Nonlinear XUV signal generation probed by transient grating spectroscopy with attosecond pulses

Nonlinear spectroscopies are utilized extensively for selective measurements of chemical dynamics in the optical, infrared, and radio-frequency regimes. The development of these techniques for extreme ultraviolet (XUV) light sources facilitates measurements of electronic dynamics on attosecond timescales. Here, we elucidate the temporal dynamics of nonlinear signal generation by utilizing a transient grating scheme with a subfemtosecond XUV pulse train and two few-cycle near-infrared pulses in atomic helium. Simultaneous detection of multiple diffraction orders reveals delays of ≥1.5 fs in higher-order XUV signal generation, which are reproduced theoretically by solving the coupled Maxwell–Schrödinger equations and with a phase grating model. The delays result in measurable order-dependent differences in the energies of transient light induced states. As nonlinear methods are extended into the attosecond regime, the observed higher-order signal generation delays will significantly impact and aid temporal and spectral measurements of dynamic processes.


Supplementary Note 1: Pulse Overlap Determination
The experimental apparatus is described in detail in the Methods and in ref. 1

. A schematic
showing the generation of XUV light and the implementation of the time delay between the XUV and NIR pulses is provided here as Supplementary Fig. 1. To ensure synchronization of the NIR pulses, we arranged the two piezo-stages in the interferometer in series. The first stage is positioned before the NIR arm is split into two by the second beamsplitter in order to control the delay of both the upper and lower beams simultaneously. Fine adjustments to the second stage in the beam path transmitted through the beamsplitter compensate for daily differences in alignment of the two NIR arms.
A commercial dispersion scan device (d-scan, Sphere Ultrafast Photonics) is utilized to measure the NIR spectrum and optimize its pulse duration before entering the vacuum chamber. The results of one such measurement are given in Supplementary Fig. 2. The NIR bandwidth extends from 550 -950 nm, which supports a near-transform limited pulse duration of 5 fs. While the duration of the pulses in the vacuum chamber will be somewhat lengthened compared to those measured with the d-scan because they do not propagate through equivalent amounts of air, experimental risetimes indicate pulse durations of approximately 6 fs. 3

Supplementary Fig. 1 | Experimental nonlinear attosecond spectroscopy apparatus
Three independently controlled beam paths allow for spatial isolation of signals that originate from different interaction pathways. BS = beamsplitter, f = focal length, Sn = tin, Al = aluminum.
The quality of spatial and temporal overlap between the three pulses is assessed using a CMOS camera (DCC1545M, Thorlabs) as shown in Supplementary Fig. 3. In order to obtain an observable signal from the XUV arm on the camera, the Sn filter is removed to allow the driving NIR pulses to propagate with the XUV. A removable pick-off mirror inserted into the beam path after the annular mirror directs all three NIR beams to a camera placed at the focus outside of vacuum. Because of the arrangement of the two stages, overlap is first found between the HHG arm and the upper NIR beam. After positioning the focal spots of the two noncollinear beams on top of each other (spatial overlap), the first delay stage is moved manually until fringes appear on the overlapping beam profiles ( Supplementary Fig. 3a). The fringes indicate that the lengths of the two beam paths are equal and thus the pulses are overlapped in time. Overlap between the HHG arm and the lower NIR beam is then established by modifying the position of the second stage. In addition to indicating time overlap, the fringes reveal the crossing angle, θ crossing , between the XUV and each of the NIR pulses. Using the equation: where λ NIR is the central NIR wavelength and x is the fringe spacing, the crossing angles are determined to be 1.0º and 0.75º for the upper and lower arms respectively.

Supplementary Fig. 3 | Evaluation of Spatial and Temporal Overlap
a, Overlap of the upper arm and the NIR light used for HHG results in fringes in the beams' spatial profile. The crossing angle is determined to be 1.0º. b, The fringe spacing at overlap of the HHG and lower arms is larger than that in (a), leading to a crossing angle of 0.75º.

Supplementary Note 2: Zeroth Grating Order Signals
The zeroth grating order in these noncollinear wave-mixing experiments essentially constitutes a transient absorption measurement. In Supplementary Fig. 4a, absorption features from the 2p state and two LISs are observed. As discussed in the helium transient absorption literature 2,3 , the exceptionally large cross section of the 1s2p resonance allows for significant resonant pulse propagation effects in a collinear geometry. Propagation effects are dependent upon the susceptibility and dephasing time of a transition, the frequency of light used for the excitation, and target gas density and propagation length. In brief, the XUV-induced oscillating dipole generates an electric field out of phase with the driving field. As this dipole-induced electric field propagates through the medium, it too will excite atomic resonances, which in turn produce radiation out of phase with the initial dipole-induced electric field. This process results in the formation of secondary pulses with durations dependent on the gas density and propagation 6 length. Meanwhile, the NIR pulse imposes a phase shift on the XUV-induced electric field so that it is no longer completely out of phase with driving field. When the timescale of the first subpulse is comparable to that of the IR perturbation, the interplay of these two effects results in additional spectral features, like those observed in the zeroth order of the 2p state. The lack of additional features in the higher lying np states can be attributed to their weaker response to the field. 8

Supplementary Note 3: Identification of Light Induced States
To assign the broad emission features between 21.6 and 22.2 eV to the appropriate LIS, we employed a joint experimental-theoretical approach.

Polarization Measurements
Dipole selection rules dictate that while both ns± and nd± LIS features appear when the polarization of the XUV and NIR pulses are parallel (θ = 0°), only the nd± LIS features remain in a perpendicular polarization (θ = 90°) 3 . We introduced a half-wave plate into the noncollinear NIR arm to modify its polarization relative to XUV in order to determine whether the LIS features observed in higher grating orders are associated with an ns or an nd dark state. As shown in Supplementary Fig. 5, the primary nonlinear LIS feature can be observed in both the perpendicular and parallel polarization geometries. Note that the presence of the wave plate increases the pulse duration and decreases the intensity of the signals, making higher order signals more difficult to achieve. To quantify the maintenance of the m = -1 LIS feature between both polarizations, the ratio between the baseline-subtracted first order LIS (nl-) and zeroth order 2s+ LIS can be compared. In the parallel polarization, both LISs are present and the nl-/2s+ ratio is 0.34. In the perpendicular polarization, the first order nl-LIS is maintained but the zeroth order 2s+ LIS disappears, resulting in an increase of the nl-/2s-+ ratio by a factor of 14.8 to 5.03.
As the feature between 21.6 and 22.2 eV is maintained in both polarizations, selection rules dictate that it must be assigned to an nd± LIS. Of the nd± LISs accessible in this experiment, the expected energy of the 3d-LIS (21.6 eV) best correlates with the observed feature, especially given that a shift to higher energies at later delays is expected due to the AC Stark shift.

Calculated Spatio-Spectral Profiles
This assignment can be verified further theoretically. As described in the methods section, the coupled time dependent Schrödinger equation (TDSE) and the Maxwell wave equation are solved numerically in the single active electron (SAE) approximation to generate spatio-spectral profiles in the far field. The effect of resonant pulse propagation effects in these wave-mixing experiments are therefore not considered here, but will be the focus of future studies. Looking specifically at the energy landscape surrounding the 2p resonance, multiple broad features distinct from the np states appear prominently in the calculated profile ( Supplementary Fig. 6).
Note that the calculations shown below have been done using slightly longer NIR pulse durations (12 fs) in order to improve the spectral resolution and more easily identify different features.
Changing the central wavelength of the NIR pulses employed in the calculation shifts the position of the broad features, as one would expect for LISs. Assuming a LIS picture, the features will shift toward their parent dark state with increasing wavelength. The feature that best corresponds with the most prominent experimental LIS shifts toward higher energies with increasing wavelength, indicating that its associated dark state is located higher in energy than the feature. Furthermore, removal of the nd dark states from the calculation substantially diminishes the intensity of this LIS feature, indicating that it must be associated with a nd dark state located ~1.5 eV away ( Supplementary Fig. 7). The most likely dark state candidate is the 3d state. Therefore, these results support the assertion that these features originate from the 3d-LIS.

Supplementary Note 4: Wave-Mixing Pathways
In a holographic picture, the generation of nonlinear signals result from diffraction of an input beam off of a sinusoidal grating. Deviations from a perfect sinusoid produce higher order nonlinear signals 6 . In a complementary picture, spatially-isolated nonlinear signals can be intuitively described by wavevector phase-matching requirements intrinsic to the perturbative interaction of one XUV photon and even numbers of noncollinear NIR photons 7 . These interactions generate a macroscopic polarization with significant contributions from higher-order terms: where ε 0 is the vacuum permittivity, χ (n) is the ℎ-order susceptibility, and is the electric field of the excitation. In an isotropic medium, these nonlinear terms result in emission from processes dependent upon odd orders of the nonlinear susceptibility. Myriad different pathways can emit at a given energy due to the broad bandwidth of the ultrashort pulses utilized in this experiment.
The broadband XUV pulse generates a coherent superposition of multiple np excited states, which is then probed by the time-coincident NIR pulses. Beating in the time dependent signal occurs when two or more NIR photons couple multiple states excited in the initial interaction to the same final state. If both of the photons originate from same NIR beam, then the resulting wave-mixing signal is produced via a ladder-type coupling pathway in which the NIR photons couple the two states through a dark state positioned between them in energy ( Supplementary   Fig. 8a). Signals resulting from this type of wave mixing process are discussed extensively in ref.  Supplementary Fig. 8b).
Significant evidence of the coupling of multiple states to the same feature can be observed in the data presented here. The 1.5 fs oscillations in the time dependence of the 4p state shown below in Supplementary Fig. 11a can be tied to the interference of a V-type pathway in which the 4p couples back to itself and a ladder type pathway involving the 2p state ( Supplementary Fig. 8c).
Furthermore, as discussed further in the following section, the persistence of the 3d-LIS features after time overlap in the first, second, and third grating orders can be explained by a 3d dark state-mediated Λ-type coupling of features associated with the strong 2p state to the LIS.

Dark State Population
An alternative explanation is that the LIS itself actually persists after overlap due to the accumulation of population in the 3d dark state. In ref. 11

Wave-Mixing Coupling Mechanisms
Although  Fig. 10b). The spacing of oscillations observed in the delay dependence resembles the strong hyperbolic side bands due the NIR-induced truncation of the 2p dipole observed on-axis 5 , perhaps indicating that these features can take part in wave-mixing processes if the cross section of their parent state is above some threshold value.

Pulse Propagation Effects
The propagation of a pulse through a dense medium with strong resonances has already been discussed within the context of the 2p state. However, as shown in ref. 16, the effects of an optically thick resonant medium on pulse propagation need not be confined to the 2p state. As the pulse propagates through the medium, the helium gas will attenuate resonant frequencies and shape the spectral phase of nonresonant frequencies. The coherent manipulation of the spectral phase can modify the time-dependent transient population and therefore the observed dynamics of the probed state. While the higher grating order LIS features are spatially separated from the XUV pulses, their time dynamics are strongly influenced by the exciting pulse. Furthermore, the wave-mixing emission itself is resonant with the LIS and therefore could be subject to reshaping.

Supplementary Note 6: Additional Examples of Temporal Dynamics of Nonlinear Signal Generation
As mentioned in the main text of the manuscript, delays in nonlinear signal generation can be observed in other states besides those shown in Fig. 4. Two additional examples are provided in Supplementary Fig. 11. Delays in nonlinear signal generation are apparent in the lowest three grating orders of the 4p state as well ( Supplementary Fig. 11a). At the photon energy of the 4p resonance, the second order feature emerges 3 ± 1 fs after the first order signal. A more pronounced delay of 4 ± 1 fs is measured between the first and third orders. Due to its lower cross section, the helium 4p state should be minimally affected by propagation effects that may affect the 2p state. As discussed in Supplementary Fig. 8, the fast modulations are due to the interference of ladder and V-type pathways.
In addition, to verify that these delays can be observed in experimental systems other than helium, we examined the near-threshold (4p 5 ( 2 P1/2)ns/nd) autoionizing states of atomic krypton between 14.0 and 14.6 eV ( Supplementary Fig. 11b). A step size of 300 as was chosen for the delay between the XUV and noncollinear NIR pulses. At each delay, 1500 laser pulses were accumulated three times to obtain an appropriate signal to noise ratio. A delay of 1.8 ± 0.4 fs is measured between the first and second grating orders at the energy of the 4p 5 ( 2 P1/2)8s/6d state.
Unfortunately, only two grating orders beyond the zeroth order can be obtained in these experiments as the cross sections of krypton's autoionizing states are an order of magnitude smaller than helium's and the XUV flux at the target is reduced significantly in the 14 eV energy range due to poor transmission through the necessary 0.1 µm indium foils (Lebow, 11-17 eV transmission). Furthermore, resonant pulse propagation effects have never been observed in these states, lending credence to our claim that these effects are not the origin of the delay.
It is important to note the integration areas leading to these plots must be carefully selected for each order, particularly for transient states that shift dramatically in energy and for features that lie close to other states in either space or energy. For example, in these experiments, the higher order 3d-LIS features shift in energy by as much as 0.3 eV over the NIR pulse duration. Phase matching requirements dictate that the divergence angle will change as well since the energy of the photons involved in the wave-mixing process are modified due to this shift. The window therefore must take into account both the shift in energy and in angle. Furthermore, the 3d-LIS lies in a congested portion of the spectra due to the nearby 2p resonance, which is spectrally distorted due to pulse propagation effects, and other LISs above and below it in energy. To obtain an accurate delay, the window should be constructed to avoid effects from these other states. In the case of the data presented here, the window integrated over for the 3d-LIS first order feature was narrowed in energy by 0.1 eV relative to the second and third orders to avoid contamination from the nearby 3s-LIS. When measured at the peak of the rise of the feature, the delay remains the same regardless of energy window, but the broader window leads to a distorted initial rise time.  Supplementary Fig. 13, the contribution of the constant amplitude grating is removed, revealing the time dependence associated with a pure AC Stark phase grating. Note that the effect of the XUV envelope is also ignored in these calculations. As in main text Fig. 5c, the modulation depth of the dipole moment increases with time in Supplementary Fig. 13a. In the far field, again higher order transient grating signals emerge later than lower order signals ( Supplementary Fig. 13b). However, without the contribution from the amplitude grating, the first order signal shifts away from the zeroth order signal in time, decreasing the delay between the first and second orders from 1.73 fs to 1.4 fs.
Slight asymmetries in the NIR angles relative to the XUV do not substantially impact these results.

Intensity Dependence
In addition to replicating the time delay between different orders observed in the experimental data and the full calculation, the combined phase and amplitude grating model also predicts that the strength of the features will not scale perturbatively with NIR intensity, as illustrated in Supplementary Fig. 14. This nonperturbative scaling can be understood by expanding the spatially-dependent term for the phase grating as a series of Bessel functions: where ∆ is the phase shift at a particular intensity in an explicitly noncollinear geometry, and k is the wavevector associated with the NIR phase grating. This expansion is valid for any value of ∆.

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First, the initial peak of the Bessel function of order shifts to larger ∆ as m increases, meaning that the higher diffraction orders appear later than the lower orders as the accumulated phase shift increases with time. Second, the Bessel functions introduce an oscillatory behavior as a function of the accumulated phase shift. Therefore, the features associated with each grating order will saturate as the intensity increases beyond a certain threshold and decrease.
This finding provides deeper insight into the strength of the grating orders in the experimental data as it explains why higher order features appear as strong as or even stronger than lower order ones under certain conditions. intensity. The lineouts are normalized to allow all orders to be viewed simultaneously. b, Without normalization, the relative intensity of the lowest three nonlinear grating orders (m = 1, 2, and 3) at a particular NIR intensity can be compared. The intensity of the m = 1 order is amplified by the contribution of a first order amplitude grating.