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Abstract
The generation of self-focusing beams of extreme ultraviolet (XUV) radiation using the focal cone high harmonic generation (FCHHG) technique is examined for high energy lasers. The FCHHG geometry is created by passing a focusing laser beam through a gas sheet prior to reaching focus and thus creating a converging beam of high harmonic radiation. This leads to a larger interaction area that increases the total area of XUV emission while not exceeding the saturation intensity of the target atoms or increasing the density of the atoms. Such a method allows for scaling of HHG to any incident laser power. An experiment was conducted demonstrating such scaling to incident 400 TW pulses, showing both the expected spectral signature of HHG and the converging cone of XUV radiation. It was found that this technique is very sensitive to spatial non-uniformity in the driving laser, which has become more prevalent in high energy laser systems.
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1. INTRODUCTION
The discovery of the HHG process in the 1980s [1,2] provided a convenient table-top coherent XUV source, significantly more compact than synchrotron radiation based sources or free electron lasers [3–5]. The HHG process does have some inherent limitations. The cutoff law [3,6–8] shows that the maximum harmonic order is determined based on the driving laser intensity and the binding energy of electrons in the target medium. Ionization of the target medium limits the intensity that can be used to produce HHG from neutral atoms and therefore limits the maximum achievable harmonic order and output intensity. Increasing the driving laser wavelength can increase the harmonic order, but this leads to a drop off of conversion efficiency [9–11].
Another limitation of HHG is the low conversion efficiency. During the HHG process only a small percentage of the driving laser energy is converted to high order harmonics. For HHG in neutral atoms the laser intensity must be below the intensity for full ionization, thereby limiting the driving laser to intensities below ${\sim}{10^{15}}\;{{\rm W/cm}^{2}}$ [12,13]. Given the limited cross sectional area of typical laser beam waists used for HHG, this maximum operating intensity constrains the maximum HHG energy which can be generated. The conversion efficiency from the laser to high order harmonics is typically on the order of ${10^{- 7}} - {10^{- 5}}$ [14–19]. However, using a well-optimized configuration, maximum conversion efficiencies reaching ${10^{- 3}}$ [20] have been achieved.
Many different techniques have been used to increase the conversion efficiency of HHG. Matching the phase of the generated harmonics to the phase of the driving laser dramatically increases the conversion efficiency [3,14,21]. A technique known as quasi phase matching, which uses two or more gas targets placed in a particular sequence, can increase the efficiency even further [20,22]. Other techniques involve temporally synthesized waveforms by combining different wavelengths, or driving the HHG process using the third harmonic [23,24]. Long focal length geometries have also been employed, giving larger focal spots to allow more effective harmonic generation [17,20,25,26].
Thus one of the challenges for HHG is the production of higher energy pulses which are capable of generating brighter XUV sources. To address this goal of generating higher overall XUV energies some research has been reported using a focal cone HHG (FCHHG) geometry [27–30]. This geometry involves placing a gas jet or cell prior to laser focus such that the beam is converging towards focus when it passes through the gas or placing a gas jet or cell after focus to produce a diverging XUV beam arising from the focal point. Unlike the typical HHG geometry of placing the gas target at laser focus, this geometry allows for a much larger interaction area of the laser with the gas jet. For the incident focal cone geometry the coherent generation of HHG leads to the generation of a converging beam of XUV radiation, allowing the XUV produced over this large interaction area to focus into a high intensity hot spot at laser focus or a high harmonic hot spot (HHHS). This method, when optimized, would allow for the generation of higher intensity XUV radiation pulses than those produced through the HHG process to date. When also accounting for the attosecond nature and smaller diffraction limited spot of HHG pulses the peak intensity could approach that of the incident laser pulse.
This technique has been reported in previous publications using millijoule pulse energies [27–29] and with a few hundred mJ pulses [30]. In the current work we demonstrate that such focal cone HHG can be extended to petawatt scale laser systems by shifting the interaction region far from the focal beamwaist allowing the generation of XUV pulses of much higher energy. This demonstrates the effectiveness of this technique for over an order of magnitude higher laser pulse energies than are achieved on lower energy driving lasers systems. However, the results also demonstrate for the first time the extreme sensitivity of such high order processes to the quality of the interaction wavefront and indeed the capability of using the non-uniformity of HHG spatial wavefront as a diagnostic of the incident laser beam quality. Initial results of this study were first reported in [31], but a more detailed summary and analysis of the results are presented here.
2. EXPERIMENTAL PROCEDURE
The experiment was carried out at the Centro de Láseres Pulsados (CLPU) in Salamanca Spain, a high power laser facility [32]. Figure 1 shows the experiment layout at the CLPU. The 800 nm VEGA 3 laser operating at pulse energies between 1 and 12 J and pulse lengths on the order of 30 fs was used. The beam is focused in vacuum into the Vega 3 chamber with a 2.5 m focal length off axis parabola. The incident laser beam diameter is 22 cm leading to an f/11.4 focal geometry. Using the paraxial approximation for a diffraction limited Gaussian beam, this leads to a beam waist of 11.6 µm and a Rayleigh length of 132 µm. The beam is then allowed to expand to the diagnostics in a secondary chamber placed further from laser focus. An X-ray camera (Sophia XO camera from Teledyne Princeton Instruments with pixel sizes of 15 µm [33]) placed 3 m from focus and XUV photodiodes (XUV 100 Photodiodes from OSI Optoelectronics [34]) placed 2.18 m from focus were used to detect the XUV signal. Thin Al filters were used at the entrances of the XUV diodes and the XUV camera to block the laser light and pass XUV radiation. A rectangular gas nozzle was placed prior to laser focus in order to investigate the focal cone high harmonic generation process. The nozzle was able to be moved in three dimensions as shown in Fig. 2, and Argon gas was used during the experiment. Additionally, a knife edge and perforated meshes, which could be inserted into the beampath, were used to investigate spatial properties of the beam while a transmission grating was used to characterize the XUV spectra.
A rectangular gas nozzle, shown in the inset of Fig. 1, was used in the experiment. The rectangular shape was used to provide a broad, relatively uniform gas sheet for the laser to pass through. A narrow wedged design, with a wedge angle of 15° fullwidth, was used in order to achieve supersonic velocities. Gas at supersonic velocities creates a slowly expanding gas sheet which is desirable for the experiment.
In order to characterize the gas density profile created by the gas jet, interferometric density measurements were carried out and compared to fluid dynamic simulations run using Ansys Fluent [35]. Argon gas was used both in the interferometry and simulations. Two-dimensional simulations were matched to the interferometry to provide appropriate scaling of the density spatial structure. Three-dimensional simulations were then run in order to obtain the line integrated density along the beam path, shown in Fig. 3. These simulations predict that the region of uniform gas density will shrink as the distance from the nozzle increases due to the gas spreading out.
Fig. 3. (a) Two dimensional plot of the gas jet line integrated number density along the beam path. (b) Vertical lineout of the line integrated density. (c) Horizontal lineout of the line integrated density.
Very large beam diameters or beams which were not centered on the gas jet overlap with spatial density gradients from the high and low density regions of the gas jet. This could then lead to spatial variations in the HHG process. As a result of this analysis most tests were done within the relatively uniform region along the centerline of the gas jet.
The expansion of the gas jet along the beam direction occurs much more rapidly as the nozzle is much narrower in this direction. The FWHM of the gas jet along the beam direction is shown in Fig. 4. Due to the gas spreading out as it moves from the nozzle there is an increase in the gas jet width. The width stays relatively small until about 10 mm from the nozzle exit at which point the gas expands at a more rapid pace.
Fig. 4. FWHM of the gas jet along the beam path as a function of distance from the nozzle exit. From the Ansys simulation.
Because the integrated gas density remains relatively constant in the uniform region of Fig. 3 it is expected that the yield across the beam will be largely the same; however, it has been found that the harmonic yield is also dependent on the steepness of the density gradient [36]. Therefore, since the gas jet is expanding as it moves from the nozzle, it is expected that there will be some variation in HHG yield as a function of height above the nozzle exit.
The experiment was carried out on a single shot basis with a period of 10 s or more between shots to allow the chamber to reach vacuum pressure ${(10 }^{-5} \;{\rm mbar})$ before the next shot. The laser shots are taken within the period of 3–5 ms after opening the gas jet nozzle, by which time the rise in chamber pressure is estimated to be less than ${10}^{-2} \;{\rm mbar}$, minimizing the absorption of the XUV radiation by ambient gas.
As shown in Fig. 2 the gas nozzle was able to be moved in 3 directions. The transverse position was chosen such that the center of the gas nozzle was in the center of the beam while the axial position and distance from the nozzle were varied. The distance between the beam and the nozzle was typically chosen to be small so as to maximize the steep flat density profile while avoiding moving the nozzle directly into the beam path.
3. SPATIAL MODULATION
Examples of the XUV images at the center of the XUV beam recorded by the X-ray camera are shown in Fig. 5. The beam passes through a 3.6 µm thick aluminum filter held in place by a circular frame resulting in the cut-circle shape on the two sides. There are also some debris spots on the camera resulting in a few masked spots within the image itself. The beam appears to consist of many bright and dark spots with a dimmer background over the full exposed region. Moreover, there appear to be interference patterns between the bright spots suggesting that each of these beams acts as a coherent light source. Figure 5(b) is the average of five shots and shows that the shape remains quite consistent even to the point where the interference patterns are well preserved when averaging.
Fig. 5. Modulated beam profile for (a) single shot and (b) five averaged shots. The laser energy was 4 J, the average pulse length was 35 fs, the gas nozzle was 90 mm from focus and 8 mm from the laser axis, and the average intensity was $2.5 \times {10^{14}}\;{{\rm W/cm}^{2}}$.
Fig. 6. Motion of the observed XUV spatial pattern while varying the horizontal pointing angle of the off axis parabola (top), and varying the vertical pointing angle on the off axis parabola (bottom). Image created from 32 shots total averaged into the six images comprising this figure. The laser energy was 8 J, the average pulse length was 40 fs, the gas nozzle was 130 mm from focus and 8 mm from the laser axis, and the average intensity at the interaction point was $2.1 \times {10^{14}}\;{{\rm W/cm}^{2}}$.
To investigate the origin of this modulation, radial scans of the gas jet position were conducted to make sure that this spatial non-uniformity was not the result of the non-uniform structure within the gas jet itself. These showed that while intensity was impacted the overall structure remained the same, therefore indicating that the spatial non-uniformity was not a result of any gas jet non-uniformities. Next, to establish if the pattern was a result of spatial non-uniformities in the filtering or the camera itself, the pointing direction of a mirror upstream of the experiment was adjusted. Figure 6 shows that adjusting the pointing direction of this mirror results in a corresponding movement of the XUV modulation patterns, thereby indicating that the source of the modulation is related to modulation in the beam profile of the incident laser beam. Due to the distance of the gas jet interaction region from the laser focus there can be considerable modulation in the beam intensity profile compared to near the beam waist where standard HHG is generated.
Images of the gas jet interaction region were taken with a visible CCD camera as shown in Fig. 7. Striations along the beam direction are clearly visible indicating that higher intensity beamlets within the main laser beam are present, providing further evidence that the observed spatial modulation is due to a non-uniform beam profile.
Fig. 7. Observation of striations in the gas jet from a camera mounted above the gas nozzle. The laser energy was 12 J, the pulse length was 46 fs, the gas nozzle was 130 mm from focus and 8 mm from the laser axis, and the intensity at the interaction was $2.5 \times {10^{14}}\; {{\rm W/cm}^{2}}$.
4. SPECTRAL CHARACTERIZATION
Thin aluminum filters transmit in the photon energy range of ${\sim}{17}$ to 70 eV and block all visible and ultraviolet light [37]. Thus with these filters placed in front of the diodes and camera it is clear that XUV light is being observed. Total aluminum filter thicknesses of 1.6–3.6 µm were employed in this experiment. Such thicknesses are relatively large for HHG experiments, but due to the high energy of the laser used they were necessary. To demonstrate that this XUV light was from HHG, a transmission grating with 500 lines/mm was placed $69.3 \pm 1\;{\rm cm} $ from the camera. In order to ensure detection of weak signals a wide grating slit was used which blurred the harmonic orders together. In order to ensure detection of weak signals a wide grating slit of ${\sim}{1}\;{\rm mm}$ width was used to obtain the overall spectrum. On some shots a bright spot in the XUV radiation pattern would only partially illuminate the slit leading to a smaller effective source size, allowing individual harmonic lines to be resolved within this envelope. The peaks of these lineouts were taken as the position of the zeroth order, and the distance on the CCD from the zeroth order peak was used with the known distance from the camera and grating frequency to determine the wavelength at any given point on the CCD. Figure 8(a) shows such an XUV spectrum on the camera where the XUV radiation only illuminates the left hand side of the slit. Figures 8(b) and 8(d) contain lineouts from Figs. 8(a) and 8(c), respectively, plotted as a function of wavelength. Figures 8(b) and 8(d) also display the calculated positions of harmonics using red dots. The plotted spectra have been corrected for filter and camera response functions. Only odd harmonics are observed as expected, and both the measured peaks and the calculated harmonic positions are in close agreement, indicating that the XUV is indeed the result of HHG.
Fig. 8. (a) CCD image of the measured spectrum for a shot with an intensity of $1.7 \times {10^{14}}\;{{\rm W/cm}^{2}}$ at the gas jet. The laser energy was 8 J, the pulse length was 95 fs, and the gas nozzle was 100 mm from focus and 6 mm from the laser axis. (b) Lineout of the right side of the image in (a). (c) CCD image of the measured spectrum for a shot with an intensity of $2.1 \times {10^{14}}\;{{\rm W/cm}^{2}}$. The laser energy was 12 J, the pulse length was 56 fs, and the gas nozzle was 130 mm from focus and 8 mm from the laser axis. (d) Lineout of the right side of the image in (c). The calculated harmonics are displayed by the red circles in (b) and (d) while the data are displayed in black lines. Data have been adjusted to account for the 3.6 µm Al filters and CCD quantum efficiency. Argon gas used to generate harmonics.
Visible in the spectra of Fig. 8(b) are harmonics between the 19th and the 35th order. The peak intensity occurs at the 23rd harmonic and falls off on either side. Figures 8(c) and 8(d) show the spectrum for a higher energy shot. The bright spot seen in the center of Fig. 8(c) corresponds to wavelengths where the overall detector response is higher. After correcting for the effects of filters and CCD response, the corresponding lineout does not show a peak at these wavelengths and instead agrees with the lower energy shot reaching a peak at the 23rd harmonic. Interestingly, despite only a relatively small difference in intensity ($4 \times {10^{13}}\;{{\rm W/cm}^{2}}$), the higher intensity shot generates a greater range of harmonics. It is possible that the harmonics extend even further; however, the transmission of the Al filter is drastically reduced at approximately 17 nm, thereby blocking subsequent harmonics [37].
5. FOCUSING CHARACTERIZATION
In addition to confirming that HHG was occurring, the focusing behavior of the FCHHG process was measured. Two methods were used to confirm this. The first was to determine the source size of the XUV. As show in Fig. 2 the closest position of the nozzle to the laser focus led to a minimum beam interaction diameter of 4.4 mm at the nozzle. Therefore, source sizes significantly smaller than 4.4 mm must necessarily indicate focusing behavior.
A knife edge, placed in the beam path, was able to provide an estimate of the XUV source size by comparing the Fresnel diffraction pattern across it with calculated Fresnel diffraction patterns of varying flat top source sizes placed at the laser focus. Figure 9(a) shows the shadowgraphic image of the knife edge formed on the camera while Fig. 9(b) shows a comparison between one of these lineouts with the calculated Fresnel diffraction pattern for source sizes between 50 and 500 µm. These calculated patterns have been adjusted to take the weighted spectra into account as the Fresnel diffraction pattern is affected by the wavelength of light.
Fig. 9. Knife edge (a) image on the CCD (the lighter region is where the knife edge obscures the beam, and the darker region is where the knife edge is not present) and (b) comparison of the measured knife edge lineout and calculated knife edge for different assumed source spot sizes. The solid black line is the measured data [with the vertical lineout taken in the black rectangle on image (a)]. The other lines correspond to the calculated diffraction pattern for spot sizes varying between 50 and 500 µm. The image is an average of ten shots where the laser pulse energy was 4 J, the average pulse length was 35 fs, the gas nozzle was 90 mm from focus (7.9 mm beam diameter) and 10 mm from laser axis, and the average intensity at the interaction point was $2.5 \times {10^{14}}\;{{\rm W/cm}^{2}}$.
The lineout is carried out on a single bright spot since the intervening dark regions do not show sufficient contrast to allow for an accurate comparison. These calculated patterns have been adjusted to take the weighted spectra into account as the Fresnel diffraction pattern is affected by the wavelength of light. The lineout agrees best with a source size of 100 µm with larger source sizes quickly losing the ringing behavior after the first initial peak. Note that these oscillations with a scale size of 200 µm are much finer than the millimeter scale size fringes seen due to interference between the individual beam spots seen in Fig. 5. The fine edge fringes are also localized to the edge region. These features are a clear indication of a sharp edge region and are not related to interference between hot spots. For this lineout the beam diameter at the interaction point in the gas jet was 7.9 mm. The fact that a source size of the order of 100 µm is measured from the diffraction pattern clearly indicates significant focusing of the XUV radiation after the generation point.
The second method to confirm the focusing was to measure the distance from the XUV source to the CCD based on the observed magnification of the mesh pattern. Shadowgraphic imaging of perforated meshes was used for this measurement. An example image of the mesh observed by the CCD camera is shown in Fig. 10. The projected distance between the holes of the mesh was measured on the CCD camera, and, knowing the hole separation and distance to the CCD, the angle at which the beam diverged can be calculated. From there the distance from the source to CCD can be calculated using the calculated angle and the measured distance between holes on the CCD. The results of this are shown in Fig. 11. The data indicate a measured source point at a distance of $3.01 \;{\rm m} \pm 1\;{\rm cm} $ from the CCD camera, in reasonable agreement with the measured distance of $3.00 \;{\rm m} \pm 1\;{\rm cm} $ to the laser focal point. This distance remains fixed as the nozzle is moved over a distance of 8 cm indicating that the cone of HHG XUV radiation is indeed converging to the geometric focal point of the incident laser radiation independent of the point in space where the generation occurs.
Fig. 10. Shadowgraphic image of the mesh. Image created by an average of five shots where the laser pulse energy was 8 J, the average pulse length was 35 s, the gas nozzle was 130 mm from focus (11.4 mm beam diameter) and 8 mm from laser axis, and the average intensity was $2.3 \times {10^{14}}\;{{\rm W/cm}^{2}}$.
Fig. 11. Distance from the CCD camera to the source as calculated from the mesh for a scan of varying gas nozzle distances to laser focus between 70 and 150 mm. The black dashed line is the least squares fit, and the black circles are the data points. The data points are averaged over a sampling of lineouts from an image on the X-ray CCD at each nozzle position, with the error bars representing the estimated measurement error. Data obtained from 45 shots where the laser pulse energy was 4 J, the average pulse length was 35 fs, the distance of the gas nozzle varied in relation to focus and distance from the laser axis, and varying the interaction point also varied the intensity between $4 \times {10^{14}}\;{{\rm W/cm}^{2}}$ close to focus to $8 \times {10^{13}}\;{{\rm W/cm}^{2}}$ far from focus.
6. CONVERSION EFFICIENCY
Conversion efficiencies were estimated from the measured signals on the XUV diode and the X-ray CCD camera. The manufacturer’s specified sensitivities were used for these calculations with corrections for the transmission by the aluminum filters and the spectral responsivities of the detectors. The conversion efficiencies of the XUV diode and X-ray CCD camera signals were averaged together. Resultant conversion efficiencies on the order of ${10^{- 7}}$ to ${10^{- 6}}$ were measured by the XUV diode and CCD camera responses. The average conversion efficiency versus backing pressure is shown in Fig. 12.
Fig. 12. Dependence of the average XUV converison efficiency on gas jet backing pressure. The black dashed line is a third-order polynomial fit to the data points. Argon gas used to generate harmonics. The upper axis gives the line integrated pressure for the various backing pressures. Data obtained from 60 shots where the laser energy was 4 J, the average pulse length was 35 fs, the gas nozzle was 90 mm from focus and 8 mm from the laser axis, and the average intensity was $2.5 \times {10^{14}}\;{{\rm W/cm}^{2}}$.
Figure 12 shows an increasing XUV conversion efficiency with the gas jet backing pressure (using argon gas) and the estimated line integrated room temperature isothermal pressure experienced by the beam (as determined by the gas jet simulations). The dominant source of error is due to the tolerance of the thin Al filters used to block the visible light (tolerance of ${\sim}25\%$). The increasing conversion efficiency versus backing pressure indicates that optimum conditions have not been reached and that higher conversion efficiencies are possible in the FCHHG geometry. In addition, the local conversion efficiency in the bright XUV spots observed on the CCD camera were significantly higher than the average values plotted in Fig. 12, reaching values of ${10^{- 5}}$.
7. DISCUSSION
There are two significant differences in the present interaction conditions compared to interactions carried out in the beamwaist of the incident laser. First the gas jet is not a perfectly uniform slab but has a density profile varying in all three dimensions together with significant thickness in space. While the integrated density stays relatively constant over several centimeters from the exit of the nozzle, the sharpness of the density profile will affect the efficiency of HHG, introducing variation in XUV intensity across the beam [36]. Second, the laser intensity is slightly increasing throughout the interaction region and interaction process. This may lead to an additional phase shift, as the point where ionization occurs in the electric field oscillations will shift slightly towards the zero crossing point as the laser propagates forward. Even a slight 1/46 wavelength shift in this interaction field point would lead to phase reversal of the 23rd harmonic relative to the copropagating beam. More detailed calculations and numerical modeling of the HHG process along the lines of papers such as [14,15,22,36] will be required to fully understand the importance of such a phase shift effect.
The HHG process is heavily dependent on the quality of the driving wavefront as has been outlined in previous research papers [38–40]. As the harmonic order increases, the required intensity for significant XUV generation also increases, leading to XUV only being generated at the peaks of the laser pulses. This makes the HHG process heavily dependent on the intensity of the fundamental laser as well as its phase. This causes aberrations in the fundamental beam to be amplified by the HHG process, leading to the creation of a heavily modulated beam. Moreover, phase distortions in the incident wavefront of the order of 1/30 to 1/62 of the incident wavelength would lead to a phase difference of 180° in the generated HHG 15th to 31st harmonics. This would lead to non-uniform phase matching conditions across the beam which would further disrupt the quality of the XUV beam. Therefore, when conducting the HHG process far from laser focus on a fundamental beam which already has some intensity modulations (as can be seen in Fig. 7), it is not surprising that spatial modulations in the XUV are observed. It is demonstrated that the non-uniform spatial structure of the XUV arises from the structure in the fundamental beam itself, moving with the pointing direction of the laser beam as it is steered. Thus, it cannot be attributed to non-uniformities in the gas jet nor to spatial non-uniformities in the detector. While such effects may indeed have an impact, the spatial modulation due to the quality of the driving laser wavefront is much stronger. The importance of achieving high wavefront quality when working away from the focal spot beam waist was recognized in a previous study [30] where a deformable mirror was used to correct the incident wavefront to the order of $\lambda /25$ rms error. Thus, the FCHHG process is expected to act as a highly sensitive monitor of minor intensity/phase imperfections in the beam.
Despite this significant spatial modulation, the measurements with both the perforated mesh and knife edge indicate that the XUV wavefront still inherits the overall spherical focusing wavefront from the incident laser beam and thus arrives at a focal point at a location very close to that of the fundamental beam. The diffraction limited size of the XUV focal spot can be much smaller than that of the incident laser wavelength. However, any optical aberrations in the incoming beam will also map onto the HHG radiation; thus, the expected focal spot of the HHG radiation would be close to the same size as the aberrated original laser spot. In the limit of a perfect spherical wavefront for the incident laser beam, it would be expected that the $n$th harmonic would have a beam waist which is $n$ times smaller than the diffraction limit of the incident laser beam. Thus, in principle very small XUV radiation spots at fairly high intensities could be generated directly in the focusing HHG beam when using high quality optical beams. In previous reports where the FCHHG technique was employed [28] the XUV beam was refocused with a 75 mm focal length mirror achieving a 600 nm beam waist radius. Calculations from that report indicated an XUV beam waist of several microns resulting from the focal cone generation. When scaled to the large interaction region and high laser energies of our case, extremely high intensity focal spots could be generated directly without any additional focusing elements. The peak instantaneous intensities would be further enhanced due to short subfemtosecond pulses of harmonic emission [41,42]. In the present experiment an accurate measurement of focal spot size for the HHG radiation was not determined. However, from the diffraction patterns measured from a knife edge a maximum spot size of the order of 100–200 µm can be determined. Given the spatial modulation observed on the CCD camera has a scale size of the order of 1 mm at a distance of 3000 mm, a source region of the order of 3000 wavelengths in size would be required to give such modulation. For the 23rd harmonic this would correspond to a source size of the order of 100 µm, approximately in agreement with the measured values.
The averaged conversion efficiencies between the CCD and XUV diodes are on the order of ${10^{- 6}}$. The peak conversion efficiencies in the beam hot spots of the CCD are on the order of ${10^{- 5}}$. Such conversion efficiencies are moderate compared to values reported in the literature of [14–20] where the interaction occurs in the beam waist of the incident laser beam. It has been demonstrated that by optimizing the generation and phase matching conditions conversion efficiencies up to the order of ${10^{- 3}}$ from laser light energy into HHG energy can be obtained [20]. Such high conversion efficiency has been obtained for optimum integrated densities of 38 mbar-mm. As seen in references such as [14,20,22], a well-optimized system for a single harmonic will experience a peak yield at a certain integrated gas density (such as 38 mbar-mm in Nayak et al. [20]) and will have a lower yield for either smaller or larger integrated gas densities. In the present study, a similar peak conversion efficiency versus integrated gas density was not observed. Instead, as seen in Fig. 12, it seems that conversion efficiencies continue to increase and level off versus gas density. This is similar to predictions and measurements in a long length hollow fiber [14] where a rise, a slight peak, and a plateau region are predicted and observed with increasing fill density for single harmonics. In our case we observe all harmonics within the Al filter pass band window, and each peak will have a peak yield at different gas densities which, when combined, could yield the plateau observed in the conversion efficiency.
Overall, it can be seen that production of high intensity high harmonic hot spots using focal cone high harmonic generation will require very uniform laser beams of high spatial quality and narrow slab gas jets. Higher quality laser beams can be obtained using adaptive optical mirrors (as demonstrated in reference [30]) and vacuum spatial filters to correct or eliminate optical beam distortions. However, approaching diffraction limited beams would probably require sacrificing a significant amount of energy in current high power laser systems. In fact it is seen that FCHHG acts as a highly sensitive monitor of the laser wavefront quality which can serve as a useful diagnostic of the laser beam quality. In addition, thin planar gas slabs could be employed by using one or more sets of skimmer blades in the expanding gas jet beam. This would probably require one or more stages of differential pumping to avoid filling the chamber with low pressure absorbing background gas.
Finally, to make use of the HHHS beams for many experiments, it might be necessary to separate the focusing HHG radiation from the high intensity incident laser radiation. This could be done using a sacrificial thin aluminum filter placed in the beam path prior to the focal point. A 400 nm Al filter would block all the laser light while transmitting on the order of 40% of the harmonic radiation. It is expected that the filter would be destroyed with each laser shot. A practical solution could be through the use of a carrier tape target with a continuous array of holes overlaid with thin (400 nm) Al windows for firing at high repetition rates [43,44]. If individual harmonic orders are desired then a sacrificial reflection grating can also be placed prior to focus, leading to an array of HHG spots of different orders well separated from the incident laser spot. For example, such sacrificial gratings can be manufactured on a continuous plastic strip using single shot KrF laser pulses [45] or by direct embossing of plastic using a master metal grating and subsequently coating with gold.
The current study shows that the FCHHG technique can now be extended to laser pulses of arbitrarily high energy by placing the interaction gas target at the appropriate intensity position. With the optimization of incident wavefront spatial quality and development of more planar gas jet sources it is expected that significantly higher conversion efficiencies could be obtained. If ${10^{- 4}}$ energy conversion efficiencies could be obtained from 10 J, 30 fs laser pulses the average HHG pulse energy would be 1 mJ with a peak power of 33 GW. If focused into 1 µm diameter focal spots this leads to average intensities of the order of $5 \times {10^{18}} \;{{\rm W/cm}^{2}}$. These energies and peak powers are higher than those obtained in typical HHG experiments and approach those available from free electron lasers which produce pulses on the order of hundreds of µJ to several mJ with peak powers of tens of GW and repetition rates of hundreds to thousands of Hz [4,46,47]. Because the individual HHG pulses have durations of the order of 0.1 fs and are much shorter than the full laser pulse the instantaneous peak powers would be another order of magnitude higher than the average values. Thus, such a source would allow one to study nonlinear XUV phenomena. However, considerable optimization of the generation process will be required to achieve these goals.
8. CONCLUSION
The experiment presented here successfully demonstrates generation of focusing high harmonics in the FCHHG geometry with laser energies up to 12 J. The XUV spectra have been measured indicating peak conversion efficiency intensities in the range of the 23rd harmonic with the highest order harmonic depending on laser intensity. The transmission grating spectra show that only odd order harmonics are observed and that harmonic peaks occur in the expected locations. Spatial analysis through the use of knife edge diffraction shows that the XUV source size is smaller than the order of 100–200 µm. Shadowgraphic imaging of a mesh foil indicated that the XUV source had a fixed distance of 3.01 m from the CCD camera. This result stays consistent within error bars as the gas jet position is varied. All these measurements were carried out at a distance of 3 m from the laser focus, thereby confirming the generation of a high quality cone of radiation emitted from the focal point of the original laser radiation. Thus, the results indicate that a converging beam of high harmonic radiation has been generated using the focal cone geometry. Measured average conversion efficiencies over ${10^{- 6}}$ were achieved corresponding to XUV pulse energies of over 10 µJ when using 10 J input pulses. However, peak conversion efficiencies in the localized bright spatial regions were significantly higher, on the order of ${10^{- 5}}$. It is expected that the use of more spatially uniform laser pulses could lead to average conversion efficiencies of this order, yielding much higher output energies, and further experiments will be required in order to assess the peak conversion efficiencies that could be obtained using the FCHHG technique.
Funding
European IMPULSE project (871161); Laserlab-Europe V (871124); MCIN/AEI/10.13039/501100011033/FEDER (PID2021-125389OA-I00); Ministerio de Ciencia, Innovación y Universidades (PID2022-140593NB, FEDER, EQC2018-005230-P, Equipment); Unidad de Investigación Consolidada de la Junta de Castilla y León (CLP087U16); Natural Sciences and Engineering Research Council of Canada (RGPIN-2019-05013).
Acknowledgment
The authors would like to acknowledge the excellent support and assistance from the laser and technical support teams at CLPU. Roberto Lera and Luis Roso acknowledge support from the Spanish Ministerio de Ciencia, Innovación y Universidades. Jose A. Pérez-Hernández acknowledges support from MCIN/AEI/10.13039/501100011033/FEDER, UE and by a “ERDF: A way of making Europe” by the European Union.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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