In this work we show that it is possible to increase the high-order harmonic yield when using wavefront-shaped laser beams. The investigation of the beam profile near the interaction region shows that the optimized beam is asymmetric and has a larger diameter. Thus, the optimized beam leads to a higher yield even if the peak intensity is lower compared to an unoptimized beam. This indicates that the wavefront of the fundamental laser beam and, accordingly, the focal profile play an important role in the efficient generation of high-order harmonic radiation.
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The availability of brilliant XUV radiation sources with high beam quality is crucial for many experiments [1–3]. With its unprecedented temporal and spatial characteristics [4,5], high-order harmonic generation (HHG) can act as suitable XUV source [6,7]. Using high-order harmonics it is possible to create the shortest pulses available today .
As in all optical nonlinear processes, efficient frequency conversion by way of HHG happens at phase-matched conditions. However, the extremely high nonlinearity and complex interplay of laser and target material  make phase-matching for HHG a problem of many parameters. Consequently, previous experiments [10, 11] showed a significant influence of the fundamental wavefront, defined as the multitude of points of equal phase in the driving laser beam profile, on the generation process. This is also supported by results given in Refs. [12, 13] and further references therein. The motivation of the experiments presented here, continuing the above mentioned studies, was to find quantifiable features in the adaptively optimized wavefront, in order to gain insight into the physical processes that improve high harmonic yield produced from such spatially shaped pulses over fundamental light from the unshaped laser output.
Using a high-resolution spatial light modulator (SLM) it is possible to manipulate the wavefront in many different ways. This method of wavefront-control offers a high number of degrees of freedom. As mentioned above, it is highly non-trivial to determine the optimal wavefront for the efficient generation of high-order harmonics ab-initio. To counter this, closed-loop optimization algorithms are typically employed . In this work a genetic algorithm similar to the one in Ref.  was used.
2. Experimental setup
The experimental setup is shown in Fig. 1. A Ti:sapphire amplifier system with a repetition rate of 1 kHz delivered laser pulses with a central wavelength of 800 nm. The laser pulses had a duration of about 30 fs and a pulse energy of about 800 μJ.
The wavefront of the laser pulses was manipulated using a reflective spatial light modulator (Hamamatsu LCOS-SLM X10468) with a spatial resolution of 800×600 pixels. The LCOS-SLM is based on a liquid-crystal design. A direct phase-only modulation with a modulation depth of 2π of the laser beam is realized via a pixel by pixel control of the alignment of the liquid crystals. This way, the wavefront of the reflected laser beam can be controlled freely. To allow for interpretation of the applied wavefronts, the phase-masks were generated using Zernike polynomials . They can thus be calculated as
The wavefront-controlled laser beam was focused with a fused silica glass lens (f = 300 mm, f/# = 15) into a sealed nickel tube with a diameter of about 2 mm, placed inside a vacuum chamber. The laser drilled holes into the front and the back of the nickel tube approximately 70 μm in diameter. Therefore, the interaction of the laser pulses, having a peak intensity of 3 × 1015 W/cm2, was limited longitudinally to approximately the tube diameter of 2 mm. As interaction medium, argon gas at a backing pressure of approximately 120 mbar was used.
In order to characterize the beam profile of the driving laser pulse at the interaction region, a focus diagnostic setup was implemented. It employs a standard SLR camera objective. The SLR objective was chosen because of its wide aperture and its very good imaging properties. The objective creates a real intermediate image of the interaction region at its back focal plane. In order to examine the beam profile at the beam waist, a standard 20× microscope objective is used as ocular lens, and a near-infrared sensitive CCD camera serves as detector. The overall magnification of this setup is 2.3× with a spatial resolution of about 5 μm. The generated high-order harmonic radiation is spectrally analyzed using a McPherson 248/310G XUV spectrometer with a 300 lines/mm grating. The spectrometer uses an imaging detection option consisting of a gated MCP and fiber-optically coupled CCD. Each spectrum is recorded by averaging over about 200 laser shots. The yield optimization is carried out using a genetic algorithm similar to the one in Ref. . The genetic algorithm searches for an optimum in the yield, applying Zernike polynomials up to the 6th radial order. The fitness function is calculated from the overall signal of the high-order harmonic spectra. In order to monitor the stability of the experimental conditions, a spectrum generated by an unshaped reference beam was recorded for every generation. This reference beam was created by applying a flat phase-mask to the SLM.
In Fig. 2 the resulting spectra of the reference (black curve) and the fittest individual (red curve) at the end of the optimization are shown. These spectra were also used to calculate the fitness. Three spectral regions, called A, B and C were defined. These three regions, marked by the blue cursors, are also shown. From these, the fitness was calculated using the fitness function10, 11]. Further studies were conducted in order to examine more closely the improvement of high-order harmonic generation by ways of wavefront-shaping the fundamental.
4. Examination of the beam profile
In order to evaluate how the optimized wavefront modifies the intensity distribution of the fundamental laser beam, the beam profile near the interaction region was recorded directly. For this, the focus diagnostic setup described above was used. The beam profile of the reference beam is shown in Fig. 3. As can be seen, it is nearly Gaussian and has a diameter of about 43 μm. Below the actual beam profile small disturbances are visible. These disturbances are caused by multiple reflections at two neutral density filters. The filters were placed directly in front of the microscope and were used to attenuate the beam. The same measurement was also done for an optimized beam. From the phase-mask applied to the SLM the intensity distribution near the focus was simulated. For this, the phase-mask, shown in Fig. 4, was assumed for the spatial phase of the incident beam. The simulation was then done by numerically calculating the Fresnel propagation integral. This way, the beam profiles of the optimized beam at different experimental conditions can be compared. In Fig. 5 the calculated and the measured beam profiles are compared. As can be seen, the calculated profile in the interaction region, shown in Fig. 5(a), is slightly distorted. The beam diameter along the longer main axis is about 46 μm. The resolution achieved by the calculation is about 10 μm and is limited by the available computational power and the limited resolution of the SLM.
The measured beam profile, shown in Fig. 5(b), was recorded using an attenuated beam with an optimized wavefront. Also no gas was present in the interaction region. This allows for comparison of the measured data with the simulation, as the calculation assumes medium-free space during propagation. The calculated beam diameter of about 46 μm is in good agreement with the measured beam diameter of about 48 μm. Similar to Fig. 3, small disturbances appear below and on the left of the actual beam profile. Again, these disturbances are caused by multiple reflections at two neutral density filters. This time, the filters were placed in front of the focusing lens. Thus, the intensity at the interaction region was very low, so no nonlinear effects occurred.
Figure 5(c) shows the beam profile measured under the same experimental conditions as were used for the optimization, i.e., with full intensity and interaction medium present. Under these conditions the beam diameter of about 53 μm is slightly larger than the beam diameter at low intensities and absence of the interaction medium. Similar to Fig. 3, attenuation of the beam occurred directly in front of the microscope, using the same two neutral density filters. This way, all nonlinear effects in the interaction medium which might affect the beam profile were present.
When comparing the different beam profiles, several interesting characteristics can be noticed. The comparison of Fig. 5(a) and Fig. 5(b) shows that the beam profile at the interaction region is influenced only by the phase-mask applied to the SLM. The calculated and the measured beam diameter agree very well. By comparing Fig. 5(b) and Fig. 5(c) the influence of nonlinear effects in the interaction medium on the beam profile can be detected. As can be seen, these effects slightly reduce the asymmetry of the laser beam, but increase the beam diameter significantly. A comparison of Fig. 5(c) and Fig. 3 clearly shows that the beam diameter of the beam optimized for an increased high-order harmonic signal is larger than the beam diameter of the reference beam, additionally showing a pronounced asymmetry. Even though this optimized beam has a reduced peak intensity compared to the reference beam, the high-order harmonic yield in the plateau from this beam is higher. These findings add complementary knowledge to results from [10, 11] where an extension of the cut-off was observed.
It has been shown that it is possible to increase the high-order harmonic yield by using wavefront-shaped laser beams. The investigation of the beam profile near the interaction region showed that the optimized beam has an asymmetric beam profile with a larger beam diameter. Thus, the optimized beam leads to a higher yield even when the peak intensity is lower compared to an unoptimized beam. This shows that the wavefront and the focal profile of the fundamental laser beam play an important role in the efficient generation of high-order harmonic radiation.
This study has been supported by TMKWB grants B154-09030 and B 715-08008. C. Kern acknowledges support from a fellowship of the Abbe School of Photonics Jena. M. Zürch acknowledges support from the FSU grant “ProChance 2009 A1”.
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