Soft X-ray harmonic comb from relativistic electron spikes A. S. Pirozhkov,1 M. Kando,1 T. Zh. Esirkepov,1 P. Gallegos,2,3 H. Ahmed,4 E. N. Ragozin,5 A. Ya. Faenov,1,6 T. A. Pikuz,1,6 T. Kawachi,1 A. Sagisaka,1 J. K. Koga,1 M. Coury,3 J. Green,2 P. Foster,2 C. Brenner,2,3 B. Dromey,4 D. R. Symes,2 M. Mori,1 K. Kawase,1 T. Kameshima,1 Y. Fukuda,1 L. Chen,1 I. Daito,1 K. Ogura,1 Y. Hayashi,1 H. Kotaki,1 H. Kiriyama,1 H. Okada,1 N. Nishimori,7 T. Imazono,1 K. Kondo,1 T. Kimura,1 T. Tajima,1 H. Daido,1 P. Rajeev,2 P. McKenna,3 M. Borghesi,4 D. Neely,2,3 Y. Kato,1 and S. V. Bulanov1,8 1Advanced Beam Technology Division, JAEA, 8-1-7 Umemidai, Kizugawa, Kyoto 619-0215, Japan 2Central Laser Facility, Rutherford Appleton Laboratory, STFC, Chilton, Didcot, Oxon OX11 0QX, UK 3University of Strathclyde, Department of Physics, SUPA, Glasgow G4 0NG, UK 4Centre for Plasma Physics, The Queen’s University of Belfast, Belfast BT7 1NN (UK) 2 5P. N. Lebedev Physical Institute, RAS, Leninsky prospekt 53, Moscow 119991, Russia 1 6Joint Institute of High Temperatures, RAS, Izhorskaja st. 13/19, Moscow 127412, Russia 0 7Laser Application Technology Division, JAEA, 2-4 Shirakata-Shirane, Tokai, Ibaraki 319-1195, Japan 2 8A. M. Prokhorov Institute of General Physics, RAS, Vavilov st. 38, Moscow 119991, Russia (Dated: January 4, 2012) n a We demonstrate a new high-order harmonic generation mechanism reaching the ‘water window’ J spectral region in experiments with multi-terawatt femtosecond lasers irradiating gas jets. A few 1 hundred harmonic orders are resolved, giving µJ/sr pulses. Harmonics are collectively emitted by anoscillatingelectronspikeformedatthejointoftheboundariesofacavityandbowwavecreated ] by a relativistically self-focusing laser in underdense plasma. The spike sharpness and stability are h explained by catastrophe theory. The mechanism is corroborated by particle-in-cell simulations. p - m PACSnumbers: 52.27.Ny,42.65.Ky,52.59.Ye,52.35.Mw,42.65.Re,52.38.Ph s a pl High-order harmonic generation, one of the most fun- 3-m)6 9 TW [Fig. 2(a-c)] 9 TW [Fig. 2(d), 3] 120 TW [Fig. 2(c)] damental effects of nonlinear wave physics, originates c4 s. from many nonlinearities, e.g. relativistic [1, 2] and ion- 9101 2 c izing matter effects [3]. Harmonics reaching the X-ray n( e0 -1 0 x(m1m)Laser He jeMt agnet Au mirrorFiltersGrating CCD i s spectral region enable nanometer and attosecond reso- y FIG. 1. (Color online) The experiment scheme (right) and lution in biology, medicine, physics, and their applica- h He plasma density profile (left) for the shots in Figs. 2, 3. p tions. Several compact laser-based X-ray sources have [ been implemented, including plasma-based X-ray lasers [4], atomic harmonics in gases [3], nonlinear Thomson 1 9 TW, 27 fs, 820 nm, f/9, and 4×1018 W/cm2 in the v scattering [5] from plasma electrons [6, 7] and electron first campaign, and 120 TW, 54 fs, 804 nm, f/20, and 3 beams [8], betatron radiation [9], relativistic flying mir- 3 rors [10], and harmonics from solid targets [1, 2, 11] 4 × 1018 W/cm2 in the second campaign, respectively. 3 Focusing laser pulses onto a supersonic helium gas jet, and electro-optic shocks [12]. Many of them [1, 5– 0 Fig.1, werecordharmonicsinthe80-360eVspectralre- 11] come from relativistic laser-matter interactions, de- . 1 termined by the dimensionless laser pulse amplitude gionintheforward(laserpropagation)direction,Figs.2, 0 3. We use flat-field spectrographs [15], comprising a a =eE /m cω >1 relatedto the laser peak irradiance 2 by0 I =0 I ea20(µm/λ )2 for linear and I = 2I for gold-coatedgrazing-incidencecollectingmirror,spherical 1 0L rel 0 0 0C 0L varied-linespace grating, and back-illuminated Charge : circular polarization. Here, e and me are the electron v Coupling Device (CCD), shielded by optical blocking fil- charge and mass, c is the speed of light in vacuum, Xi ω , λ and E are the laser angular frequency, wave- ters – two 0.16 µm multilayer Mo/C, two 0.2 µm Pd, or 0 0 0 one 0.2 µm Ag on 0.1 µm CH substrate. The acceptance ar l1e.n3g7t×h1a0n1d8Wpe/ackm2el[e2c]t.ric field, respectively, and Irel = angleis3×10−5 srinthefirstand3×10−6 srinthesec- ond campaign. In the latter case, the spectrograph has In this Letter we experimentally demonstrate bright two channels for observation at 0◦ (forward direction) ExtremeUltraviolet(XUV)andsoftX-rayharmonicsre- and 0.53◦ in the laser (linear) polarization plane. The solveduptoafewhundredordersfromgasjetsirradiated wavelength calibration is performed in-place using spec- bymulti-terawattfemtosecondlasers. Weproposeanew tra of Ar and Ne plasmas. The harmonics energy and mechanism of high-order harmonics generation. photon numbers are conservatively estimated using ide- We have performed two experimental campaigns us- alizedspectrographthroughputs,i.e. theproductsofthe ing J-KAREN [13] and Astra Gemini [14] lasers. The calculated [16] collecting mirror reflectivities and filter laser pulse power P , duration τ, wavelength λ , off-axis transmissions, measured diffraction grating efficiencies, 0 0 parabolicmirrorf-number, andirradianceinvacuumare and manufacturer-provided CCD efficiencies. 2 0 CCD Counts 250 0 CCD Counts 150 (a) (a) 70 80 90 100 150 200 ħω(eV) 0 CCD Counts 400 λ(nm) 16 15 14 13 12 11 10 9 8 7 6 5 4 (b) 70 75 110 115 120 125 130nH=/f stnuoc denniB1246800000000000090 95 100 (b1)4f5 = 01.58085105, 5ħf1 =6 01.313685 enVH*1 =7 0126175 ħ012 (00e00Vstnuoc denniB) stnuoc deλ(n7468m0000)000 166ħ080f1 =5 11.4391081 3e1V0102 1810ħ10 = 295 e1V580 71002060 n(Hc5=ħ)ω/(eVf 4) Ve)r/(dJs ЧЧ101001(c) 9 TW denniB3101(stnuoc)210890 T 1ħ5W142f =8 105.5613370ħ1508 =1 3012n.68ħH02=8( 1ee1/V3V64f)2 stnuoc denniBnniB11225050000000000000 80 100ħ = 9.2 eV120 140(ħd) (eV ) dħ0.1 80 100 120 140ħ (eV) E/ d 0.01 FIG. 3. (Color) Modulated spectra with (a) resolved and (b) nearly unresolved harmonic structure for P = 9 TW, Ve)r/JsЧ0.810(d) 100 9 1T2W0 140 Wa1t6er0 Wħin d(eoVw) ndeen=ote4d.7b×y d1a0s1h9ecdmb−ra3.ckeLtisnein-o(uat)sa(ncd) a(bn)d, r(eds)pseh0ctoiwvelsyp.ectra (d 0.01 KC Ч photons/sr and 40µJ/sr, respectively. These amount to ħ d E/ 150 200 250 300 ħ (e3V50) 4×109 photons and 90 nJ, assuming 1.5◦ angular ra- d dius inferred from our particle-in-cell (PIC) simulations, FIG. 2. (Color) Typical single-shot comb-like spectra. (a) whichagreeswithsimilarspectralintensitiesobservedat RawdataforP0 =9TW,ne =2.7×1019 cm−3. (b)Lineouts 0and0.53◦ channels. Fig.2(d)showsthespectrumupto fortworegionsmarkedin(a)bydashedbrackets. Dottedver- thespectrographthroughputcut-offof360eV.Theemis- tical lines for harmonics of the base frequency ω =0.885ω . f 0 sion reaches the ‘water window’ region (284-543 eV), re- The highest resolved order is n∗ = 126. (c) Spectra for the shotshownin(a)(red)andforPH =120TW,n =1.9×1019 quired for high-contrast femtosecond bioimaging, where 0 e cm−3 obtainedfromthe2nd (blue)and3rd (green)diffraction the9TWlaserproduces(1.5+−00..64)×1010 photons/srand orders. Inset: harmonicsinthe2nddiffractionorder;similarly 0.8+0.3µJ/sr, corresponding to 3×107 photons and 1.7 −0.2 forthe3rd order. (d)Thespectralregionembracingthe‘wa- nJ for the same 1.5◦ radius. The uncertainties are due terwindow’forP =9TW,n =4.7×1019 cm−3. Thedrop 0 e to the 10% filter thickness tolerance and CCD noise. at the carbon K absorption edge, K , is due to hydrocarbon C A large number of resolved harmonics, e. g. n∗ (cid:39) contamination. Dashedcurvesin(c,d)forbackgrounddueto H theCCDdarkcurrent,read-outnoise,andscatteredphotons. 126 in Fig. 2(a,b) and >∼ 160 in Fig. 2(c, inset), strictly bounds the laser frequency change during the harmonic emission process, δω/ω ≤1/(2n∗ ), otherwise the orders H In both campaigns the laser power significantly ex- n∗ and n∗ +1 overlap. The laser frequency decreases H H ceeds the relativistic self-focusing threshold, P ≈ due to an adiabatic depletion of the laser pulse losing sf 17 GW×(n /n ), where n = mω2/4πe2 ≈ 1.1 × energy on plasma waves, while the number of photons cr e cr 0 1021cm−3(µm/λ )2 is critical density. Thus both lasers’ is conserved [17–19]. For slow depletion, the frequency 0 pulses self-focus to tighter spots reaching higher irradi- downshift rate equals the energy depletion rate, δω/ω = ance determined by the plasma density and laser power. −δx/L [17–19]. ForFig.2(a,b),thedepletionlengthis dep The comb-like spectra comprising even and odd har- L ≈2.7 mm estimated from the observed 70% energy dep monicordersofsimilarintensityandshapearegenerated transmission through the 0.9 mm plasma. Estimate [19] bybothlinearlyandcircularlypolarizedpulsesinabroad gives a similar value, L ≈ 8.7(n /n )3/2λ ≈ 3.4 mm. dep cr e range of plasma electron densities from n (cid:39) 1.7×1019 The condition δω/ω <0.4% (or, alternatively, the phase e to 7×1019 cm−3. The harmonic base frequency ω is errorof<25mrad)givestheharmonicemissionlengthof f downshifted from the laser frequency ω0, Fig. 2(b,c), in <∼12µm. We note that the emission length can be much correlation with the transmitted laser spectra. longer than the longitudinal size of the moving source. The data obtained with different lasers demonstrate Inmanyshotswithlinearpolarization(40%withthe9 the effect’s reproducibility and robustness and the pho- TWlaser),theharmonicspectrumexhibitsdeepequidis- ton yield scalability with the laser power, Fig. 2(c). For tant modulations, Fig. 3. They are visible with dis- the120TWlaser,thephotonnumberandenergyinunit cerniblehighordersandevenwithnearlyunresolvedindi- solid angle within one harmonic at 120 eV are 2×1012 vidual harmonics in some shots, Fig. 3(b,d), where blur- 3 λ0z, ne Highs ohuarrcmeonic (a) <py> (b) 0 y 1 x Folds(A) 2 Electron spike Bow Cusps(A) 3 wave n n e e Folds Laser FIG. 4. (Color) (a) 3D PIC simulation. Electron density n (top, upper half removed) and its cross-section at z = 0 e (bottom). The electromagnetic energy density for ω ≥ 4ω 0 (red arcs). (b) Catastrophe theory model. Singularities in FIG. 5. (Color) 2D PIC simulation. (a) The electron den- the electron density created by foldings of the electron phase sity (blue), laser envelope (curves for a = 1,4,7,10), and space (x,y,(cid:104)p (cid:105)), where p is averaged over the laser period. y y electromagnetic energy density, W , for frequencies from 60 H to 100ω (red). (b) The upper spike emission spectrum for 0 the dashed rectangle in (a). (c) The electron density profile ring can be caused by a greater downshift of the laser 10 laser cycles earlier than (a) and the right spike structure frequency during a longer emission of harmonics, con- (d). sistently exhibiting a few times greater photon number. Since in some cases the modulation depth is ∼100%, we conclude that the modulations result from interference when a linearly polarized (along y axis) laser pulse self- between two almost identical strongly localized coherent focused to the amplitude of a = 6.6 and FWHM waist 0 sources separated in time and/or space. In shots with of10λ ,reachesapointwheren =1.8×1018cm−3. The 0 e circularly polarized pulses (90 shots with the 120 TW simulation window size is 125λ ×124λ ×124λ with 0 0 0 laser), large-scalespectralmodulationsarenotobserved. the resolution of dx = λ/32, dy = dz = λ/8; the num- The unique properties of the observed harmonics mis- ber of quasi-particles is 2.3 × 1010; ions are immobile. match previously suggested mechanisms of high-order The electromagnetic energy density of a high-frequency harmonics generation. Atomic harmonics are excluded (ω ≥4ω ) field reveals the source of high-order harmon- 0 becausebothlinearlyandcircularlypolarizedlaserpulses ics,Fig.4(a,redarcs): theyareemittedbytheoscillating producebothevenandoddharmonicorderswithaweak electron density spikes formed at the joining of the bow sensitivity to gas pressure. Betatron radiation is not rel- wave and cavity boundaries. While emission is also seen evant because the base frequency of its harmonics is de- from the cavity walls [12], at higher frequencies much termined by the plasma frequency and electron energy, stronger radiation originates from the electron spikes. not the laser frequency. Nonlinear Thomson scatter- The high-order harmonics emitted by the electron ing cannot provide the observed photon numbers even spikes are seen in 2D PIC simulation, performed in the under the most favorable assumptions. For the 9 TW 87λ ×72λ windowwithdx=λ /1024,dy =λ /112and 0 0 0 0 shot, Fig. 2(a,b,c), the numerical calculation of single- 6×108quasi-particles. InFig.5,thelaserpulseacquiring electron radiation spectra [7,20] gives at 100 eVat most the amplitude of a = 10, the duration of 16λ and the 0 0 2×10−10 nJ/eV sr, assuming the self-focused laser am- waist of 10λ meets plasma with n = 1.7×1019cm−3. 0 e plitude of a = 7, pulse duration of 30 fs, and the ob- Starting from the 7th order, harmonics are well dis- 0 servation angle of 15◦ close to optimum [7], where the cernible up to the 128th order, as allowed by the sim- high-frequencynonlinearThomsonscatteringcomponent ulation resolution. The emission is slightly off-axis with ismostintense. Thenumberofelectronsencounteredby the angle decreasing with increasing harmonic order. the laser pulse is N = πd2∆xn /4 ≈ 6×109, where A strong localization of electron spikes, their robust- e sf e n =2.7×1019cm−3,d ≈5µmistheself-focusingchan- ness to oscillations imposed by the laser and, ultimately, e sf neldiameter[21],and∆x≈12µmistheharmonicsemis- theirsuperiorcontributiontohigh-orderharmonicsemis- sion length (see above). This gives at most 1 nJ/eV·sr, sion is explained by catastrophe theory [27]. The laser 200 times smaller than experimentally observed. For pulsecreatesamulti-streamelectronflow[28,29]stretch- other spectra shown in Fig. 2(c,d) even less fraction cor- ing and folding an initially flat surface formed by elec- responds to nonlinear Thomson scattering. tronsintheirphasespace,Fig.4(b),[23,30]. Thesurface We performed two- (2D) and three-dimensional (3D) projectionontothe(x,y)planegivestheelectrondensity PIC simulations of harmonics generation using the code where outer and inner folds are mapped into singular REMP [22]. As seen in simulations [23, 24], the laser curvesoutliningthebowwaveandcavityboundaries,re- pulseundergoesself-focusing[2,18],pusheselectronsout spectively. Catastrophetheoryhereestablishesuniversal evacuating a cavity in electron density [18, 25] and gen- structurally stable singularities, insensitive to perturba- erating a bow wave [26]. Fig. 4(a) represents the case tions. The bow wave and cavity boundaries produce the 4 ‘fold’ type singularity (A , according to Arnold’s classi- wave formation with electron density spikes, and collec- 2 fication [27]), where the density grows as (∆y)−1/2 with tive radiation by a compact electric charge. Catastrophe decreasing distance to the boundary, ∆y. At the joint theory explains the electron spike sharpness and stabil- of the two folds, the density grows as (∆y)−2/3, forming ity. The maximum harmonic order scaling with the laser a higher order singularity – the ‘cusp’ (A ), Fig. 5(c,d), intensity will allow reaching the keV range. Our results 3 [23, 31]. Stronger singularities exist [29], however they openthewaytoacompactcoherentX-raysourcebuilton are not stable against perturbations. Located in a ring a university laboratory scale repetitive laser and accessi- surrounding the cavity head, the cusp is seen in sim- ble,replenishableanddebris-freegasjettarget. Thiswill ulations as an electron spike. For linear polarization, impactmanyareasrequiringabrightX-ray/XUVsource theharmonicemittingringbreaksupintotwosegments, for pumping, probing, imaging, or attosecond science. Fig. 4(a). The spike oscillations imposed by the laser We acknowledge the financial support from MEXT generate high-order harmonics. The cusp singularity en- (Kakenhi20244065,21604008,21740302,and23740413), suresatightconcentrationofelectriccharge,makingthe JAEA President Grant, and STFC facility access fund. emission coherent, i. e. the intensity is proportional to the particle number squared N2, similarly to the coher- e ent nonlinear Thomson scattering. However, the cusp consists of different particles at every moment of time, [1] U. Teubner and P. Gibbon, Rev. Mod. Phys. 81, 445 in contrast to a synchronous motion of the same par- (2009). ticles. We note that for constructive interference it is [2] G. A. Mourou et al., Rev. Mod. Phys. 78, 309 (2006). sufficientthatthesourcesizeissmallerthantheemitted [3] A. McPherson et al., J. Opt. Soc. Am. B 4, 595 wavelength in the direction of observation only. The es- (1987); P.B.Corkum,Phys.Rev.Lett.71,1994(1993); timated number of electrons within the singularity ring, F. Krausz and M. Ivanov, Rev. Mod. Phys. 81, 163 N ∼106,providesasignallevelclosetotheexperiment. 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