By focusing a high repetition rate (50kHz), compact, femtosecond laser system with low pulse energy (25μJ) using a tight-focusing geometry, we demonstrate fully phase matched high-order harmonic generation for the first time at very high repetition rates, resulting in EUV light with full spatial coherence. The result is a practical, single-box, coherent source useful for applications in metrology, ultrafast spectroscopy, imaging and microscopy. The soft x-ray flux can be improved further by increasing the laser pulse energy and/or repetition rate.
©2009 Optical Society of America
High harmonic generation (HHG) is a unique source of femtosecond-to-attosecond duration short wavelength light that is opening up new studies of electron dynamics in atoms, molecules, and materials with ultrahigh time resolution [1,2]. When the HHG process is fully phase matched, the generated harmonics are brightest and emerge as fully spatially coherent extreme ultraviolet (EUV) beams [3–8]. High spatial coherence is desirable for many applications in metrology, lithography, coherent microscopy, and holography ; while high repetition rates are desirable for applications such as coincidence imaging or photoelectron spectroscopies [1,2].
However, to date it has not been possible to generate fully phase matched harmonics with full spatial coherence at very high repetition rates (>> kHz). This is because compact, high repetition rate lasers (50 – 100 kHz) have low per pulse energies, which must be focused tightly to reach sufficient intensity to ionize the gas and to generate high harmonics. Close to the focus, the driving wave has a complex wavefront and a strong Gouy phase shift, which strongly influences phase matching conditions . High harmonic generation is also accompanied by a dramatic dynamic change in the index of refraction of medium due to ionization. Moreover, for each particular harmonic order, two separate electron trajectories generate the same photon energy. All these effects make phase matching in a tight focusing geometry difficult, and as a result, phase matched, fully coherent, HHG at very high repetition rates >> kHz has not been possible to date.
Past work demonstrated the generation of high-order harmonics using a low pulse energy (7 μJ), high repetition rate (100 kHz) laser , using a tight focusing (f/8) geometry. However, in that work, the rapid fall-off of HHG flux with increasing harmonic order was characteristic of a non phase matched geometry, and the estimated flux in this work was correspondingly low. Other past work demonstrated that fully phase matched, spatially coherent, HHG beams can be generated when phase matching is achieved by guiding the driving laser beam in a long, gas-filled, hollow waveguide [7,8,12]. In this case, the laser beam forms a stable eigenmode with uniform spatial phase, while full phase matching over an extended distance results in a near-perfect HHG mode. However, scaling this waveguide geometry to lower pulse energies requires the use of smaller diameter capillaries, where the much higher optical dispersion and guiding loss for the propagating laser light make phase matching progressively more difficult .
In this paper, we show that by increasing the pulse energy by a moderate factor (3 × ) and the gas pressure (~40 × ) compared with previous work , we can implement fully phase-matched high harmonic conversion at very high (50 kHz) repetition rates for the first time . As a result, we generate excellent mode quality, fully spatially coherent, HHG beams at very high repetition rates for the first time. We get a photon flux of 2 × 109 photons per second at a photon energy of 45 eV, using a compact driving laser that integrates the pump lasers and ultrafast oscillator and amplifier system into a single 2’ × 6’ box. This photon flux is at the theoretical limit for the density-length gas medium we can use. The observed HHG flux scales quadratically with gas pressure, as expected for phase matched high harmonic conversion. Finally, we also show that the HHG flux scales approximately linearly with repetition rate between 10 and 50 kHz, demonstrating that gas recombination times and heating effects are not a limiting factor.
2. Experimental setup
A schematic of our experimental setup for generating harmonics at very high repetition rates is shown in Fig. 1 . An ultrafast regenerative laser amplifier employing downchirped pulse amplification  (Wyvern, KMLabs Inc.) provides 45-fs-duration pulses, with an pulse energy of 25 μJ, at 50 kHz repetition rate, at a wavelength of ~800nm and with good beam quality (M2 <1.3). The laser beam is focused into a gas cell using a ~f/15 focusing lens. The gas cell consists of a hollow glass capillary with inner diameter 400μm and outer diameter 1.2mm, through which two 100 µm holes were drilled in the walls using a CO2 laser (see Fig. 1). One end of the capillary is pressurized with Ar, while the other is sealed. The design of the gas cell functions as a good spatial filter for the incoming laser beam with a very high damage threshold. Based on a flow simulation, 90% of gas can be confined in the laser-gas interaction region. The gas cell also significantly decreases the loading for the vacuum system. The lens is translated to precisely control the focus position in the gas cell to obtain optimal phase matching conditions. The focus diameter of 22 μm results in a peak intensity of 2.6 × 1014 W/cm2, over a confocal parameter distance of 950 µm. At these laser intensities, Ar is approximately 18% ionized at the temporal peak of the pulse. The scattered laser light and harmonics orders below the 11th are blocked by two aluminum filters (each of them is 200 nm). The spatial profile of the harmonic beam was recorded with an x-ray CCD camera (Andor, DO420-BN) at a distance of 50 cm past the focus.
To measure the spatial coherence of the HHG beam, a pinhole pair (50-μm pinholes placed 150 μm apart) was placed 20 cm after the focus, where the HHG beam was sampled at 15% of its beam diameter. The interference pattern was then recorded at a distance of 90 cm after the pinhole pair. The depth of modulation of the fringes depends on the correlation between the local phase of the wavefront of the HHG beam at the two points where it is sampled by the pinhole pair. The HHG spectrum was measured by inserting a concave ruled reflectance grating (1200 grooves/mm, Newport 52025BK) to separate and partially refocus the individual harmonics. In this setup, the x-ray CCD camera was placed at the focal plane, approximately 16 cm from the grating.
3. Results and discussion
Figure 2 (a) shows an image of the HHG beam 50 cm after the gas cell, for an Ar pressure of 200 torr, and with the focal position approximately 10 μm after the gas cell. The diameter of the HHG beam is 1.25 mm at the 1/e2 points, with a slight ellipticity due to imperfections in the sidewalls of the gas cell. The HHG beam size is consistent with a small HHG source diameter of ~8 μm (laser focus diameter = 22 μm), and with a diffraction limited HHG beam divergence of ~5 mrad. The harmonic photon flux is estimated at ~2 × 109 photons/s, which was obtained by summing the accumulated number of counts on the CCD camera, while considering the quantum efficiency of the CCD camera and the transmission of the Al filters.
The interferogram of the EUV beam is shown in Fig. 2 (b), taken with an acquisition time of 300 s (1.5 × 107 laser shots). The interference pattern contains two circular Airy distribution patterns with fringes in between. The two Airy distributions are separated by ~750 μm on the CCD camera in the far field. The visibility of the central fringe is 85%. The slight decrease in fringe visibility is primarily due to the measurement, and not to less-than full coherence of the beam. Simulations shown in Fig. 2(c) assuming the central wavelength of the measured spectrum, 100% spatial coherence, and a beam divergence of 5 mrad indicate the expected fringe visibility in this geometry is 87%. Since the EUV spectrum contains 4 harmonic orders, the visibility of off-center fringes is expected to diminish quickly. The measured contrast thus illustrates both the long-term wavefront stability as well as the nearly 100% spatial coherence of the HHG beam.
To optimize HHG conversion efficiency, a coherent build-up of the HHG signal must occur over some macroscopic pressure-length product (phase-matching) . In a free-focus geometry, the phase mismatch is the sum of a pressure-dependent positive contribution from neutral atoms, a negative contribution from the free-electron dispersion, and a pressure-independent negative contribution from Gouy phase, in addition to any intrinsic atomic phase contribution [3,5,10,13]:3]. Therefore, to optimize phase matching, the distance between the laser focus and the gas cell was varied by moving the lens position along the propagation direction. Figure 3 (a) shows the conversion efficiency for the 29th harmonic order (45 eV) as a function of laser focus position, at a backing pressure of 200 torr of Ar. Two different phase matching regions can be identified, where the different contributions to the total ∆k equation compensate each other (∆k = 0). On axis, collinear phase matching is seen when the laser is focused 50 μm before the gas cell, while non-collinear off-axis phase matching is seen when the laser is focused 10 μm after the gas cell.
Another important method to increase the HHG flux would be to increase the backing pressure, since in this tightly focused geometry, phase matching is only weakly dependent on pressure. Thus, in theory the HHG flux can be increased by increasing the pressure (the length of the medium cannot be increased due to the short confocal parameter). Figure 3 (b) shows the HHG spectrum through two 200 nm Al filters as a function of pressure. The HHG flux at the 27th and 29th orders increases quadratically with gas pressure (density), which is consistent with phase matched HHG signal buildup as a significantly higher number of atoms contribute coherently to the emitted HHG signal. In contrast, the 35th order is 100 times weaker and increases linearly with pressure, indicating incoherent build-up. As discussed below, phase matching is possible for harmonic orders less than order 33rd, for which ionization in the medium balances the neutral atom dispersion. Therefore, these observations support our claim of fully phase matched high harmonic conversion at photon energies below 50 eV.
The lineouts (c) of Fig. 3 also show that the HHG flux at the 27th, 29th and 35th orders saturate at pressures above 190, 190, and 150 torr respectively, due to a combination of defocusing of the laser by the generated plasma and reabsorption of the harmonics by the gas. Comparison of the observed saturation density-length product (200(torr)·0.04(cm) = 8 (torr∙cm) at 45 eV) with the theoretical absorption-limited harmonic generation (~5·P·L = ~5/σ45eV = ~130 (torr·cm)) [16–18] reveal that the saturation pressure here (200 torr) is ~15 times smaller than the theoretical absorption-limited pressure. We estimate that at a pressure of 250 torr, the plasma-induced defocusing length may be as short as 550 μm, or ~1.2‰ the Rayleigh range [19,20]. In a tight focusing geometry, the harmonic yield at 45 eV is likely limited by the ionization-induced defocusing of the fundamental laser beam, which reduces the peak laser intensity, restricting the maximum achievable HHG signal. The other possible limitation is that if the beam diverges from the focus while still in the high-pressure region, the HHG light generated only near the focus will be partially reabsorbed before reaching the vacuum.
By tightly focusing the laser beam ~10 μm after the center of the gas cell, at an optimum pressure of ≈200 torr, the maximum HHG conversion efficiency of ~2 × 109 photons per second at 45 eV is achieved. This conversion efficiency is at the theoretical limit of efficiency for this case of a short confocal parameter (several hundred microns) and density-length product (see Table 1 .). For comparison, using driving laser energies of ≈0.5 mJ in a few cm long waveguide (150 µm diameter), photon flux of ~1011 photons per second at 45 eV is achieved at repetition rates of ≈1 kHz. In this case, the increased density-length product possible for higher pulse energies compensates for the lower repetition rate. Similarly high density-length products will be possible at 50 kHz repetition rates in the future by using slightly higher laser pulse energies around 250 µJ .
Figure 4 shows the full HHG spectrum at argon pressures of 50 and 250 torr. The selective enhancement (quadratic growth) of the 27th and 29th orders with increasing gas pressure indicates phase matching. The dashed green line of Fig. 4 shows the phase-matching cutoff, which is the highest HHG photon energy that can be generated before the ionization of medium exceeds the critical ionization level: ηcr = (1 + Natm ·λL 2 ·re / (2·π·∆δ))−1 (3.8% in Ar) [5,13,22]. For ionization levels above critical, the free-electron dispersion is too high to be compensated by the neutral atom dispersion. Since the geometric phase (Gouy phase shift) has the same sign as the free-electron contribution to the dispersion, phase matching of the HHG process is thus not possible above photon energies of 50 eV.
Figure 5 shows the total HHG flux as a function of laser repetition rate, demonstrating that it scales approximately linearly from 10 to 50 kHz. The slight deviation from a linear dependence is the result of a slightly changing thermal lensing in the Ti:sapphire regenerative amplifier, that shifts the focal position [15,23]. This is supported by a slight change in the divergence angle of the HHG beam, from 6.5 to 5 mrad, as the repetition rate is increased.
Improvements in the overall HHG flux at very high repetition rates can be obtained by a moderate increase in pulse energy, combined with further increases in the repetition rate of the driving laser. In the current geometry, although the HHG process is fully phase matched at the theoretical limit for the density-length product used, the very short confocal parameter that must be used in order to obtain the required laser intensity at low pulse energy, combined with pressure limitations due to ionization-induced defocusing in a tight focusing geometry, limit the obtainable efficiency.
Based on the very significant (100-10,000 × ) increase in flux that we obtained by using 25 µJ energy compared with previous work using 7µJ pulse energies (see Table 1.), as well as our previous work generating harmonics in waveguides using laser pulse energies in the 100-500 µJ regime , we expect that a phase-matched waveguide geometry should be able to reach optimum, absorption-limited, efficiency using ~100 – 300 µJ driving laser pulse energies. Up to 1013 (~100 μW) photons per second at 45 eV can be expected by focusing ~250 μJ laser pulses at repetition rates of 100 kHz into hollow waveguides with ~100 μm inner diameter. Such a system is a feasible next step for demonstrating a compact, single-box, ultrafast Ti:sapphire driven, fully coherent HHG source. Other laser technologies such as fiber lasers can also obtain very high average power at high repetition rate, and may be suitable for high-flux HHG sources in the future. However, the available pulse durations and pulse temporal contrast are issues that have a major impact on the efficiency of HHG, and phase matching of harmonics generated by fiber lasers has not been demonstrated to date .
The range of photon energies generated under full phase-matching conditions can also be extended further by using mid-infrared driving lasers, to well beyond the experimental phase matching cutoffs of 50 eV in Ar, 100 eV in Ne, and 130 eV in He using 800 nm driving pulses [22,25,26]. Recent experimental work using long wavelength driving lasers has demonstrated full phase matching in the water window around 330 eV, while in theory, full phase matching of the HHG process in the hard x-ray region of the spectrum around 1 keV will be possible in the future.
We have demonstrated fully phase matched, fully spatially coherent, high harmonic beams at very high repetition rates (50 kHz) for the first time. The generated harmonic photon flux is at the theoretical limit for the density-length products used. We have also demonstrated that the phase matched HHG flux scales linearly with the laser repetition rate from 10 to 50 kHz. By focusing modest laser pulse energies (25 µJ) in to a short gas cell at high pressures, full phase matching of the HHG process is possible, limited only by ionization-induced defocusing of the laser beam in a tight focusing geometry. This compact, single-box, laser system delivers HHG flux at 45 eV with 2 × 109 photons per second, which will be useful for applications in photoelectron spectroscopy, coincidence molecular imaging, metrology, imaging, and microscopy.
The authors gratefully acknowledge funding from the NSF Engineering Research Center for EUV Science and Technology, the Department of Energy, and Kapteyn-Murnane Laboratories Inc. for contributing use of the laser for this experiment.
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