# Transverse Electromagnetic Mode Conversion for High-Harmonic Self-Probing Spectroscopy

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## Abstract

**:**

_{01}Transverse Electromagnetic Mode. The two bright foci produce two synchronized HHG sources. One of them is used to probe on-going dynamics in the generating medium, while the other serves to heterodyne the signal. We first compute overlap integrals to investigate the mode conversion efficiency. We then establish a clear relation between the laser phase-front curvature and the far-field overlap of the two HHG beams. Both Fresnel diffraction calculations and an experimental lens position scan are used to reveal variations of the phase front inclination in each source. We show that this arrangement offers $\frac{{\lambda}_{XUV}}{100}$ precision, enabling extremely sensitive phase measurements. Finally, we use this compact setup for TSI and measure phase variations across the molecular alignment revival of nitrogen and in vibrating sulfur hexafluoride. In both gases, the phase variations change sign around the ionization threshold of the investigated molecule.

## 1. Introduction

_{01}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.

## 2. Phase-Plate Mode Conversion

_{21}, 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 ${\omega}_{0}$ and a TEM

_{01}mode. Both electric fields are considered collimated. We find that a maximum of 90.94% of the input Gaussian beam is converted into the TEM

_{01}mode of waist $\omega =0.58\times {\omega}_{0}$. 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 TEM

_{0i}modes with waist $\omega =0.58\times {\omega}_{0}$. As expected, the overlap integral with TEM

_{02i}modes is zero, while TEM

_{04j+1}and TEM

_{04j+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 TEM

_{00}mode with waist ${\omega}_{0}$ to a TEM

_{01}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 TEM

_{01}beam of waist ω. A maximum of 90.94% is reached for $\omega =0.58\times {\omega}_{0}$. (

**b**) Decomposition of the spatial profile of the beam after the phase plate in the TEM

_{0i}basis. Each coefficient is calculated as the overlap integral between the GSB and the TEM

_{0i}mode of waist $0.58\times {\omega}_{0}$. TEM

_{02i}modes do not contribute. TEM

_{04j+1}(blue dots) and TEM

_{04j+3}(red dots) modes contribute with opposite signs. (

**c**) Three–dimensional view of the GSB electric field.

## 3. Far-Field Overlap

**Figure 3.**(

**a**) Phase front bidimensional maps (first row) and lineouts along y=0 (second row) of a Gaussian laser beam shaped with a 0-π phase plate 10mm upstream of the focus of a 1-m lens (left) (${\text{z}}_{f}$–10mm), at the focus (center) (${\text{z}}_{f}$) and 10mm downstream of the focus (right) (${\text{z}}_{f}$+10mm). The beam diameter is 32.4mm at 1/e

^{2}. It is apertured down to 15mm and has a 10-mm hole centered on-axis. This annular beam geometry corresponds to the one used with the LUCA laser. We checked that the results presented here are still valid for the Gaussian geometry used for the Aurore laser. (

**b**) Phase front inclination at peak intensity across the laser focus in, respectively, the upper (solid line) and lower (dashed line) source.

_{9}to H

_{17}as a function of the laser focus with respect to the gas jet position. These measurements confirm that the two sources generally emit harmonics in different directions. Interferences are only observed when the focus is close to the gas jet position. This condition is even more strict for higher harmonics. In addition, for H

_{15}and H

_{17}generated close to the focus, one can clearly identify out-of-axis contributions that can be related to long-trajectory HHG. Although this part of the emission also features fringes, the phase in the short-trajectory HHG can still be analyzed, since the two contributions are spatially separated.

**Figure 4.**(Color on-line) Far-field profile of H

_{9}to H

_{17}of the Aurore laser at CELIAgenerated with a 0-π phase-plate when the gas jet is moved through the focus. The peak intensity has been normalized for each harmonic and each position of the gas jet. The gas target is nitrogen.

_{6}). This broad-band spectrum presented in Figure 5a results from the juxtaposition of the first and second diffraction orders of the XUV-grating. H

_{9}to H

_{17}were obtained from the first order of diffraction and H

_{19}to H

_{35}from the second order. The relative amplitude between the two spectra was adjusted following the results presented in [29]. Figure 5b shows the amplitude of the one-dimensional Fourier transform of the above spectrum along the vertical dimension. The dashed line emphasizes the linear relation between the interferogram spatial frequencies and the photon energies:

**Figure 5.**(Color on-line) (

**a**) Interference pattern in the SF

_{6}HHG spectrum reconstructed from the first (H

_{9}to H

_{17}) and second order (H

_{19}to H

_{35}) of diffraction of the grating; (

**b**) Fourier transform decomposition in spatial frequencies of the interferogram presented in (a). The dashed line reveals the Young double-slit linear dispersion. Spatial frequencies are given in ${\tilde{k}}_{0}=\frac{2\pi}{{\lambda}_{0}}\times \Gamma $ units (see Equation (1) in the text).

_{17}to H

_{29}generated in nitrogen with LUCA (100mJ, 20Hz, 60fs, CPAlaser system available at CEA-Saclay) and both the phase (Panel b) and amplitude (Panel c) of the Fourier transform of these fringe patterns. The imaging system used in this experiment has a lower resolution than the one used at CELIA, and the spatial frequency in H

_{29}is close to the highest that can be resolved. In the next section, we use the harmonics generated in nitrogen to investigate the precision on phase measurements and the stability of our phase-plate mode conversion interferometer.

**Figure 6.**(Color on-line) (

**a**) The fringes’ lineout of H

_{17}(red) to H

_{29}(blue) generated in nitrogen with the LUCA laser. Each harmonic order was integrated over a 0.2-photon unit interval centered at the harmonic frequency; the harmonic spatial profiles have been arbitrarily shifted for clarity; (

**b**) phase and (

**c**) amplitude of the Fourier transform of each fringe pattern presented in (a); the phase has been wrapped, so that its variations around the amplitude peak are clearly visible for each harmonic order. The amplitude has been normalized at the non-zero spatial frequency peak. (

**d**) The red circles represent the precision of the interferometer evaluated as 2π times the inverse of the standard deviation of the phase over the spatial frequency width of each amplitude peak. The black “x”-shaped crosses indicate the long-term stability of the interferometer estimated as 2π times the inverse of the standard deviation of the phase for about ten thousand shots.

## 4. Resolution and Stability of the Interferometer

## 5. Inversion of the Harmonic Phase at a Threshold

_{2}highest occupied molecular orbital (HOMO) recombination dipole. This result was almost immediately challenged by phase measurements using other techniques, like gas mixing [12], transient grating spectroscopy [13] and RABBIT(Reconstruction of Attosecond Beating by Interference of two–photon Transitions) [2]. However, to our best knowledge, no further attempts to measure the phase variations inside the revival of nitrogen using TSI were reported. In Figure 7, we show phase measurements inside the revival of N

_{2}for H

_{9}to H

_{17}using our mode-conversion interferometer and a two-pulse alignment stacker [32]. The first striking feature of Figure 7 is that even after only one kick, the first quarter, first half and third quarter revivals (resp. at 2.06, 4.12 and 6.18ps) are clearly seen in all harmonic orders studied here. The second kick allows one to couple more energy into the rotational wave packet initiated by the first kick, to increase the degree of alignment and therefore the contrast of the revival trace, which corresponds here to a deeper phase modulation. Being able to follow nitrogen revival through the variations of the high harmonics phase reflects the high stability of our experimental arrangement. We expect that being able to measure phase variations inside molecular revival with good enough resolution will greatly help to understand specific issues, like fractional revivals [33,34].

**Figure 7.**The phase of H

_{9}to H

_{17}generated in nitrogen as a function of the alignment pump-HHG probe delay. Two kickers (indicated with arrows on the first row) separated by one full rotational period (8.24ps) are used to increase the degree of molecular alignment.

_{9}and H

_{11}is maximum at half-revival, whereas it is minimum for H

_{13}to H

_{17}. Interestingly, H

_{11}(which might be below the N

_{2}effective ionization potential due to the ponderomotive shift [35] in our experimental conditions) is the least contrasted. This could be the result of a destructive interference between two processes contributing to the harmonic emission, like HHG and multiphoton perturbative harmonic generation.

_{6}were the very first evidence of time-resolved vibrational quantum beats in molecules followed by HHG self-probing spectroscopy. More recently, phase measurements in vibrationally-excited SF

_{6}using a transient excitation grating [43] and our mode-conversion interferometer [44] were reported for H

_{13}to H

_{27}. In Figure 8, we present the results extended to H

_{9}and H

_{11}, that is to harmonic orders below and at a threshold of SF

_{6}HOMO (15.7eV), HOMO-1 (16.9eV), HOMO-2 (17.2eV) and HOMO-3 (18.3eV). Our previous studies showed that these orbitals may give non-negligible contributions to the harmonic emission [44]. The four panels of Figure 8 show the oscillations of the phase (first row) and intensity (second row) of H

_{9}(red) to H

_{25}(blue) as a function of the pump-probe delay, filtered at two different vibrational mode frequencies: 774cm

_{−1}(${\nu}_{1}$-A

_{1g}mode, first column) and 524cm

_{−1}(${\nu}_{2}$-T

_{1g}mode, second column). The first piece of information in Figure 8 is that all harmonic orders show intensity and phase modulations at these two vibrational mode frequencies. Also weaker, the oscillations in the phase signal filtered at 524cm

_{−1}are real and significant. For the three other panels, the oscillations are in phase in all harmonic orders, except one, which shows close to out-of-phase less contrasted oscillations. This is the case for H

_{11}(harmonic generated at a threshold of SF

_{6}HOMO-2) in the phase filtered at 774cm

_{−1}, for H

_{13}in the intensity filtered at 774cm

_{−1}and for H

_{15}in the intensity filtered at 524cm

_{−1}. As proposed in [44], the dephasing occurring at H

_{13}and H

_{15}in the oscillations of the intensity could be linked to the shape resonance in the B-channel of SF

_{6}(corresponding to ionization from and recombination to HOMO-2). The new result presented here is the dephasing in the oscillations of the phase at H

_{11}, which we believe is a similar effect to the one observed at the threshold in the revival of nitrogen.

**Figure 8.**(Color on-line) Phase and intensity of H

_{9}(red) to H

_{25}(blue) generated in SF

_{6}as a function of the pump-probe delay filtered at two vibrational frequencies (774 and 524 cm

_{−1}). All curves are arbitrarily shifted for clarity. The dashed lines help to visualize the relative phase of the oscillations.

## 6. Conclusions

_{01}mode of waist 0.58$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{\omega}_{0}$. We emphasized the relation between the IR laser phase-front curvature and the far-field overlap of the two XUV beams when using this mode-conversion device to create two spatially-separated HHG sources at the focus of a lens. We demonstrated the high resolution of this interferometer (${\lambda}_{XUV}/100$ down to 27nm) and the good stability of the overall setup (the standard deviation of the phase measurement is 46.5mrad at H

_{27}).

_{2}revival and vibrational quantum beats in SF

_{6}through TSI phase measurements. At the threshold of these two molecules, similar results are observed in the behavior of the phase variations. The harmonic generated at the threshold of the molecule shows an inverted behavior with respect to the harmonic generated above the threshold. More theoretical investigations are needed to determine whether this is an intrinsic or phenomenological feature of HHG in molecules.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Camper, A.; Ferré, A.; Lin, N.; Skantzakis, E.; Staedter, D.; English, E.; Manschwetus, B.; Burgy, F.; Petit, S.; Descamps, D.; Auguste, T.; Gobert, O.; Carré, B.; Salières, P.; Mairesse, Y.; Ruchon, T. Transverse Electromagnetic Mode Conversion for High-Harmonic Self-Probing Spectroscopy. *Photonics* **2015**, *2*, 184-199.
https://doi.org/10.3390/photonics2010184

**AMA Style**

Camper A, Ferré A, Lin N, Skantzakis E, Staedter D, English E, Manschwetus B, Burgy F, Petit S, Descamps D, Auguste T, Gobert O, Carré B, Salières P, Mairesse Y, Ruchon T. Transverse Electromagnetic Mode Conversion for High-Harmonic Self-Probing Spectroscopy. *Photonics*. 2015; 2(1):184-199.
https://doi.org/10.3390/photonics2010184

**Chicago/Turabian Style**

Camper, Antoine, Amélie Ferré, Nan Lin, Emmanouil Skantzakis, David Staedter, Elizabeth English, Bastian Manschwetus, Frédéric Burgy, Stéphane Petit, Dominique Descamps, Thierry Auguste, Olivier Gobert, Bertrand Carré, Pascal Salières, Yann Mairesse, and Thierry Ruchon. 2015. "Transverse Electromagnetic Mode Conversion for High-Harmonic Self-Probing Spectroscopy" *Photonics* 2, no. 1: 184-199.
https://doi.org/10.3390/photonics2010184