Inhomogeneous High-order Harmonic Generation in Krypton Clusters
H. Ruf, C. Handschin, R. Cireasa, N. Thiré, A. Ferré, S. Petit, D. Descamps, E. Mével, E. Constant, V. Blanchet, B. Fabre, Y. Mairesse
aa r X i v : . [ phy s i c s . a t m - c l u s ] F e b Inhomogeneous High Harmonic Generation in Krypton Clusters
H. Ruf , C. Handschin , R. Cireasa , N. Thir´e , A. Ferr´e , S. Petit , D.Descamps , E. M´evel , E. Constant , V. Blanchet , B. Fabre , Y. Mairesse Universit´e de Bordeaux - CNRS - CEA, CELIA, UMR5107, F33405 Talence, France a Universit´e de Toulouse, 118 route de Narbonne, F-31062 Toulouse, Franceb CNRS, Laboratoire Collisions Agregats Reactivite, IRSAMC, F-31062 Toulouse, France High order harmonic generation from clusters is a controversial topic: conflicting theories exist,with different explanations for similar experimental observations. From an experimental point ofview, separating the contributions from monomers and clusters is challenging. By performing aspectrally and spatially resolved study in a controlled mixture of clusters and monomers, we are ableto isolate a region of the spectrum where the emission purely originates from clusters. Surpringly, theemission from clusters is depolarized, which is the signature of statistical inhomogeneous emissionfrom a low-density sources. The harmonic response to laser ellipticity shows that this generation isproduced by a new recollisional mechanism, which opens the way to future theoretical studies.
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High harmonic generation (HHG) [1] refers to the in-teraction of high intensity laser light with matter, whichleads to the emission of broadband coherent radiationin the extreme ultraviolet domain. HHG from clustersis considered a promising light source, showing higheremission frequencies [2, 3]. HHG is also a spectroscopictool to extract structural and dynamical information onthe emitting medium from the properties of the harmonicradiation (spectrum, phase and polarization state). Thistechnique, which has been used to probe atoms [4, 5] andsmall molecules [6], relies on the basic mechanism of HHGdescribed as a three step model [1, 7, 8]: First, a boundelectron escapes in the strong laser field through tunnelionization. Second, the electron is driven away then ac-celerated back towards the parent ion. Finally the elec-tron recombines radiatively with the parent ion. Thisrecombination encodes the structure of the medium inthe emitted light. The extension of this technique to thecase of clusters would allow investigation of strong fieldprocesses in many-body systems, the role of multielec-tron effects and monitoring of cluster dynamics throughthe harmonic signal [9, 10].The exact mechanism of HHG from clusters is still de-bated. Various extensions of the three step model havebeen proposed [11–14]. The dominant channel gener-ally considered is ionization and recombination to thesame atom (atom-to-itself). Since clusters are dense me-dia, there is also a possibility of recombination to neigh-bouring ions [12, 13]. This atom-to-neighbour emissioncan produce incoherent radiation due to a lack of phase-locking between the two atomic wavefunctions [14]. An-other contribution to harmonic emission may come froma wavefunction partially delocalized over the whole clus-ter, from which electrons tunnel out of and to whichthey recombine coherently (cluster-to-itself). In additionto these recollisional mechanisms, very different mech-anisms could also co-exist as is the case in overdenseplasmas [15] or bulk crystals [16]. From an experimen- tal point of view a particular difficulty consists in dis-entangling the harmonics produced by different species(monomers and different size clusters) and possibly dif-ferent mechanisms.In this letter we present a detailed experimental studyof HHG from clusters which aims at disentangling theharmonic signal from clusters and identifying the mecha-nisms at play. By studying the spectral and spatial profileof the harmonics as a function of pre-expansion gas tem-perature, we identify one region corresponding to pureemission from clusters. A simulation verifying the re-sults from a polarization measurement confirm that onlyfew emitters contribute to this region, which is consistentwith the low density of large clusters in the generatingmedium. The fast decay of the harmonic signal with laserellipticity indicates that these harmonics are produced bya recollision mechanism which strongly depends on thecluster size, as expected in a cluster-to-itself picture.We generate high harmonics in a supersonic gas jet,employing a valve (Even-Lavie [17]) pulsed at 1 kHz witha d = 150 µm diameter trumpet nozzle and a jet expan-sion half angle of α ∼ ◦ . The backing pressure waskept constant at p = 20 bar, while the pre-expansionnozzle temperature T varied. Cluster formation canbe described by the empirical Hagena scaling parameterΓ ∗ = k ( d/tanα ) . T . p [18], with a condensation parame-ter k equal to 2890 for krypton [18]. The average clustersize can be estimated by [19]: ¯ N = 33 (Γ ∗ / . . Inour experiment the cluster size was varied by changingthe nozzle temperature between 350 K ( ¯ N ≈ N ≈ δN , of thecluster size distribution is broad and leads to significantcontribution of small cluster, which can be estimated by δN/ ¯ N = 45% [19].We used the Aurore Ti:Sa-Laser system from CELIAwhich delivers 7 mJ, 35 fs, 800 nm pulses at 1 kHz. Thelaser was focused using a 37.5 cm focal spherical mirrorat 4.8 mm from the nozzle exit, which ensures proper jet D i v e r g e n ce A ng l e ( m r a d ) Wavelength (nm)(a)(b) D i v e r g e n ce A ng l e ( m r a d ) Harmonic Order
Hagena Parameter (x 10 )
13 12 11 10 9 8 H a r m on i c S i gn a l ( a r b . un it s ) Temperature (K) P l a s m a S i gn a l ( a r b . un it s ) (c)
55 50 45 40 30 2535
FIG. 1: (a) High harmonic spectrum at 350 K. The verticallines correspond to electronic transitions in krypton ions. (b)Normalized difference signal of a spectrum at 350 K and aspectrum at 510 K. The signal from the pink areas dominatesat 350 K. thermalization. The harmonic spectrum was dispersedusing an abberation-corrected concave gold grating. Theimaging detector consisted of dual microchannel platescoupled to a phosphor screen and a 12 bit CCD-camera.Each image was averaged over 15000 laser shots.Figure 1(a) shows a harmonic spectrum obtained ata laser intensity of 1 × W/cm . The horizontal linearound zero divergence angle is due to scattered lightfrom lower harmonics. In addition to the harmonic spec-trum, we observe isotropic radiation as vertical lines, cor-responding to plasma lines [20, 21], which originate fromelectronic transitions in excited krypton ions. Plasmalines at 38.7 nm and lower wavelengths correspond toelectronic transitions in at least five times ionized kryp-ton ions [22]. A strong line observed at 43.4 nm refersto an electronic transition in Kr [23]. The laser inten-sity employed is not high enough to field ionize kryptonatoms five (I p =78.5 eV) or seven times (I p =125.8 eV). Inturn, impact ionization within clusters leading to inner shell ionzation [24] and leaving the freed electrons in aquasy bound cloud behind, can be responsible for theformation of such highly ionised Kr [11, 25].Apart from plasma lines, the spectrum from Fig. 1(a)does not show specific features that could be the signa-ture of the presence of clusters in the medium. In order toidentify the contribution of clusters we performed a dif-ferential measurement: we monitored the spatially andspectrally resolved evolution of the signal as the nozzletemperature increased from 350 K to 510 K, which de-creases the cluster average size. Fig. 1(b) reveals clearregions with distinct behaviors: a low energy compo-nent (up to harmonic 25) which increases with temper-ature, and an off-axis component (dominant from har-monic 23 to 31) which presents a higher cutoff and de-creases monotonously with temperature (Fig. 1(b)) as theplasma line. In the following we will focus on the lattercomponent, whose behavior is correlated to that of theplasma lines and thus to the presence of large clusters.In order to characterize the different mechanisms thatmay be at play in the harmonic emission, we performed apolarimetry measurement [26]. We used an unprotectedsilver mirror in combination with a gold grating as a fixedpolarizer and rotated the laser polarization with a zero-order half waveplate. The signal of each pixel of ourdetector is monitored as a function of the angle of thehalf waveplate resulting in a Malus’ law with a cos de-pendence. After subtracting the background, a Fouriertransform is performed to extract the amplitude of theoscillatory component, which is then normalized to thesum of the Fourier transform. This procedure providesthe amplitude of the oscillation and therewith the de-gree of linear polarization. Fig. 2(a) depicts the spatio-spectrally resolved degree of linear polarization obtainedat 350 K. As expected, the plasma lines are clearly unpo-larized. More surprisingly, the same region we attributedto cluster emission in Fig. 1(b) shows a remarkably lowdegree of linear polarization. A low degree of linear polar-ization is very unexpected for high harmonic generationby a linear laser field [26]. Note that we systematicallyobserved this low degree of linear polarization in differentconditions and various species (CO , argon and mixturesof CO /krypton), but not in the absence of clusters usingan effusive gas jet.The polarization direction of the high harmonics is setby the recollision direction of the electrons and the elec-tronic structure of the ground state to which these elec-trons recombine. Within the strong field approximation,in linear polarization the recollision direction is parallelto the laser field. In a centrosymmetric medium the har-monics are thus necessarily polarized along this direction.To go beyond the strong field approximation, we haveperformed Classical Trajectory Monte Carlo (CTMC) [5]calculation of the recollision angles in atoms at intensi-ties of 1.2 × W/cm . These calculations take intoaccount the influence of the ionic core on the electron (a) Harmonic Order
17 19 21 23 25 27 29 D i v e r g e n ce A ng l e ( m r a d ) Atoms FIG. 2: (a) Polarimetry map depicting the oscillation am-plitude of the the Malus’ law for each pixel at 350 K. (b)(b) Simulated harmonic spatial profile for a single laser shot(left), 1000 shots (middle), and degree of linear polarizationfor 1000 shots (right). The first row corresponds to HHG inatoms and the second row to HHG in large clusters. trajectories, which broadens the distribution of recolli-sion angles. After summing the recolliding electrons overthe polar angle, we obtain a recollision angle distribu-tion peaked around ± ◦ [5, 27]. In clusters, given thehigh charge states observed in our experiments, we ex-pect the ionic potential to play a more important role[28]. Futhermore screening effects can inhomogenize theelectric field over the cluster [29]. These two effects willlead to a broader recollision angle distribution.For understanding the origin of the apparent depo-larization of harmonic emission, we performed simplesimulations of the macroscopic generation process as-suming an infinitely thin medium. We randomly dis-tribute N emitters on a square grid of 100*100 µ m whichrepresents the generating medium. Each emitter lo-cated in ( x, y ) radiates an electric field ~E q ( x, y ) = I ( x, y ) q Eff / e iα q I ( x,y ) ~u ( x, y ), where I ( x, y ) is the fun-damental intensity distribution, q Eff the effective non-linearity of the harmonic emission (typically 5), α q theintensity dependence of the harmonic phase ( α ≈ × − cm /W for the end of the plateau), and ~u ( x, y )is unit vector with a direction randomly picked in thedistribution of recolliding angles. For clusters the lat-ter is assumed to be three times broader than the one obtained for atoms by using CTMC. We calculate theelectric field resulting from the coherent sum of the con-tributions from the N emitters in the far field by Fouriertransforming the near field profile. Two extreme casesare considered: emission from an ensemble of clustersand from an ensemble of monomers with a density of10 cm − . Assuming that only one atom out of 10 emits high harmonics (due to the recombination prob-ability) and a medium thickness of 100 µ m, we get 10 atoms to distribute on our grid. For clusters, the den-sity is much lower: assuming a medium with 80% of theatoms forming clusters of 30000 monomers, the densityis ≈ × cm − [19], which leads to a few hundred tothousand emitters with an emission probability the sameor higher than that for atoms. In the following we willconsider 1000 emitters and we checked that the resultswere qualitatively similar using 100 to 10000 emitters.These simulations show that while in the case of atomicemission the harmonic far field profile is well defined in asingle laser shot, the profile obtained from clusters is veryinhomogeneous and fluctuates significantly from shot toshot because there are too few emitters to obtain a niceconstructive interference on axis and destructive off axis(Fig. 2(b)). The polarization state from clusters (notshown) is also very inhomogeneous, showing importantpolarization angles and ellipticities. When averaging over1000 shots, the spatial intensity profile becomes muchsmoother and the polarization angle is quasi homoge-neously equal to zero. We also calculate the degree oflinear polarization of the resulting light, by defining aMalus law out of the Stokes parameters and extracting itsnormalized oscillatory component. While the polariza-tion appears perfectly linear in monomers, the emissionfrom clusters appears depolarized. These results confirmthat the unpolarized cutoff harmonics (Fig. 1(b)) area fingerprint of HHG from large clusters, which have alow density in the jet and thus produce inhomogeneousharmonic emission.In order to further elucidate clusters as a class of emit-ters, we studied the sensitivity of the high harmonicsignal to the ellipticity ǫ of the generating laser field.The main polarization axis was kept vertical by employ-ing a fixed zero-order quarter waveplate behind an ad-justable zero-order half waveplate. The ellipticity wasvaried from -0.45 to 0.45 recording 49 points. As ǫ in-creases, we observe an exponential decay of the wholeharmonic spectrum, which is consistent with a recolli-sion picture of HHG. The decay rate β of the harmonicintensity I with ellipticity can be evaluated by a Gaussianfit: I ∝ exp ( − βǫ ). We extract this decay rate for eachpixel of the detector and obtain the results shown in Fig.3(a). As in Fig. 1(b)) the same areas can be identified,the off-axis cutoff with β >
45 and on-axis plateau with β <
45. Remarkably the nozzle temperature increasesthe decay rate in the first case, while it hardly affects β in the second case (Fig. 3(b)). When extracting the
15 17 19 21 23 25 27 29 253035404550556065 D i v e r g e n ce A ng l e ( m r a d ) -6-4-20246 (a) Harmonic OrderTemperature (K)
410 430 450 470 490 5102530354045505560
B(cid:0)(cid:1)(cid:2) (b)
H(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9) (cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19) (cid:20)(cid:21)
FIG. 3: (a) Harmonic decay rate β at 510 K. (b) Evolution of β with temperature. The values are an average over the areasdepicted above. slope for each pixel, a positive slope is observed only forthe off-axis cutoff region.The exact mechanism of high harmonic generation inclusters, responsible for the emission of the highest har-monics, has yet to be determined. First, let us mentionthat the laser intensity and electron density are too lowto produce significant coherent wake emission from theplama [15]. Second, the fast decay of the harmonic signalwith ellipticity indicates that the process is recollisionaland is thus different from what was recently reported inthe case of bulk crystals [16], which still shows a signif-icant signal at ǫ = 0 .
5. Third, recombination to neigh-bouring ions [14] can also be excluded as it would leadto lower β -values. We therefore suggest another mecha-nism of HHG in clusters, namely tunnel ionization froma partly delocalized electron wavefunction and recombi-nation to this wavefunction (cluster-to-itself). Even lo-calized electronic states in a Van der Waals clusters [30],could be driven by the strong laser field from one siteto another, ending up in a partially delocalized wave-function after a few optical cycles [31]. This processis accompanied by a significant amount of ionization ofthe cluster (as observed from the plasma lines) whichwill increase its ionization potential and consequently thehigher cutoff observed for the cluster emission. Record-ing the harmonic signal versus ǫ , one obtains the cross-correlation between the recolliding wavefunction and theground wavefunction. In the ”cluster-to-itself” picturethe electrons of interest are expected to tunnel mostlyfrom the surface of the cluster, so that the initial width ofthe electron wavepacket after tunneling is proportional tothe cluster size. The wavepacket spreads laterally during acceleration outside the cluster – the smaller the initialwavepacket, the stronger the spread. The decay rate, β ,is determined by the width of the recolliding wavepacketand the extension of the initial wavefunction, i.e. thecluster size. For large enough clusters the latter will bedominant, so that the decay rate is expected to decreasewith increasing cluster size. This is consistent with theobserved temperature dependence of H27 in Fig. 3(b).In conclusion, by performing a 2D spectro-spatial anal-ysis, we are able to disentangle several contributionsto the harmonic signal from a mixture of clusters andmonomers. The high-energy off-axis emission showsthree clear features: a higher cutoff of the harmonic emis-sion, a higher decay rate with ellipticity when the averagecluster size is reduced, and a very low degree of linear po-larization. We attribute these effects to a high harmonicgeneration mechanism in which delocalized electrons tun-nel from a cluster and recollide coherently to the wholecluster. This hypothesis needs to be further investigatedusing the appropriate theoretical tools. This work couldbe extended by performing a complete polarimetry studyof the harmonic emission in elliptical laser fields, whichwas recently shown to be a sensitive probe of the in-fluence of the ionic potential in HHG [27]. This wouldenable us to determine the process responsible for thegeneration of the lowest harmonics and possibly furtherdifferentiate the contribution from monomers and fromatom-to-atom emission in clusters. 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