Two-color XUV+NIR femtosecond photoionization of neon in the near-threshold region

The measurement was performed at the CAMP endstation [20] at the beamline BL1 of the FEL FLASH at DESY [10]. The FEL was operated at a mean photon energy of 25.2 eV (49.2 nm) with an inherent bandwidth of 0.3–0.4 eV (FWHM). The XUV pulses with an average pulse energy of ∼4 μJ (at the experiment) and a pulse duration of ∼100 fs (FWHM) were focused to a focal spot with a diameter of <50 μm. It is important to note that FLASH is operated in self-amplified spontaneous emission (SASE) mode [10]. Therefore, initial stochastic fluctuations in the electron density lead to a spiky intensity distribution of the XUV pulse rather than a well-defined Gaussian pulse. Moreover, this substructure changes substantially from pulse to pulse, as illustrated by simulated [21] SASE FEL pulses shown in figure 1. For an XUV photon energy of about 25 eV each spike has a duration of about 15–20 fs (FWHM) [22–24] leading to 4–5 spikes within the 100 fs pulse. Invoking the Fourier transform relation, the same spiky structure is present in the XUV spectral distribution, which can be measured more easily, in contrast to the temporal shape. Spectra acquired with the FLASH high-resolution spectrometer [25] showed indeed 4–5 peaks on average.

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The FEL was operated at a (single) pulse repetition rate of 10 Hz to overlap each of the XUV pulses with the NIR dressing field produced by the Ti:sapphire based pump–probe laser system at FLASH1 [26]. The 10 Hz laser system provides 70 fs (FWHM) laser pulses centered at a wavelength of 810 nm which were focused to a spot of ∼100 μm diameter. The pulse energy was ∼100 μJ to get peak intensities of ∼1 × 10¹³ W cm⁻². The relative jitter between optical laser and FEL pulses was on the order of 100 fs FWHM, thus in the range of the pulse durations. In addition spatial jitter and drifts changed the intensity distributions during the measurement. Both, the FEL and NIR laser, were linearly and horizontally polarized.

The XUV and NIR pulses were overlapped spatially and temporally in the center of the CAMP endstation [20] interacting with a collimated neon gas jet (5 mm diameter) which had a density of ∼10⁹ atoms cm⁻³. The photoelectrons from the process Ne 2p⁶ $\to$ Ne⁺2p⁵ + e⁻ with a kinetic energy of 3.6 eV (Ne 2p binding energy: 21.6 eV) were recorded using the long (upper) side of the double-sided flat-electrode velocity map imaging (VMI) spectrometer in the CAMP endstation, equipped with an MCP/phosphor screen and a CCD camera [20]. The alignment procedures for optimizing the VMI as well as finding spatial and temporal overlap have been described in [27, 28]. The VMI collects the complete 4π solid angle of emitted electrons, measuring the projected electron angular distribution in the FEL polarization plane [29]. The extraction voltages (∼220 V cm⁻¹) were set to cover the kinetic energy range up to 15 eV. The VMI images were recorded at 10 Hz.

Due to the spatial and temporal jitter, the recorded data fluctuated from no sidebands (no overlap) to maximum number of sidebands (best overlap). Thus, the 'effective' intensity was altered for each shot. To get sufficient statistics, more than 200 000 images were recorded in total and used for the analysis. The images were sorted according to the intensity ratio between the sidebands and the main photo line which are clearly separated in the raw images (see figure 2). Five effective intensity bins were chosen, covering the whole range between 'no sidebands' and 'maximum number of sidebands'. To reconstruct the slices of the 3D photoelectron angular distributions out of the 2D-VMI images, different methods of the inverse Abel transformation have been applied. The BASEX method [30], direct integration of the Abel integral [30] and an iterative approach [31] led to identical results. The result for bins 2–5 is shown in figure 2.

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In order to ensure that the experimental system was functioning correctly and to verify the data analysis codes, the well-known angular distribution for the Ne 2p photoelectron emission at ℏω ∼ 25 eV without the NIR field was analyzed. The angular distribution anisotropy parameter (β₂) was determined to be −0.25 ± 0.06 which is in good agreement with the literature values of −0.23 ± 0.05 [32]. Without the NIR laser field, the Ne 2p photoelectron line as well as other reference measurements using Kr as the target gas, leading to emission lines in the range from 1 to 11 eV [33], showed no sign of tilting as a function of emission angle. Thus, we can rule out that the measured angular dependencies result from inhomogeneous fields in the VMI spectrometer.

The ionization of an atom is described within the single active electron approximation, i.e. we assume that the orbitals of all atomic electrons except the active one are frozen. This approximation is valid for moderately strong laser fields, much less than 1 a.u.; i.e. laser intensity I ≪ 10¹⁶ W cm⁻², when the polarization of the core orbitals by the NIR field can be ignored. We suppose that both NIR and XUV pulses are linearly polarized and their polarization vectors are parallel. Then the ionization process is axially symmetrical with respect to the direction of polarizations which we assign as z-axis.

The motion of the active electron is described by the TDSE where the interaction of the electron with the electric fields of the two pulses is taken in the dipole approximation. The interaction of the electron with the atomic core is described by the time-independent Hartree–Slater potential [34]. The big difference in the magnitude of the XUV and the NIR fields and in the frequencies of the pulses allows one to use the first order perturbation treatment and the rotating wave approximation for the description of ionization by the XUV field. The influence of the strong NIR field is fully taken into account by the TDSE solution. By expanding the active electron wave function in spherical harmonics one gets a system of equations for partial wave functions. (For details of derivation see [13].) Note that due to the axial symmetry of the problem the projection of orbital angular momentum of the electron is conserved, not mixed by the fields, thus the calculations are performed for each projection separately.

The system of partial equations has been solved numerically. The time evolution of the partial wave functions was calculated with the split-propagation technique using the Crank–Nicolson propagator [35]. (For details see [12].) The partial amplitudes of photoionization are calculated by projecting the partial functions onto the corresponding continuum functions, eigenfunctions of the ionic Hamiltonian: (for further details see [12]). The calculated complex amplitudes determine the double differential cross section (DDCS). Examples of calculated DDCS are presented in figure 3.

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FLASH is operated in SASE mode, leading to radiation pulses consisting of a sequence of coherent spikes showing a similar temporal duration [10, 22]. In the present case, the pulse duration was on average about 100 fs (FWHM) consisting of several (on average ∼5) stochastically distributed 15–20 fs (FWHM) spikes. The spike distribution changes for each single SASE pulse. Since TDSE calculations are rather time consuming, the unpredictable nature of the SASE process prohibits extensive calculations using realistic SASE pulses. The DDCS calculated for short XUV pulses (as explained in the above section) were used as the basis to simulate the experimental results. The number of created electrons is proportional to the FEL intensity (i.e. number of XUV photons). Therefore one can add up the DDCSs calculated by solving the TDSE for defined NIR intensities and weigh them with the FEL intensity present at the corresponding NIR intensity. Since we are interested in the average DDSC for the FEL SASE case, the procedure can be simplified. Averaging the spiky single FEL pulses sufficiently (several hundreds) the resulting shape approaches a Gaussian FEL intensity distribution (see figure 1). Thus the adding and weighing DDSCs can be done by using the Gaussian XUV FEL average pulse duration.

In order to simulate the photoelectron angular distribution for all five intensity bins, we used the delay between XUV and NIR pulses. The main reason for the shot-to-shot fluctuations in the experiment (besides the changing pulse structure) is the relative timing jitter between XUV and NIR pulses. Thus, we can simulate the experimental data by introducing a delay in the averaging procedure which is 0 fs for the bin representing the highest NIR intensity (bin 5) and is larger for the other bins (e.g. 60 fs for bin 3) as is sketched in figure 1.

The actual delay values for each bin can be determined from the information of the 'effective' NIR intensity in the bin. Fortunately, the binned experimental spectra contain sufficient information to determine an 'effective' NIR intensity present during the interaction (I_eff) by using the ponderomotive shift of the main photoelectron emission line. As e.g. shown in [1, 6] the NIR field increases the ionization potential by the dynamical Stark effect, leading to a reduction of the kinetic energy of the photoelectrons by ${U}_{p}={I}_{L}/4{\omega }_{L}^{2}$ (with the NIR intensity I_L and NIR frequency ω_L).

The energetic shift of the main line can be used to determine the averaged NIR intensities acting during the interaction, thus defining an 'effective' laser intensity for each bin. One has to note that the FEL spikes are significantly shorter as compared to the NIR pulse, thus during each single FEL pulse the created electrons are interacting essentially with whole range of NIR intensities (compare to figure 1).

For the highest intensity bin we get a shift of the angularly integrated main line of 0.24 eV due to the ponderomotive potential corresponding to an NIR intensity of ${I}_{{\rm{eff}}}\sim 4\times {10}^{12}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$, see figure 2. This is, as expected, lower than the actual peak intensity of the NIR laser field due to the averaging of the laser intensities.

As first step, TDSE calculations have been performed with well-defined, fully coherent XUV and distinctly longer NIR pulses (scenario A). This way we can study the interaction of a single FEL spike theoretically and determine the influence of the NIR intensity. In a second step, these single-spike photoelectron emission distributions are used to simulate the FEL pulse by averaging over several fluctuating XUV spikes (scenario B).

The parameters were chosen reasonably close to the experimental condition. The XUV photon energy was set equal to the experimental one, i.e. 25.2 eV, which yields a Ne(2p) photoelectron of 3.6 eV kinetic energy. The pulse duration was 10 fs (FWHM), which is slightly shorter as the expected SASE single spike but numerically less elaborate. Under the given conditions, calculations showed that the calculated DDCS is almost identical if the XUV pulse length and the NIR pulse length are set 1.5 times longer. The laser pulse was sufficiently longer (factor 3) than the XUV pulse to simulate the interaction with a rather well-defined NIR intensity, close to the peak intensity. The calculations were performed for an NIR intensity range from 5 × 10¹¹ to 1 × 10¹³ W cm⁻². Here, we discuss the results of TDSE calculations for some representative peak intensities of 2.0, 4.0 and 6.0 × 10¹² W cm⁻² as shown in figure 3 for scenario A.

In the case of near-threshold two-color photoionization two new features are evident. First, at sufficiently large intensity (figures 3(b) and (c)) the sidebands show clearly an angular dependence: the energy of the sideband is smaller at 0° than at 90°. The effect increases with increasing intensity. Interestingly, the sideband position at 90° (perpendicular to the NIR polarization) does practically not change for different intensities (see blue dashed line in figure 3), while at lower angles (close to the polarization direction) the kinetic energy shifts to smaller values. Note that this angular dependence is opposite to that observed in the experiment. The seeming contradiction is resolved when the averaging due to the SASE spikes is accounted for (see discussion in section 4.2 and figure 4 for scenario B).

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Presumably, the tilting is related to the rescattering of slow electrons from the ionic core. The slow electrons driven by the strong NIR field can revisit the residual ion and after scattering they can interfere with the directly emitted ones. The interference of the rescattered wave packet, modified by the time-dependent NIR field and the emitted wave packet, can strongly modify both, the angular distribution and the energy spectrum. As shown in [36, 37] for photoionization by attosecond XUV pulses, rescattering drastically changes the electron spectra in forward and backward directions with respect to the NIR field polarization. At the same time, in the perpendicular direction the influence of rescattering is practically negligible. For fast photoelectrons this effect is negligible and the position of the sidebands is independent of the emission angle. Note that in case of ATI, the dependence of the ATI peak position on the emission angle, has been observed (e.g. [14, 15]) but has not been discussed in detail.

The second interesting feature of the sidebands in the near-threshold region is 'filamentation', that is splitting of the sidebands. Indeed, at large peak intensities (above 3 × 10¹² W cm⁻²) the sidebands split in two or sometime three lines (see figure 3(c)). In our calculations the splitting can be seen not only in sidebands, but also in the main line. Analogous splitting of ATI peaks is a well-known fact confirmed both experimentally [15–17] and theoretically [17–19, 38]. The first observation of a narrow fine structure of the individual ATI peaks was reported by Freeman et al [16] for subpicosecond laser pulses. It was interpreted as a resonant enhancement in the ionization process produced by ponderomotive shifts of states. An alternative explanation of the origin of the ATI peak fine structure was suggested by Eberly et al [39, 40]. In their model calculations the substructure of the ATI peaks appears because of multiphoton ionization of weakly transiently excited atomic states which are dipole-connected to the ground state.

A similar interpretation can be applied in our case of near-threshold two-color ionization. The wave packet of a slow electron created by the XUV pulse, is driven by the NIR electric field. It can be diffracted by the residual ion, thus transiently exciting Rydberg states which can lead to the fine structure of the sidebands. Such complicated dynamics which involves also interference of the undistorted wave packet and the rescattered one was analyzed by solving TDSE in [12, 36, 37] for near-threshold ionization by attosecond XUV pulses in the presence of NIR laser field. In the present calculations, the solution of the TDSE for femtosecond pulses includes the above mentioned phenomena and results in the sideband fine structure as displayed in figure 3. For the present experimental situation, the averaging over a large range of intensities washes out the filamentation effect (as seen in figure 4).

Using the averaging of the calculated DDCS, as described in section 3.2, the angularly resolved photoelectron spectra can be simulated and compared to the measurements. In figure 4 the angularly resolved photoelectron spectra are shown for two different effective NIR intensities ((a), (b) ${I}_{{\rm{eff}}}\sim 4\times {10}^{12}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ (bin5 in figure 2) and (c), (d) ${I}_{{\rm{eff}}}\sim 2\times {10}^{12}\,{\rm{W}}\,{\mathrm{cm}}^{-2}$ (bin3 in figure 2)). Different features can be observed and will be discussed below.

In the experiment, the first sideband at the low kinetic energy side (−1st) has a rather constant intensity distribution as a function of the emission angle. While on the higher energy side as the order of the sidebands increases, one can see from figure 4 that the angular distribution becomes more concentrated around 0°. The higher order sidebands (3rd and higher) show essentially no contribution for emission angles exceeding 45°. These features have already been observed for photoelectron and Auger sidebands at larger electron energies [2, 3, 5] and can be easily explained within simple models of sideband formation. Considering long NIR pulses, the intensity of the sideband of the order n is proportional to ${J}_{n}^{2}({A}_{L}k\cos \theta /{\omega }_{L})$ [8, 11] where J_n is the Bessel function, A_L is the amplitude of the vector potential of the NIR field, ω_L is its frequency, and $k=\sqrt{E}$ is the electron wave number. In our case, the parameter $\alpha ={A}_{L}k/{\omega }_{L}$ is small. For small arguments (${A}_{L}k\cos \theta /{\omega }_{L}\ll 1$) the Bessel function is proportional to ${\left(\alpha \cos \theta \right)}^{n}$, thus the intensity of the nth sideband is proportional to $\cos {\theta }^{2n}$. Therefore, with increasing n the angular distribution is more and more concentrated around θ = 0.

A much more interesting observation is the dependence of the electron kinetic energies of the sidebands (on the higher energetic side of the main line) on the emission angle. Looking at the experimentally measured spectra in figures 4(b), (d), one notes, that the mean kinetic energy of the main line and the higher energetic sidebands, predominantly the +1st sideband, are shifted to lower kinetic energies at higher angles.

To compare the simulations based on SASE pulses and the experimental results in more detail, the insets in figure 4 show the enlarged +1st sideband with a (green) line indicating the observed tilt. The green line represents a slope of −0.75 eV between 0° and 90° for the experimental case and −0.35 eV for the simulation. Though the SASE simulation is based on a very crude and simplified model, the main features observed in the experiment can be reproduced. The simulation shows the tilt in the same direction as the experiment, however, the effect is less pronounced in the simulation, which can be attributed to the insufficient modeling of the (unknown) SASE fluctuations. As pointed out before, the shift in the simulations using short XUV pulses at a well-defined NIR intensity (figure 3) shows a tilt in the opposite direction.

How does the averaging change the sign of the tilt and influence electrons differently at small and large angles? A closer look to the photoelectron spectral distribution for the short XUV pulses interacting with well-defined NIR intensities (figure 3) can explain the observations. The main fraction of the electrons in one sideband (particularly pronounced in the +1st one) is emitted within a rather small angular region. This effect becomes more pronounced for higher NIR intensities (compare figures 3(b) and (c)). With increasing intensity this emission maximum of the sidebands shifts to larger angles (closer to 90°) and to lower kinetic energies. Figure 5 shows the emission angle and peak kinetic energy of the emission maximum of the first sideband from the TDSE calculations as a function of NIR intensity. Thus, summing all NIR intensity contributions according to our SASE model (figure 4), we indeed get the 'upwards' tilting of the sidebands which can qualitatively explain the experimental finding.

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This example emphasizes the importance of a detailed modeling of the process in particular when pulses containing a complex substructure, as in the case of SASE, are involved. Thus, averaging over pulse structure fluctuations can alter the behavior of certain effects completely. To finally reveal the sideband features near threshold, i.e. tilting and filamentation, predicted by theory, experiments using high harmonic generation or seeded FELs will be required.