Hergott et al. Vol. 20, No. 1 /January 2003 /J. Opt. Soc. Am. B 171
Application of frequency-domain interferometryin the extreme-ultraviolet
range by use of high-order harmonics
Jean-Francois Hergott, Thierry Auguste, Pascal Salieres, Laurent Le Deroff, Pascal Monot,Pascal d’Oliveira, David Campo, Hamed Merdji, and Bertrand Carre
Departement de Recherche sur l’Etat Condense, les Atomes et les Molecules, Direction des Sciences de la Matiere,Commissariat a l’Energie Atomique, Service des Photons, Atomes et Molecules, Centre d’Etudes de Saclay,
91191 Gif-sur-Yvette, France
Received June 6, 2002; revised manuscript received September 17, 2002
1. INTRODUCTIONWith the rapid and recent advances in the technology ofshort-pulse high-power lasers, many experimental andtheoretical studies have been carried out on the genera-tion of high-order harmonics of intense laser pulses thatinteract with gases.1,2 Progress has been made both onthe covered spectral range (the water window, corre-sponding to wavelengths shorter than 4.4 nm, wasreached with harmonic orders as high as 300)3–5 and onthe generated harmonic beam energy (energies in excessof 1 mJ for the 15th harmonic have recently beendemonstrated6). The properties of the harmonic beamshave been characterized (spatial7–9 and temporal10–12 co-herence, beam quality,13 pulse duration14–16), demonstrat-ing that harmonic radiation presents unique properties ofcoherence and femtosecond duration in the extreme-ultraviolet (XUV) range. Note that the possibility of gen-erating attosecond pulses has recently beendemonstrated.17,18 Finally, the tunability and high rep-etition rate (up to 5 kHz) of the harmonic radiation makeit a versatile XUV tabletop source that can be easilyimplemented near, e.g., high-intensity lasers for system-atic studies of laser–matter interactions.
A growing number of experiments have been reportedthat use harmonic ultrashort pulse duration to performtime-resolved studies of ultrafast dynamics in atomic19,20
and molecular21–23 spectroscopy and in solid-state24,25
and plasma26 physics. On the other hand, only a few ex-periments make use of the good coherence properties.XUV interferometry, developed so far with synchrotrons27
or x-ray lasers,28–30 would benefit significantly from thegeneration of high-order harmonic properties. Thanks totunability, one can choose the appropriate probe wave-length, for example, far from resonances; or, in plasmas,one can probe different density layers by switching fromone harmonic to another. The ultrashort pulse durationwould allow one to follow ultrafast processes such as op-tical field-induced ionization (OFI) or to prevent fringeblurring that is due to the rapid expansion of a plasma, asobserved in interferometry experiments that use x-raylasers.28 Finally, XUV interferometry can use a uniqueproperty of harmonic generation: the mutual coherenceof two harmonic sources. It is possible to generate twophase-locked harmonic pulses from two phase-locked la-ser pulses.31–33 In this scheme we avoid splitting theXUV harmonic beam into two subbeams by amplitude orwave-front division, an operation that implies the use of acomplicated and expensive XUV beam splitter with poorefficiency. It is much easier to split the laser beam by useof conventional optics, then generate two identical har-monic beams, and finally recombine them after theprobed medium.
The possibility for mutual coherence in the generationof high-order harmonics has been used in two interferom-etry experiments of plasma diagnostics. In the firstexperiment,34 performed in collaboration with the LundLaser Center, Sweden, we generated two phase-lockedharmonic beams [harmonic 11 (H11)], spatially separatedby focusing the two halves of a laser beam at different lo-cations in a gas jet. We created the plasma on the path of
2003 Optical Society of America
172 J. Opt. Soc. Am. B/Vol. 20, No. 1 /January 2003 Hergott et al.
one XUV beam by irradiating an aluminum foil with a 70-mJ, 300-ps laser pulse and used the second XUV beam asthe reference. The interference fringes observed in thefar field when the beams overlap are locally shifted, al-lowing a two-dimensional mapping of the electron densityas high as 2 3 1020 cm23. In the second experiment32 wegenerated two phase-locked harmonic pulses separated intime by focusing two (phase-locked) laser pulses at thesame location in a gas jet. The XUV pulses were shownto interfere in the spectral domain, i.e., after dispersionon a grating. It is well known that the spectrum of thetotal field, which is the sum of two identical pulses sepa-rated by delay t, displays an interferometric modulationof the single-pulse spectrum, that is, a fringe pattern.This is the temporal analog of the Young two-slit experi-ment in which diffraction is replaced by dispersion.35 Itprovides the basis for Ramsey fringe spectroscopy andmore generally for frequency-domain interferometry(FDI), which is routinely used in the visible and infraredspectral ranges to probe the dynamics of ultrafastphenomena36–40 in solid-state and plasma physics. Inthe reported experiment32 this technique was, for the firsttime to our knowledge, successfully transposed in theXUV range and applied to the time-resolved diagnostic ofa laser-produced plasma.
In this paper we give a full account of this experimenttogether with complementary studies performed sincethat time. In Section 2 we use XUV FDI to diagnose thephase locking between two harmonic pulses generated inthe same medium. We report a detailed study of thefringe contrast as a function of the generation param-eters: gas pressure, laser intensity, and focus position,harmonic order, delay, and relative intensity of the two la-ser pulses. This gives insight into the dynamics of thegeneration and ionization processes. In Section 3 we useXUV FDI to measure the electron density of a plasma cre-ated by OFI of a high-pressure helium jet with an intenseshort-pulse laser. By varying the delay between thepump and the probes, one can follow the temporal evolu-tion of the electron density, in principle, with a femtosec-ond resolution. In our experimental system—underdense plasma, not optimal time-resolution—thephysical significance of the density measurements is lim-ited. Our primary purpose has been to demonstrate thattime-resolved plasma diagnostic is feasible with the XUVFDI technique. In Section 4 we briefly discuss the possi-bilities initiated by this study.
2. STUDY OF PHASE LOCKING BETWEENTWO HARMONIC PULSESA. Experimental SetupWe carried out the experiments on the UHI10 laserfacility, which is a two-beam, 10-Hz, Ti:sapphire system(l 5 800 nm). Briefly, the low-energy ultrashort pulseproduced by a modified commercial Ti:sapphire oscillatoris stretched up to 300 ps by an aberration-free Offnerstretcher. After four amplification stages, the pulse en-ergy is approximately 1.2 J (600 mJ after recompression).The pulse is then recompressed down to 60 fs in a vacuumchamber directly connected to the experimental chamber;this defines the main beam (pump) with 10-TW peak
power. A small amount of the energy is picked up be-tween the third and the fourth amplifier and sent to a sec-ond compressor in air. This second beam (probe) has anenergy of 4 mJ after recompression.
The high-order harmonics were generated in a gas jetby two infrared pulses (low-energy beam) separated intime. We obtained the delayed pulses by using calibratedbirefringent quartz plates whose axes were rotated at 45°from the laser polarization. For a short incident laserpulse, the difference between group velocities on the ordi-nary and the extraordinary axes leads to a time splittingthat is proportional to the plate thickness. Plates withdifferent thicknesses were used to produce delays be-tween 120 and 600 fs. The birefringent plates wereplaced before the compressor of the low-energy beam toavoid self-phase modulation at higher intensity. A polar-izer set after the birefringent plate was used to projectboth components (ordinary and extraordinary) on thesame axis so that they could interfere. Note that thissetup is highly stable (compared with a Michelson inter-ferometer) and consequently allows for integration of thesignal over thousands of shots without blurring thefringes, the latter being sensitive to dephasing betweenthe two infrared pulses. As a matter of fact, a dephasingof half of a fundamental period divided by the harmonicorder (a few hundred attoseconds) would shift the har-monic pulses out of phase, inducing a change of the fringepattern from bright to dark fringes and conversely. As acounterpart of stability, the birefringent plates do not al-low continuous tuning of the delay.
The infrared pulses were focused with an f/110 bicon-vex lens at intensities up to 5 3 1014 W/cm2 onto a jet ofargon or xenon. The jet nozzle was a 0.3 3 3 mm slit,the largest dimension of the slit being parallel to the laserpropagation axis. The backing pressure was between150 and 900 Torr; the pressure in the jet, measured with aMach–Zehnder interferometer, was approximately a fac-tor of 10 lower.
The spectral analysis of the harmonics was performedwith a grazing-incidence flat-field XUV spectrometer in-cluding a gold-coated toroidal mirror and grating. Thedetection in the dispersion plane was carried out eitherwith a photomultiplier placed behind a slit or with micro-channel plates coupled to a phosphor screen and an 8-bitcharge-coupled device camera. In the first case, a fringepattern was obtained as a step-by-step scan of the trans-mitted frequency. Each point in the spectrum was aver-aged over 40 shots with an allowed dispersion of 7% in la-ser energy. The slit in front of the photomultiplier was16 mm wide, which ensured a spectral resolution betterthan 0.1 Å. In the second case, the fringe pattern wasobtained in a single shot. The microchannel plates weretilted to a grazing incidence of 8° to increase the effectivespectral resolution by a factor of 7. Indeed, the observa-tion of spectral interferences requires both a high-dispersion spectrometer, and a high-resolution detectorsince the period of the fringes decreases quadraticallywith the harmonic order and is inversely proportional totime delay t between the two pulses. In the wavelengthdomain, the period of the modulation is given bydl 5 l2/ct. As an example, for l 5 72.7 nm (H11) andt 5 450 fs, the resolution of the system (spectrometer
Hergott et al. Vol. 20, No. 1 /January 2003 /J. Opt. Soc. Am. B 173
plus detector) should be much better than dl 5 0.39 Å toresolve the fringes.
B. Experimental ResultsLet us consider first, in Fig. 1, the experimental spectra ofH11 generated in argon by a single 60-fs laser pulse(dashed curve) and by two phase-locked pulses (solidcurve) focused at 2 3 1014 W/cm2 (see caption for detailedexperimental conditions). In the latter case, the delaybetween the pulses was 120 fs, leading to highly con-trasted fringes in the spectrum with a modulation ampli-tude of 90%. It appears clearly that the envelope of thefringe pattern is approximately the same as the one ob-tained with a single pulse. Note that these spectra arenormalized, but that the maximum (spectral) intensity ofthe two-pulse spectrum should be four times larger thanthat of the single-pulse spectrum because of constructiveinterference.
In the following, we discuss the influence of the differ-ent physical variables (harmonic order, delay) or genera-tion parameters (absolute and relative intensities of thepulses, gas pressure, and focusing) on the two-pulse spec-trum in the case of generation in argon.
1. Variation with Order and DelayFigure 2 shows the experimental spectra of (a) H11, (b)harmonic 15 (H15), (c) harmonic 19 (H19), (d) harmonic23 (H23) generated by two pulses delayed by 120 fs.Regular contrasted fringes are measured for all orders.As expected, the fringe period decreases quadraticallywith the harmonic order. It is close to the resolutionlimit of our detection system for H23 (dl 5 0.34 Å).Partially because of the finite resolution, the contrast de-creases with increasing order: it drops from 70% for H11to 25% for H23. From H15 to H23, there is no modula-tion of the spectrum in the far blue wing. The nonmodu-lated part broadens with the harmonic order, reaching al-most half of the spectrum for H23.
Fig. 1. Experimental spectra of (H11) generated by a single60-fs laser pulse (dashed curve) and by two 60-fs pulses (solidcurve) focused at 2 3 1014 W/cm2 in an argon gas jet (120-fs de-lay, 600-Torr, backing pressure, 3-cm jet/focus position).
We checked that the fringe period varies as the inverseof the time delay between the two pulses. Figure 3 showsthe evolution of the fringe period for harmonic 13 for a de-lay between 120 and 600 fs. On the one hand, one cansee that the measured fringe period actually decreases as1/t. On the other hand, the fringe contrast remains un-changed when the delay increases, even for t much longerthan the pulse duration. The experiment therefore dem-onstrates a robust mutual coherence between the twoharmonic pulses.
2. Variation with Generation ParametersThe influence of the laser intensity on the fringe patternwas investigated. The experimental spectra of H15 ob-tained for laser intensities of 2 3 1014, 3.5 3 1014, and5 3 1014 W/cm2 are shown in Figs. 4(a), 4(b), and 4(c), re-spectively (120-fs time delay). When we increased the la-ser intensity, the blue wing of the spectrum gained ampli-tude and no longer showed modulations whereas the redwing still showed modulations but with a reduced con-trast, which dropped from 65% at 2 3 1014 W/cm2 to 25%
Fig. 2. Experimental spectra of (a) H11, (b) H15, (c) H19, (d)H23, all generated by two laser pulses delayed by 120 fs and fo-cused at 2 3 1014 W/cm2 (600-Torr, backing pressure, 0 jet/focusposition).
Fig. 3. Fringe period as a function of time delay t between twolaser pulses. The circles represent the experimental points, andthe solid curve represents a 1/t fit.
174 J. Opt. Soc. Am. B/Vol. 20, No. 1 /January 2003 Hergott et al.
at 5 3 1014 W/cm2, resulting in a strong blue–red wingasymmetry in the spectra. At even higher intensity, thefringes completely disappeared and the spectrum broad-ened. This trend was also observed for other harmonics,clearly for H19 and to a lesser extent for H11.
The fringe pattern also depends on the laser focus po-sition with respect to the gas jet. When the laser focusmoved from 3 cm in front of the jet (condition of Fig. 1) tothe center of the jet (Fig. 2), the fringe contrast was re-duced from 90% to 70% for H11. In the case of H15, weobserved a clear broadening of the spectrum, as shown inFig. 5. An additional peak appears on the blue side for
Fig. 5. Variation of the fringe pattern of H15 with the jet/focuspositions of z 5 3 cm (solid curve) and z 5 0 (dashed curve).
focusing in the jet (dashed curve) compared with focusing3 cm before the jet (solid curve).
The contrast was found to decrease when the gas pres-sure increased, which is illustrated in Fig. 6 with thespectra of H19 obtained at 150 Torr (dashed curve) and600 Torr (solid curve). The contrast dropped from 40% toapproximately 20% when the pressure increased from 150to 600 Torr.
3. Variation with Characteristics of the Two PulsesWe first investigated the influence of the relative inten-sity of the two laser pulses. We changed the intensity ra-tio, fast over slow pulse, by rotating the retarding plate:it varies as (tan a),4 where a is the angle between the slowaxis of the birefringent plate and the laser polarization(the first pulse is more intense than the second fora . 45°). We started from a situation in which a is 45°,the intensity of both pulses being 3.5 3 1014 W/cm2. Fig-ure 7 shows the fringe patterns of H15 obtained for twosymmetric values of the rotation angle: 40° (solid curve)and 50° (dashed curve), corresponding to an intensity ra-tio of 1/2 (I1 5 2.3 3 1014 W/cm2, I2 5 4.6 3 1014 W/cm2) and 2 (1 ↔ 2), respectively. It is clear that the twosituations are not equivalent: When the second pulse isthe most intense (solid curve) the fringe pattern still ex-hibits contrasted fringes, whereas when the first pulse ismore intense (dashed curve) the spectrum is much largerwithout clear modulation. For a 5 42° (I1 /I2 5 0.7) itwas even possible to increase the contrast slightly com-pared with a 5 45°. Another interesting feature wasobserved in the case of H11, illustrated in Fig. 8 (time de-lay of 450 fs). The fringe pattern obtained for a 5 45°and intensities of 3 3 1014 W/cm2 is represented by adashed curve. As expected, the fringe period wasreduced by approximately a factor of 3 compared with thecase of a 120-fs delay (Fig. 2). When a increased to 49°(I1 5 3.8 3 1014 W/cm2, I2 5 2.2 3 1014 W/cm2), the
Fig. 6. Evolution of the fringe pattern of H19 with backing pres-sures of 150 Torr (dashed curve) and 600 Torr (solid curve)(5 3 1014-W/cm2 intensity, 1-cm, jet/focus position, 120-fs timedelay).
Hergott et al. Vol. 20, No. 1 /January 2003 /J. Opt. Soc. Am. B 175
fringe contrast was strongly reduced on the red wing ofthe spectrum. For a 5 41°, it is on the blue side thatthe contrast is reduced.
In addition, we investigated the influence of a phaseshift introduced between the two laser pulses. A half-wave plate was placed between the retarding plate andthe polarizer so that, by dephasing the laser pulses, wewould induce a controllable phase shift between the twoXUV pulses. First, we expected that, when the half-waveplate axes were aligned at 45° between that of the bire-fringent plate, both XUV lasers pulses would identicallyaffect the plate, i.e., would not relatively dephase. Then,once the half-wave plate axes were aligned with that ofthe birefringent plate, the two infrared pulses would rela-
Fig. 7. Fringe patterns obtained for H15 at two angles of thebirefringent plate: 40° (solid curve) and 50° (dashed curve)(600-Torr, backing pressure, 1-cm, jet/focus position, 120-fs timedelay).
Fig. 8. Fringe patterns obtained for H11 at two angles of the bi-refringent plate: 45° (dashed curve) and 49° (solid curve) (600-Torr, backing pressure, 1-cm, jet/focus position, 450-fs time de-lay).
tively dephase p ; this would lead to a phase shift ofq 3 p on the harmonic pulses, with q as the odd har-monic order, thus inducing a fringe shift of half of a periodon the interferogram. Actually, it turned out that otherphase modulation effects were introduced by the plate, sothat relative phase shifting of the XUV pulses betweenthe two above cases was achieved for rotation of the half-wave plate of less than 45°, which is illustrated in Fig. 9for the conditions of Fig. 7 and a 5 45°. From the initialsituation, spectrum as a solid curve, the XUV pulses arerelatively phase shifted by kp, with k odd, spectrum as adashed curve, by rotation of the half-wave plate. The twospectra are clearly out of phase.
C. DiscussionThe study presented in Subsection 2.B shows that (1) twophase-locked harmonic pulses can be produced that inter-fere in the frequency domain, (2) the contrast of thefringes decreases as the harmonic order, the laser inten-sity, and the gas pressure increase, and (3) there is anasymmetry in the spectra between the blue and the redsides that shows a different behavior as a function of gen-eration parameters.
To explain the above features, one has to examine thephysics of harmonic generation carefully. Let us firstconsider point (2): The reduced contrast can proceedfrom two phenomena, either different intensities or a deg-radation of mutual coherence of the two harmonic pulses.Both effects can be induced by ionization of the medium,as is shown below. High-order harmonic generation is in-trinsically linked to ionization, and a high efficiency gen-erally implies a high degree of ionization. The second la-ser pulse thus interacts with a medium that has beenpartially ionized by the first one. Consequently, har-monic generation from the second pulse can be stronglyreduced due to (a) depletion of the generating medium, (b)defocusing of the laser beam by the free-electron densitygradient, and (c) phase mismatch induced by the free elec-trons between the laser-driven nonlinear polarization and
Fig. 9. Fringe patterns obtained for H11 at two angles of a half-wave plate introduced before the polarizer. Between the twocases, the relative phase of the two harmonic pulses differs by kpwith k odd (450-fs time delay).
176 J. Opt. Soc. Am. B/Vol. 20, No. 1 /January 2003 Hergott et al.
the macroscopic harmonic field in the medium. The in-tensity I2 of the second harmonic pulse is then muchsmaller than I1 and the contrast of the fringes subse-quently decreases as 2AI2 /I1. Moreover, the temporallyvarying free-electron dispersion induces a blueshift of thelaser frequency that is reflected as a broadening of theharmonic spectrum on the blue side. This temporalmodulation of the phase of the harmonic emission couldresult in a loss of mutual coherence between the two har-monic pulses and therefore to a blurring of the fringesand a reduced contrast. Note also that the temporallyvarying atomic dispersion that is due to ionization couldplay a role in the reduced contrast by affecting the phasesof both the laser and the harmonic fields, which could beof particular concern for the harmonic orders close to theionization threshold or an atomic resonance, where theatomic dispersion is high. However, the free-electron dis-persion that affects the laser pulses and the subsequentgenerating processes is in general the dominant cause ofcontrast degradation.
It is thus clear that an increase in laser intensity in-duces a fast decrease of the fringe contrast because of thehigher ionization (see Fig. 4). Similarly, the reduced con-trast observed when increasing the order (see Fig. 2) re-flects the fact that higher-order harmonics are efficientlygenerated later in the laser pulse at higher intensity,when ionization grows rapidly to its maximum and sub-sequently induces more perturbation in the generationprocess. A higher pressure magnifies the free-electrondispersion effects and also determines a lower contrast(see Fig. 6). The influence of the relative intensity of thetwo laser pulses can be easily understood on the same ba-sis: when the first pulse is the most intense, high ioniza-tion is achieved at the end of the pulse, so that generationconditions are strongly asymmetric between the two har-monic pulses; the interference disappears. In contrast,when the second pulse has the highest energy, it can com-pensate for the defocusing of the laser beam and balancethe intensities of the two harmonic pulses, resulting in ahigher contrast (see Fig. 7).
To interpret point (3), asymmetry between the blue andthe red sides in the harmonic spectrum, one should takeinto account an additional process, namely, the intrinsicharmonic chirp. It has been shown10,12 that in a strong-field low-frequency regime the phase of the harmonic di-pole varies quasi-linearly with the laser intensity('2hqIL , with hq . 0). This dephasing relative to thedriving field corresponds to the semiclassical action ac-quired by the electron along the quantum orbit that leadsto harmonic emission. To understand its effect, let usconsider the time-dependent frequency of the harmonicfield, v(t), derived at first approximation of perfect phasematching, i.e., assuming that the harmonic field phase isequal to the wq
NL phase of nonlinear polarization1:
v~t ! 5 2]wq
NL
]t
5 qvF1 1e2
2mce0v2 Ez0
z ]Ne~z8, t !
]tdz8G 1 hq
]IL
]t.
(1)
The second term in Eq. (1) is the blueshift of the laserpulse that is due to the time-dependent electron density,integrated along the optical path in the gas, which is sub-sequently reflected in the harmonic frequency (see furtherin the text). The third term, proportional to the time de-rivative of the intensity corresponds to the intrinsic chirp:on the leading edge of the pulse @(]IL /]t) . 0#, the har-monic emission is shifted to the blue, whereas on the fall-ing edge @(]IL /]t) , 0#, it is shifted to the red. One canthus state that fringes on the blue (red) side of the spec-trum come from interference of the harmonic fields emit-ted on the leading (falling) edges of the two laser pulses.
When the laser intensity is high enough and I1 > I2 ,most of the ionization occurs on the leading edge of thefirst pulse [dashed curve in Fig. 10(a)]. This is evenstrengthened if the high free-electron density gradient in-duces a defocusing of the laser beam, since the reduced in-tensity will not ionize the medium further. Conse-quently, the leading edges of the first @(]Ne /]t) . 0# andsecond harmonic pulses @(]Ne /]t) ' 0# will differ by bothamplitude and chirp and will present poor mutual coher-ence. In contrast, the two trailing edges are identicallyaffected, i.e., reduced amplitude but no shift. One thusexpects that on the blue side of the fringe pattern the har-monic intensity increases and contrast decreases,whereas on the red side the intensity is reduced but the
Fig. 10. Simulation of the fringe pattern of H15 generatedin argon. (a) Harmonic double pulse temporal profile witht 5 400 fs (solid curve) and temporal evolution of the ionizationfront (dashed curve). (b) Spectral fringe pattern without (solidcurve) and with (dashed curve) temporal evolution of the electrondensity.
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contrast remains high. This is illustrated in Fig. 10 by asimple model. When we take into account only the inten-sity difference between the two harmonic beams, the con-trast of the fringes for H15 is reduced homogeneously inthe whole spectrum [solid curve in Fig. 10(b)]. In con-trast, when the dephasing that is due to the ionizationfront is included in the calculation, the contrast remainshigh on the red side, whereas it is strongly reduced on theblue side. Note also the blueshift of the central wave-length that is due to the ionization-induced blueshift ofthe first infrared pulse. The behavior observed in Figs. 2and 4 is thus consistent with a negative chirp of the har-monic emission.41–43 More advanced simulations are inprogress, including ionization and, accordingly, all thetime-dependent dispersion terms that enter the phases ofthe laser and the harmonic fields, to determine their rela-tive contribution to the contrast reduction. Preliminaryresults have been published in Ref. 32.
3. APPLICATION OFEXTREME-ULTRAVIOLET FREQUENCY-DOMAIN INTERFEROMETRY TO PLASMADIAGNOSTICHere, we demonstrate the feasibility of a time-resolvedplasma diagnostic by using FDI in the XUV. To ourknowledge it constitutes the first XUV interferometrymeasurement on a subpicosecond time scale.
A. Experimental Setup and ResultsWe have used the XUV FDI technique to measure thetime-dependent electron density of a plasma created byOFI of helium gas. The experimental setup is shown inFig. 11. A fraction of the energy of the 10-TW laser beam(160 mJ after compression) was focused to a vacuum in-tensity of 1018 W/cm2 with an f/6 off-axis parabolic mirrorseveral millimeters before a high-pressure pulsed heliumjet. A cylindrical nozzle with a 1-mm-diameter outletwas used. The atomic density profile and its maximumabsolute value as a function of the backing pressure weremeasured with a Mach–Zehnder interferometer. Thedensity profile was approximately Gaussian; a maximumdensity of 2.4 3 1019 cm23 was obtained for 18-bars back-ing pressure. The overall cross-sectional diameter of theplasma measured by infrared shadowgraphy was 270 mm.
Fig. 11. Experimental setup for the application of XUV FDI tothe diagnostic of a helium plasma.
Since the laser intensity was far above the ionization in-tensity of both He I and He II (Ii ; 1016 W/cm2), com-pletely stripped ions were produced on the full length ofthe jet along the pump beam axis. The plasma wasprobed with H11 at 45° from the propagation axis of the10-TW laser beam (note that only practical constraints—the parabolic mirror is not drilled—have determined thenoncollinear pump–probe geometry). The reasons forchoosing H11 generated in Xe were twofold: first, thehigh XUV flux obtained makes it possible to overcome theself-emission of the plasma. Second, the photon energyof H11 (17 eV) is below the first excited state of He (21eV), which prevents absorption of the harmonic beams bythe neutral gas that surrounds the plasma. The XUVpulses were focused onto the plasma with an f 5 1-m to-roidal mirror. The focal spot size was measured by use ofa knife-edge technique. Figure 12 shows the derivativeof the signal measured as a function of the knife-edge po-sition; the dashed curve is a Gaussian fit with a 110-mmfull width at half-maximum. Since the harmonic focalspot is smaller than the plasma size, we have assumedthat the phase shift integrated over the optical path washomogeneous in the transverse spatial dimension. Notethat, unlike classical interferometers that give two-dimensional spatial information, FDI is one dimensionalin space, with the other dimension of the interferogrambeing the spectrum. Thus, all the fringes carry the sameinformation (averaged over one spatial dimension), whichcreates the sensitivity in this type of interferometer.
The time delay between the two phase-locked harmonicpulses was fixed at t 5 300 fs (constant fringe period),but delay Dt between the 10-TW laser pulse that gener-ates the plasma and the XUV pulses was varied by use ofa delay line to probe the ionization dynamics. For thesake of definiteness, we set the zero delay as the one forwhich the pump pulse crosses the jet exactly between the
Fig. 12. Measurement of the spot size of H11 at the focus of thetoroidal mirror by use of the knife-edge technique.
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probe pulses; Dt is positive (negative) when the pump isbefore (after) the probe midpoint (the formal definition ofzero delay is of limited significance because of the coarsetemporal resolution). Note that only the temporal varia-tion of the refractive index (electron density) betweenprobe 1 and probe 2 is measured, which implies that thisvariation is not negligible at the t time scale. Accord-ingly, from the reference situation in which there is nophase shift between the pulses (Dt , 2t/2), a shift is ex-pected when the plasma is produced either before probes(Dt . t/2) but still evolves at probe time or between thetwo pulses. Figure 13(a) shows the fringe pattern mea-sured as a function of the pump–probe delay. Each ver-tical stripe that corresponds to one delay is obtained fromspatial integration of a single-shot image of the fringes.From the reference (Dt , 2300 fs, no plasma), fringesare clearly shifted when the pump is between the probepulses (2300 fs , Dt , 300 fs), that is, when electrondensity increases rapidly between probe times 1 and 2.Note finally that a blurring of the fringes occurs aroundthe 2150-fs delay. The average normalized shift, dl/Dl,where dl is the negative wavelength shift measured inFig. 13(a), is plotted in Fig. 13(b) (filled circles); also plot-ted is the calculated shift (dashed curve) from a simplemodel discussed in the next subsection.
B. DiscussionThe time-dependent electron density averaged along thepropagation direction can be, in principle, deduced fromthe experimental curve given in Fig. 13(b). The mea-sured fringe shift is proportional to the relative phasechange w2(Dt 1 t/2) 2 w1(Dt 2 t/2) between the twophase-locked harmonic pulses, where w1(Dt 2 t/2) [withrespect to w2(Dt 1 t/2)] is the phase shift of the first(with respect to the second) harmonic pulse when itcrosses the medium at time Dt 2 t/2 (with respect toDt 1 t/2), i.e., at delay Dt with respect to the midpoint ofthe probes. Therefore, the fringe shift is given by
dl
Dl~Dt ! 5
w2~Dt 1 t/2! 2 w1~Dt 2 t/2!
2p. (2)
We obtained the relation between the fringe shift and thespace-averaged electron density by writing the w1,2(t)phase shift as a function the refractive index of theplasma:
w1,2~t !
2p5
@n~t ! 2 1#L
l' 2
Ne~t !L
2lNc, (3)
where L and Nc are, respectively, the plasma length(along the probe axis) and the critical density at the har-monic wavelength. Using Eq. (2), we can now rewritethe space-averaged electron density as
Ne~Dt 1 t/2! 5 Ne~Dt 2 t/2! 22lNc
L
dl
Dl~Dt !. (4)
At large negative delays (Dt , 2800 fs), the harmonicprobes cross the medium far before the creation of theplasma so that Ne(Dt 2 t/2) ' 0. From this initialvalue, using Eq. (4) and the fringe shift in Fig. 13(b), wecan reconstruct the time evolution of the space-averaged
plasma electron density from negative to positive delaysDt. The reconstructed Ne(t) is plotted in Fig. 13(c) (filledcircles). Dispersion of the points increases for positivedelays that are due to the summation technique, so that athree-point averaged curve is also plotted (solid curve).The electron density increases rapidly around the zero de-lay and saturates at approximately 7 3 1019 cm23. How-ever, the increase is not as fast as expected. Indeed, inthis OFI regime, the ionization front should be shorterthan the pump pulse duration (;50 fs). Moreover, thetemporal resolution that is due to the harmonic probe du-ration should be better than the pump pulse duration.15,16
In fact, the stretching of Ne temporal evolution is intro-duced by the noncollinear geometry of the pump and the
Fig. 13. Variation of (a) the spatially integrated fringe patternand (b) the average fringe shift (filled circles) with delay Dt be-tween the ionizing pump and the H11 probe pulses. The dashedcurve in (b) was calculated from a simple model (see text). (c)Space-averaged electron density deduced from (b) as a function oftime (filled circles). The solid curve represents the results ofthree-point averaging.
Hergott et al. Vol. 20, No. 1 /January 2003 /J. Opt. Soc. Am. B 179
probe beams (probe at 45° from the ionization front),which mixes time and space, limiting the temporal reso-lution to 200 fs.
We briefly give more insight into the significance ofthe Ne(t) averaged density that we determined in Eq. (4).In the inset in Fig. 10, Z is the coordinate along thepropagation axis of the pump beam, z and x are thecoordinates along the propagation axis, and x is perpen-dicular to the probe beams. At first approximation, thetime-dependent local electron density is a function ofNe(t 2 Z/v) of time t and the Z coordinate (assuming theionized medium is homogeneous in the dimension perpen-dicular to the Z axis), where v is the group velocity. Now,a probe ray, at abscissa x in the probe beam (1 or 2) crosssection, undergoes a phase shift w i(t), which can be ex-pressed as
w i~x, Dt 7 t/2! 5 2p
lNcE
0
L
Ne~t 2 Z/v !dz, (5)
where Z 5 z cos u 1 x sin u ; t 5 z/v 1 Dt 7 t/2 is thevariable time at which the medium is probed along the Zpath by the infinitely short probe pulse (assuming thesame group velocity v for the probe beams). One there-fore obtains
w i~x, Dt 7 t/2! 5 2p
lNcE
0
L
NeF z
v~1 2 cos u! 2
x
vsin u
1 Dt 7 t/2Gdz. (6)
For u 5 45°, it is clear that phase shifts w i(x, t) in Eq.(6) include contributions of different points (z) in the me-dium, i.e., of different times in the evolution of the elec-tron density. The resulting temporal resolution is of theorder of L(1 2 cos u)/v ' 270 fs, larger than the durationof the pulses. This would not be the case for a collineargeometry (u 5 0°), which makes it possible to measureNe(Dt 1 t/2) 2 Ne(Dt 2 t/2) with the pulse duration asthe actual resolution.
A simple model that takes into account the geometry ofthe experiment is indeed able to reproduce the evolutionof the fringe shift in Fig. 13(b). The harmonic probeswere assumed infinitely short. The temporal variation ofthe gas ionization was primarily described by a step func-tion at time zero; a smooth turn on with different slopeswas also tested. Actually, it was observed that the mainfeatures of the curve are determined by the geometry ofthe experiment rather than the detailed ionization dy-namic. The result for a steplike ionization front is dis-played in Fig. 13(b) (dashed curve). The best fit to theexperimental data was obtained for an average electrondensity of 6 3 1019 cm23 at maximum ionization. It isworth noting that this value is in close agreement withthe one obtained from Eq. (4) and is consistent with thegas density in the jet if we assume double, i.e., full, ion-ization of He.
The fact that the ionization front is fast, on the timescale of the probe pulse duration, has an interesting con-sequence when it coincides with one of the probe pulses.Indeed, the rapidly varying free-electron dispersion in-duces a phase modulation on the harmonic pulse, result-
ing in a loss of mutual coherence with the second har-monic pulse, as discussed in Section 2; the contrast of thefringe pattern is then reduced. This is actually observedin Fig. 13(a) as fringe blurring around 2150 fs (the pumpcoincides with probe 2); it is much less clear at oppositedelay (pump coincides with probe 1), probably because ofthe finite resolution and the nonsymmetrical roles ofpulses 1 and 2 in the interference process.
Finally, the long-lasting behavior observed in Fig. 13(b)for positive delays (Dt ' 500 fs) is likely due to an in-crease of the plasma radius that is due to impact ioniza-tion of the surrounding gas by high-energy electrons.Note that this cannot be attributed to density fluctuationsbecause the plasma period is only 13 fs. This cannot aswell be due to the plasma recombination because, first,the fringe shift would be positive and, second, recombina-tion is expected to occur on a longer time scale.
4. SUMMARYIn summary, we first used the frequency-domain interfer-ometry technique in the XUV to study the mutual coher-ence of two harmonic pulses generated in the same me-dium by two time-delayed phase-locked laser pulses. Wefound conditions in which good phase locking between twoharmonic pulses can be obtained, as demonstrated by acontrast of the fringe pattern as high as 90%. However, astrong distortion of the fringe pattern (low contrast andlarge spectral asymmetry) was observed when we in-creased the laser intensity, the gas pressure, and the har-monic order. We have shown that this results from thedeleterious effects of the medium ionization on the har-monic emission: a reduced efficiency (caused by mediumdepletion, laser defocusing, free-electron-induced phasemismatch) and a loss of mutual coherence (free-electron-induced phase modulation). Because of the intrinsicchirp of the harmonic emission, the temporal asymmetryintroduced by ionization results in a spectral asymmetryof the fringe patterns. We then demonstrated the feasi-bility of time-resolved plasma diagnostic by using theXUV FDI technique and by studying the ionization dy-namics of a high-pressure helium gas jet irradiated by anintense short-pulse laser. The experimental configura-tion imposed an effective time resolution of 200 fs, so that,to our knowledge, we report the first XUV interferometrymeasurement on a subpicosecond time scale. This opensvaluable possibilities for the ultrafast diagnostic of denseplasmas: the ultimate resolution of the technique, givenby the harmonic pulse duration, is currently a few tens offemtoseconds or less; the spectral range for the probe ex-tends down to 10 nm. Experiments are in progress onoverdense plasmas (Ne . 1021 cm23), laser producedfrom various solid films of 0.1–1-mm thickness, collineargeometry allowing an improved time resolution of 50 fs.Finally, the XUV FDI technique takes optimal advantageof the harmonic source properties (temporal, spatial, andmutual coherence, short wavelength, short-pulse dura-tion, tunability) as other XUV interferometry schemes al-ready do, e.g., those based on intrinsic temporal and spa-tial coherence. Besides plasma diagnostics, thesetechniques should efficiently serve in various fields for
180 J. Opt. Soc. Am. B/Vol. 20, No. 1 /January 2003 Hergott et al.
which time-resolved XUV interferometry is relevant, suchas laser–solid and laser–surface interactions.
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