp [−ikz′′ − ilφ] =∑n,m e

−ikz−ilφ−i kk1 p(B1+B2))

×ein(k1z+l1φ)+im(k2z+l2φ+KB1p)

×Jn[A1(mKB1 − kk1 (B1 +B2))

]Jm

(−kA2B2k1

)(2)

where K = k2/k1. Averaging over z picks out the domi-nant microbunching frequencies,

k ≡ ak1 = nk1 +mk2, (3)

where a = n + mK. This is precisely the same fre-quency conversion relation as EEHG. Here, however, thereis also an l-mode conversion relation, found by averaging

4.6 4.7 4.8 4.9 5.0 5.1 5.20.000

0.001

0.002

0.003

Bunching Spectrum

λ [nm]

|b|2

l=-1

l=-1

l=-1

l=-1

l=-1

l=-1

Figure 2: Bunching spectrum of l = −1 helical mode inσt=10 fs beam optimized for n = −1, m = 50 with l1 =1, l2 = 0, A1 = A2 = 3, B1 = 16.94, B2 = 0.36,λ1 = λ2 = 240 nm

4.6 4.7 4.8 4.9 5.0 5.1 5.20.0000

0.0005

0.0010

0.0015

λ [nm]

Bunching Spectrum

l=50

l=51

l=49

l=48

l=47l=52

|b|2

Figure 3: Bunching spectrum optimized for n = −1, m =50. Both the OAM mode and frequency are up-convertedwith l1 = 0, l2 = 1,A1 = A2 = 3,B1 = 29.5,B2 = 0.61,λ1 = λ2 = 240 nm

over the azimuthal coordinate φ. Assuming an axisymmet-ric, uncorrelated initial e-beam distribution fi(r, φ, p) =N0(2π)

−1/2e−p2/2f0(r), the φ integral in (1) is straight-

forward, and the final azimuthal density mode excited inthe beam has the same up-conversion as the frequency:

l = nl1 +ml2. (4)

One can therefore simultaneously convert both the fre-quency and azimuthal mode number of the microbuncheddistribution in EEHMG. The final bunching factor is ex-pressed in general as,

b(l)n,m =

∣∣∣∣e−ξ2/2 ∫ Jn (A1ξ) Jm (−aA2B2) f0(r)rdr∣∣∣∣(5)

where ξ = −(nB1 + aB2). If the transverse variation ofthe fields is negligible over the e-beam such that l1, l2 = 0,A1 = A1 and A2 = A2, this reduces to the expression in[1] for standard EEHG. With A2 = B2 = 0, the generalformula for HGHMG is recovered [14].

EXAMPLES

Among the numerous possible combinations of helicalmode numbers l1 and l2 and harmonic numbers n and m,two examples illustrate the primary aspects. First consideran l1 = 1 modulation in the first stage of Fig 1 from afield of the form A1(r) = A1(

√2r/w0)

|l1|e−r2/w20 , where

w0 is the laser spot size. The e-beam is then modulated inthe second stage by a simple Gaussian laser with l2 = 0.From Eq. (4) the final azimuthal mode number is given byl = nl1, and thus does not depend on the potentially largeharmonic number m of the frequency up-conversion. Aswith EEHG, for m� 1, the bunching factor is maximizedwhen n = −1 and B1, B2 have the same sign. Thus, thefinal mode number is l = nl1 = −1. Figure 2 shows themicrobunching bunching spectrum (calculated from Eqs. 1and 2 for a finite beam with rms length cσt) from a helicalmodulation in the first stage. The spectrum is composedof different frequency spikes, each of which corresponds toan l = −1 helical modulation. In a radiator, this beam willemit light with a helical phase at the frequency (or frequen-cies) within the radiator bandwidth.

It is interesting to note that the final mode l has the oppo-site sign as l1 by virtue of the choice n = −1. This abilityto transform between different azimuthal mode numbers isa key feature of the 3D echo recoherence effect. Consider areversal of the previous arrangement, where now, the heli-cal modulation is performed in the second modulator. Thefirst modulation has no transverse dependence (l1 = 0),so that the final azimuthal mode number is then l = ml2,which can be made large during harmonic upconversion form� 1. The spectrum shown in Figure 3 displays the resultfrom this arrangement where each frequency peak is nowcharacterized by a distinct l value. Thus, a narrowband ra-diator can be tuned to pick out a single peak to emit a co-herent vortex with a specific value of topological charge.Large l modes are not amplified in FELs due to their largeemission angles and weak coupling, but can be emitted su-perradiantly in an undulator or from a CTR foil.

It is noted that, not only does the long term memory ofthe correlated phase space have to be preserved to obtainthe high-harmonic values, but the highly correlated heli-cal structure is also highly susceptible to wash out fromtransverse motion of the electrons. Optimization of thisscheme requires precise tuning of the betatron phase ad-vance (which should be a multiple of π ) in order to reestab-lish the 3D structure. Note that odd-π phase angles mir-ror the x and y electron positions across the z axis andthus change the sign of odd l modes (even modes are un-changed).

ACKNOWLEDGMENTS

Work supported by U.S. DOE under Contract Nos. DE-AC02-76SF00515 and DE-FG02-07ER46272.

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