Echo-enabled tunable terahertz radiation generation with a laser-modulatedrelativistic electron beam

Zhen Wang, Dazhang Huang, Qiang Gu, and Zhentang Zhao*

Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China

Dao Xiang†

Key Laboratory for Laser Plasmas (Ministry of Education), Department of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, China(Received 3 June 2014; published 2 September 2014)

A new scheme to generate narrow-band tunable terahertz (THz) radiation using a variant of the echo-enabled harmonic generation is analyzed. We show that by using an energy chirped beam, THz densitymodulation in the beam phase space can be produced with two lasers having the same wavelength. Thisremoves the need for an optical parametric amplifier system to provide a wavelength-tunable laser to varythe central frequency of the THz radiation. The practical feasibility and applications of this scheme aredemonstrated numerically with a start-to-end simulation using the beam parameters at the Shanghai DeepUltraviolet Free-Electron Laser facility (SDUV). The central frequency of the density modulation can becontinuously tuned by either varying the chirp of the beam or the momentum compactions of the chicanes.The influence of nonlinear rf chirp and longitudinal space charge effect have also been studied in ourarticle. The methods to generate the THz radiation in SDUV with the new scheme and the estimation of theradiation power are also discussed briefly.

DOI: 10.1103/PhysRevSTAB.17.090701 PACS numbers: 41.75.Fr, 41.60.Cr

I. INTRODUCTION

As has been widely used in radar, security, communi-cation, medical imaging, etc., THz radiation has drawn a lotof attention all over the world. Many studies have beendone on the generation of THz radiation [1–4]. Accelerator-based free electron laser (FEL) technology, such as thefourth generation light source, can produce radiation withhigh intensity, high peak power, and coherence, whichmight be a great candidate for the generation of THzradiation. In recent years, a new scheme of employing theecho effect previously observed in hadron accelerators wasproposed [5] and experimentally tested [6] to generate highharmonics of FEL radiation, and the bunching coefficientsof this scheme were also analyzed in detail [7] to give us aclearer picture of the mechanism behind it. Providing theadvantages of the echo-enabled harmonic generation FELsuch as compact size, tunable frequency, etc., a method ofusing two lasers to modulate the electron beam to generatedensity modulation at THz frequency has been proposedand analyzed in Refs. [8–10]. In this scheme, the relativisticelectron beam is first sent into a modulator to interact with a

laser with the wave number k1. Then the beam is trans-mitted into the following modulator to interact with anotherlaser with the wave number k2. The energy modulationwill be converted into density modulation when the beampasses through a dispersion section. In this situation, thedensity modulation at wave number k ¼ nk1 þmk2 can beachieved, where n and m are nonzero integers. By varyingthe wavelength of those two lasers and the parameter ofdispersion, one can vary the central frequency of the beamdensity modulation conveniently. In this scheme, an opticalparametric amplifier is needed to provide two differentlasers (e.g., 800 nm and 1560 nm in this scheme) tomodulate the electron beam, respectively.As briefly mentioned in [9], one may generate the

continuously tunable density modulation with two lasershaving the same wavelength, which is different from thescheme mentioned above, if there is a chicane between thetwo modulators, similar to the echo-enabled harmonicgeneration scheme. A chirped electron beam is modulatedin themodulator at wave number k0. Thewave number of thebeam density modulation will be compressed to k1 ¼ C1k0and its high harmonics when the beam passes through thefirst dispersion sectionwith smallR56. Then the beamwill besent into another same modulator to interact with a laser atthe samewave number k0. After the beam passes through thesecond dispersion section, the energy modulation is con-verted into density modulation at the final wave numberk ¼ nC1C2k0 þmC2k0, where n and m are nonzero inte-gers. It is easy to find out that one may vary the final

*zhaozhentang@sinap.ac.cn†dxiang@sjtu.edu.cn

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published article’s title, journal citation, and DOI.

PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS 17, 090701 (2014)

1098-4402=14=17(9)=090701(7) 090701-1 Published by the American Physical Society

frequency of density modulation continuously by varyingthe chirp of the beam or R56 of the first dispersion section.In this article, the result of the generation of density

modulation is discussed theoretically in Sec. II. Then inSec. III, the simulation results of the scheme to generateTHz density modulation are presented. The THz generationin SDUV via the new scheme is briefly discussed inSec. IV. Summaries and conclusion are given in Sec. V.

II. GENERATION OF DENSITY MODULATIONIN THEORY

As introduced in Sec. I (for more details, see Ref. [8]),the layout of the scheme to generate THz density modu-lation in a relativistic electron beam with two differentlasers at wave numbers k1 and k2 is shown in Fig. 1(a1),which consists of two modulators and a dispersion section.The beam is modulated by two different lasers at wavenumbers k1 and k2 when the beam passes through twomodulators, respectively. After the beam passes through thedispersion section, the energy modulation is converted todensity modulation at wave number

k ¼ nk1 þmk2: ð1Þ

Consider an initial Gaussian energy distribution as

f0ðpÞ ¼N0ffiffiffiffiffiffi2π

p e−p2=2; ð2Þ

where N0 is the number of electrons per unit length, p ¼ðE − E0Þ=σE represents the dimensionless energy deviationwith central energy E0 and slice energy spread σE. After thebeam passes through the dispersion section, the distributionfunction becomes

ffðz; pÞ ¼N0ffiffiffiffiffiffi2π

p exp

�−1

2½p − A1 sinðk1z − BpÞ

− A2 sinðk2z − KBpþ ϕÞ�2�; ð3Þ

where A1 ¼ ΔE1=σE, A2 ¼ ΔE2=σE represent the energymodulation amplitude of each modulator and B ¼R56k1σE=E0, K ¼ k2=k1.

The bunching factor bn;m at each harmonic, defined ashe−ikn;mzi is

bn;m ¼ jJn½ðnþ KmÞA1B�× Jm½ðnþ KmÞA2B�e−ð1=2Þ½ðnþKmÞB�2 j: ð4Þ

We show the phase space evolution of the beam at eachsection in Fig. 2 with λ1 ¼ 800 nm, λ2 ¼ 900 nm, A1 ¼ 5,A2 ¼ 5, and R56 ¼ 2 mm to illustrate the dynamics occur-ring when the beam passes through the beam line. We caneasily see that energy modulations at wave number k1 andk ¼ nk1 þmk2 are achieved at the exit of modulator 1 andmodulator 2, respectively. The energy modulation is con-verted to density modulation after the beam passes throughthe dispersion section and the microbunching forms at THzfrequency if we take the suitable parameters. The asym-metry of the beam density distribution comes from the lackof electrons at both sides of the phase space.As a comparison, we show the new scheme to generate

THz density modulation, which is briefly mentioned in [9],in Fig. 1(a2). It is easy to find out that there is a dispersionsection between two modulators, which is similar to theecho-enabled harmonic generation scheme. Based onEq. (1), the density modulation in the beam is achievedat the exit of the second dispersion section at the wavenumber

k ¼ nC1C2k1 þmC2k2; ð5Þ

where C1 and C2 are the bunch compression factorprovided by the first and the second dispersion section,and the distribution function is [11]

FIG. 1. Layout of two schemes to generate THz densitymodulation in a relativistic electron beam.

FIG. 2. Phase space evolution of the electron beam at variouslocations, where p is the dimensionless energy derivation inEq. (2). (a) Exit of modulator 1; (b) exit of modulator 2; (c) exit ofdispersion; and (d) final density profile.

WANG et al. Phys. Rev. ST Accel. Beams 17, 090701 (2014)

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ffðξ; pÞ ¼N0ffiffiffiffiffiffi2π

p exp

�−1

2fp − A2 sinðkξ − KB2pþ ϕÞ

− A1 sin½ξ − ðB1 þ B2Þp

þ A2B1 sinðKξ − KB2pþ ϕÞ�g2�; ð6Þ

where B1;2¼Rð1;2Þ56 k1σE=E0, C1¼1=ð1þhRð1Þ

56 Þ, and C2 ¼ð1þ hRð1Þ

56 Þ=½1þ hðRð1Þ56 þ Rð2Þ

56 Þ�.The bunching factor bn;m is thus written as

bn;m ¼ jJnf−A1½nB1 þ ðnþ KmÞB2�g× Jm½−ðnþ KmÞA2B2�e−ð1=2Þ½nB1þðnþKmÞB2�2 j: ð7Þ

In this scheme, the R56 of the first dispersion is verysmall (about 1–2 orders of magnitude smaller than the R56

of the second dispersion). That is to say we can ignore nB1

in Eq. (7) for nB1 ≪ nB2 so that Eq. (7) becomes the samewith Eq. (4). It is easy to point out that for an electron beamwithout energy chirp, the bunching factors of those twoschemes are almost the same. The central frequency ofdensity modulation in the beam for those two scheme arethe same as well for h ¼ 0 leading to C1 ¼ 1 and C2 ¼ 1.The advantage of this new scheme over the old one is

that we can use one laser with the beam split into twoidentical pulses to modulate the electron beam twice in twomodulators, respectively, when there is energy chirp in thebeam. As a result, the complexity of the whole apparatus isreduced. In the following, we will discuss the detail of thenew scheme.Consider an initial Gaussian energy distribution function

with a linear energy chirp h as

f0ðp; ξÞ ¼N0ffiffiffiffiffiffi2π

p exp

�−ðp −HξÞ2

2

�; ð8Þ

where H ¼ hk0σE=E0

is the dimensionless energy chirp and

ξ ¼ k0z, and h ¼ dγγdz is defined as the energy chirp along

the beam.The derivation in Appendix A of Ref. [11] shows the

bunching factor at the harmonic factor a after the beam,without an energy chirp, passing through two modulatorsand two dispersions as

b¼ 1

N0

����Z þ∞

−∞dpe−iapBf0ðpÞhexpf−iaξ− iaA1Bsinξg

×expf−iaA2B2 sinðKξþKB1pþKA1B1 sinξþΦÞgi����:

ð9Þ

Again, K ¼ k1=k2 and k1;2 are the seed laser wave numbersof each modulator, A1;2 ¼ ΔE1;2=σE and B ¼ B1 þ B2.

According to Ref. [12], by changing the integrationvariable from p to p0 ¼ p −Hξ, for the two identicalwavelengths k1 ¼ k2, it is computed that K ¼ 1 in Eq. (9).Therefore we conclude that the angular bracket denotingaveraging over ξ does not vanish only if

a ¼ nþmð1þHB1Þ1þHB

; ð10Þ

and the bunching factor becomes

bn;m ¼����Jm

�nþmð1þHB1Þ

1þHBA2B2

Jn

�A1ðnBþmB2Þ

1þHB

����× exp

�−ðnBþmB2Þ22ð1þHBÞ2

�; ð11Þ

where Jn;m is the Bessel function of the first kind.According to Eq. (5), when k1 ¼ k2 ¼ k0, we have

k ¼ nC1C2k0 þmC2k0: ð12Þ

It is straightforward to see that with n ¼ 1, m ¼ −1, andðC1 − 1ÞC2 ∼ 10−2, if a 1047 nm laser is employed tomodulate the beam, an about 3 THz density modulation canbe generated and its THz radiation spectrum can be easilymeasured by the interferometer.Based on the preliminary simulation with ELEGANT

[13], we make the beam energy chirp h ¼ 11.22 m−1,

Rð1Þ56 ¼ 0.477 mm, and Rð2Þ

56 ¼ 40 mm, where R56 is themomentum compaction of each dispersive section. Thebunching factor b1;−1 vs the energy modulation amplitudesof each modulator is calculated and shown in Fig. 3. It iseasy to point out that the bunching factor maximizes itsvalue when A1 ¼ 5 and A2 ¼ 10.5.Similarly, we make A1 ¼ 5, A2 ¼ 10 and vary R56 of

two dispersions to calculate the value of the bunchingfactor. The central frequency of density modulation in thebeam varies as well when the R56 of two dispersionschange. We show the result of our calculations in Fig. 4,from which we can see that it is easier to vary the centralfrequency of density modulation in the beam by varying

FIG. 3. The bunching factor vs laser energy modulationamplitudes.

ECHO-ENABLED TUNABLE TERAHERTZ RADIATION … Phys. Rev. ST Accel. Beams 17, 090701 (2014)

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R56 of the first dispersion section than that of the sec-ond one.As we show in Fig. 2, we use an electron beam with

linear energy chirp to interact with the 1047 nm laser inthose two modulators and show the phase space evolutionin Fig. 5 in this scheme with two modulators and twodispersions. The beam density distribution and its Fourier

transform images are shown in Fig. 6. The mechanism ofallowing a chirped electron beam to interact with two lasersof the same wavelength can be explained as follows: thebeam is modulated by the laser in the first modulator andgenerates energy modulation at wave number k0; then theenergy modulation is compressed to C1k0 with the com-pression of the whole beam; the modulated beam interactswith the laser again in the second modulator to produce asuperimposed energy modulation; finally the energy modu-lation is converted to density modulation at wave numberðC1 − 1ÞC2k0 when the beam passes through the seconddispersion.

III. GENERATION OF DENSITY MODULATIONIN A START-TO-END SIMULATION

The beam line of SDUV FEL facility, which is used tocarry out the Echo-Enabled Harmonic Generation experi-ment [6], is shown in Fig. 7. It consists of the injector, linearaccelerator, two modulators, two dispersions, and a radi-ator. The beam at peak current 50 A is accelerated to160 MeV with 5 S-band (2856 MHz RF frequency) linacstructures. The bunch compressor chicane in the linac isturned off in this scheme. The wavelength of the seed laserin those two modulators is 1047 nm.We use the particle tracking code ASTRA [14] to

generate a Gaussian beam with the charge of 200 pCand the peak current of about 50 A from the photocathoderf gun and is accelerated to about 30 MeV with an S-bandlinac structure, considering the strong space charge effectin the injector. Simulation with ELEGANT code beginswhen the beam passes through L1 (consists of 2 S-bands)and finishes after the beam passes through the seconddispersion section (BC2, seen in Fig. 7). After beingaccelerated to 160 MeV, the beam with energy chirp

063

670

6367

0.6367

0.6367

127

341

2734

1.2734

1.2734

191

011

9101

1.9101

1.9101

254

682

54682.5468

2.5468

318

353

1835

3.1835

3.1835

3.8202

3.8202

382

023

82024.4569

4.4569

445

69

5.0936

5.0936

509

365.7303

573

03

6.367

636

7

7.0038

700

387.64058.2772

8.9139

R56(2)(mm)

(b)(a)

R56(1

) (mm

)

10 20 30 40 50 60

0.2

0.4

0.6

0.8

1

FIG. 4. Bunching factor and central frequency (THz) of densitymodulation vs R56 of each dispersion. (a) Bunching factor vs R56

of each dispersion. (b) Central frequency of density modulationvs R56 of each dispersion.

FIG. 5. Phase space evolution of the electron beam with linearenergy chirp at various locations, where p is the dimensionlessenergy derivation in Eq. (2). (a) Exit of modulator 1; (b) exitof modulator 1, closer look; (c) exit of modulator 2; (d) exit ofmodulator 2, closer look; (e) exit of dispersion 2; and (f) exit ofdispersion 2, closer look.

−0.4 −0.2 0 0.2 0.40

5

10

15

z (mm)

)b()a(

dens

ity d

istr

ibut

ion

5 10 150

2

4

6

8

10

12

frequency (THz)

ampl

itude

FIG. 6. Beam density distribution and its fast-Fourier-transform(FFT) image (small peaks in FFT image come from the uniformelectron beam with finite length). (a) Beam density distribution.(b) FFT image of beam density.

FIG. 7. Layout of SDUV FEL facility for generation of THzdensity modulation in a chirped electron beam.

WANG et al. Phys. Rev. ST Accel. Beams 17, 090701 (2014)

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h ¼ 11.22 m−1 is sent into the modulator to interact with a1047 nm laser. To gain the maximal bunching according tothe theory (seen in Fig. 2), it is optimal to regulate the laserpower at ΔE1 ¼ 25 keV while the slice energy spread ofthe beam is about 5 keV. The same is true with the secondmodulator at ΔE2 ¼ 50 keV. The phase space of the beamis shown in Fig. 8 and the main parameters in the simulationare listed in Table I. The images of phase space are differentfrom those in Fig. 5 because the head of the beam is on theleft in ELEGANT.According to Fig. 3, the optimal energy modulations is

A1 ¼ 5 and A2 ¼ 10. In the following simulation, we makea comparison among the laser off, A1 ¼ 5, A2 ¼ 5 andA1 ¼ 5, A2 ¼ 10. The beam currents and their Fouriertransform images are shown in Fig. 9, from which we couldsee that the scheme could provide fundamental frequencyof THz when A2 ¼ 5 and higher harmonic when A2 ¼ 10to satisfy the needs of various users. In Fig. 9, we could alsosee that the wavelength of density modulation in the leftside is longer than that in the right side. The reason is thatthere is not a higher harmonic linearizer (e.g., an X-bandRF cavity) to linearize the longitudinal phase space of thebeam in the SDUV facility so that a nonlinear curvatureexists in the longitudinal phase space, and as a result theelectrons in the head are compressed differently to that inthe tail when the beam passes through the second

dispersion. The spectrum is not single frequency comparedto that in Fig. 6.All the simulations above with ELEGANTare conducted

without considering the longitudinal space charge (LSC)effect. According to Refs. [15–18], the LSC effect, as themain effect driving microbunching instability, may degrade

FIG. 8. Longitudinal phase space of the beam at two importantlocations. (a) Exit of modulator 1; and (b) exit of modulator 2.

TABLE I. Main parameters in the ELEGANT simulation.

Parameter Value

Bunch charge (nC) 0.2Beam energy (MeV) 160Transverse beam size (rms, mm) 0.2Slice energy spread (keV) 5Laser wavelength in U1 nd U2 (nm) 1047Laser power in U1 (MW) 4.5Laser power in U2 (MW) 20Laser waist size (mm) 1.7Period of U1 and U2 (cm) 5Number of periods of U1 and U2 10Energy modulation amplitude in U1 (keV) 25Energy modulation amplitude in U2 (keV) 50R56 in U1 (mm) 0.477R56 in U2 (mm) 40

(a) (b)

(c) (d)

(e) (f)

FIG. 10. Comparison between LSC effect off (left) and on(right). (a) Slice energy spread at the entrance of modulator 1,LSC off. (b) Slice energy spread at the entrance of modulator 1,LSC on. (c) Beam current at the exit of dispersion 2, LSC off.(d) Beam current at the exit of dispersion 2, LSC on. (e) Longi-tudinal phase space at the exit of dispersion 2, LSC off. (f) Longi-tudinal phase space at the exit of dispersion 2, LSC on.

5 10 150

0.2

0.4

0.6

0.8

1

frequency (THz)

(b)(a)

ampl

itude

FIG. 9. Beam current and its fast-Fourier-transform image.(a) Beam currents for A2¼5 (red) and A2¼10 (blue) when A1¼5and initial current without laser modulation (black). (b) FFTimage of beam current for A2 ¼ 5 (red) and A2 ¼ 10 (blue)when A1 ¼ 5.

ECHO-ENABLED TUNABLE TERAHERTZ RADIATION … Phys. Rev. ST Accel. Beams 17, 090701 (2014)

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the quality of the electron beam when the beam passesthrough the bunch compressor chicane. A comparison ismade to illustrate the influence on the slice energyspread and the current of the beam with the LSC effect.The result is shown in Fig. 10, from which we could seethat energy modulation is accumulated (spikes in theslice energy spread with LSC on) because of the initialdensity modulation coming from shot noise and drive laser,and the peak current of the electron beam at the exit ofdispersion 2 reduces. The position of the peaks move alittle and the number of the peaks changes (from 7 to 6);this is because the energy chirp of the beam is reducedbecause of the longitudinal space charge force (the electronsin the head gain energy and the electrons in the tail lose theirenergy; see the phase space in Fig. 10), leading to thedifference between the LSC effect off and on. So the LSCeffect may degrade the quality of the beam. Nevertheless,considerable density modulation at THz frequency is stillproduced with the LSC effect taken into account.

IV. GENERATION OF THZ RADIATION

Transition radiation (TR), undulator radiation, diffrac-tion radiation, and synchrotron radiation are all goodmethods to generate THz radiation. Considering the THzdensity modulation in the electron beam, transition radi-ation and undulator radiation are the better choices for us.In SDUV, the electron beam energy is about 160 MeV andthe undulator is dedicated to generate sub-μm FEL radi-ation; therefore, it is difficult to match the resonancecondition because the period and the magnetic field ofthe undulator are both too large.Subsequently, TR is the only, but even better, choice

because it is relatively easy and convenient to operate in ourscheme. As an example, a round TR target made of ametallic foil with radius r is placed at the end of the linac inSDUV, and the THz radiation is generated when the beampasses through the target [8]. The TR spectral angulardistribution for a single electron with a round TR targetfollows:

d2UdΩdω

¼ e2

4π3ϵ0cβ2sin2θ

ð1 − β2cos2θÞ2 ½ð1 − ΥðζÞ�2; ð13Þ

where

ΥðζÞ ¼ ζ

βγ sin θ½J1ðγ sin θζÞK2ðζÞ − γθJ2ðγ sin θζÞK1ðζÞ�;

ð14Þand Υ ¼ 2πr=γλθ is the polar angle, K1;2 is the modifiedBessel function of the second kind.For the beam with energy 160 MeV, charge 200 pC, and

bunch length 2 ps, and the round TR target with radius5 mm, we calculated that the peak radiation intensity wasabout 0.12 μJ=THz and peak power at 4.5 THz was

approximately 0.06 MW. The low charge of the beam leadsto a low peak power of the radiation. The shape of theradiation spectrum is identical to the FFT curve shown inFig. 9(b).Compared to the regular methods of THz generation

such as the optical rectification of a laser or production bygyrotrons, although the peak power provided by our newmethod is relatively low, it brings us a fast and convenientway to adjust the radiation frequency in a considerablywide range, which is requested in the research areas such asbiophysics, structure biology, material science, and so on.Therefore, we believe this new method will open a newdoor for THz research and application.

V. CONCLUSIONS

Considering the wide applications of THz radiation inrecent years, we analyzed the scheme to generate tunableTHz density modulation by a 1047 nm seed laser at theSDUV FEL facility in theory and simulation, which turnedout to be feasible for future experiments at the SDUVfacility. With the scheme, the central frequency of thedensity modulation of the beam can be continuously tunedin at least 1–5 THz, which could be measured by aninterferometer conveniently. The influence of nonlinear rfchirp and longitudinal phase space charge effect have alsobeen studied. Our results show that this new scheme isfeasible and easily applied in the Shanghai Deep UltravioletFree-Electron Laser facility.

ACKNOWLEDGMENTS

We wish to acknowledge the help of many colleagues inSINAP for the discussions on the analytical and simulationresults, and many useful suggestions from the experts in theother institutes. The work is partially supported by theMajor State Basic Research Development Program ofChina (2011CB808300) and National Natural ScienceFoundation of China (NSFC), Grants No. 11275253 andNo. 11327902.

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