Molecular frame photoemission in dissociative ionization of H₂ and D₂ induced by high harmonic generation femtosecond XUV pulses

Probing ultrafast electronic and nuclear dynamics in excited neutral or ionic molecules in the gas phase is at the core of a vast field which now extends from femto- to atto-science. The ultimate goal is to combine understanding and control of such dynamics to produce a snapshot mapping of the electronic density or wavefunction at the molecular scale with femto to attosecond temporal resolution.

Alternative experimental approaches, taking advantage of remarkable advances in ultrashort light sources and imaging techniques, have been explored in the last few years. In the first approach, the probe is provided by the interaction of the molecule with a strong laser field at the optical cycle timescale. It includes tunnel ionization which reveals the molecular orbital 'shape' and symmetry (Jaron-Becker et al 2003, Alnaser et al 2005, Pavicic et al 2007), and electronic wavepacket recollision in a 'self-probing' scheme, leading to either high-energy electrons in above-threshold ionization and electron diffraction (Meckel et al 2008, Cornaggia 2010, Boguslavskiy et al 2012), or attosecond emission and high harmonic generation (HHG) in the extreme-UV range (XUV) (Itatani et al 2004, Li et al 2008, Wörner et al 2010, Haessler et al 2010, Mairesse et al 2010, Torres et al 2010). In the HHG process, spectral amplitude and phase of the emitted light gives access to the molecular dipole which couples the bound valence wavefunction to a large set of continuum states, providing a basis for the time-resolved tomographic reconstruction of molecular orbitals (Itatani et al 2004, Torres and Marangos 2007, Haessler et al 2010, Vozzi et al 2011, Le et al 2009). The self-probing scheme is especially adapted for studying electronic/nuclear dynamics which is triggered by the strong field itself, therefore, acting simultaneously as pump and probe, e.g., in multi-orbital coherent evolution after tunnel ionization (Mairesse et al 2010, McFarland et al 2008, Smirnova et al 2009, Diveki et al 2012). Accordingly, it intrinsically puts limit to the possibility of exploring field-free dynamics. In the second 'external probing' approach, the excitation and probe of a target molecule are provided by a combination of ultrashort XUV-laser pulses with a controlled delay at the as timescale. Several studies have investigated electronic/nuclear dynamics after photoionization (PI) by high harmonic pulses using laser pulses as a coherent probe or a control field, e.g., in H₂ and D₂ (Gagnon et al 2008, Sansone et al 2010, Kelkensberg et al 2009, 2011), O₂ (Siu et al 2011), or Br₂ (Li et al 2010). At even higher intensity and shorter wavelength in the XUV range, coherent dynamics is also investigated in the same type of pump/probe studies using free electron lasers (FELs) (Johnsson et al 2009, Rudenko et al 2010, Cryan et al 2012).

To understand the dynamics at the deepest level in both approaches, it is of particular relevance to follow the electronic motion in the molecular frame. To that purpose, the use of impulsive alignment of the molecular target plays a key role in a number of laser-based experiments. When photoionization is involved, the molecular frame photoelectron angular distribution (MFPAD) is one of the most 'complete' electronic observables which describe photoemission in the presence of nuclear motion (see Reid 2012 for a recent review). MFPADs sensitively depend on the detailed symmetry of the initial bound state and the final ionic state, of the quantum interferences between ionization channels, as on the phase and polarization properties of light. It is therefore highly relevant as a 'probe' observable of ultrafast coherent dynamics in molecules, where multiple pulse excitation and attosecond phase control allow for extensively varying state symmetries and channel interferences. The advantage of molecular alignment for the determination of MFPADs has been demonstrated, e.g., in a strong field ionization of polyatomic molecules such as OCS (Holmegaard et al 2010), time-resolved imaging of electron dynamics during the non-adiabatic photodissociation of CS₂ (Bisgaard et al 2009), and most recently for the photoionization of N₂, O₂ and CO induced by ultrashort XUV–HHG sources (Rouzée et al 2012).

On the other hand, it is well established that dissociative photoionization (DPI) allows one to determine MFPADs by means of electron–ion coincidence momentum spectroscopy, in the axial recoil approximation, primarily for high (MHz) repetition rate light sources such as synchrotron radiation (SR) (e.g., Landers et al 2001, Lafosse et al 2002, Yagishita et al 2005, Liu et al 2008, Reid 2012 and references therein). Although the typical rate for femtosecond lasers (of mJ energy) and HHG XUV sources in the kHz range is more challenging for coincident experiments, time-resolved MFPADs have been obtained using electron–ion vector correlations for DPI of small polyatomic molecules such as NO₂ induced by the first harmonics of femtosecond IR lasers (Davies et al 2000, Vredenborg et al 2008). Moreover, in experiments coupling a kHz HHG light source and a COLTRIMS spectrometer, time-resolved electron–ion kinetic energy correlations characterizing, e.g., autoionization of O₂ (Sandhu et al 2008, Cao et al 2011), and angular dependence of single-photon ionization of N₂ and CO₂ into resolved ionic states (Thomann et al 2008) have been reported.

In this paper, we demonstrate the measurement of MFPADs in the dissociative ionization of H₂ and D₂, after resonant excitation of doubly excited states (DES) belonging to the Q₁ and Q₂ series, by femtosecond XUV pulses at 31.9 eV produced from HHG, using the electron–ion coincidence vector correlation (VC) method (Lafosse et al 2000) and taking advantage of a complete analytical description of the MFPADs (Lucchese et al 2002, Dowek and Lucchese 2012). We validate the MFPADs measured using the HHG pulses, by comparing them with those measured at the SOLEIL synchrotron, in the same spectral region. This work was first motivated by a recent theoretical description of single-photon DPI in H₂ and D₂ as a function of the XUV pulse duration between 1 and 10 fs. These calculations predict significant effects of the pulse duration on the kinetic energy release (KER) of the atomic fragments (Sanz-Vicario et al 2006), as well as an evolution of the MFPADs as a function of the KER at the femtosecond scale (Peréz-Torres et al 2009). In a single-photon process, these effects relate to the overall spectral width of the electronic/nuclear wavepacket which is excited in the Q₁ and Q₂ regions and evolves accordingly in the autoionization/dissociation channels (González-Castrillo et al 2011). In this relatively simple case, it was interesting to parallel the HHG-based experiment performed with 'broadband' XUV pulses at 1 kHz 'low' repetition rate and limited statistics in VC, with the SR-based study serving as a reference. Second, our study on steady-state H₂ and D₂ has clear objective to prepare dynamical studies using dissociative (or non dissociative) PI as a probe, where time-resolved MFPAD could give detailed and reliable information on the electronic wavepacket evolving in the nuclear field. Photoelectron angular distributions in XUV + IR two-colour PI of atoms, investigated previously using a related experimental setup, have shown to be a sensitive probe of interferences between ionization pathways involving different numbers of photons (Guyétand et al 2008a and 2008b).

As the simplest molecule, hydrogen is often considered as a prototype system in molecular physics. Despite its structural simplicity, the detailed understanding of its dynamics launched by interaction with light is still challenging, both experimentally and theoretically. This is particularly the case after resonant XUV excitation of the Q₁ and Q₂ series of repulsive DES embedded in the continuum of the H₂⁺(²Σ_g⁺) and H₂⁺(²Σ_u⁺) ionic states, respectively. Among different possible reactions in these open channels, DPI: H₂ + hν → H⁺ + H(1s) + e⁻, is especially interesting since electronic autoionization (AI) and nuclear dissociation dynamics are coupled, both occurring within a few femtoseconds (Sanz-Vicario et al 2006, Sansone et al 2010).

Following a vertical excitation in the Franck–Condon region (FC) of the H₂ ground state (X ¹Σ_g⁺) induced by one photon absorption in the 28–35 eV range, DPI involves several pathways describing the evolution of the H₂* system as a function of the internuclear distance R. It can be represented semi-classically using potential energy curves, as shown in figure 1: direct ionization (DI) into the dissociative continuum of the molecular ion ground state H₂⁺(X ²Σ_g⁺), DI into the first excited molecular ionic state H₂⁺(²Σ_u⁺), AI into the H₂⁺(X ²Σ_g⁺) ionic ground state via the first DES Q₁ ¹Σ_u⁺, which is mostly contributing to AI for the Q₁ series, and AI via the second DES Q₂ ¹Π_u, which is mostly contributing for the Q₂ series, into both the H₂⁺(X ²Σ_g⁺) ground state and the H₂⁺(²Σ_u⁺) ionic state. All these pathways lead to the only populated first dissociation limit H⁺ + H(1s) + e⁻, (see L₁ label in figure 1.) and were recognized in pioneering studies (Ito et al 1996, Latimer et al 1995). When they are kinematically indistinguishable, producing a photoelectron and a photoion fragment of comparable energies, it has been shown that quantum mechanical interferences occur, which involve both resonant and non-resonant ionization (Sanchez and Martin 1997a, 1999a) as well as autoionization into ionic states of different u-g symmetry, taking place at different internuclear distances (Martín et al 2007).

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Complete experiments, relying on coincident electron–ion momentum spectroscopy, have been performed in order to characterize the outcome of such DPI reactions (Lafosse et al 2003, Martín et al 2007). They provide for each (H⁺, e) DPI event, the (V_H+, V_e) velocity VC. Within the axial recoil approximation, they enable one to record molecular-frame (MF) photoemission as a function of the KER transferred to the H⁺ + H dissociating pair, or the photoelectron energy which is complementary for a given photon excitation energy and dissociation limit. Using linearly polarized SR, the MFPADs revealed symmetry breaking in terms of backward–forward (BW–FW) emission asymmetries, resulting from quantum interferences involving different reactions pathways (Lafosse et al 2003, Martín et al 2007, Fernàndez and Martìn 2009). Quantum interferences also govern the occurrence of strong oscillations of the β_e asymmetry parameter characterizing the photoelectron angular distribution relative to the polarization axis as a function of the photoelectron energy, predicted theoretically (Fernàndez and Martìn 2009), and recently demonstrated in the measurements of DPI of H₂ and D₂ (Reddish et al 2012). Inducing DPI by a circularly polarized SR also revealed remarkable properties of the molecular frame circular dichroism in photoionization of H₂ (Dowek et al 2010), a fingerprint of the invoked quantum interferences. A detailed comparison of the circular dichroism measured as a function of the KER with state of the art calculations using the time-dependent Schrödinger equation supports the existence of two autoionization regimes in the femtosecond range (Dowek et al 2010), consistent with previous evaluations of the AI lifetimes of the Q₁ and Q₂ DES (Sanchez and Martin 1997b, 1999b).

The paper is organized as follows. In section 2, we describe the experimental setup which has been used to perform the experiments at the Saclay Laser-Matter Interaction Center (SLIC, CEA-Saclay) and at the SOLEIL synchrotron, including the VC electron–ion momentum spectrometer. In the description of light sources, more emphasis is put on the XUV HHG beam line, which was combined here for the first time with the VC spectrometer. In section 3, we report the experimental results characterizing the DPI channels populated by the interaction of XUV femtosecond pulses with H₂ and D₂ targets in terms of the electron–ion kinetic energy correlation diagrams (KECDs). The KECD is the first 2D observable which demonstrates that the acquisition of (H⁺, e) coincident events is achieved and that the (V_H+, V_e) correlation is properly determined. In section 4, these results are compared with reference measurements performed at SOLEIL at six photon excitation energies in the 31.6–32.7 eV range: namely we compare the 1D ion yield spectra displaying the ion yield as a function of the ion fragment kinetic energy, as well as the 1D β_H+/D+ ion asymmetry parameter dependence as a function of the ion kinetic energy. Section 5 is dedicated to the discussion of the MFPADs, relying on the determination of four F_LN(θ_e) 1D functions and their expansion in Legendre polynomials, where θ_e is the polar emission angle of the photoelectron in the MF frame, according to the general analytical expression of the MFPAD developed previously (Lucchese et al 2002). This method is shown to be very helpful to achieve the feasibility of doubly differential measurements relying on electron–ion coincidence momentum spectroscopy, when using a light source of a few kHz repetition rate. Here again, comparing the MFPADs induced by the H21 harmonic femtosecond pulse with reference data measured at SOLEIL provides a support for the data achieved.

In the experiment, an XUV beam of photon energy around 32 eV is crossed with an H₂/D₂ supersonic beam in the source volume of an electron/ion momentum imaging spectrometer, producing dissociative photoionization through resonant excitation in the Q₁ and Q₂ DES region. In the first part, DPI is produced by femtosecond XUV pulses from laser-driven high harmonic generation; experiment was installed by the PLFA laser at CEA-SLIC facility. In the second part, DPI is produced by the SR on the XUV DESIRS beamline at SOLEIL (Nahon et al 2012). We describe the VC setup and method common to the two parts, and then focus on the HHG source.

2.1. Electron–ion vector correlation

The electron–ion velocity spectrometer used in this work, described in detail previously (Lafosse et al 2000, Lebech et al 2002), is schematized in figure 2. The interaction region, at the crossing of the supersonic jet of randomly oriented hydrogen or deuterium molecules and the light beam, is located at the centre of the spectrometer. Positive ions and electrons are extracted in opposite directions from the interaction region by a dc uniform electric field of few tens of V cm^–1 and guided to two time and position sensitive delay-line anode detectors (PSDs) (RoentDek xxxx) through an intermediate region where two focusing electrostatic lens sets (Λ_i and Λ_e) are applied. The extraction field was chosen such that the spectrometer ensures a 4π steradian collection of ion fragments and electrons for the E_e-KER region of interest. Here, typically an extraction field of E_extr = 100 V cm⁻¹ or 150 V cm⁻¹ was used, ensuring both a 4π collection of the ions (H⁺ or D⁺) and a 4π collection of the electrons (e⁻) for E_e < 9 eV and E_e < 14 eV kinetic energy, respectively. The time signals from the ends of the delay lines are encoded as stop signals in an eight-channel time-to-digital converter (TDC) (DTPI xxxx) with an encoding step of 250 ps, where four channels are dedicated to the positive ion detector delay lines and four to the electron detector delay lines. The TDC outgoing signals provide the ion position and time of flight (TOF) for the ion fragment and the electron position, while the use of a time-to-amplitude converter (TAC) is preferred to measure the TOF of the electron with an optimal resolution. The (V_x, V_y and V_z) velocity components are derived from the position coordinates and TOF for each particle of the (H⁺, e) coincident event leading to the (V_H+, V_e) VC.

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Only the timing of the event acquisition differs depending on whether the 1 kHz XUV light source or the MHz SR is used. In the HHG induced ionization, we use a logical signal synchronous with the 1 kHz laser pulse as the common start for the eight channels of the TDC and the TAC. The stop of the TAC consists of the signal detected on the front micro-channel plate (MCP) of the electron detector. At SOLEIL, it is more efficient to use the signal detected on the front MCP of the electron detector as a common start for the CTN-TDC and the TAC, the logical signal synchronous with the SOLEIL light pulse serving as the stop for the TAC.

Performing coincidence measurements requires the occurrence of a maximum of one event per pulse, which imposes that on average the count rate per pulse must be on the order of a few per cent (Stert et al 1999, Davies et al 2000, Vredenborg et al 2008). When an SR light source is used, the few MHz typical repetition rate enables us to perform coincidence measurements with a ∼10⁻³ rate of events per pulse, producing adequate acquisition statistics in a limited time. Under the present conditions, of a laser-based source with a repetition rate of 1 kHz, we determined that an acquisition rate of the order of few tens coinc s^–1 (or 0.1 event/pulse) ensures a true-coincidence acquisition mode. Therefore the data reported in this paper are of very different statistical quality: an experiment of DPI of H₂ or D₂ induced by H21 femtosecond pulses kept in stable conditions during few hours corresponds typically to about 200 000 coincident events recorded, among which less than 10 000 pertain to the DPI channels (H⁺, e) or (D⁺, e).

The VC spectrometer was mounted onto the CIEL ultrahigh-vacuum (UHV) chamber (Guyétand et al 2008a), working at a standing pressure of 10⁻⁹ mbar for the experiments using the PLFA HHG source at SLIC, or installed in the SAPHIRS set-up (Richard-Viard et al 1996) for those using the SR on the DESIRS beamline at SOLEIL. The continuous molecular supersonic expansion (backing pressure about 1.5 bars) produced through a 70 µm continuous nozzle was shaped by two skimmers (ϕ₁ = 200 µm, ϕ₂ = 300 µm) in CIEL and one skimmer (ϕ = 1 mm) in SAPHIRS, corresponding to a beam diameter at the interaction centre on the order of 1 mm and 2–3 mm, respectively. The ratio achieved between the supersonic and thermalized components of the molecular beam in these measurements was typically of ρ ≈ 2 to 3 for H₂ and ρ ≈ 4 for D₂. The dimensions of the light beam at the focus point were typically of the order of 200 µm in the horizontal direction (x) and 100 µm in the vertical direction (z) at SOLEIL; the focal spot diameter is ∼250 µm at SLIC (see below). The main instrumental widths contributing to the resolution in the determination of the velocity components, and subsequently energies and angles of the emitted charged particles, are the δy length of the interaction region along the light beam axis, and the δt jitter of the electron TOF estimated here of the order of δt ≈ 170 ps, affecting V_ye, V_yi and V_ze, respectively.

For the measurements discussed in this paper, the SOLEIL SR facility, delivering 40 ps (FWHM) light pulses, was operated in the single-bunch mode which corresponds to a 0.85 MHz repetition rate. In this mode, one estimates the number of photons to be in the range of 10⁺⁵–10⁺⁶ photons per pulse, giving around 10⁺¹¹–10⁺¹² photons per second for the 1000 resolving power used here. With the goal to perform benchmarking reference experiments for the coincidence measurements using the H21 harmonic, the DESIRS undulator based high-resolution beamline was run below the optimal resolution: using a 400 grooves mm^–1 grating and entrance and exit slits of 100 µm, resulted into a 30 meV spectral resolution (FWHM). The light beam was linearly polarized (P) parallel to the VC spectrometer axis z as in the experiments performed with the femtosecond XUV source described in section 2.2.

2.2. Femtosecond XUV source

The femtosecond XUV source is schematized in figure 3. The PLFA laser chain is based on chirped pulse amplification (CPA) of a titanium–sapphire (Ti:Sa) oscillator in a three-stage amplifier at 1 kHz repetition rate. It delivers pulses centred at λ_NIR ≈ 806 nm with a full width at half-maximum (FWHM) of 25 nm, i.e., a pulse duration of ∼45 fs, and a beam quality factor M²_PLFA ∼ 1.5. Laser pulses of effective 6–7 mJ energy were focused with a lens of 3 m focal length (f-number ∼100) into a 4 cm long cell, containing argon (Ar) gas at 1–4 mbar pressure. For NIR focal spot of diameter ∼250 µm, the intensity driving HHG was close to ∼10⁺¹⁴ W cm⁻², i.e., the saturation intensity in Ar. The generating NIR beam was then removed, first, using a silica plate (not shown in figure 3 for clarity), working at ≈11° grazing incidence angle and reflecting R_Plate ∼ 50% of XUV harmonics, and second together with low harmonics (order ≤ 11), using a 200 nm thick aluminium (Al) filter transmitting T_Filter ∼ 10% of high XUV harmonics. H21 selection was ensured by a set of B₄C/Si plane and spherical multilayer mirrors working near normal incidence at 4°, especially designed to achieve high reflectivity R_H21 ≈ 30% for H21 and to reject the lower and higher order harmonics (< 2% for two mirrors), as shown in figure 3(b); the multilayer mirrors have been deposited at the CEMOX platform for x-ray optics (CEMOX xxxx).

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The XUV light pulses from HHG have been characterized spectrally, first in the course of the experiment using the electron spectrometer, and second after the experiment (HHG in the same conditions) using an XUV light spectrometer. Both measurements are compatible and determine a central photon energy E_H21 = 31.9 ± 0.3 eV (note that it is slightly lower than the expected value 32.27 eV for H21, which may be due to HHG in the 'red wing' of slightly chirped IR pulses, Altucci et al 1998, Tcherbakoff et al 2003), and a spectral width of 0.35 ± 0.05 eV FWHM. In the generating conditions, the XUV pulse duration is estimated to be in the 15–25 fs FWHM range (Mairesse et al 2005, Sola et al 2006); it is larger than the Fourier limit (∼7 fs) because of the harmonic chirp introduced by the intensity-dependent phase of the harmonic emission (Salières et al 1999). Note that, in order to prepare pump/probe studies with XUV pulses of adjustable duration, the XUV pulse duration has been tentatively varied using the polarization gating technique (Tcherbakoff et al 2003, Sola et al 2006). However, in this work we only monitored the variation of H21 spectral profile between the with/without gate configurations, i.e., a variation of the spectral width by ∼200 meV; this is not sufficient to characterize the pulse duration. Actually, under our experimental conditions, the intensity temporal gradient was lower in the gate than without, leading to a reduced harmonic chirp; temporal confinement and reduced chirp in the gate partially compensate and result in the limited variation of the spectral profile which is measured. Finally, under our experimental conditions of limited statistics, we found that the DPI measurements did not significantly vary between the with/without gate configurations.

The number of photons was estimated to a few 10⁺⁸ photons per pulse, thus giving around 10⁺¹¹ photons per second. The XUV beam was focused in the interaction chamber by means of two multilayer mirrors (plane and spherical with 50 cm focal length at ∼normal incidence) for spectral selection and focusing of H21. This leads to an XUV focal spot of diameter ∼250 µm, corresponding to the size of the harmonic source after imaging by the XUV optics, and confocal parameter in the milimetre range.

The measurements of DPI of H₂ or D₂ induced by H21 femtosecond pulses could be performed in stable conditions for few hours: in the first series reported, they correspond typically to an acquisition of about 200 000 coincident events. Examination of the ion TOF spectra, not presented here, shows that the dominant contribution arises from non-dissociative photoionization of H₂ or D₂ with typically 20 coinc s^–1 (≈96%), DPI of the parent molecule amounts to about 0.6 coinc s^–1 (≈ 3%) and the residual gas component (mainly H₂O⁺, OH⁺ and H⁺) could be kept below 0.2 coinc s^–1 (≈1%). These numbers reflect the respective photoionization cross sections and target densities, taking into account the effective collection efficiency for each molecular target. The coincidence rate of the order of 20 coinc s^–1 corresponds typically to a rate of 0.1 event per pulse.

The electron–ion kinetic energy correlation diagram (KECD) is the first 2D observable which demonstrates that the acquisition of (H⁺, e) coincident events is achieved and that the (V_H+, V_e) correlation is properly determined. The KECDs derived from the (V_H+, V_e) and (V_D+, V_e) vector correlations are displayed in figures 4(a) and (b), respectively. They represent 2D histograms of the (H⁺/D⁺, e) coincident events as a function of the ion fragment kinetic energy E_H+ or E_D+ (half of the KER of the (H⁺ + H)/(D⁺ + D) fragments) in abscissa and the electron energy E_e in ordinate. Their general structure has been observed and discussed earlier for SR excitation energies inducing resonant DPI of the Q₁ and Q₂ DES (Lafosse et al 2003, Martín et al 2007, Elkharrat et al 2009). For H21 incident light, the (H⁺/D⁺, e) coincident events are distributed along the only populated dissociation limit L₁ (H(1s) + H⁺) at E_Diss ≈18.076 eV, with a major contribution of direct ionization into the dissociative component of the H₂⁺(²Σ_g⁺) ground state, where most of the excess energy is released to the photoelectron, and to a lower extent into the H₂⁺(²Σ_u⁺) dissociative first excited state of the molecular ion, where most of the excess energy is released to the atomic fragments. These structures correspond to the (12.8 eV ≤ E_e ≤ 13.8 eV, 0 eV ≤ E_H+/D+ ≤ 0.5 eV, peak D) and (0 eV ≤ E_e ≤ 1.3 eV, 6.2 eV ≤ E_H+/D+ ≤ 6.9 eV, peak E) range, not including the broadening due to the instrumental resolution. The structures identified at intermediate E_e and E_H+ energies (band A) are the signature of AI processes involving mainly the Q₁(¹Σ_u) and Q₂(¹Π_u) lowest DES in each series. These states contribute differently to the ionization process as a function of the KER for the H₂ and D₂ isotopes, which are identical in terms of electronic properties but dissociate at different nuclei velocity (Elkharrat et al 2009). Other structures contribute to the KECD: on the one hand, weak peaks are seen along the E₊ = 0 eV axis which correspond to DI of the H₂/D₂ molecule induced by the lower harmonics significantly attenuated by the multilayer mirrors in the incident light beam (see figure 3), harmonic H13 providing the largest contribution for E_e ≈ 1.5 eV due to its lower attenuation (peak d). On the other hand, the KECD in figure 4(a) also shows a weak contribution of (H⁺, e) events produced by DPI of H₂O absent in the KECD of the (D⁺, e) events. In any case, selecting the coincident events along the L₁ diagonal filters the events coming from DPI of H₂ and D₂.

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In order to qualify the DPI measurements performed by irradiation with the broadband H21 femtosecond pulses, we compare these results with similar data recorded at the SR SOLEIL, mapping the 31.6–32.7 eV photon excitation energy region at six discrete energies.

4.1. Spectral dependence of the H⁺/D⁺ ion fragment energy spectra

For a quantitative comparison of the electron–ion kinetic energy distributions, it is convenient to plot the 1D ion yield as a function of the E_H+/E_D+ ion energy, as shown in figure 5. We select the 2–8 eV E₊ ion energy range where the most interesting dynamical features occur, for which a 4π collection of both (H⁺/D⁺, e) particles is achieved, and normalize the set of spectra in this energy region. The 1D histogram is obtained after selection of the (H⁺/D⁺, e) coincident events along the L₁ diagonal and projection onto the E_H+/D+ axis.

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Although the statistical quality of the two sets of measurements is quite different, with about 3000 events in each set for the H21 data, compared to 150 000 events for the SR data acquired in a comparable time, the comparison is very consistent. The two maxima at E_H/D+ ≈ 3.5 eV and E_H/D+ ≈ 6 eV in the I(E_H+) or and I(E_D+) spectra, with a relative importance specific of the H₂ or D₂ isotope, are well identified in the H21 fs experiment. Furthermore, the clear evolution of the ion yield spectra with photon energy at SOLEIL, in particular for the D₂ target, supports the excitation energy assigned to the H21 harmonic as ranging between 31.68 and 31.98 eV.

4.2. Spectral dependence of the angular distribution of the H⁺/D⁺ ion fragments

The second step of comparison between the outcome of the H21 induced DPI reactions and the set of reference measurements performed at SOLEIL is presented in figure 6. It displays the H⁺/D⁺ ion fragment angular distribution relative to the polarization axis in terms of the 1D β_H+/D+ ion asymmetry parameter dependence as a function of the ion kinetic energy, where −1 ≤ β ≤ 2. Here again, the evolution of the β_H+ and β_D+ asymmetry parameters for DPI induced by the H21 fs pulses is quite similar to that measured for an SR excitation energy hν ≈ 31.9 eV. The positive β values characterize the overall dominant role of the parallel transition at this excitation energy in the E_H+ ⩾ 3.5 eV range for H₂ and 2 eV ≤ E_D+ ⩽ 5.5 eV for D₂, involving primarily the resonant excitation and subsequent AI of the Q₁(¹Σ_u⁺) state, as discussed in section 5. The minimum observed for β_H+ and β_D+ around E_D+ ≈ 6 eV, more pronounced for D₂, indicates the contribution of the perpendicular transition assigned to AI of the Q₂(¹Π_u) state. The different evolution of β_H+ and β_D+ as a function of the KER for the H₂ and D₂ isotopes is consistent with the fact that AI of the Q₁(¹Σ_u⁺) state dominates at lower KER values for D₂ than for H₂. The significant spectral dependence of the β_H+/D+ curves, which decrease with increasing photon energy, illustrates the increasing contribution of resonant excitation and subsequent AI of the Q₂(¹Π_u) state when the photon excitation energy gets higher.

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Within the axial recoil approximation (Zare 1967) which applies for the studied fast dissociation of the H₂ /D₂ and H₂⁺ /D₂⁺ repulsive states, the angular analysis of the measured (V_H+, V_e) and (V_D+, V_e) vector correlations for a given selection of coincident events gives access to the MFPADs, for any orientation of the molecular axis with respect to the polarization axis. The generalized MFPAD $I(\chi ,\theta _{\mathop{\rm e}\nolimits} ,\phi _e )$ is a function of the polar angle χ which refers to the molecular orientation relative to the polarization axis and the electron emission direction (θ_e, ϕ_e) characterized by a polar and an azimuthal angles in the MF frame. The polar angle θ_e is relative to the molecular axis z_MF and ϕ_e accounts for the MF angular dependence relative to the polarization. The evolution of these observables as a function of the ion fragment energy, or KER, provides the most sensitive probe of the photoionization dynamics and its coupling with the nuclear motion. Using the formalism developed previously, gathering the dynamical information in terms of four F_LN(θ_e) functions, with L = (0, 2) and N = (0,1, 2) when PI is induced by linearly polarized light, is an efficient mean to describe MF photoemission (Lucchese et al 2002, Dowek and Lucchese 2012). The F_LN(θ_e) functions are partial-wave expanded as follows:

$$\begin{equation} F_{LN} (\theta _e ) = \sum\limits_{L'} {C_{L'LN} } P_{L'}^N \left( {\cos \theta _e } \right), \end{equation} \tag{ 1 }$$

where the $C_{L^\prime LN}$ coefficients express directly in terms of the photoionization dipole matrix elements (Lucchese et al 2002). In this study, the F_LN quantities are a function of both the electron emission polar angle with respect to the molecular axis and the E_H+/D+ (E₊) ion kinetic energy. The general expression of the MFPAD writes in the simple form:

$$\begin{eqnarray} &&\fl I(\chi ,\theta _e ,\phi _e ,E_ + ) = F_{00} (\theta _e ,E_ + ) + F_{20} (\theta _e ,E_ + )P_2^0 ( {\cos \chi }) \nonumber\\ &&+\, F_{21} (\theta _e ,E_ + )P_2^1 ( {\cos \chi })\cos (\phi _e ) \nonumber\\ &&+\, F_{22} (\theta _{\mathop{\rm e}\nolimits} ,E_ + )P_2^2 ( {\cos \chi })\cos (2\phi _e ) \end{eqnarray} \tag{ 2 }$$

where the χ and ϕ_e dependence is in terms of the four P_L^N(cosχ) associated Legendre polynomials and the simple cos(ϕ_e) and cos(2ϕ_e) functions, respectively. The F_LN functions are derived from a fit of the measured I(χ, θ_e, ϕ_e, E₊) angular distribution according to equation (1), using the whole statistics of the measurement in the KER region of interest. Here again we focus on the 2–8 eV E_H+/D+ energy range where the most interesting dynamical features occur. Once the F_LN functions are determined, the MFPADs I(χ, θ_e, ϕ_e, E₊) are obtained using equation (2) with the same statistical quality for any given orientation χ of the molecular axis. This method is therefore particularly appropriate for the present experimental conditions, when DPI is induced by the 1 kHz HHG light source.

For the data reported in this paper, where photoionization was induced by linearly polarized light P, we select MFPADs for the χ = 0° and χ = 90° molecular orientations relative to P, which represent the parallel and perpendicular components of the PI transition. The $I_{\chi = 0^\circ } (\theta _e ,E_ + )$ MFPAD has a cylindrical symmetry, and the $I_{\chi = 90^\circ } (\theta _e ,\phi _e ,E_ + )$ is characterized by a cos(2ϕ_e) azimuthal dependence. Together with the 3D MFPADs for the χ = 0° and χ = 90° molecular orientations, it is convenient to display the cuts of the two MFPADs in the plane ϕ_e = 0°/ϕ_e = 180° defined by the molecular and polarization axes, for a quantitative evaluation of the statistical quality of the data, according to

$$\begin{equation} I_{\chi = 0^\circ } (\theta _e ,E_ + ) = F_{00} (\theta _e ,E_ + ) + F_{20} (\theta _e ,E_ + ), \end{equation} \tag{ 3 }$$

and

$$\begin{eqnarray} &&\fl I_{\chi = 90^\circ } (\theta _e ,E_ + ) = I_{\chi = 90^\circ }^{\phi _e = 0^\circ - 180^\circ } (\theta _{\mathop{\rm e}\nolimits} ,E_ + ) \nonumber\\ &&\hphantom{\fl I_{\chi = 90^\circ } (\theta _e ,E_ + )}= F_{00} (\theta _e ,E_ + ) - \case{1}{2}F_{20} (\theta _e ,E_ + ) + 3F_{22} (\theta _e ,E_ + ).\nonumber\\ \end{eqnarray} \tag{ 4 }$$

5.1. F_LN functions and MFPADs for DPI of H₂ and D₂ for E₊ ≈ 5.5 eV ion kinetic energy

Figure 7 displays linear plots of the F₀₀(θ_e), F₂₀(θ_e), F₂₁(θ_e) and F₂₂(θ_e) functions measured for DPI of H₂ induced by H21 fs pulses, selecting coincident events corresponding to a central ion kinetic energy E_H+ = 5.5 eV (5 ≤ E_H+ ≤ 6 eV). We select this E₊ ion fragment energy as an example to illustrate the MFPADs, on the one hand because it is close to a maximum in the ion yield spectra for both targets, and on the other hand because the β_H+ and β_D+ asymmetry parameters display a minimum in this energy range, which corresponds to a rather balanced contribution of parallel and perpendicular transitions. The characteristic oscillatory behaviour of the F_LN(θ_e) functions displayed in figure 7 are well identified, with fingerprints quite similar, within the statistical error bars, to those measured for the same E_H+ energy selection after excitation by a photon of hν = 31.98 eV at the SR, also shown in figure 7. F₂₂(θ_e) is of positive sign for the studied processes corresponding to a PI transition from a Σ⁺ neutral state into an ionic state of same Σ⁺ symmetry (Lucchese et al 2002). We recall that, due to the spectral width of the H21 pulses mentioned earlier, the angular results in figure 7(a) correspond to the integration over a broader excitation energy bandwidth for the H21 fs source (≈350 meV) than for the SR even when broadened to a 30 meV bandwidth.

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The MFPADs derived from these F_LN functions using equations (2)–(4) for DPI of H₂ induced by H21 and SR are plotted in figures 8(a), (b) and (c), (d) for a molecule aligned parallel and perpendicular to the polarization axis, respectively, with the H⁺ emission velocity being oriented in the upper direction. The 3D MFPADs are presented for the H21 induced DPI in (a) and (c). The minor differences observed in the F_LN functions in figures 7(a) and (b) lead to some differences in the MFPADs for H21 fs pulses and SR excitation modes as plotted in figure 8; however, the general features are quite comparable. The corresponding MFPADs for DPI of D₂ are displayed in figures 9(a), (b) and (c), (d) for the same E_D+ ≈ 5.5 eV kinetic energy.

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For H₂, the MFPAD induced by H21 for a parallel transition (χ = 0°) (figure 8(a)) displays the typical pattern illustrating the dominant contribution of a scattering p-partial wave, with a notable backward–forward (BW–FW) asymmetry favouring electron emission in the direction of the emitted H⁺ ionic fragment. Such anisotropies previously observed at neighbour photon excitation energies (Lafosse et al 2003) are assigned to quantum interferences between undistinguishable reaction pathways, here between direct ionization through the H₂⁺(²Σ_u⁺) ionic state and AI from the Q₁ (¹Σ_u⁺) state through the H₂⁺(²Σ_g⁺) ground state (Fernàndez and Martìn 2009). These emission anisotropies are known to vary significantly as a function of the KER of the atomic fragments and photon energy. For a perpendicular transition (χ = 90°) (figure 8(c)), the MFPAD is characterized by the dominant contribution of a d-partial wave and also displays a strong emission asymmetry as identified previously (Martín et al 2007). In this case, it is due to the interference between two channels of H₂⁺(²Σ_g⁺) and H₂⁺(²Σ_u⁺) ionic core, respectively, resulting from AI of the Q₂ (¹Π_u) state (Martín et al 2007, Fernàndez and Martín 2009). Consistent with the similarity between the F_LN functions measured for DPI induced by H21 and SR shown in figure 7, the MFPADs for the (χ = 0°) and (χ = 90°) at an SR photon energy hν = 31.98 eV reported in figures 8(b) and (d) display the similar features as those induced by H21.

Considering the MFPADs for DPI of D₂ for E_D+ = 5.5 eV in figure 9, the main difference with the H₂ target is observed for the parallel transition where the measured distribution displays an inverse BW–FW emission asymmetry, favouring now the emission of the photoelectron in the direction opposite to that of the D⁺ ion fragment. It is notable that the measured emission anisotropies discussed in figures 8 and 9 appear larger for the reported H21 data than for those obtained using SR, although the histogramming integration bands are broader (1 eV) for H21 than for SR (0.5 eV). These effects possibly related to the spectral broadband of harmonic light could be explored in further detail, e.g., using a 'narrow' polarization gate to significantly vary the pulse duration and its spectral width. As mentioned in section 2.2, the operation of a polarization gate to control the H21 spectral profile has not been fully conclusive in our conditions, and will be repeated in a further campaign; we therefore do not discuss extensively the possible effects of the XUV spectral width on MFPAD in the present results.

Due to the low statistics for DPI of H₂ and D₂ induced by the 1 kHz H21 light source, we discuss in the following section an appropriate method to visualize the main features of the MFPADs evolution with the ion fragment kinetic energy, as well as their comparison with the results of the spectrally resolved investigation performed at SOLEIL.

5.2. Evolution of the MFPADs as a function of the ion kinetic energy using Legendre polynomial expansion

An efficient way to synthesize the evolution of the MFPADs as a function of the E₊ ion kinetic energy is to take advantage of the partial-wave expansion of the F_LN(θ_e) functions as described in equation (1), and plot the evolution of the C_L'LN dynamical coefficients. Consistent with the presentation of the MFPADs for the χ = 0° and χ = 90° molecular orientations in section 4.1, we present here the coefficients of the Legendre polynomial expansion of the I_{χ = 0°} (θ_e) and I_{χ = 90°} (θ_e) MFPADs. For simplicity, I_{χ = 90°} (θ_e, E₊) is reduced in this section to its main contribution:

$$\begin{equation} I_{\chi = 90^\circ }^{} (\theta _e ,E_ + ) = F_{00} (\theta _e ,E_ + ) - \case{1}{2}F_{20} (\theta _e ,E_ + ) \end{equation} \tag{ 5 }$$

which corresponds to the expression of $I_{\chi = 90^\circ } (\theta _e ,\phi _e ,{\rm E}_ + )$ averaged over ϕ_e.

Under such conditions, both I_{χ = 0°} (θ_e) and I_{χ = 90°} (θ_e) expand in terms of $P_{L^\prime }^0 \left( {\cos \theta _e } \right)$ Legendre polynomials involving then a set of D_L' single index dynamical parameters, according to

$$\begin{equation} I_{x = 0^\circ /x = 90^\circ } (\theta _{\mathop{\rm e}\nolimits} ,E_ + ) = \sum\limits_{L' = 0}^{2l_{\max } } {D_{L'(0^\circ /90^\circ )} (E_ + )P_{L'} [\cos (\theta _{\mathop{\rm e}\nolimits} )]} , \end{equation} \tag{ 6 }$$

where l_max is the maximum angular momentum considered in the partial wave expansion. Here l_max = 3 corresponds to the inclusion of up to f partial waves. The evolution of the D_L' coefficients, where L' is even, characterizes the outcome of the dominant partial waves as a function of the KER, whereas those with odd L' values, of smaller amplitude, describe the BW–FW emission anisotropies. In this paper, we focus on the presentation of the D_L' coefficients, for even L' values, which are the most robust in particular for DPI induced by the H21 fs pulses. BW–FW asymmetries derived for the DPI processes induced by the SR will be discussed separately. Figure 10 displays the evolution of the D_L'(0°) and D_L'(90°) coefficients, with L' = 0, 2,4,6, for DPI of H₂ (a), (b) and D₂ (c), (d) recorded at SOLEIL at a photon excitation energy hν = 31.98 eV. As discussed above, the isotope effect when moving from H₂ to D₂ is most clearly observed for the parallel transition, where the maximum of the D_0(0°) cross section is reached for a higher E₊ kinetic energy for H₂ (E_H+ ≈ 6 eV) than for D₂ (E_D+ ≈ 3.5 eV), with a dominant contribution of a pσ partial wave in both cases. The D_0(90°) cross sections are quite similar for H₂ and D₂: they both display two maxima, the strongest around E₊ ≈ 6 eV, with a large contribution of a dπ partial wave reflected here by a negative D_4(90°) parameter.

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These characteristic features are quite well identified at the same level of the analysis for the DPI reactions induced by H21 fs pulses. This is illustrated in figure 11 for the major D_L'(0°) and D_L'(90°) coefficients, where they are compared with those measured at the closest photon excitation energy hν = 31.98 eV at SOLEIL.

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We have presented a comparative study, in a VC experiment, of DPI in H₂ and D₂ after single photon excitation in the Q₁ and Q₂ DES region by XUV pulses around 32 eV, provided either by a femtosecond HHG source at a 1 kHz repetition rate, or by synchrotron radiation at SOLEIL (ps pulses at MHz repetition rate) which mainly serves as a reference. In the two experimental configurations, we measure a series of laboratory frame and molecular frame observables, such as the kinetic energy distribution of the ion/electron, asymmetry parameters, the molecular frame PADs which are analysed in terms of the F_LN(θ_e) functions and Legendre polynomials, and their evolution with ion kinetic energy. Despite a significantly lower statistics, detailed consistency between HHG and SR results demonstrates the feasibility of MFPAD determination in DPI using the femtosecond XUV harmonic source.

This study has clear objective to establish MFPAD as a highly sensitive and reliable photoionization-based probe of electronic/nuclear dynamics induced by a combination of as to fs XUV and laser pulses (Kelkensberg et al 2011, Tzallas et al 2011, Wörner et al 2010). These multiple pulse studies can still be referred to as pump/probe, although the complex sequence of coherent excitation in various ionization/fragmentation channels somewhat mixes the pump/probe distinction. It is the strength of time-resolved VC and MFPAD to reveal and monitor the details of this rich dynamics, with an fs to sub-fs time resolution. While VC in ground state molecules already requires long acquisition time for sufficient statistics at kHz repetition rate, it is even more demanding in multiple pulse experiment. This will be notably and straightforwardly improved using now available high repetition rate lasers–-tens of kHz to MHz-–delivering up to mJ femtosecond pulses to drive HHG (Boullet et al 2009, Rothhardt et al 2012, Heyl et al 2012), as well as using long-term actively stabilized optical arrangement for pulse superposition (Böttcher et al 2008).

Other improvements of VC studies in the XUV/fs range will come from impulsive laser alignment of the target molecules in the e⁻/ion spectrometer, so that non-dissociative PI can be explored (Thomann et al 2008). Moreover, circularly polarized XUV pulses (Vodungbo et al 2011) of 10 fs duration with enough photons per pulse to pump the molecular system, combined with ∼1 fs probe will allow studying the time evolution of circular dichroism in H₂ (Dowek et al 2010) and D₂.

Among the several cases where MFPAD constitutes a unique probe observable, it is worth noting that it can support a tomographic reconstruction of molecular orbitals from molecular dipole measurements over a large continuum space (Santra 2006). Actually, it will be of high interest in future works to perform and compare time-resolved imaging of molecular orbitals by the two routes which have been proposed, on the one hand, the HHG-based 'self-probing' which is still debated (Itatani et al 2004, Haessler et al 2010, Vozzi et al 2011), on the other hand, the PI-based 'external probing' which has still to be demonstrated. Finally, either on HHG or at FEL sources, XUV/XUV pump–probe experiment where VC are measured will soon open a completely new field on nonlinear processes in the extreme-UV range.

We gratefully acknowledge fruitful discussions with P Salières, and A Palacios, H Bachau, F Martin and coworkers. We are very grateful to JC Houver for the development of data analysis software for this research program. We acknowledge the SPAM Laser team members for their help in operating the PLFA laser and M Billon for his technical assistance for the femtosecond experiment. We are very grateful to L Nahon responsible for DESIRS beamline, G Garcia coordinator of SAPHIRS setup and their co-workers for their cooperation. We acknowledge the staff of synchrotron SOLEIL for operating the machine. We are grateful to E Bouisset, S Damoy, C Lacan from ISMO for their technical support. We acknowledge the support of C Elkharrat during the PLFA experiments, and the contribution of M Morand in a studentship training period. This work was supported by the Réseau Thématique de Recherche Avancée 'Triangle de la Physique' under contract «Dynelec» number 2008-046T, the Agence Nationale pour la Recherche under contract «AttoWave» number ANR-09-BLAN-0031-01, the LASERLAB-II transnational access network through ALADIN-JRA under contract RII3-CT-2003-506350. PB acknowledges the Centre National de la Recherche Scientifique for a delegation position.