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PBPL Publications: 2009-00025

PUBLICATION

HELICAL ELECTRON-BEAM MICROBUNCHING BY HARMONIC COUPLING IN A

HELICAL UNDULATOR

E. Hemsing, P. Musumeci, S. Reiche, R. Tikhoplav, A. Marinelli, J.B. Rosenzweig, A. Gover

Abstract

Microbunching of a relativistic electron beam into a helix is examined analytically and in

simulation. Helical microbunching is shown to occur naturally when an e beam interactsresonantly at the harmonics of the combined field of a helical magnetic undulator and an

axisymmetric input laser beam. This type of interaction is proposed as a method to generate astrongly prebunched e beam for coherent emission of light with orbital angular momentum at

virtually any wavelength. The results from the linear microbunching theory show excellentagreement with three-dimensional numerical simulations.

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Helical Electron-Beam Microbunching by Harmonic Coupling in a Helical Undulator

E. Hemsing,1 P. Musumeci,1 S. Reiche,1 R. Tikhoplav,1 A. Marinelli,1,2 J. B. Rosenzweig,1 and A. Gover3

1Particle Beam Physics Laboratory, Department of Physics and Astronomy University of California Los Angeles,

Los Angeles, California 90095, USA2

Universita degli Studi di Roma La Sapienza, Via Antonio Scarpa 14, Rome, 00161, Italy3

Faculty of Engineering, Department of Physical Electronics, Tel-Aviv University, Ramat-Aviv 69978, Tel-Aviv, Israel(Received 13 August 2008; published 29 April 2009)

Microbunching of a relativistic electron beam into a helix is examined analytically and in simulation.

Helical microbunching is shown to occur naturally when an e beam interacts resonantly at the harmonics

of the combined field of a helical magnetic undulator and an axisymmetric input laser beam. This type of

interaction is proposed as a method to generate a strongly prebunched e beam for coherent emission of

light with orbital angular momentum at virtually any wavelength. The results from the linear micro-

bunching theory show excellent agreement with three-dimensional numerical simulations.

DOI: 10.1103/PhysRevLett.102.174801 PACS numbers: 41.60.Cr, 42.50.Tx, 42.60.Jf, 42.65.Ky

A relativistic electron beam (e beam) that is subject tothe free-electron laser (FEL) instability becomes micro-

bunched in density and velocity at the resonant interactionwavelength, and at harmonics. The modulated e beam canthen be used to emit radiation in various kinds of radiationschemes that preserve the characteristic frequency andgeometry of the microbunching in the e beam. In general,the microbunching interaction generates a purely longitu-dinal modulation, such that the subsequent radiative pro-cess produces emission at, or near, the fundamentaltransverse optical mode. The range of modulation wave-lengths obtainable in modern devices make this techniqueappealing for the generation of light with wavelengths thatspan many orders of magnitude. However, in some casesone may wish to produce radiation with the modulated

beam that has a specific higher-order phase or intensitystructure. Depending on the emission process, the e beammust be prepared with the correct microbunching structurein the modulator. Here we examine a simple modulatorarrangement that generates a helically microbunched ebeam, such that the electrons are arranged in a helix (ormultiply twisted helices) about the longitudinal axis. Sucha beam can be used for generation of coherent light thatcarries orbital angular momentum (OAM) as a result of theimprinted helical e beam density distribution on the opticalphase.

Light that carries OAM is a subject of intense researchfor a myriad of applications [15]. Coherent OAM modesallow the possibility of light-driven micromechanical de-vices or the use of torque from photons as an exploratorytool [6]. Laguerre-Gaussian (LG) modes of free-spacepropagation are of particular interest since they are knownto possess a well-defined value of angular momentum l@per photon due to an azimuthal component of the linearmomentum [7]. Emission of OAM light with an e beammay be accomplished through a variety of radiation pro-cesses, including harmonic emission from an FEL [8] or

through coherent transition radiation [9] or superradiantFEL emission [10,11] with helically microbunched beams.

Modes of this type may be particularly relevant for studywith modern optical and next-generation x-ray FELs withthe ability to probe the structure of matter down to A lengthand attosecond time scales. For future FEL light sources,the ability to directly generate intense higher-order LGmodes in situ would further extend the experimental andoperational capabilities.

In this Letter, we propose a scenario for generating ahelically microbunched e beam which utilizes harmoniccoupling in a helical undulator. We show that microbunch-ing of this type may be easily performed with an axisym-metric EM seed input in the modulator section of an opticalklystron [12]. The helical beam can then be used as a

source of OAM radiation in a downstream radiator. Alinear analysis of the harmonic bunching process is devel-oped in which the input field and density modulation aretreated as superpositions of orthogonal modes [13,14], andthe beam couples at harmonics via field gradients in theinput laser beam. Simple symmetry arguments are em-ployed to clearly delineate the spatial dependence of theharmonic fields that couple to the e beam. The method alsoillustrates the geometric structure of the resonant phase forarbitrary harmonics, the mode coupling for a cylindricallysymmetric e beam, and the resulting density and velocitymodulations excited in the e beam during the interaction.Selection rules are also derived for coupling betweenazimuthal modes in the e beam (l) and in the field (l0) atharmonics (h). These rules apply both to the modulatorsetup described here and to harmonic radiation in a FELwith a cold beam.

The interaction between the e beam and the input radia-tion field in an undulator operating as a buncher can bedescribed analytically using standard linearized FEL equa-tions in the small-signal regime if the radiation fields areinjected with sufficient power (or the interaction length is

PRL 102, 174801 (2009) P H Y S I C A L R E V I E W L E T T E R Sweek ending1 MAY 2009

0031-9007=09=102(17)=174801(4) 174801-1 2009 The American Physical Society
http://dx.doi.org/10.1103/PhysRevLett.102.174801http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://dx.doi.org/10.1103/PhysRevLett.102.174801

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short enough) that the total field energy is unaffected by thee beam throughout interaction. The fields are assumed tobe dominantly transverse. The electric field is given by themodal expansion

E?r; t ReX

q

Cqz~E?qr?e?eikzq!z!t; (1)

where ~E?q

is an eigenfunction of an infinite, ideal wave

guide (or optical fiber), Cqz is the mode amplitude, e? isthe field polarization vector and kzq! is the axial wavenumber of the mode q at the frequency !. The TEM modesare orthogonal and normalized, with mode power

Pqq;q0 kzq!=20!ReR

~E?q ~E?q0d

2r?, where

0 1=c20 is the permeability of free-space. The totalpower in the input field is then PT

PqjCq0j2Pq.

The e beam is described in a linear fluid model withdensity distribution

nr; t n0fr? Ren1rei!z=v0t; (2)where n0 is the electron density, v0

zc is the longitu-

dinal e beam velocity, fr? is the transverse densityprofile of the e beam and n1r is the spatial densityperturbation. The transverse variation of the charge densityis assumed small compared with the longitudinal variation.With the Lorentz force equation for a single componentplasma fluid in the presence of both the magnetostatic

undulator field Bw Refj~Bwjeweikwzg and the electro-magnetic input fields [Eq. (1)], the density evolution equa-tion in the cold-beam limit is written as [10]

@2

@z2 2pfr?

n1r g?2pfr?

0cK

2e!

Xq0

Cq0 z

kzq0 kw2 ~E?q0 r?ei1q0 z;

(3)

where p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

e2n0=2z 0mev

2

0

qis the relativistic longitu-

dinal plasma wave number and hq0 !=v0 kzq0 !

hkw is the detuning of the input mode relative to the ebeam energy at the harmonic h. The harmonic frequencies

at resonance are !h 2hkwc2z , with 2z 1 2z1,2 2z1 K2, K ej~Bwj=meckw is the undulator pa-rameter, j~Bwj is the field amplitude of the undulator andw

2=kw is the undulator wavelength. We neglect the

nonlinear contribution of the input field in the small-signalapproximation Kf=K2 ( 1 where Kf ejE?j=!mec.Polarization alignment between input EM field and theelectron motion in the undulator is given by g? e? ez ew, where the polarization vector of the helicalundulator is

e w ex iey=ffiffiffi

2

p; (4)

which corresponds to either right () or left () circularpolarization along z. Maximal coupling from polarization

alignment (g? 1) is obtained when the optical fieldpolarization is e? ez ew which describes a left-circularly polarized wave (positive helicity [15]) matchedwith a right-circularly polarized undulator.

The e beam is coupled to the input field modes throughthe ponderomotive fields, given by the right-hand sideof Eq. (3). Since the resonant interaction is sustainednear the synchronous condition only for the first harmonic

(1

q0 0), there is no higher-harmonic coupling of theelectron beam to first order. Coupling at higher harmonicscan be excited in a cold beam through the higher-orderresonant interaction between the electrons and the gra-dients of the EM fields. This contribution is calculated by

Taylor expanding the field modes ~E?q about the averagecentroid trajectory of an electron r?,

~E?qr? X1n0

1

n!Re~r?weikwz rn ~E?qr?; (5)

where the transverse coordinate of an electron is r? r? Re~r?we

ikwz

and

r is the gradient operator whichacts on r?. The electron wiggling amplitude is ~r?w K=kwzez ew. Insertion of Eq. (5) into Eq. (3) in-troduces additional oscillatory wiggling terms whichcouple to higher-harmonic resonances. In the regime wherej~r?wj is much smaller than the characteristic transverse ebeam size r0, we approximate r? r? in Eq. (5) andobtain an expression for the Taylor expanded harmonicfield components that are resonant with the integer har-monics h,

~Eh?qr? eih1

@r

i

r@

h1 X1

m

0

1mm!

m

h

1

!

iK

2ffiffiffi

2p

kwz

2mh1r2m? ~E?qr?; (6)where r2? 1r @rr@r 1r2 @2 is the transverse Laplacianoperator. The higher-order terms (m > 0) in the Taylorseries can be neglected if the wiggle amplitude of the elec-trons is small compared to the optical beam size. WithEq. (6), harmonic interactions are described in Eq. (3) by

replacing ~E?q0 r?ei1q0 z with ~E

h?q0 r?e

ihq0 z.

It is convenient to express the density perturbation n1ras a sum over the expansion eigenmodes such that theorthogonality of the basis can be used to compactly writethe density evolution Eq. (3) in terms of spatial modulationamplitudes. The density modulation expansion is,

n1r kw0

e

Xj

ajz~E?jr?: (7)

The substituted harmonic terms in Eq. (3) are then multi-

plied by ~E?qr? and integrated over the transverse coor-dinates, yielding an expression for the evolution of thedensity mode amplitudes at harmonics:


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d2

dz2aqz 2p

Xj

Fq;jajz Xq0Dhq;q0Cq0 ze

ihq0 z:

(8)

The coupling of the e beam with arbitrary transversedensity distribution fr? to the harmonic fields is givenby the coupling coefficient

D hq;q0 g?

2

pcKkzq0! kw2

2!kwFh

q;q0 (9)

and the overlap of the e beam and the input fields is

Fhq;q0

Rfr?~Eh?q0 r?~E?qr?d2r?R j~E?qr?j2d2r? ; (10)

where Fq;j F1q;j with m 0 in Eq. (6). Equation (8)describes the density bunching evolution of the e beam atthe harmonics with arbitrary initial conditions on thebunching aq0, velocity modulation daq0=dz and inputfield amplitudes Cq0. The second term on the left handside of Eq. (8) is the contribution of the longitudinal spacecharge. Note that with the coupling turned off (D

hq;q0 0),

Eq. (8) describes the coupled harmonic oscillations of thedensity modes due only to longitudinal space-charge ef-fects. This effect can become important, especially at lowerenergies, and may be useful for calculation of the densityand velocity bunching amplitudes of the e beam over a driftbefore or after the undulator.

The coupling selection rules and resulting e beam mod-ulations [Eq. (7)] that are excited in the interaction with aharmonic field input mode can be examined with a basiswith the form

~E?qr? Rlpreil; (11)where the mode index takes on two values, q p; lcorresponding to the radial (p) and azimuthal (l) modes.Assuming an axisymmetric e beam profile fr? fr,the integral over in Eq. (10) is straightforward,

Fhp;l;p0;l0 / 2l0;lh1: (12)

This shows how the coupling between azimuthal densitymodulation modes in the e beam, l, and the azimuthalmodes in the EM field, l0, depend on the harmonic numberh and the direction of undulator polarization, . In thesimple case of an input EM mode with arbitrary l

0intro-

duced to the undulator at the fundamental frequency (h 1), the corresponding azimuthal mode l l0 is excited inthe e beam. At the second harmonic frequency (h 2), theaxisymmetric EM field mode l0 0 will generate a heli-cally bunched e beam with azimuthal mode number l 1 for an RH undulator and l 1 for an LH undulator.Thus, an initially axisymmetric e beam and axisymmetricinput EM field mode can be used to produce a helicallymicrobunched e beam at the second harmonic (Fig. 1). It isinteresting to note that the slowly varying amplitudes of the

helical density distribution suggest an intriguing analoguewith the manifestation of the Berry phase along the azimu-thal coordinate [16,17].

The e beam density bunching is parametrized by thebunching factor [18]. To accommodate helical bunching ofthe e beam, we define a modified bunching factor blzwhich incorporates the amplitude of bunching due to thelth discrete azimuthal mode:

blz 1

n0R

fr?d2r?Z

n1reild2r?: (13)

This gives blz l;0 when the density perturbation isn1r n0fr?, i.e., complete longitudinal bunching atthe l 0 e beam mode. Helical e beam density bunchingcurves at the second harmonic are shown in Fig. 2 for threedifferent values of the total input power of an l0 0Gaussian free-space mode at 2c=!2 10:6 m.In each case, there is good agreement between the solu-

tions to Eq. (8) with an LG basis [10] and 3D numericalparticle tracking simulations performed with TREDI [19]

FIG. 1 (color online). Buncher device where the axisymmetricinput field imprints a helical density and velocity modulation onthe e beam through the harmonic interaction in a helical undu-lator.

FIG. 2 (color online). Injection of a Gaussian input mode ontoa Gaussian e beam for a RH undulator excites l 1 densitybunching at the second harmonic ( 10:6 m). The associ-ated helical bunching curves for a 350 m spot size Gaussianmode with waist at z 0:15 m on a r0 350 m e beam are ingood agreement with TREDI simulations (points) with 26,K 0:65, w 2 cm.


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for dominant bunching at the l 1 density mode.Significant bunching is observed near the exit of the35 cm RH undulator with only 15 MW of input power.Bunching into modes l 1 is negligible. Note that theanalytic description assumes r0 ) j~r?wj but still workswell in this case where r0 $ 5j~r?wj. The departure of ourlinear theory from simulations (which include nonlinearaspects) is evident for very large bunching (jblj > 35%), asin the case of 125 MW of input power.

The effects of space charge are ignored in Fig. 2, but canplay a crucial role in the evolution of the microbunching.

Figure 3 includes space-charge effects in the case of injec-tion of l0 1, l0 0, and l0 1 free-space LG modeswith a Gaussian e beam in a RH undulator. The corre-sponding axial velocity modulation modes (l 2, l 1, and l 0 modes excited in the e beam, respectively)manifest in the growth of the bunching factors downstreamof the 20 cm undulator, which reach a maximum and thendecrease due to space charge.

Maximum peak microbunching is achieved with theinjection of the natural l0 1 field, which bunches thebeam into the l 0 mode, i.e., longitudinally separatedmicrobunches (b0). This l

0 1 optical mode excites largerbunching than the other modes at the 2nd harmonic be-

cause it has larger resonant coupling to the e beam. Thisrelates directly back to the 2nd harmonic FEL scenariowhere this optical mode achieves the highest gain [8]. Ingeneral for FELs with this coupling, a helical undulatorwith right (left) circular magnetic field polarization ampli-fies an l0 h 1 (l0 1 h) azimuthal mode with left(right) circular polarization. The spin and orbital compo-nents of the photon angular momentum in the emittedphotons in the FEL add since the projection of each ontothe axis of propagation has the same sign. The jl0j > 0 EM

modes described here are characterized by a null intensityon-axis, as expected [2022]. Generation of intense lightwith OAM by helical beams in FELs is the subject of more

detailed future work, and should include transverse beta-tron motion and emittance effects beyond the cold-beam

model presented here.This research was supported by grants from Department

of Energy Basic Energy Science Contract No. DOE DE-

FG02-07ER46272 and DE-FG03-92ER40693, and Officeof Naval Research Contract No. ONR N00014-06-1-0925.

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

5

10

15

20

25

30

FIG. 3 (color online). Longitudinal space-charge effects in-cluded in bunching evolution for the injection of l0 1, 0,and 1 modes onto a Gaussian e beam for a RH undulator at h 2. The optical spot size is 500 m for each mode, with the waistpositioned halfway through the 20 cm undulator (vertical line).The beam displays the onset of plasma oscillations downstream.

Input power is 15 MW and the e beam current is 300 A with 26, K 0:65, w 2 cm.


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