Colloquium: Optimal control of high-harmonic generation

High-harmonic generation provides an attractive light source of coherent radiation in the extreme-ultraviolet (XUV) and soft-x-ray regions of the spectrum and allows for the production of single attosecond pulses or pulse trains. This Colloquium covers the control of high-harmonic spectra by temporal and spatial pulse shaping of the driving laser pulses and its implications on time-resolved XUV spectroscopy and attosecond pulse shaping. It summarizes important steps for extending existing pulse shaping techniques and control schemes from the near-infrared or visible part to shorter wavelengths. Using adaptive pulse shaping of the driving laser pulses, several groups have demonstrated control of the high-harmonic spectrum, including the author’s work on the complete control over the XUV spectrum of high-order harmonics, generated in a gas-filled hollow fiber. It is possible to achieve both the enhancement and the suppression of single or several selected harmonic orders. These arbitrarily shaped soft-x-ray spectra will allow for important modifications of the resulting harmonic pulses in the temporal domain. This constitutes first steps towards direct attosecond pulse shaping in the soft-x-ray domain. Moreover, high-harmonic generation in a hollow-core fiber can be enhanced by coupling into a single fiber mode using a feedback-controlled adaptive two-dimensional spatial light modulator.

Figure 1

(Color online) Illustration of the three-step model for high-harmonic generation: (1) tunnel ionization of the electron, (2) acceleration in the laser electric field, and (3) recombination and emission of a high-energy photon. The energy of the emitted photon depends on the ionization potential of the atom and on the kinetic energy of the electron upon its return to its parent ion.

Figure 2

(Color online) Different classes of electron trajectories during the propagation phase of high-harmonic generation, plotted in the position-velocity plane: The trajectories start at the atom located at (0, 0). Depending on the phase $\Phi$ of the electric field at the instant of ionization, they can return to the core at position 0 with different kinetic energies, visualized by the intersection with the velocity axis. The traces $a(\Phi =45°)$ and $c(\Phi =3°)$ correspond to the short and long trajectory, respectively, leading to the same final energy. Class $b(\Phi =17°)$ is a cutoff trajectory with the highest kinetic energy $\left(3.17U_p\right)$, $d(\Phi =0°)$ starts at the peak of the electric field where most electrons are produced but returns to the core with zero kinetic energy. Trajectory $e(\Phi =-45°)$ never returns to its parent atom.

Figure 3

Typical high-harmonic spectrum. The temporal periodicity of high-harmonic emission with half the laser oscillation period leads to a harmonic spacing of twice the fundamental frequency. The spectrum can be divided into three parts (which have been exaggerated in this schematic diagram for clarity): the perturbative regime at low orders, the plateau for intermediate orders, and the cutoff at the highest orders.

Figure 4

(Color online) Pressure and mode dependence of high-harmonic generation in hollow fibers. (a) The excitation of one fiber mode shows the shift in maximum harmonic emission towards higher energies for increasing pressure. (b) If two different fiber modes are excited, two distinct local maxima can be seen. The maximum emission of $34\mathrm{nm}$ radiation at lower and higher pressure is governed by phase matching of the $E{H}_{11}$ and $E{H}_{12}$ fiber modes, respectively.

Figure 5

Designed high-harmonic spectra to illustrate the challenges for further high-harmonic generation experiments: extension of the cutoff energy, increase in the efficiency of the high-harmonic conversion process, and the selection of single harmonics or range of harmonics.

Figure 6

(Color online) Working principle of adaptive control. The incoming laser pulse is first shaped in a laser pulse shaper and is then used to perform the experiment in the quantum system. The experimental signal is analyzed and a fitness is assigned to the corresponding laser pulse. Utilizing this ranking, the evolutionary algorithm can automatically and iteratively find the optimum laser pulse shape.

Figure 7

(Color online) Geometries for high-harmonic generation in atomic or molecular gaseous media. (a) Generation in a free-focusing geometry (gas jet), selectivity of single harmonics is not possible. (b) Generation inside a gas-filled hollow capillary, selectivity of different single harmonics can be achieved with a high contrast with respect to neighboring orders.

Figure 8

Optimizations of harmonics performed in a free-focusing geometry. (a) Negative outcome of a selective optimization of harmonics in region A compared to region B. No selectivity could be achieved. (b) Cutoff extension in a free-focusing geometry. The optimization of the high-harmonic cutoff position is due to the clean-up of higher-order phase distortions by the deformable mirror, thus increasing the laser peak intensity, leading to higher cutoff photon energies.

Figure 9

(Color online) Experimental setup for temporal pulse shaping. Ultrashort laser pulses from a regenerative amplifier undergo further spectral broadening and compression using self-phase modulation in an argon-filled hollow capillary and a prism compressor. A deformable mirror as the symmetry plane of the prism compressor allows temporal pulse shaping. The shaped pulses generate high-harmonic radiation in a second argon-filled hollow fiber. The resulting spectrum is recorded by a CCD camera equipped soft-x-ray spectrometer and analyzed by the computer which runs a closed-loop evolutionary algorithm to optimize the spectral shape. The upper inset depicts the modification of the optical paths of the different frequency components as introduced by the deformable mirror. The inset on the right shows a typical optimization run for the selection of a single harmonic. The fitness value increases as a function of the number of generations.

Figure 10

(Color online) Optimizations of different single harmonic orders in a hollow fiber. Harmonics can be selected with a high contrast ratio with respect to neighboring orders.

Figure 11

(Color online) Selection of extended frequency ranges of high-harmonic orders. (a) Reference spectrum obtained with an unmodulated laser pulse. (b) Optimization of the long-wavelength spectral part while simultaneously reducing the short-wavelength region and (c) vice versa.

Figure 12

Suppression of harmonics. (a) Starting with a typical harmonic spectrum in the plateau region, we can suppress (b) one or (c) two harmonic orders. These modifications of the spectra imply major modulations of the harmonic time structure on an attosecond time scale.

Figure 13

(Color online) Optimization of high-harmonic emission in a hollow fiber using spatial laser pulse shaping. (a) Optimization of the total yield results in an increase by a factor of 5. (b) Selective enhancement of the 23rd harmonic (gray-shaded area) as the optimization goal results in an increase by two orders of magnitude. However, no selectivity is achieved.

Figure 14

(Color online) Spatially resolved HHG spectra: (a), (b) high pressure and (c), (d) low pressure. Two fiber modes contribute to the signal in (a) before the optimization while the optimized pulse (b) results in an overall increase in intensity of about ten. The excitation of a single mode is clearly visible. At low pressures the change in excited fiber modes from (c) before the optimization to (d) after the optimization becomes more evident.

Figure 15

High-harmonic spectra and their corresponding Fourier transform assuming a flat spectra phase. (a) The unshaped spectrum show the periodicity of $T∕2$ for the harmonic pulses in the time domain as expected. (b) One or (c) two suppressed harmonics result in an increased effective spacing of frequency components and thus in a sub-half-cycle modulation of the harmonic time structure.