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International audience; Optische Strahlen mit Orbital Angular Momentum (OAM) sind ein sehr aktives Forschungsfeld für ihre prospektiven Anwendungen, insbesondere bei kurzen Wellenlängen. Wir betrachten hier solche Strahlen, die durch hochharmonische Erzeugung (HHG) in einem seltenen Gas erzeugt werden, und analysieren die Charakterisierung ihrer Hochladungs-Wirbelstruktur durch einen EUV (Extreme UltraViolet) Hartmann-Wellenfrontsensor. Wir zeigen, dass solche HHG-Strahlen in der Regel aus einer Reihe von zahlreichen Vortex-Modi bestehen. Die Empfindlichkeit der Intensität und Phase des HHG-Strahls auf die infraroten (IR) Laseraberrationen wird mit einem deformierbaren Spiegel untersucht.; Optical beams carrying Orbital Angular Momentum (OAM) are a very active field of research for their prospective applications, especially at short wavelengths. We consider here such beams produced through high-harmonic generation (HHG) in a rare gas and analyze the characterization of their high charge vortex structure by an EUV (Extreme UltraViolet) Hartmann wavefront sensor. We show that such HHG beams are generally composed of a set of numerous vortex modes. The sensitivity of the intensity and phase of the HHG beam to the infrared (IR) laser aberrations is investigated using a deformable mirror.; Les faisceaux optiques portant un Moment Angulaire Orbital (OAM) font l'objet de recherches très actives,en raison de l'intérêt de leurs applications potentielles, particulièrement à courtes longueurs d'onde. Nous nous intéressons ici à des faisceaux produits par Génération d'Harmoniques d'Ordre Elevé (HOE) dans un gaz rare, et analysons la structure du vortex de haute charge grâce à la caractérisation complète obtenue par un capteur de front d'onde de type Hartmann dans l'Extrême UltraViolet (EUV). Nous montrons que ces faisceaux HOE sont généralement constitués d'un ensemble de nombreux modes de vortex. Un miroir déformable nous permet d'étudier la sensibilité de l'éclairement et de la phase du faisceau HOE vis à vis des aberrations du laser infrarouge (IR).

We present in this paper ongoing studies on seeded plasma-based soft X-ray lasers using the LASERIX facility. After an overview of the source setup and characteristics, we will discuss two prospects that might increase the interest of these coherent soft X-ray sources. We have successfully generated harmonic beams carrying Orbital Angular momentum. We briefly present this experiment and discuss the remaining challenges for amplifying these kinds of beams in a plasma-based soft X-ray laser. In the second part, we present numerical results on harmonic seed amplification in conditions leading to Rabi oscillations. This regime is known to be a promising way to reach femtosecond pulses in the present plasma amplifiers. We will discuss the possibility of increasing the stability of such a regime by the introduction of a continuous refractive loss in the gain region.

In this paper, we present the characterization of high charge vortex structure of an OAM beam produced through high-harmonic generation in a rare gas using an XUV Hartmann wavefront sensor. We show that the phase-matched absorption limited high-harmonic generation is able to retain the high-charge vortex structure of the XUV beam even in a rather long (1 cm) generation medium. Additionally, our recent results on the influence of infrared driver wavefront quality on the characteristics of XUV vortex beam are presented.

We demonstrate that high harmonic generation when driven by vortex driver contaminated with various orbital angular momentum (OAM) modes, the upconverted EUV harmonic deviates from perturbative conversion law to exhibit a spectrum of OAM orders.

Secondary sources development such as X-rays for medical applications demand diode-pumped high-energy, high-power and high repetition rate picosecond lasers leading to a race to the kW milestone. Year after year, the pulse energy increased to few hundreds of millijoules at 1 kHz [1], but all the kW systems are based on the Yb:YAG thin disk technology which until now, to the best of our knowledge, has come short to deliver the full kilowatt level at 1 kHz. On the other side, cryogenically cooled bulk Yb:YAG amplifiers has shown its capabilities to reach 1 J at 500 Hz with 3.8 ps pulses [2].

We demonstrate for the first time, to the best of our knowledge, the ability of extreme ultraviolet (XUV) Hartmann wavefront sensors to characterize high charge vortex beams produced by high-order harmonic generation up to the order of 25. We also show that phase matched absorption limited high harmonic generation is able to maintain the high charge vortex structure of the XUV beam even in a rather long (1 cm) generation medium.

The scheme of the XUV/X-ray free electron laser based on the optical undulator created by two overlapped transverse laser beams is analyzed. A kinetic theoretical description and an ad hoc numerical model are developed to account for the finite energy spread, angular divergence and the spectral properties of the electron beam in the optical lattice. The theoretical findings are compared to the results of the one- and three-dimensional numerical modeling with the spectral free electron laser code PLARES., Comment: submitted to Phys. Rev. ST Accel. Beams

International audience; We study temporally localized electron wave packets, generated using a train of extreme ultraviolet (XUV) attosecond pulses to ionize the target atoms. Both the electron wave packets and the attosecond pulse train (APT) are characterized using the same technique, based on interference of two-photon transitions in the continuum. We study, in particular, the energy transfer from a moderately strong infrared (IR) field to the electron wave packets as a function of time delay between the XUV and the IR fields. The use of an APT to generate the electron wave packets enables the generation at times not accessible through tunneling ionization. We find that a significant amount of energy is transferred from the IR field to the electron wave packets, when they are generated at a zero-crossing of the IR laser field. This energy transfer results in a dramatically enhanced above-threshold ionization even at IR intensities that alone are not strong enough to induce any significant ionization.

A multi-terawatt, ultra-short laser pulse was used to create a large, well-defined relativistic ionization front, and the collision of a probe laser pulse with it was studied. Due to the good definition of the ionization front it was also possible to isolate the probe frequency up-shift due to the collision, unlike in previous experiments. A good agreement with the theoretical model for photon acceleration was found. Furthermore, we report novel measurements of ionization front velocities, based on the photon acceleration effect, in excess of 0.99c.

International audience; Following the theoretical predictions, the observation of two-photon processes by interaction of vacuum ultraviolet (VUV) radiation with inner-shell levels of atoms requires focused intensities in the 10^13-10^14 W/cm^2 range. Our aim is to reach this regime in order to study non-linear optics at these wavelengths. We first optimized the high harmonic conversion efficiency in argon by studying the best experimental conditions for phase-matching, concentrating on focus geometry related to laser energy, cell length and position relative to the focus. We then studied the resulting harmonic beam focusability by a toroidal mirror ( f=10 cm) and made an image of the harmonic focus. We conclude with an evaluation of the focused intensity that we are able to reach experimentally.

International audience; Nous avons utilisé une technique récente de mesure d'impulsions ultra-courtes afin d'étudier la dynamique de l'ionisation et la propagation dans un milieu gazeux. Nous présentons les mesures de variation de l'intensité, de la phase temporelle et du spectre d'une impulsion laser intense (10^16W/cm^2), ultracourte (35 fs) après son interaction avec un milieu gazeux. Les variations de l'indice de réfraction du plasma sont rapides et modifient la structure spatiale et temporelle de l'impulsion lors de sa propagation dans le gaz. Ces variations d'intensité et de phase temporelles peuvent être utilisées pour connaitre la dyriamique de l'interaction laser-plasma. En particulier, cette expérience permet de mettre en évidencc les effets sur la génération d'un milieu de gain pour le laser X de type OFI (Optical Field Ionization).

International audience; We report a systematic study of high order harmonic generation with an infrared laser apertured by an iris, as a function of the aperture size. Measurements were made of harmonic generation efficiency for different gas species, laser energies and focal geometries. Harmonic efficiencies as a function of aperture show a characteristic peaked shape, which is independent of gas species and harmonic order. A one dimensional, time dependent simulation of harmonic generation in a gas cell, taking into account experimentally measured transverse coherence of the laser, closely reproduces the observed behaviours. We show that the aperture diameter which maximizes harmonic yield is the result of a compromise between considerations of focal geometry and ionization (which favour small apertures); and harmonic dipole amplitude and phase (which favour large apertures).

In pulsed spectrometries, temporal transients are often analyzed directly in the temporal domain, assuming they consist only of purely exponentially decaying sinusoids. When experimental spectra actually consist of Gaussian or Voigt profiles (Gauss–Lorentz profiles), we show that the direct methods may erroneously interpret such lines as the sum of two or more Lorentzian profiles. Using a Nelder and Mead Simplex method, modified by introducing new means to avoid degeneracies and quenchings in secondary minima, we demonstrate that a large number of different solutions can be obtained with equivalent accuracy over the limited acquisition time interval, with final peak parameters void of physical or chemical meaning.

The high harmonics produced by focusing an intense femtosecond laser in a gas are theoretically shown to be locked in phase. The physics of this locking is discussed and a new method based on quantum interference in two-photon, two-color ionization allowing to retrieve the relative phase of harmonic pairs is described. The main result is that the 5 harmonics of orders 11-19 produced in argon generate a train of subfemtosecond pulses with a period of 1.35 fs and a duration of 250 attoseconds.

We investigate experimentally the optical tunneling times associated with frustrated total internal reflection of a light beam. Using as physical clocks the lateral shifts and angular deviations suffered by the transmitted and reflected beams, we measure both components of a complex tunneling time: the phase time and a semiclassical time. The phase time is shown to imply superluminal velocities and to depend strongly on boundary conditions not linked to the tunneling process. By contrast, the semiclassical time yields subluminal velocities and is related solely to tunneling.

International audience; We describe theoretically and numerically the interaction of a laser- or LINAC-accelerated beam of relativistic electrons with a high intensity optical lattice, resulting from the superposition of two transverse laser pulses. The bunch is trapped and guided within the potential channels of the optical lattice, leading to betatron oscillations. We describe the emission from individual particles and from the bunch, and analyze its spectrum, considering the dominant incoherent radiation as well as possible effects of partial coherency. Analysis of the emitted radiation should provide useful information on the characteristics of the electron beam, and its interaction with the optical lattice.

A reverse Goos-Hanchen displacement between TE and TM polarizations is predicted and experimentally isolated. The necessary reversal of the evolution of the difference between the phase retardances for the TE and TM reflections vs. the angle of incidence is obtained thanks to a Fabry-Perot–like structure. The experimental measurements are in good agreement with numerical calculations that take fully into account the Gaussian nature of the light beam.

We demonstrate that we can use the extreme ultraviolet radiation produced by high order harmonic generation to perform photoionization experiments. With harmonics from the 11th to the 69th of a 140 fs Cr:LiSAF laser operating at 825 nm, we measure the relative photoionization cross sections of xenon, krypton, argon and neon over the range 10 to 110 eV. With narrow bandwidth harmonics produced by a tunable, 1 ps dye laser, we observe the autoionizing states between the 4p5 ionization thresholds in krypton.

We present a simple, analytic, and fully quantum theory of high-harmonic generation by low-frequency laser fields. The theory recovers the classical interpretation of Kulander et al. in Proceedings of the SILAP III Works hop, edited by B. Piraux (Plenum, New York, 1993) and Corkum [Phys. Rev. Lett. 71, 1994 (1993)] and clearly explains why the single-atom harmonic-generation spectra fall off at an energy approximately equal to the ionization energy plus about three times the oscillation energy of a free electron in the field. The theory is valid for arbitrary atomic potentials and can be generalized to describe laser fields of arbitrary ellipticity and spectrum. We discuss the role of atomic dipole matrix elements, electron rescattering processes, and of depletion of the ground state. We present the exact quantum-mechanical formula for the harmonic cutoff that differs from the phenomenological law Ip+3.17Up, where Ip is the atomic ionization potential and Up is the ponderomotive energy, due to the account for quantum tunneling and diffusion effects.

International audience; We present experimental and numerical results on high-order-harmonic generation with a flat-top laser beam. We show that a simple binary tunable phase plate, made of two concentric glass plates, can produce a flat-top profile at the focus of a Gaussian infrared beam. Both experiments and numerical calculations show that there is a scaling law between the harmonic generation efficiency and the increase of the generation volume.

The authors present high-order harmonic generation results obtained with different laser systems, a 1 ps 1053 nm Nd-glass laser, a 2 ps 616 nm synchronously pumped dye laser, a 36 ps 1064 nm mode-locked Nd-YAG laser and the second harmonics (308 nm and 532 nm) of the latter two systems. They investigate the influence of the laser pulse width, the excitation wavelength (from the near infrared to the ultraviolet) and the atomic medium on the number of photons produced and on the maximum energy attained. Harmonic generation also depends strongly on the focusing conditions. By using simple arguments and results of numerical calculations in xenon, they show that the conversion efficiency in general follows a simple b3 power law, b denoting the laser confocal parameter, up to a transition regime where the coherence length of the process becomes equal to the medium length. By applying the b3 scaling as a normalization factor, they can then compare experimental results obtained in different focusing geometries. Their experimental data show that the optimization of the photon energy produced is favoured by using long incident wavelengths and light atomic systems with a high ionization energy.

A scheme for an X-ray free electron laser is proposed, based on a Raman process occurring during the interaction between a moderately relativistic bunch of free electrons, and twin intense short pulse lasers interfering to form a transverse standing wave along the electron trajectories. In the high intensity regime of the Kapitza-Dirac effect, the laser ponderomotive potential forces the electrons into a lateral oscillatory motion, resulting in a Raman scattering process. I show how a parametric process is triggered, resulting in the amplification of the Stokes component of the Raman-scattered photons. Experimental operating parameters and implementations, based both on LINAC and Laser Wakefield Acceleration techniques, are discussed., 22 pages

We report an experimental approach we used in order to time-resolve coherent optical phonon oscillations in a bismuth (111) oriented bulk crystal by extreme ultraviolet (XUV) and soft x-ray femtosecond pulses based on high-order harmonic generation in rare gases.

International audience; We explore experimentally and numerically the physics underlying the optimization of high-order harmonic generation by intense laser pulses, whose temporal profile is tailored by a learning genetic algorithm. Based on a large set of optimization data obtained under different generation parameters, we show that the algorithm converges toward a class of very special profiles on the leading edge of the pulse. The behavior of the harmonic signal is then compared with theoretical simulations based on the time-dependent Schrödinger equation, allowing one to identify separately the role of microscopic and macroscopic phenomena in the temporal dynamics of high-harmonic generation and optimization.

International audience; We use semiconductor (Si) and metallic (Al, Zr) transmission filters to shape, in amplitude and phase, high-order harmonics generated from the interaction of an intense titanium sapphire laser field with a pulsed neon gas target. Depending on the properties of the filter, the emitted attosecond pulses can be optimized in bandwidth and/or pulse length. We demonstrate the generation of attosecond pulses centered at energies from 50 to 80 eV, with bandwidths as large as 45 eV and with pulse durations compressed to 130 as.

International audience; Afin de contrôler la génération d'harmoniques d'ordre élevé dans des conditions optimisées, nous avons réalisé des expériences préliminaires de contrôle spectral et spatial du laser infrarouge. Nous avons démontré la possibilité de modifier la longueur d'onde des harmoniques grâce à la modification de la phase spectrale du laser infrarouge par un modulateur acousto-optique « Dazzler ». Nous montrons que cette méthode permet d'obtenir un spectre plus large et un flux de photon intégré beaucoup plus important qu'en utilisant une simple modification de la dérive de fréquence du laser. Nous présentons également les premiers résultats de mise en forme de la tache focale du laser par optique adaptative, et l'effet de ce contrôle spatial sur la génération d‘harmoniques.

International audience; Nous présentons des résultats expérimentaux de spectre d'harmoniques anormales d'ordre élevé générées à basse intensité laser (2 10^13 W/cm^2) dans l'argon. Ce spectre ne peut pas être expliqué par le modèle classique à trois étapes qui prédit les caractéristique du spectre . Les ordres observés dans notre mesure vont bien au delà de la limite de la coupure donnée par l'intensité du laser dans le milieu gazeux. De plus, nous n'observons pas les ordres très bas. Nous présentons une étude systématique de ces spectres, ainsi que des simulations numériques issues de la résolution de l'équation de Schrödinger dépendante du temps.

International audience; We present a method to shape the intensity distribution of an intense laser beam in the focal region, by means of a bimorph adaptive mirror. We emphasize the advantages in terms of flexibility with respect of previous methods making use of specially designed retarding phase plates; however, the nature of the mirror imposes constraints on the accessible wavefronts, whose consequences are analyzed. We present an experimental demonstration of the feasibility of the scheme on a high intensity femtosecond laser system.

International audience; We demonstrate and evaluate the importance of an intrinsic chirp inherent to attosecond pulse creation accompanying high-order harmonic generation in recently published experimental data by Dinu et al. [Phys. Rev. Lett. 91, 063901 (2003)]. We present an analytical model, from which the atomic origin of the harmonic chirp is clearly understood. Moreover, the behavior of the chirp as a function of experimental parameters such as laser intensity is inferred. The comparison between our model and the experimental data provides us with useful information about the conditions in which the high-order harmonics is generated.

International audience; We present experimental harmonic spectra showing anomalously high orders generated at low laser intensities (around 2 × 10^13 W cm^−2) in argon, krypton and xenon. These high orders cannot be explained with the usual three step model that predicts the existence of a sharp cut-off after the region of constant efficiency called the plateau: the observed orders are far above the cut-off limit inferred from the laser intensity in the medium. The systematic study of these spectra are presented, and some possible theoretical models are explored.

International audience; We present a direct measurement of the coherence length in the high-order-harmonic generation process up to orders as high as 25. The harmonic signal generated in an argon gas cell by a 805-nm, 6-mJ, 30-fs laser is studied as a function of medium length and shows very high-contrast coherence fringes. From a one-dimensional time-dependent model, we unravel the conditions for which such Maker fringes can appear in high-order harmonic generation.

International audience; We present a direct method of studying the focusability of an intense, short-pulse extreme-ultraviolet (XUV) beam obtained by high-harmonic generation. We perform near-field imaging of the focal spot of five high-harmonic orders strongly focused by a broadband toroidal mirror. To visualize the focal spot directly, we image the fluorescence induced by an XUV beam on a cerium-doped YAG crystal on a visible CCD camera. We can thus measure the harmonic spot size on a single image, together with the Strehl ratio, to evaluate the quality of focusing. Such techniques should become instrumental in optimizing the focusing conditions and reaching intensities required for exploring attosecond nonlinear optics in the XUV range.

Summary form only given. X-rays produced from intense subpicosecond laser systems have the unique properties to provide subpicosecond X-ray pulse duration and to be fully synchronized with the laser system. This allows the probing by X-rays of ultrafast phenomena, previously limited to all-visible pump-probe investigation. Ultrafast X-ray science is important to biologists, chemists and physicists because it will allow them to follow atomic motions, which ultimately drive processes like (bio-) chemical reactions and phase transitions, on the time-scale at which they occur. The last 2 years have shown important advances, both in the study of coherent radiation in the sub-keV range by high harmonics or X-ray lasers, and in the development of optical pump-keV X-ray probe ultrafast spectroscopy using synchrotrons and laser-produced plasmas.

International audience; Nous présentons une étude complète de l'influence de la diaphragmation du faisceau infrarouge sur la génération d'harmoniques d'ordres élevés. Nous avons mesuré l'efficacité de conversion obtenue dans différents gaz et pour différentes énergies laser en fonction de l'ouverture du diaphragme. Les courbes ainsi déduites présentent un caractère assez général : le maximum d'efficacité de conversion se détache nettement des valeurs obtenues pour des ouvertures de diaphragme trop importantes : un facteur 25 d'amélioration est obtenu entre le cas sans diaphragmation et le cas optimal. Une simulation basée sur un code uni-dimensionnel de calcul d'accord de phase dépendant du temps permet de reproduire fidèlement les résultats expérimentaux. Nous pouvons ainsi conclure que l'ouverture optimale du diaphragme est celle qui réalise le meilleur compromis entre la géométrie focale dont découle l'accord de phase (qui aurait tendance à favoriser des diaphragmes fermés) et l'amplitude et la phase du dipôle harmonique (qui sont favorisées par des diaphragnies ouverts).

International audience; We investigate the relevance of the absorption limit concept in the optimization of high harmonic generation. Thanks to the first direct observation of the coherence length of the process from high-contrast Maker fringes, we unravel experimental conditions for which the harmonic dipole response is enhanced when phase matching is realized within the absorption limit, leading to record conversion efficiencies in argon. Moreover, we show that harmonic generation in guided or freely propagating geometries are equivalent in the loose focusing regime. This analysis is generalized to other advanced phase-matching schemes, thereby predicting the possibility to boost the conversion efficiencies using light noble gases.

It was shown in 1999 that D2 cluster explosion under ultrashort intense laser irradiation can lead to ion energies sufficient to drive nuclear fusion reactions [Ditmire et al. Nature 398, 491]. We show how the use of molecular clusters allows to further enhance the ionic acceleration, allowing one to reach optimal fusion cross sections. A new low density high energy regime is described, in which fusion occurs via a spallation-like process. The process increases with laser intensity up to the relativistic threshold, at which the neutron production is overwhelmed by gamma production resulting from electron acceleration.

We present experimental results concerning the observation and acceleration of negative and positive ions produced from high‐intensity laser beam interactions with deuterated and hydrogenated clusters. The 30‐fs, 10‐Hz rep. rate, 30 TW Ti‐Sa laser of LOA was focused on clusters produced from a pulsed gas nozzle. The novelty of our experimental work was the gas composition (CD4 and CH4), which was used for the cluster formation. Both kinds of ions, negative and positive, were observed with approximately equal energies. The maximum energy of both kinds of ions as well their energy distribution was measured using a Thomson parabola mass spectrometer. Negative and positive ions of H, D, and C were observed with a maximum energy up to 70 keV for a maximum laser beam intensity of 1018 W/cm2. The mass spectrometer allowed us to correlate both the positive and negative ion energy spectrums with the laser beam intensity in order to study the process of cluster explosions, ion acceleration, and negative ion formation in the fs regime. We summarize the more important and new results as follows: (1) only single positive and negative ions were measured on the mass spectrometer; (2) the number of negative ions is equal to the number of positive ions; (3) similar energy spectrums for positive and negative ions were measured; (4) the positive and the negative ions are accelerated in the same radial direction, outwards from the interaction volume; and (5) similar energy spectrums for positive and negative ions at different laser energies (the laser intensity increases by a factor of 5) were measured. There are two possible “scenarios” concerning the process of negative ion formation. The first process is related with the formation of different ion species during the Coulomb explosion of the clusters. The second process for negative ion formation is the double electron attachment caused by collisions between the positive ions with the backing gas. A number of arguments confirm that the second process is more probable. The fact that from the mass spectrometer measurements the value for the more probable energy of the positive ions is the same as the energy at the maximum of the double electron attachment cross section leads to the second process of negative ion formation. For a complete description of our observation, however, we need an efficient physical process for negative ion formation, which will be the subject of our future work. The process of cluster explosion accelerates the D ions sufficiently to produce neutrons from DD nuclear fusion reactions. The efficiency of both the negative ions and the neutron alternative source will be discussed.

International audience; Following the interaction of superintense, short pulse lasers and plasmas, ions can be accelerated to velocities sufficient to drive nuclear fusion reactions, in particular, by the process of Coulomb explosion of clusters [T. Ditmire et al., Nature (London) 398, 491 (1999)]. We show here how short bursts of neutrons can be produced using a jet of low-density deuterated methane clusters. Ion velocity distributions were simultaneously measured by a Thomson parabola mass spectrometer, demonstrating deuteron energies up to 120 keV. We show that, in such conditions, nuclear fusion will occur not only in the hot plasma core, but also in the cold outer region by collision processes.

International audience; In principle, the temporal beating of superposed high harmonics obtained by focusing a femtosecond laser pulse in a gas jet can produce a train of very short intensity spikes, depending on the relative phases of the harmonics. We present a method to measure such phases through two-photon, two-color photoionization. We found that the harmonics are locked in phase and form a train of 250-attosecond pulses in the time domain. Harmonic generation may be a promising source for attosecond time-resolved measurements.

A laser–plasma interaction experiment was performed in order to match effective conditions for wake-field electron acceleration. A 30-fs pulse was made to interact with a preformed plasma generated via the exploding foil technique from a 1-μm-thick plastic film. The irradiance of the femtosecond pulse in the plasma was 1020 Wcm−2. The interaction conditions were investigated via imaging and spectroscopy at the fundamental and the second harmonic of the laser frequency, both forward and backward. Our data clearly show that conditions suitable for electron acceleration are achieved close to the propagation axis and can be easily reproduced from shot to shot. In contrast, significant growth of instabilities occurs at the boundaries of the interaction region. These observations are consistent with a preliminary evidence of forward acceleration of high-energy electrons. Optical, X-ray, and γ-ray data obtained for different positions of the foil target with respect to the laser focal plane further support this promising scenario.

This paper discusses the generation of high-order optical harmonics from solid targets using laser pulses of 35 and 120 femtoseconds. Harmonics up to the 35th order were observed. High conversion efficiency has been achieved, e.g., ${10}^{\ensuremath{-}6}$ to the $10\mathrm{th}$ harmonic. It is demonstrated that the harmonic emission is highly directional and that the harmonic efficiency decreases rapidly with increasing plasma scale length.

We report the experimental- demonstration of phase-matched generation of high harmonics of orders 33 to 45 over 2 mm, in a gas cell filled with 30 torr of neon. Increasing the medium length from 1 to 4 mm is shown to enhance this specific group of harmonics by a factor of 50. To explain these results, we recall how phase-matching conditions are generalized in strong fields. By a phase-matching map formalism, we show how the spatial gradient of the intensity-dependent phase allows one to satisfy those conditions over millimeters.