Enhancement of high-order harmonics in a plasma waveguide formed in clustered Ar gas

Generation of high-order harmonics (HHs) is intensified by using a plasma waveguide created by a laser in a clustered gas jet. The formation of a plasma waveguide and the guiding of a laser beam are also demonstrated. Compared to the case without a waveguide, harmonics were strengthened up to nine times, and blue-shifted. Numerical simulation by solving the time-dependent Schrödinger equation in strong field approximation agreed well with experimental results. This result reveals that the strengthening is the result of improved phase matching and that the blue shift is a result of change in fundamental laser frequency due to self-phase modulation (SPM). © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (190.2620) Harmonic generation and mixing; (270.6620) Strong-field processes; (320.7110) Ultrafast nonlinear optics. References and links 1. J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. 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Introduction
Generation of high-order harmonics (HHs) is a popular method to get coherent extreme ultraviolet (XUV) light, which has many applications in science and technology [1][2][3][4]. However, many applications require higher HH photon flux of HHs. However, the increase of laser intensity for a higher HH flux is limited by ionization-induced laser-defocusing in gas samples [5][6][7]. A waveguide configuration is one approach to overcome plasma-induced laser beam defocusing. By using a plasma waveguide preformed in gas-filled capillary discharges, the harmonic flux would be increased and the cutoff could be extended [8][9][10]. A dual-cell scheme can also increase the HH conversion efficiency [11].
Another approach is the quasi-phase-matching technique in a modulated and gas-filled hollow-core waveguide. This process increases HH output by a factor of 2 10 to 3 10 compared to that in non-phase-matched case [12]. Photon flux of HHs has also been increased by using a guided configuration at a laser intensity that greatly exceeds saturation intensity [13]. The enhanced harmonic yield has been obtained using synthesized two-color laser pulses and optimized waveguide configuration [14]. Spatially-prepared filamentation in Ar gas cell can intensify HH by nearly an order of magnitude [15].
Atomic clusters are more efficient than monatomic gas as sources of coherent ultraviolet light [16][17][18][19]. They absorb laser light efficiently, which results in heated electrons, rapid production of ions and x-ray emission [20]. The properties of an atomic cluster for HH have been investigated in detail [21]. The dependence of HH on cluster size was investigated [22,23], in which the contributions from clusters and gas monomers was also investigated, showing that clusters are more efficient radiators. Hence the combination of a waveguiding technique and cluster target should further increase HH photon flux.
In this letter, we demonstrate the enhancement of HHs in a plasma waveguide configuration formed in a clustered Ar gas. A gas jet with a slit nozzle is used to generate Ar clusters [24, 25] at a backing pressure of 10 bars. We demonstrate guiding in an end-pumped waveguide generated in a clustered gas [26-28]. Plasma waveguide formation using a prepulse is investigated by analyzing time-resolved interferograms from a Michelson interferometer (MI). The electron density of a waveguide is extracted from the interferogram by Abel inversion. By the interaction of another delayed femtosecond laser pulse with the plasma waveguide, the HH is generated and enhanced by a factor of nine at the 27th-order harmonic, compared to the case without a waveguide. Numerical simulation by solving the time-dependent Schrödinger equation in strong field approximation showed the enhancement and the blue shift of HHs that agree well with experiment results. This result shows that the blue shift of HH is caused by a blue shift of the fundamental laser frequency as a consequence of self-phase modulation (SPM) [29].

Experimental setup
The experiment [ Fig. 1] used a 10-Hz chirped pulse amplification (CPA) laser at 790 nm with a pulse duration of 35 fs FWHM and a pulse energy of 35 mJ. The beam was split in a 2:3 ratio into a low-energy pulse (P1) and a high-energy pulse (P2) by a beam splitter. P1 is a prepulse that is used to create a plasma waveguide in a clustered Ar gas. P2 is a drive pulse, which is delayed. P1 and P2 were combined by a beam combiner, then focused on a clustered Ar gas by a lens (L) with a focal length of 50 cm. The beam radius at focus was about 125 μm. Clusters are formed by a supersonic slit nozzle (0.5 mm × 5 mm) [ Fig. 1 inset] installed above the pulsed valve (Parker series 99 with 0.5 mm diameter orifice). The nozzle and gas jet can be precisely positioned by a motorized X-Y-Z stage to a laser focus in vacuum. The average cluster size was about several nanometers under Ar gas backing pressure of 10 bar [30]. Each cluster had a radius of ~1.7 nm, (composed of an average of ~450 atoms), as estimated using Rayleigh scattering [31]. The gas jet was synchronized by a DG535 (Stanford Research System) with the pre-pulse. The calculated peak intensities were 3.0 × 10 14 W/cm 2 for P1 and 4.5 × 10 14 W/cm 2 for P2. The leakage light (P3) from the delay-line mirror was used to probe the waveguide by using a Michelson Interferometer (MI). Two lenses were used to image the plasma waveguide on a CCD in the MI. The interferogram encodes the information of the phase shift due to a cylindrically symmetric plasma. By varying the probe time with respect to P1, the evolution of the laser plasma waveguide was investigated. To get the HH spectrum, HHs were collected and focused by a toroidal mirror onto the slit of a flat-field XUV spectrometer [32], and the infrared light was blocked by a 0.8-μm-thick Al film. We investigated the waveguide formation by P1 without P2, by using a transverse P3 at different time delays D t , to optimize the time for waveguide formation. Interferograms [ Fig.  2(a)] of the Ar plasma waveguide created by P1 were captured at 3 ≤ D t ≤ 26 ns. The electron densities [ Fig. 2(b)] of the waveguide were extracted from the interferogram by Abel inversion  at the red lines in Fig. 2(a). Without P2, a good waveguide was formed ~20 ns after the interaction of P1 with the clustered gas. At D t = 20 ns, the profile [ Fig. 3(a)] of the plasma waveguide from P1 was not completely open along the propagation direction [ Fig. 3(a)]. The Injection of P2 into the waveguide at D t = 20 ns opened it completely [ Fig. 3(b)]. The beam profiles of the drive pulse P2 passing through the waveguide for different delays are shown in Fig. 4. The intensity of the output beam was weak before the waveguide was formed. The reason is that the drive pulse is absorbed and refracted out when a waveguide is not properly formed; the beam gets stronger as waveguide is being formed. The highest throughput of 70% was achieved at D t = 20 ns. The beam profile and strong intensity of P2 confirms that the pulse propagated through the waveguide. In subsequent measurements we used D t = 20 ns to guide P2. To distinguish the contribution of P1 to HH intensity, the spectra [ Fig. 5] were measured for three cases: (1) P1 only, (2) P2 without a waveguide and (3) P2 with a waveguide; 1500 shots were accumulated for each spectrum. When only P1 was used, the channel is not completely open along the propagation direction, as seen in Fig. 3(a). The electron density of the 2nd half of the channel is in the order of a few x 10x 17 cm −3 . Ar density is then expected to be in the order of 10 18 cm −3 . The transmission in the spectral region of our interest is negligibly small. In case of P2 with a higher intensity, a waveguide seems to be open all the way to the end. In this case. The phase matching conditions are better met with longer and more uniform wave guide [33]. The use of P2 with a waveguide clearly intensified HH whose order ≥ 19, compared to the intensity of HH without a waveguide.

Experiment
Phase matching condition was met for a longer distance with a waveguide than without one. With a uniform waveguide [ Fig. 3 (b)], the electric field of the HHs can be added together in phase in different parts of the waveguide, therefore the intensity of HH dispersive medium grows with the medium length. The enhancement was different at different orders because phase-matching conditions differ among orders due to electron dispersion [34]. The highest enhancement by a factor of nine occurred in the 27th-order harmonic. The HH frequency also shifted toward blue, as also reported in [21]. The HH spectrum split and broadened [ Fig. 5(a)]; these effects occur when the driven laser intensity exceeds the ionization threshold (saturation intensity) of an Ar cluster. They are attributed to propagation effects in a rapidly ionizing medium [35], and to temporally-changing phase-matching conditions (transient phase matching) [36].

t t d t i dt a t d p t t A t iS t t a t d p t t A t E t w t dt
where 1 ( ) E t is the electric field of an external laser, ( ) A t the corresponding vector potential, d the bound-continuum transition element matrix, s P and S are respectively the saddle point stationary of canonical momentum and quasi-classical action, and ( ) a t is the amplitude of the ground state wave function which is calculated in the ADK model [38] The last term considers the ground-state depletion. The harmonic spectrum of the single-atom response is obtained by the Fourier transform of the dipole acceleration.
The macroscopic propagations of both the fundamental and the HH beam in a plasma waveguide are included by solving the non-adiabatic three-dimensional coupled wave equations [39]. The propagation of the fundamental beam in a plasma waveguide can be described in a moving coordinate frame by density like the plasma channel causes the variation of the refractive index which introduces self-phase modulation (SPM). The frequency shift due to the SPM in an ionizing gas is proportional to the gas density [29], and yields the blue shift of high-order harmonics [41]. The splitting at the density of 1x 10 19 cm −3 shown in Fig. 7 is estimated to be ~8.6x10 −3 eV. This splitting would lead to the splitting of 0.2 eV for the 25th harmonic, which is still ~2x smaller than the experimental observation shown in Fig. 5. Hence the splitting in HH has a different origin. Further investigation will be carried out.

Conclusion
Appropriate formation of a waveguide in clustered Ar gas is demonstrated. Beam profiles emerging from a waveguide with different delays demonstrate that the throughput of the drive pulse increases as waveguide is being formed. HH was intensified by the plasma waveguide in which the intense laser pulse can make a longer propagation without defocusing in the waveguide, and because the phase matching conditions are better met than in the case without the waveguide. HH was intensified by a factor of nine at the 27th-order harmonic. Numerical simulation of HH generation in a plasma waveguide agreed well with experimental observation of the enhancement and blue shift of HH. The blue shift of HH is due to the blue shift of the fundamental laser frequency induced by SPM which resulted from the temporal change of electron density. This method is a simple way to intensify HH for applications that need high HH photon flux.