Frequency-Resolved High-Harmonic Wavefront Characterization Essay

1. Introduction

Self-probing spectroscopy based on high-order harmonic generation (HHG) has proven to greatly benefit from phase measurements, both for structural and dynamic studies of the generating medium [1,2,3]. Indeed, the infrared (IR) laser-driven electron rescattering process [4,5,6] leading to the emission of harmonic combs in the extreme ultra-violet (XUV) allows one to probe atoms and molecules in situ with unprecedented attosecond time resolution and Ångström precision. In particular, the phase of the attosecond emission gives access to photo-recombination Wigner delays [2,7,8] and phase difference in multi-channel ionization of aligned molecules [3,9]. Specific techniques have been developed to measure phases in the XUV domain [1,3,10,11,12,13] and unravel this subtle information. All of these interferometric techniques require the synchronization of two light or matter waves (the one to be characterized and a reference) with sub-femtosecond precision. In HHG two-source interferometry (TSI), the two light waves are overlapped in the far-field to form fringes on a phosphor screen (see Figure 1). The position on the screen of the latter one directly reflects the phase difference between the two emissions. In a recent paper [14], we reported the use of a 0-π phase plate to convert a Gaussian beam into a TEM01 Transverse Electromagnetic Mode and created two bright spots at the focus of a lens. These two foci were used as two spatially separated and synchronized HHG sources. We qualified this setup for TSI self-probing spectroscopy with full three-dimensional simulations of the generation process and experimental phase measurements in aligned carbon dioxide. Here, we give a more thorough insight into this 0-π phase plate interferometer, how it works, its precision and its stability. The spatial interferogram analysis procedure is also described. We then report the phase measurements using this setup and demonstrate that subtle information about nitrogen alignment revivals and sulfur hexafluoride Raman vibrations can be followed through harmonics phase variations. These results reveal phase variations with opposite behaviors below and above a threshold. This method could be used to assign HHG to one specific ionization channel in molecules having several orbitals contributing to the non-linear process.

Figure 1. (Color on-line) A 0-π phase plate is used as a mode convertor for HHG phase measurement by twp-source interferometry (TSI). The inset below the gas jet shows an experimental picture of the two bright spots created at the focus of a lens with a 0-π phase plate on the LUCAlaser. One of the two sources is used as a probe of the dynamics induced by a pump beam that is overlapped only with this source. The other HHG source remains unexcited and is used as a reference. A set of micro-channel plates and a phosphor screen are placed in the flat-field of a variable groove grating that allows one to disperse the different harmonic orders in one dimension and resolve their spatial profile in the other dimension. The overlap of the two emissions in the far-field results in interference fringes. Their position directly reflects the phase difference between the two HHG sources.

Figure 1. (Color on-line) A 0-π phase plate is used as a mode convertor for HHG phase measurement by twp-source interferometry (TSI). The inset below the gas jet shows an experimental picture of the two bright spots created at the focus of a lens with a 0-π phase plate on the LUCAlaser. One of the two sources is used as a probe of the dynamics induced by a pump beam that is overlapped only with this source. The other HHG source remains unexcited and is used as a reference. A set of micro-channel plates and a phosphor screen are placed in the flat-field of a variable groove grating that allows one to disperse the different harmonic orders in one dimension and resolve their spatial profile in the other dimension. The overlap of the two emissions in the far-field results in interference fringes. Their position directly reflects the phase difference between the two HHG sources.

2. Phase-Plate Mode Conversion

Interferometric precision is more and more demanding as the central frequency of the signal increases. In particular, interferograms in the XUV-domain [15] require highly stable experimental arrangements (for instance, the wavelength of H21, the 21st harmonic order of a 800-nm laser, is about 38nm). In TSI, the reference beam is used to heterodyne the signal, id est, the signal is transferred to a lower frequency. Recently, several TSI schemes were developed using amplitude division [15,16,17,18] and wave-front division [1,19]. All-transmissive schemes are more adapted to this kind of experiment than a Michelson or Mach–Zehnder interferometer, since they intrinsically offer higher stability. In a recent paper [20], TSI was implemented to determine the amount of orbital angular momentum carried by HHG using diffractive optical element (DOE) beam splitting [21]. In [14], we instead based our two source setup on the spatial mode conversion of the driving laser beam. In the absence of a chirp-pulse amplification (CPA) digital laser [22] providing on-demand mode femtosecond pulses, we used a 0-π phase plate to implement the mode conversion [23,24,25,26]. As shown in Figure 2a, we computed the overlap integral between the Gaussian phase-plate-shaped beam (GSB; see Figure 2c) of waist and a TEM01 mode. Both electric fields are considered collimated. We find that a maximum of 90.94% of the input Gaussian beam is converted into the TEM01 mode of waist . This value is slightly smaller than the 93% reported in [26], but still rather close. This conversion rate could be improved by using a cylindrical lens to adjust the waist of the transmitted beam in the direction orthogonal to the 0-π step. In Figure 2b, we show the overlap integral between the GSB and higher TEM0i modes with waist . As expected, the overlap integral with TEM02i modes is zero, while TEM04j+1 and TEM04j+3 contribute with opposite signs and a decreasing amplitude as j increases. These higher order contributions are revealed at the focus of a lens as diffraction peaks in the surroundings of the two bright spots of interest. However, the intensity in these undesired diffraction patterns is not strong enough to induce the HHG non-linear non-perturbative process (see [14]).

Figure 2. (Color on-line) (a) Conversion efficiency from a TEM00 mode with waist to a TEM01 mode with waist ω using a 0-π phase step as a mode convertor. The conversion efficiency is computed as the overlap integral between the Gaussian phase-plate-shaped beam (GSB) and a TEM01 beam of waist ω. A maximum of 90.94% is reached for

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