We present the control of high-harmonic generation (HHG) in hollow fibers using adaptive pulse shaping techniques. The shaping capabilities of our spatial light modulator (SLM) are demonstrated by the excitation of specific fiber modes inside a hollow fiber with a helium-neon laser. Afterwards spatially shaped ultrashort pulses are used to generate phase-matched high-harmonic radiation in a fiber. We show that by controlling the mode structure, we can manipulate the spatial and spectral properties of the generated harmonics.
©2006 Optical Society of America
When an intense ultrashort laser pulse interacts with matter, the radiation can be converted into large integer multiples of its fundamental frequency in a process called high-harmonic generation (HHG) [1, 2]. The generated harmonics can extend up to very high orders spanning a wide wavelength range of XUV or soft x-ray radiation. Due to the coherent nature of this process the timescale of the generated XUV bursts can reach down to the attosecond regime [3, 4].
To exploit these features for applications such as time-resolved spectroscopy a large number of harmonic photons is desired, preferably in a spectrally narrow region consisting only of a single isolated harmonic. In most experiments a gas jet is used as conversion medium in a moderately focused geometry. This way, a large number of harmonics up to the cut-off frequency can be generated. However, all XUV photons are more or less evenly distributed among the plateau harmonics limiting the yield of isolated harmonics. Different experiments have already been carried out to enhance the harmonic output using adaptive pulse shaping techniques, but the overall structure of the broad harmonic plateau is still present [5–7]. Of course there is the possibility to cut out unwanted parts of the spectrum, but the conventional use of gratings  introduces phase distortions and only a small fraction of suitable XUV photons will reach their target.
The more elegant alternative to spectral filtering is to only generate the radiation that is needed in the first place by manipulating the frequency conversion process itself. This could be done using designed driver pulses to exert control over the single atom response of the conversion medium. But this effect has not been observed in gas jets so far .
However, it has already been successfully demonstrated that one can take advantage of phase-matching effects if the generation takes place inside a gas-filled hollow fiber. The generated radiation is then limited to a narrow spectral region . This type of control can be complemented by also tailoring the driving laser pulse itself. In addition to the enhancement of the overall harmonic efficiency and the isolation of a single harmonic , more general possibilities to shape the XUV spectrum have therefore been shown. Recent experiments have successfully demonstrated that the spectral shape of high harmonics can be controlled by temporal-only pulse shaping .
Generating harmonics in hollow fibers will give all dominantly participating fiber modes special importance. Here we show that with a certain level of control over the excitation of these modes the spectral properties of high harmonics can be influenced. Our first results show that by spatial pulse shaping the overall harmonic signal can be enhanced and the harmonic spectrum be modified. We see that different fiber modes involved in the generation process evidently leave their imprints on the spectral and spatial behavior of the generated harmonics. The combination of temporal and spatial pulse shaping of the driving laser pulse will be another big step leading to higher control over the coherent XUV radiation.
2. Phase matching
When a laser beam travels inside a guiding structure a combination of discrete waveguide modes is excited . The transverse components of these modes are standing waves confined by the inner walls of the waveguide. The wavevector k therefore consists of a transverse and a longitudinal component that is shorter compared to the freespace k-vector. The phase velocity along the waveguide will therefore become frequency-dependent. The k-vector inside a gas-filled fiber is then approximately given by:
where the first term corresponds to simple vacuum propagation, and the second and third terms result from dispersion of the gas and of the plasma respectively . Na is the atom density, δ depends on the neutral gas dispersion, Ne the electron density, and re the classical electron radius. The last term represents the influence of the waveguide with a as the inner radius of the hollow fiber and unm as the mth root of the Bessel function Jn-1(z) corresponding to a discrete propagation mode in the fiber. Phase-matching for the harmonic order q is achieved when the phase velocity of the driving laser pulse matches that of the harmonic. In terms of the wavevector k the following condition must be met:
Such a phase-matching configuration with a hollow fiber can be used to favor a selected spectral region for harmonic generation [9, 10, 13]. In Eq. (1) there are a number of different adjustable parameters by which the phase-matching condition can be engineered: wavelength, gas pressure, gas species, wave-guide size, and spatial mode. Conventionally only the pressure or gas species is adjusted to select the spectral region where phase-matching applies , but one can see from Eq. (1) that the excited fiber mode(s) are also of importance. The excitation of different fiber modes changes the phase velocity of the driving pulse and leads to different intensity distributions inside the fiber. The excitation of more than one mode at a time favors harmonics of different spectral regions (for each fiber mode unm a different frequency satisfies the phase-matching condition in Eq. (2)). Furthermore the phase relationship between different modes will possibly influence the phase structure of the harmonics generated under these circumstances. Which fiber modes are excited after the driving laser has been coupled into the fiber is determined by the spatial amplitude and phase profile of the driving laser pulse at the entrance of the fiber. However, it should already be noted at this point that near the critical power of self-focussing inside the fiber [14, 15] different fiber modes do not longer travel independently and start to mix (see Section 4). This changes the mode distribution as a function along the fiber. Therefore we need an optimization algorithm that is able to manipulate the spatial properties of the driving laser to be able to fulfill Eq. (2) at a certain point inside the fiber.
3. Optimization of fiber modes
To have control over the spatial phase of the laser pulse we use a liquid-crystal-based two-dimensional spatial light modulator (Hamamatsu PAL-SLM ). It works in a reflective mode with an active area of 20 × 20 mm2 (reflectivity ≈ 90%). Inside the PAL-SLM is a 2D liquid-crystal display (LCD) consisting of 1024 × 768 pixels working as a pixel mask controllable by an XVGA signal from a computer. By means of a built-in laser diode this pixel mask is imaged onto the backside of the active area. As a consequence, the refractive index changes for linearly polarized light impinging on the front side (read light) depending on the intensity of the laser diode light (write light) . The imaged LCD pattern is blurred by about 2–3 pixels. Due to this smoothing there are no gaps between neighboring pixels. Depending on the polarization of the incoming laser light the PAL-SLM can be used for phase-only modulation or for amplitude modulation. The maximum achievable phase modulation is higher than 2π for wavelengths below 950 nm. For all results presented here, the PAL-SLM was used in its phase-only modulation mode. The wavefront distortion of the unbiased PAL-SLM was measured in an interferometric setup and can be stated to be about λ/10.
To test the shaping capabilities of the PAL-SLM we homogeneously illuminated the active area of the pulse shaper with a helium-neon laser (Fig. 1). The spatially shaped reflection is focussed into a 10 cm long hollow fiber (140 μm inner diameter), equivalent to the fiber that is used for the optimization experiments of the high harmonics. The fiber output is recorded by a CCD camera and read out by a computer. This computer also controls the phase modulation of the PAL-SLM so that a feedback-looped optimization algorithm can be applied. To be able to optimize the fiber output we defined a fitness function as the integral overlap between the picture on the CCD camera and a predefined bitmap mask which is used as the optimization goal. The higher the fitness, the higher is the correspondence of the fiber output with the bitmap mask.
We use an evolutionary algorithm  that starts with a fixed number of random phase patterns as a first generation. The phase patterns are represented by a square array of n × n pixel blocks (the pixels of the PAL-SLM are grouped together to limit the number of effective parameters to speed up the optimization process). Each pixel block represents a gene that can change its phase by mutation or crossover. In consecutive generations the fittest phase patterns are used again without alterations, the rest are replaced by mutated versions of the fittest individuals of the previous generation or by a crossover of two of the fittest. This way, the fitness asymptotically reaches a local optimum after a number of generations. As the active area of the PAL-SLM takes about 400 ms to change its phase front, the optimization process can take several hours to converge.
Prior to each optimization run, the coupling into the fiber was adjusted manually with a flat phase profile on the PAL-SLM until a complex superposition of fiber modes was achieved resulting in a rather complex intensity distribution on the CCD camera. This was used as the starting point for the optimization. The first target masks we used represent the fundamental fiber mode (Fig. 2(a)), a double mode (Fig. 2(b)) and a triple mode (Fig. 2(c)) as a result of interfering lower-order modes. The corresponding optimized fiber outputs can be seen below the target masks (Fig. 2(d)–(f)). After the optimal phase profiles for different fiber modes have been found it is possible to switch between them without the need for additional optimization runs.
The optimization to a more complex ring mode is shown in Fig. 3. Next to the fiber output obtained for a flat phase profile of the PAL-SLM (Fig. 3(b)) there is an additional snapshot of an already slightly optimized result (Fig. 3(c)) where most of the parts of the ring structure (Fig. 3(c)) are already reproduced (see also movie in Fig. 3(d)).
Under the assumption of an initially flat phase profile of the laser beam and a gaussian radial intensity distribution we can estimate how the spatial pulse shaper affects the beam. Figure 4 shows the intensity (Fig. 4(a)) and phase distribution (Fig. 4(b)) at the fiber entrance (indicated by the circle) obtained by Fourier transforming the modulated spatial beam profile corresponding to the phase pattern that couples into the ring mode (fiber output of Fig. 3(d)). The intensity has its maximum in the center of the entrance hole and shows a more or less radially symmetric distribution (waist size of about 0.16a). The corresponding phase profile in (Fig. 4(b)) is relatively flat (-0.5 ± 0.3 rad) in the region of high intensity. The ideal waist size that is needed to couple most efficiently into the fundamental hybrid mode, EH11, of the fiber is given by w 0 = 0.64a . The smaller waist size of Fig. 4(a) is more suited to excite a higher order radially symmetric mode (Fig. 3(d)).
The estimated intensity and phase distributions of the triple mode (Fig. 2f) are shown in Fig. 5. The intensity is divided into three different regions analogous to the distribution of the fiber output. The corresponding phase profile consists of a plateau-like structure. In each of the three high-intensity regions the phase varies only slightly but the phase offsets are different (left region 0.8±1.0 rad, upper right region 3.4± 0.4 rad, lower right region -1.6 ±0.4 rad).
These optimization results look promising enough to take the next step: target masks with arbitrarily placed spots that can not be easily constructed as a superposition of only a few low-order modes (Fig. 6).
The selective excitation of specific fiber modes and complex combinations thereof clearly shows that the PAL-SLM works excellently to influence the electric field distributions inside the fiber. During each optimization the laser focus did not miss the fiber entrance. Therefore there will be minimal or no damage done to the fiber by application of ultrashort pulses under comparable experimental conditions. In the next section we will use the presented technique to shape ultrashort 800 nm femtosecond laser pulses to exert control over high-harmonic generation in a hollow fiber.
4. Optimization of high-harmonic generation
For these experiments we used a regeneratively amplified Ti:sapphire laser system delivering pulses of about 80 fs at a central wavelength of 800 nm and a repetition rate of 1 kHz. These pulses are spectrally broadened by self-phase modulation in a gas-filled hollow fiber. A prism compressor is used to shorten the pulses to about 20 fs duration. The laser pulses hit the active area of the PAL-SLM that was set to work with a 15 × 15 pixel block pattern. With a beam diameter of about 10 mm approximately 60 of these pixel blocks are illuminated with a peak intensity of about 1010 W/cm2 (Recent experiments with 30 fs pulses reveal that the damage threshold for the PAL-SLM is at least higher than a peak intensity of 3.5×1010W/cm2). The reflected beam (300 mJ) is then focussed through a flat AR-coated window inside a vacuum chamber into a 10 cm long hollow fiber (inner diameter 140 μm) that is divided into three parts  (Fig. 7) with small gaps between them. This allows the middle part to be filled with argon at a constant pressure of 170 mbar while the pressure in the outer parts drops down to vacuum level. The middle fiber part is the place where the harmonics are generated. The 800 nm laser pulse and the lower-order harmonics are blocked by a 0.3 μm thick aluminum filter behind the fiber. The transmitted soft-x-ray radiation is characterized with an extreme-ultraviolet (XUV) CCD-camera-based spectrometer (grazing-incidence monochromator (Jobin-Yvon, LHT30) with back-illuminated CCD (Princeton Instruments Digital CCD System, SX-400/TE)). The distance of the high-harmonic source and the spectrometer slit is about one meter and the beam divergence ranges from about 3 to 12 mrad. The recorded spectra are evaluated by a fitness function, and an evolutionary algorithm is applied to optimize the spectral shape of the generated radiation.
As a first step we used the overall harmonic yield as fitness function to optimize the spatial phase of the laser pulse. The fitness function was defined as the integrated yield over the harmonic orders 17 through 23. Prior to each optimization run the setup was adjusted manually for maximum harmonic output. As reference signal we used the fitness obtained with a flat phase profile on the PAL-SLM and optimal manual adjustment of the laser focussing point within the hollow fiber. The overall harmonic yield after running the evolutionary algorithm increased by a factor of about 5 compared to the yield of the reference signal (Fig. 8). The phase profile on the PAL-SLM converged to a nontrivial structure that can not be achieved by means of conventional optical components.
The pulse shaper changes the spatial intensity and phase profile of each pulse at the entrance of the fiber, changing the modes propagating inside the fiber. At high intensities the initially excited modes will start to interact more and more with each other and couple nonlinearly making a theoretical approach exceedingly difficult. An estimate of the excited fiber modes is difficult since the fiber modes that are excited at the entrance or at the exit of the fiber do not have to be the same ones that are present in the middle part of the fiber where high harmonics are predominantly generated. Therefore we use the spatial structure of the generated high-harmonic radiation as an alternative method to draw conclusions about the mode structure inside the fiber at the point where the harmonics are generated. We use the full two-dimensional area of the x-ray CCD camera inside the spectrometer to be able to image the spatial structure of the high harmonics at the spectrometer entrance slit. Figure 9 shows the spatially resolved spectra of an optimization of the 23rd harmonic . Due to the limited size of the x-ray CCD, only the spatial upper part of the spectra can be recorded at a time.
In Fig. 9(c) a bimodal structure is visible that was obtained at a pressure of 170 mbar by manually coupling into the fiber (flat phase profile on PAL-SLM). Each harmonic is split in two with different center wavelengths. This can be explained by assuming that two different fiber modes generate harmonics with different amounts of blueshift . This situation was taken as a starting point for a new optimization of the output of the 23rd harmonic. To check the level of possible control over the spectral shape of the harmonic spectrum we changed the fitness function to A2/B (compare Ref. ), where A denotes the yield of the 23rd harmonic (region A in Fig. 9(b)) and B the integrated yield of the neighboring harmonic orders (region B in Fig. 9(b)). The 23rd harmonic could be enhanced by about two orders of magnitude, but the neighboring orders are still significantly present. An equivalent optimization performed in a previous experiment with temporally shaped femtosecond laser pulses revealed far better results  indicating that additional temporal pulse shaping will be crucial for complete control of HHG. The bimodal structure disappeared indicating a clear single mode harmonic output beam. To rule out the effect that just the laser intensity inside the fiber was increased simply by an improvement of the coupling into the fiber we can state that there is less blueshift in the optimized spectrum compared to the unoptimized one. Therefore we can be sure that the enhancement of the cutoff is not due to the single-atom response where an increased intensity would also lead to the production of higher harmonic orders.
The unoptimized and optimized phase profiles of the PAL-SLM that were used to obtain the results of Fig. 9 were used again to generate harmonics at a much lower argon pressure (20 mbar) inside the hollow fiber. The corresponding spatially resolved spectra are shown in Fig. 10(c) and (d). At this low pressure phase-matching effects are not important and nonlinear mode coupling effects become negligible so that spatial changes in the generating pulse directly carry over to the harmonic beam. The harmonic output is low, but substantial differences in the beam shape between neighboring harmonics and corresponding orders of Fig. 10(c) and (d) can be seen indicating the influence of different fiber modes during the high-harmonic generation process.
5. Summary and outlook
The results of Fig. 9 show that it is clearly advantageous to have all laser intensity concentrated in one mode at the point where harmonics are generated inside the fiber. As has already been mentioned before, the excited fiber mode distribution changes as it propagates inside the fiber due to nonlinear mode coupling effects, being a complex function of the propagation coordinate z and the initial beam profile of the incoming laser pulse . Under these circumstances adaptive pulse shaping represents a convenient method as an adaptive fiber mode filter to find the correct initial mode distribution that eventually transforms into the desired single fiber mode. Future work will cover further investigations of the process of HHG with a setup with combined temporal and spatial shaping capabilities. Additional work is done to get more information on the phase structure of the engineered XUV bursts using XFROG  opening the possibility to demonstrate a way to transfer the pulse shaping methods of IR-femtosecond pulses to the XUV-attosecond regime.
This work has been supported by the DFG (SPP 687 1–2), the Austrian Science Fund (F016 ADLIS), the German-Israeli Cooperation (GILCULT), and the ‘Fonds der chemischen Industrie’. C. Winterfeldt acknowledges support from the Studienstiftung des deutschen Volkes.
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