High-order harmonic generation

Stig Borgström, Mette Gaarde, Per Jönsson, Jörgen Larsson, Eric Mevel, Anders Persson, Sven-Göran Pettersson, Tomas Starczewski, Sune Svanberg and Claes-Göran Wahlström, Carlo Altucci^*, Bertrand Carré^*, Anne L'Huillier^*, Joanna Muffett^*, Roland Smith^* and John Tisch^*

^*Visiting scientists

High-order harmonic generation, using high-power, short-pulse lasers, has been found to be a promising way of generating coherent radiation in the extreme ultraviolet (XUV) and soft X-ray spectral regions. This radiation has unique properties of short pulse length, high peak power and high spectral brightness. With a tunable laser, the generated radiation is also tunable. Besides its basic research interest (as a probe of the behaviour of an atom strongly perturbed by an electromagnetic field), the harmonic generation process is a novel source of coherent radiation in this short-wavelength range. In the present section we will describe some of our work aimed at understanding the fundamental physics involved in the generation process and on the characterisation of the radiation generated. In Section B3 the application of high-order harmonic radiation in atomic spectroscopy will be discussed.

Study of the cut-off law

The formation and the extent of the characteristic plateau in the harmonic spectra have been investigated by performing systematic studies of the intensity dependence of individual harmonics [A1]. Numerical calculations of the single-atom response, valid in the regime of tunnelling ionisation, have shown that the cut-off energy, i.e. the photon energy of the highest harmonic in the plateau, is well approximated by the simple formula I_p+3U_p. I_p is the ionisation energy and U_p the ponderomotive energy, proportional to the intensity. (The limit is hence given by the maximum value of the intensity seen by the non-linear medium; the saturation intensity for ionisation.) In our experiments, we found that the cut-off energy in the macroscopic field also increased linearly with laser intensity. The constant of proportionality, however, is reduced due to intensity-dependent phase- matching effects [A2]. Harmonic generation in He and He⁺ have also been investigated through numerical simulations [A3]. This was done for the KrF wavelength and at intensities beyond the corresponding saturation intensity.

Characterisation of the radiation

In order to make use of high harmonic radiation in different applications, its main features, such as spectral, temporal and spatial profiles, should be well characterised. First, ionisation of the non-linear medium leads to time-dependent changes in the refractive index. This causes a spectral broadening and a blue shift of the laser radiation propagating through the medium. We have investigated how this affects the harmonics generated in the ionising medium. The spectral profile of the 15th harmonic in xenon is illustrated in Fig. A1 for three different intensities. The spectral broadening and the shifts of the harmonics were found to be proportional to the broadening and the shift of the fundamental laser radiation [A1].

Fig. A1. Normalised spectral profiles of the 15th harmonic in xenon at 25 mbar, recorded at three different peak intensities.

Secondly, we performed temporal measurements of the harmonic radiation generated in an ionising gas [A4]. We studied, in particular, the occurrence in time of the fifth harmonic radiation relative to the generating laser pulse. In order to obtain an absolute time reference, we simultaneously generated the harmonics in unsaturated, non-linear crystals and in the ionising gas, recording both pulses with the same streak camera. We found that at low intensities, the harmonic radiation pulses coincided in time with the laser pulses, but as the laser intensity increased above a certain threshold intensity, an intensity-dependent shift in time between the laser pulse and the harmonic pulse was observed. This behaviour was reproduced in numerical simulations.

Finally, the harmonic spatial profile and its dependency on various parameters, such as focusing, ionisation and intensity-dependent phase variations of the atomic dipoles, was studied theoretically [A5]. It was found that the spatial and spectral coherence depend strongly on the focusing geometry. These properties can be controlled and optimised by moving the laser focus position relative to the nonlinear medium [A6].

Optimisation of the efficiency with respect to the gas pressure

In order to determine the optimum gas pressure to be used when harmonic radiation is generated, we have studied in detail how the time- and space-integrated harmonic signal depends on the gas density [A7]. We found that at low pressures (a few mbars), the harmonic signal increased approximately quadratically with the pressure, as expected for a coherent process. At pressures of a few tens of mbars the yield saturated, and even decreased with further increase in pressure. The saturation pressure, and the pressure for maximum yield, were found to be order dependent for the high-order harmonics in the cut-off regime.

High-order harmonics from alkali-metal ions

To generate harmonic radiation with a very short wavelength (high cut-off energy), one should choose atoms or ions with high ionisation potential as the non-linear medium. The singly ionised alkali-metal atoms look like rare gas atoms, but with much higher ionisation potentials. We have studied high-order harmonic generation in laser-produced Na⁺ and K⁺ ions [A8]. However, the highest harmonic orders observed (~27) were not as high as expected from estimates based on the saturation intensities for these ions and on the focused intensities in the absence of the medium. The limited extents of the plateaus are explained, at least partly, as being the consequence of ionisation-induced defocusing of the high-power laser beam, reducing the peak intensity obtained inside the medium. The spatial far-field distribution of the harmonic radiation was also found to exhibit complex structures which varied with the focusing conditions.

Fig. A2. Experimental setup to study harmonic generation in laser-produced alkali-metal ions.