Echo-enabled harmonics up to the 75th order fromprecisely tailored electron beamsE. Hemsing1*, M. Dunning1, B. Garcia1, C. Hast1, T. Raubenheimer1, G. Stupakov1 and D. Xiang2,3*

The production of coherent radiation at ever shorter wave-lengths has been a long-standing challenge since the inventionof lasers1,2 and the subsequent demonstration of frequencydoubling3. Modern X-ray free-electron lasers (FELs) use relati-vistic electrons to produce intense X-ray pulses on few-femto-second timescales4–6. However, the shot noise that seeds theamplification produces pulses with a noisy spectrum andlimited temporal coherence. To produce stable transform-limited pulses, a seeding scheme called echo-enabled harmonicgeneration (EEHG) has been proposed7,8, which harnessesthe highly nonlinear phase mixing of the celebrated echophenomenon9 to generate coherent harmonic density modu-lations in the electron beam with conventional lasers. Here,we report on a demonstration of EEHG up to the 75th harmonic,where 32 nm light is produced from a 2,400 nm laser. We alsodemonstrate that individual harmonic amplitudes are con-trolled by simple adjustment of the phase mixing. Resultsshow the potential of laser-based manipulations to achieveprecise control over the coherent spectrum in future X-rayFELs for new science10,11.

The well-known echo effect is a highly nonlinear process that wasoriginally described in 1950 by E.L. Hahn in the context of precessingnuclear spins9. Echoes have been observed in many areas of physics,including cyclotron beams12, plasma waves13, photons14, hadroncolliders15, molecular rotors in gas16 and in classical mechanicalsystems17. Broadly described, echoes arise in a medium that is firstexcited by a perturbation that then damps completely due to dephas-ing. The inherent damping mechanism is non-dissipative, however,and the perturbations remain deeply encoded as highly filamentarystructures in the phase space. Following an excitation by a secondperturbation, the phase mixing is partially reversed as the systemevolves, and the echo signal appears as a recoherence effect. This isthe mechanism behind the proposed echo-enabled harmonic gener-ation (EEHG) technique, used to produce fully coherent X-ray pulsesin a free-electron laser (FEL)7,8. EEHG exploits the fact that thefine phase-coherent echo structures have much higher frequencycontent than the original perturbations, so the temporal coherenceof modulations imprinted by conventional lasers can be passed toshorter wavelengths, with the electrons serving as the harmonicupconversion medium.

The EEHG concept is illustrated in the schematic of our exper-imental layout in Fig. 1. The E = 120 MeV relativistic electronbeam co-propagates with a laser pulse (λ1 = 800 nm) in a magneticundulator, U1 (see Methods). The resonant interaction imprints asinusoidal energy modulation with amplitude ΔE1 in the electronbeam phase space (Fig. 1a). Linear longitudinal dispersionthrough the magnetic chicane C1, given by the transport matrixelement R(1)

56 , allows electrons with different energies to moverelative to one another as they traverse different path lengths. This

phase-mixing transformation turns the sinusoidal modulation intoa filamentary distribution with multiple energy bands (Fig. 1b).A second laser (λ2 = 2,400 nm) then produces an energy modulation(ΔE2) in the finely striated beam in the U2 undulator (Fig. 1c).Finally, the beam travels through the C2 chicane where a smallerdispersion R(2)

56 brings the energy bands upright (Fig. 1d).Projected into the longitudinal space, the echo signal appears as aseries of narrow current peaks (bunching) with separation λ = λ2/h(Fig. 1e), where h is a harmonic of the second laser that determinesthe character of the harmonic echo signal.

EEHG has been examined experimentally only recently andat modest harmonic orders, starting with the 3rd and 4thharmonics18,19, the 7th harmonic20 and most recently at the15th harmonic21 of infrared lasers. These studies demonstratedthat the EEHG harmonics are generated with small relativeincreases in the beam energy spread compared with high-gainharmonic-generation (HGHG) schemes22–24 and also that thespectrum is much less sensitive to initial electron-beam phase-space distortions. However, to reach from ultraviolet seedlasers to the soft X-ray wavelengths where the most compellingscientific applications are found, harmonic numbers h ≥ 50 arerequired, which in turn requires exquisite preservation of thefine-scale phase-space memory.

Here, we report the first results of EEHG experiments in theh = 60 and h = 75 harmonic regimes, which also highlight the fineand unique control the echo mechanism provides over the coherentemission spectrum. The results suggest the possibility to preciselytailor the pulses in future X-ray FELs for new science applications10,11.

From EEHG theory, the optimal harmonic bunching bh is tunedand scales according to the ratio of dispersions, h ≈ λ2R

(1)56 /λ1R

(2)56 .

Accordingly, the C1 chicane was set to R(1)56 = 12.5 mm, close to its

peak value. In the first experiment, the dispersion of the secondchicane was approximately R(2)

56 = 600 μm to produce bunchingnear the 60th harmonic of the 2,400 nm laser. The bunched beamwas boosted to 192 MeV in a short linac (Fig. 1) such that theλ ≈ 40 nm harmonic wavelengths were radiated at the 3rd harmonicof the U3 emission spectrum (see Methods). Figure 2a shows thecharacteristic undulator radiation pattern seen in the quadraticdependence of the wavelength on the vertical forward angle (seeMethods), and the bright peaks are the coherent harmonic echosignals. The image is the average of over 200 consecutive shots,but is representative of typical single shots. The spectrum of eachshot, taken from the vertical projection, is shown in Fig. 2b.Analysis of the spectral stability over all shots gives a 0.025 nm(r.m.s.) variation of the central wavelength of the 59th harmonicsignal (0.063% relative). The observed bandwidth of each harmonicpeak for each shot is equal to the measured 0.15 nm resolution ofthe extreme ultraviolet (EUV) spectrometer (see Methods), butsimulations indicate the true bandwidth to be 0.011 nm (full-width

1SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA. 2Key Laboratory for Laser Plasmas (Ministry of Education), Department ofPhysics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China. 3Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao TongUniversity, Shanghai 200240, China. *e-mail: ehemsing@slac.stanford.edu; dxiang@sjtu.edu.cn

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Figure 1 | Schematic layout of the echo beam line and evolution of the electron-beam phase space after successive transformations. a–c, The electronbeam is first modulated in energy by a laser (a), dispersed by transit through a magnetic chicane (b) and modulated again by a second laser (c). d,e, Thefinal phase space distribution (d) has a series of vertical energy stripes that project to sharp peaks in the electron beam current distribution (e).

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Figure 2 | Echo harmonics of the 2,400 nm laser in the 60th harmonic range. a, Undulator emission. b, Individual spectra over multiple consecutive shots.c, Spectrum of the incoherent spontaneous radiation (single shots in light red, average in dark red) and the coherent echo signal (single shots in light blue,average in dark blue).

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at half-maximum, FWHM). Simulations also show that the 1 pslasers produce echo bunching in a 0.4 ps (FWHM) portion of the∼1 ps (FWHM) electron beam. The envelope of the average spec-trum is shown in Fig. 2c (blue line), together with the spectrumof incoherent spontaneous radiation from the beam when both seedlasers are blocked (red line), which has a small bump at the 3rd undu-lator harmonic. Numerical simulations of the bunching indicate goodagreement with the observed EEHG spectrum for ΔE1 = 38 keV andΔE2 = 84 keV. These energy modulation amplitudes are consistentwith those observed experimentally at the electron energy spectrometer.It is noted that, with laser 1 blocked, no harmonic signals wereobserved, which confirms that the signal is due to the combination

of modulations that produce an echo rather than HGHG bunchingfrom a single laser.

To access the highest harmonics in our set-up, dispersion in thesecond chicane was lowered towards its minimum value aroundR(2)56 = 500 μm and the EUV spectrometer was moved to view the

75th harmonic regime at 32 nm. Coherent emission in this caseis peaked at the 4th harmonic resonance of U3 and thus isemitted slightly off-axis by the nature of even harmonic undulatoremission. The measured spectra are shown in Fig. 3a. With laserson, the echo harmonics are clearly visible (blue lines), and with thelasers off, only spontaneous radiation is observed (red). In Fig. 3b,the averaged spontaneous background (Fig. 3a, dark red) is

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Figure 3 | Echo in the 75th harmonic range. a, Measured electron beam emission spectrum with echo lasers on (single shots in light blue, average in darkblue) and with lasers off (single shots in light red, average in dark red). b, Coherent signal with spontaneous emission spectrum subtracted (black) and theharmonic amplitudes of |bh|2 from numerical simulations (red).

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Figure 4 | Tuning harmonics with dispersion. a, Individual echo harmonics of the λ2 = 800 nm laser tuned as a function of the second dispersion R(2)56 .b, At certain dispersion values, specific harmonics are dominant and nearby harmonics are almost completely suppressed. c,d, Matching results fromnumerical simulations for |bh|

2, where the narrow bandwidth is resolved.

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subtracted from the averaged echo spectrum (Fig. 3a, dark blue),and results from numerical simulations for the bunching are com-pared with the measured coherent spectrum. Simulations indicatethat energy modulations of ΔE1 = 61 keV and ΔE2 = 104 keVproduce a harmonic spectrum that is strikingly similar to thedata, both in the relative amplitudes of the harmonics and inthe variation of the envelope of the harmonic peaks. Again,these values are consistent with experimental measurements. Aswas also the case with the experiments on the 60th harmonic,the observed shot-to-shot amplitude fluctuations were due primar-ily to charge fluctuations (as seen in the variation of the incoherentsignal) and systematic intensity fluctuations in the second laserpulse (see Methods). With laser 1 blocked in this configuration,the harmonics of the optimized HGHG signal were fully sup-pressed, which confirms the efficiency of the echo harmonicgeneration process.

It is important to note that the coherent signal at these wave-lengths and at this beam energy is calculated to be strongly sup-pressed by longitudinal smearing effects of beam betatron motionin U3, due to the finite beam emittance and the fixed U3 strongfocusing channel. This probably accounts for the relative smallnessof the coherent signal compared to the spontaneous background inboth the 60th and 75th harmonic experiments. This effect becomesstronger at shorter wavelengths, which precluded investigation ofstill higher harmonics in our set-up, but is significantly reducedin optimized soft X-ray FELs with larger beta functions andhigher energy beams (see Methods).

The variation of the harmonic envelope observed in Fig. 3b,where certain harmonics are suppressed (for example, the 72nd)while nearby harmonics are enhanced (for example, the 74th), isa newly observed and unique feature of the beam echo effectcompared to HGHG. This effect is predicted from EEHGtheory, which indicates that the individual harmonics can be pre-cisely tuned by the dispersion (see Methods). Results of a separateexperiment to explore this harmonic tunability are presented inFig. 4, where the measured spectra in the neighbourhood of the20th harmonic from two 800 nm seed lasers are shown inFig. 4a and b, respectively. Here, ΔE2 was held stable while thevalue of R(2)

56 was varied. Individual harmonics within the envelopeare seen to be optimized at slightly different dispersion values. Forinstance, at R(2)

56 = 660 μm, bunching at 40 nm can be optimizedwhile that at 42 nm can be suppressed; the 2 nm difference ismore than two orders of magnitude smaller than the wavelengthof the lasers used to create the fine structure. This precisecontrol over the coherent beam distribution is also reproducedin simulations with ΔE1 = 29 keV and ΔE2 = 27 keV, as shownin Fig. 4c,d.

The experiments thus show that the proposed beam echo effectcan be used to produce ultrahigh harmonic bunching and that itenables precise control via practical means over the fine-scalebunching spectrum. Demonstration of EEHG at the 75th harmonicsuggests that it may be straightforward to use this technique toseed the amplification of fully coherent and tunable soft X-rays inthe water window (2–4 nm) above the gigawatt level in a singlestage from ultraviolet lasers in future FELs. Furthermore, two-stage EEHG, with soft X-rays from the first stage seeding thesecond stage, might eventually allow extension to hard X-ray wavelengths.

MethodsMethods and any associated references are available in the onlineversion of the paper.

Received 18 March 2016; accepted 21 April 2016;published online 6 June 2016

References1. Maiman, T. H. Stimulated optical radiation in ruby.Nature 187, 493–494 (1960).2. Schawlow, A. L. & Townes, C. H. Infrared and optical masers. Phys. Rev.

112, 1940–1949 (1958).3. Franken, P. A., Hill, A. E., Peters, C. W. & Weinreich, G. Generation of optical

harmonics. Phys. Rev. Lett. 7, 118–119 (1961).4. Emma, P. et al. First lasing and operation of an angstrom-wavelength

free-electron laser. Nature Photon. 4, 641–647 (2010).5. Ishikawa, T. et al. A compact X-ray free-electron laser emitting in the

sub-angstrom region. Nature Photon. 6, 540–544 (2012).6. McNeil, B. W. J. & Thompson, N. R. X-ray free-electron lasers. Nature Photon. 4,

814–821 (2010).7. Stupakov, G. Using the beam-echo effect for generation of short-wavelength

radiation. Phys. Rev. Lett. 102, 074801 (2009).8. Xiang, D. & Stupakov, G. Echo-enabled harmonic generation free electron laser.

Phys. Rev. ST Accel. Beams 12, 030702 (2009).9. Hahn, E. L. Spin echoes. Phys. Rev. 80, 580–594 (1950).10. Hemsing, E., Stupakov, G., Xiang, D. & Zholents, A. Beam by design:

laser manipulation of electrons in modern accelerators. Rev. Mod. Phys.86, 897–942 (2014).

11. Schneider, J. R. Photon science at accelerator-based light sources. Rev. Accel. Sci.Technol. 3, 13–37 (2010).

12. Hill, R. M. & Kaplan, D. E. Cyclotron resonance echo. Phys. Rev. Lett.14, 1062–1063 (1965).

13. Gould, R. W., O’Neil, T. M. &Malmberg, J. H. Plasma wave echo. Phys. Rev. Lett.19, 219–221 (1967).

14. Kurnit, N. A., Abella, I. D. & Hartmann, S. R. Observation of a photon echo.Phys. Rev. Lett. 13, 567–568 (1964).

15. Spentzouris, L. K., Ostiguy, J.-F. & Colestock, P. L. Direct measurement ofdiffusion rates in high energy synchrotrons using longitudinal beam echoes.Phys. Rev. Lett. 76, 620–623 (1996).

16. Karras, G. et al. Orientation and alignment echoes. Phys. Rev. Lett.114, 153601 (2015).

17. Chebotayev, V. P. & Dubetsky, B. Ya. A classical model of the photon echo. Appl.Phys. B 31, 45–52 (1983).

18. Xiang, D. et al. Demonstration of the echo-enabled harmonic generationtechnique for short-wavelength seeded free electron lasers. Phys. Rev. Lett. 105,114801 (2010).

19. Zhao, Z. et al. First lasing of an echo-enabled harmonic generation free-electronlaser. Nature Photon. 6, 360–363 (2012).

20. Xiang, D. et al. Evidence of high harmonics from echo-enabled harmonicgeneration for seeding X-ray free electron lasers. Phys. Rev. Lett.108, 024802 (2012).

21. Hemsing, E. et al. Highly coherent vacuum ultraviolet radiation at the 15thharmonic with echo-enabled harmonic generation technique. Phys. Rev. STAccel. Beams 17, 070702 (2014).

22. Yu, L. H. Generation of intense UV radiation by subharmonically seeded single-pass free-electron lasers. Phys. Rev. A 44, 5178–5193 (1991).

23. Allaria, E. et al. Highly coherent and stable pulses from the FERMI seeded free-electron laser in the extreme ultraviolet. Nature Photon. 6, 699–704 (2012).

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AcknowledgementsThe authors thank D. McCormick, K. Jobe, I. Makasyuk, T. Beukers, C. Hudspeth, J. Cruz,V. Yakimenko and the rest of the Test Facilities Group for their support. The authors alsothankM. Guehr for his guidancewith the design and construction of the EUV spectrometerand E. Johnson for the VISA undulator. This work was supported by the US DOE Office ofBasic Energy Sciences under award no. 2012-SLAC-10032 using the SLAC NLCTA facility,which is partly supported by US DOE Office of High Energy Physics under contractno. DE-AC02-76SF00515. The work of D.X. was supported by the Major State BasicResearch Development Program of China (grant no. 2015CB859700) and by the NationalNatural Science Foundation of China (grant no. 11327902).

Author contributionsG.S. conceived the original EEHG theoretical concept, which, with D.X., was laterdeveloped further. E.H., B.G. and M.D. carried out the experiments. B.G. and M.D.designed and built the EUV spectrometer. E.H. and B.G. performed the data analysis andperformed simulations. D.X., C.H. and T.R. provided guidance on the experiments and onpotential applications. E.H. and D.X. wrote the paper, with contributions from allother authors.

Additional informationReprints and permissions information is available online at www.nature.com/reprints.Correspondence and requests for materials should be addressed to E.H. and D.X.

Competing financial interestsThe authors declare no competing financial interests.

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MethodsExperiments were conducted at the Next Linear Collider Test Accelerator (NLCTA)at SLAC National Accelerator Laboratory.

Electron beam. The NLCTA electron beam was generated in a photocathode RF gun(RF frequency at 2.856 GHz) at a repetition rate of 10 Hz and was boosted toE = 120 MeV with two X-band linac structures (RF frequency at 11.424 GHz). Echomanipulations were performed at this energy. The electron beam charge was ∼20 pCin the experiment, the FWHM bunch length was ∼1 ps, the slice energy spread wasσE ≃ 1–2 keV, and the projected normalized transverse emittance was ∼1.7 μm. Thebeam current was kept low to avoid lasing in the undulator, so that the undulatorharmonics and off-axis emission angles provided a broadband diagnostic of theharmonic bunching spectrum. The measured r.m.s. energy variation about the final190 MeV central energy was ∼190 keV (0.1%). At this level, the R(1)

56 = 12.5 mmdispersion in the first chicane led to a 40 fs r.m.s. electron beam arrival time jitter inthe downstream undulators.

Laser beam. The two seed lasers originated from a single 800 nm (∼1 ps FWHM)laser pulse that was sent through a beamsplitter. One pulse seeded the U1 undulatorat λ1 = 800 nm and the other pumped an optical parametric amplifier (OPA) toproduce a λ2 = 2,400 nm pulse (∼1 ps FWHM, ≤10 μJ) in the U2 undulator. Theenergy of the 2,400 nm laser had considerable fluctuations due to the pointing stabilityand long transport of the 800 nm laser, which led to relatively large shot-to-shotfluctuations of the echo signals in Figs 2 and 3. The OPA could also be bypassed toallow modulation of the beam in U2 with the 800 nm seed laser. By using the two800 nm lasers, the echo signals in Fig. 4 had much better energy stability.

Echo beamline. The echo beamline consisted of three undulators (a series ofmagnets with alternating fields), two chicanes (a series of bending magnets thatchange the electron’s path length according to its energy) and one acceleratorstructure. The resonant wavelength of each undulator was λr = (λu/2huγ

2)(1 + K2/2 +θ2γ2), where λu is the undulator period, K is the normalized undulator parameter, huis the undulator harmonic, γmc2 is the beam energy in the undulator, and θ is theforward emission angle. U1 had 10 periods, each with a length of λu = 3.3 cm and anundulator parameter of K = 1.82. The beam then transitted the C1 chicane, and thenco-propagated with the second laser pulse (λ2 = 2,400 nm if from the OPA, λ2 = 800 nmif the OPA was bypassed) inside the U2 undulator (10 periods with a period lengthof 5.5 cm and K = 2.76). The tuning of U2 was such that the 2,400 nm laser pulse wasresonant with the 120 MeV (γ = 235) beam at the 1st harmonic (hu = 1) of theundulator and the 800 nm pulse was resonant with the 3rd harmonic. After passingthrough the C2 chicane, the beam was accelerated up to a maximum of 192 MeV bya 1.05-m-long x-band linac. Radiation was produced by the beam inside the U3undulator (110 periods with a period length of 1.8 cm (L = 198 cm total length) andK = 1.26). The U3 was a 2 m section of the VISA-II Halbach-type pure-permanentmagnet undulator with eight distributed strong focusing focus, drift, defocus, drift(FODO) cells, each of 24.75 cm length25. At E = 190 MeV (γ = 372), the on-axisresonance of the U3 undulator was 117 nm at the hu = 1 fundamental. Four intra-undulator retractable beam profile screens imaged by charge-coupled device (CCD)cameras allowed electron beam orbit and beta-matching correction with externalsteering magnets and upstream quadrupoles, respectively.

Each four-dipole chicane was remotely adjustable but had a minimumdispersion to allow the electron beam to pass cleanly by the laser injection/rejectionmirror assembly, which was located between the second and third dipoles. This setthe geometric minimum R56 to ∼500 μm to avoid beam scraping, but there is aknown uncertainty of 100 μm in the recorded dispersion at these small values. Eachchicane also had two magnetic quadrupoles for horizontal dispersion correction:one positioned halfway between the first and second dipole, and one halfwaybetween the third and fourth dipole.

Laser and electron beam overlap. Effective interaction between the lasers andelectron beam was achieved when the electron and laser beam overlapped bothspatially and temporally in the modulators. Spatial overlap between the 800 nm laserand the electron beam was achieved by steering the laser to the same position as thebeam on the screens upstream and downstream of the undulators. Temporal overlapwas achieved with two steps. First, an optical transition radiation screen downstreamof each undulator was used to reflect out the laser and undulator radiation, whichwas detected by a fast photodiode. By referencing the signals to an external trigger,the laser and electron beam could be synchronized to within ∼30 ps. More precisetiming was then carried out by using a scanning delay stage and measuring thecoherent transition radiation from the electron beam downstream of C1. The secondlaser at 2,400 nm was made to overlap with the beam in U2 in a similar way.Unattenuated, the 2,400 nm idler pulse profile could be seen with the CCD beamprofile cameras for transverse overlap with the electrons. A bandpass filter centred at2,400 nm was used in the experiment to ensure that the beam was only modulated bythe 2,400 nm laser in U2 and not the 1,200 nm signal pulse from the OPA.

EUV spectrometer. Coherent EUV radiation emitted by the beam in U3 wasmeasured with a high-resolution, custom-built, ultrahigh-vacuum spectrometer withsensitivity in the 11–62 nm wavelength range26. Radiation was first reflected off aninsertable gold-coated mirror at 15° incidence ∼0.5 m downstream of U3 and sentthrough an extraction port off the beam line. The light then passed through a slitwith independently adjustable blades and then struck a Hitachi 001-0640aberration-corrected concave grating with 1,200 G mm–1 at an incidence angle of4.7°. The blaze angle was 3.7° and the grating imaged the slit at a flat field focus469 mm from the centre. The combined assembly allowed 60–80% efficiency overthe wavelengths of interest. This grating was fixed-mounted in a custom high-vacuum chamber connected to a large bellows that allowed the light to travel to the40 mm MicroChannel Plate (MCP) detector (Model BOS-40, Beam ImagingSolutions, ≤2 kV power supply). Behind the MCP was a P-43 phosphor screen,powered by a ≤5 kV power supply, which was then directly imaged with a CCDcamera. The MCP/phosphor/camera assembly was mounted to a linear motion stagethat allowed 120 mm of translation parallel to the grating focal plane to capture thefull wavelength range, including the zeroth order. Calibration of the spectrometerwas carried out using the known grating equations in combination with theharmonic radiation spectrum from the undulator and the echo harmonics from the800/2,400 nm and 800/800 nm laser seeding configurations. The spectrometer canbe set to capture different harmonic emissions of the U3 undulator.

Simulations. Numerical simulations used measured realistic beam distributions,laser energy modulations and chicane dispersions. Electrons were tracked throughthe echo beamline and the final beam distribution was used to calculate thebunching at various wavelengths. The dispersion in C1 was taken as R(1)

56 = 12.5 mmand the electron beam length was 0.43 ps r.m.s. for all simulations. For the Echo75 simulations in Fig. 3b, R(2)

56 = 484 μm (in good agreement with the measuredvalue considering the uncertainty of the dispersion near the minimal value),ΔE1 = 61 keV and ΔE2 = 104 keV. With a seed laser pulse length of 1 ps FWHM,simulations indicate that the coherent bunching extends over 0.4 ps of the electronbeam and that the FWHM bandwidth of |bh|

2 is 0.007 nm at 32 nm. In Fig. 4,the parameters used in the simulation are ΔE1 = 29 keV, ΔE2 = 27 keV andR(2)56 = 590–730 μm.

Bunching factor. From EEHG theory, the bunching amplitude at the bunchingfrequency ωn,m = nω1 +mω2 is

bh = e−ξ2 /2Jn −ξΔE1 /σE

( )Jm −ωn,mΔE2R

(2)56 /cE

[ ](1)

where ω1,2 = 2πc/λ1,2 are the seed laser frequencies, ξ = (σE /cE)(nω1R(1)56 + ωn,mR

(2)56 )

and Jn,m are Bessel functions. In practice, the modulations and dispersions aretuned so that bh is maximized at the desired echo harmonic of the second laserh = ωn,m/ω2 =m + n(ω1/ω2) for a given n and m, where m is typically large and n < 0with |n| small. Optimized echo harmonics are given in general approximately by theratio h ≈ −nλ2R

(1)56 /λ1R

(2)56 . In all of our experiments, the tuning was such that n = −1.

Smearing of bunching. The beam betatron motion in U3 resulted in longitudinalsmearing effects that reduced the bunching factor at high harmonic frequencies.This is due to the fact that particles with larger betatron amplitudes have longer pathlengths. The longitudinal smearing effect is estimated as Δz =

��2

√ϵnL/γ⟨β⟩ ≈ 5 nm

which is comparable to λ/2π, where L = 2 m is the length of undulator U3,ϵn ≈ 0.5 mm mrad is the (assumed) electron beam slice transverse emittance,⟨β⟩ = 65 cm is the average beta function in U3 and λ is the wavelength of theharmonic radiation. The beta function in the undulator is fixed by the permanentmagnetic focusing at the 190 MeV beam energy, so the beta function of the beamcannot be increased to mitigate this effect without mismatching the beam. In theabsence of this smearing effect, simulations indicate a bunching factor of severalpercent, depending on the harmonic tune. Given that the effective bunching due tonoise is bSN ∼ 1/

���Ns

√∼ 4 × 10−4 (where Ns is the number of electrons in a slippage

length Lλr/λu), this would result in an increase of the coherent emission by aboutfour orders of magnitude above the incoherent background. Similarly, for optimizedFELs operating in the water window, the few percent echo bunching factor is alsoabout two orders of magnitude higher than shot noise bunching, which enables theproduction of fully coherent radiation.

References25. Carr, R. et al. Visible-infrared self-amplified spontaneous emission amplifier free

electron laser undulator. Phys. Rev. ST Accel. Beams 4, 122402 (2001).26. Garcia, B., Dunning, M. P., Hast, C., Hemsing, E. & Raubenheimer, T. O. Facility

upgrades for the high harmonic echo program at SLAC’s NLCTA. In Proc. 37thInt. Free Electron Laser Conf. (FEL 2015) (eds Kang, H.-S. et al.) 422–426(JACoW, 2015).

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