XFEL SASE pulses can enhance time-dependent observables

To realistically account for temporal intensity profiles of SASE pulses in the simulations, it is necessary to model the pulses' stochastic nature. For this we employ the partial-coherence method shown to emulate the temporal structure of SASE pulses seen in experiments [70, 71]. This allows us to generate temporal intensity profiles, $\mathcal{I}(t)$ in equations (2) and (5), that display substantial shot-by-shot variation, yet still have a well-defined Gaussian-shaped average.

The partial-coherence method defines the temporal intensity profile as $\mathcal{I}(t) = |\mathcal{E}_{\mathrm{pc}}(t)|^2$, where the partially coherent electric field amplitude is [72],

$$\begin{align} \mathcal{E}_{\mathrm{pc}}\left(t\right) = N_{\mathcal{E}} \mathcal{G}_\mathrm{f}\left(t\right) \mathcal{E}_{\mathrm{in}}\left(t\right).\end{align} \tag{ 6 }$$

Here, $N_{\mathcal{E}}$ is a normalisation factor that ensures that the total energy of the pulse takes a particular value and does not fluctuate across different shots. The electric field $\mathcal{E}_{\mathrm{in}}(t)$ is incoherent,

$$\begin{align} \mathcal{E}_{\mathrm{in}}\left(t\right) = \frac{1}{2\pi} \int\ \mathcal{E}_0\left(\omega\right)\mathrm{e}^{\mathrm{i}\phi\left(\omega\right)+\mathrm{i} \omega t}\ \mathrm{d}\omega,\end{align} \tag{ 7 }$$

with an initial electric field in frequency-space, $\mathcal{E}_0(\omega)$, which serves as a starting point and can be expressed as,

$$\begin{align} \mathcal{E}_0\left(\omega\right) = \int\ \cos{\left(\omega_\mathrm{c}t\right)}\mathrm{e}^{-t^2/t_\mathrm{c}^2 - \mathrm{i} \omega t}\ \mathrm{d}t.\end{align} \tag{ 8 }$$

The exponential term in equation (8) is the coherence function containing the coherence time $t_\mathrm{c}$ and defines, via its Fourier transform, the frequency spectrum of the pulse. The cosine accounts for the rapid oscillation of the electric field with carrier frequency $\omega_\mathrm{c}$.

Moreover, $\phi(\omega)$ in equation (7) is a phase factor which takes a different value for each point on the frequency grid in the range $- \pi \unicode{x2A7D} \phi(\omega) \unicode{x2A7D} \pi$. This has the effect of giving each frequency component in the pulse a random phase and, after the Fourier transform, results in an incoherent electric field that has a random shape in the time domain.

Finally, $\mathcal{G}_\mathrm{f}(t)$ in equation (6) is a Gaussian filter function,

$$\begin{align} \mathcal{G}_\mathrm{f}\left(t\right) = e^{-4 \mathrm{ln}2 \left(t - \tau_\mathrm{f}\right)^2/d_\mathrm{f}^2},\end{align} \tag{ 9 }$$

which defines the shape that the average pulse will take. Here, $\tau_\mathrm{f}$ is the centre of the Gaussian and $d_\mathrm{f}$ is its full-width half-maximum (FWHM). The purpose of the filter function is to confine $\mathcal{E}_{\mathrm{in}}(t)$ to a certain region in the time domain, taking the completely incoherent electric field and making it partially coherent. An example of a pulse generated by the partial-coherence method is shown in figure 1.

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To sort and select the SASE pulses, it is necessary to define parameters that reliably characterise the shape of each individual intensity profile. Here, we present definitions for an effective pulse duration $d_\mathrm{p}$ and a pump–probe delay time τ, which are analogous to the parameters that define an idealised Gaussian pulse.

For a transform-limited pulse with a Gaussian-shaped intensity profile, the duration $d_\mathrm{p}$ is usually defined as its FWHM. We define the respective times at which the intensity of the Gaussian pulse equals half its maximum as $t_\mathrm{L}$ and $t_\mathrm{R}$. For a SASE pulse, however, these times cannot be rigorously defined via the value of $\mathcal{I}(t)$ because $\mathcal{I}(t)$ can, in principle, take any value at any point in time. Instead, $t_\mathrm{L}$ and $t_\mathrm{R}$ are defined via the integral of $\mathcal{I}(t)$ from $-\infty$ to $t_\mathrm{L}$, $A_\mathrm{L} = \int_{-\infty}^{t_\mathrm{L}} \mathcal{I}(t)\textrm{d}t$, and the corresponding integral from $t_\mathrm{R}$ to $\infty$, $A_\mathrm{R} = \int^{\infty}_{t_\mathrm{R}} \mathcal{I}(t)\textrm{d}t$. For a normalised Gaussian-shaped pulse, these integrals take the values $A_\mathrm{L} = A_\mathrm{R} \approx 0.1195$. In analogy, for a SASE pulse, we determine $t_\mathrm{L}$ and $t_\mathrm{R}$ numerically such that $A_\mathrm{L}$ and $A_\mathrm{R}$ take the same value as for a Gaussian pulse. The pulse duration is then simply calculated as the difference $d_\mathrm{p} = t_\mathrm{R}-t_\mathrm{L}$. Notably, this definition reproduces the standard FWHM duration for a Gaussian pulse. The approach is illustrated in figure 1.

The delay time τ can be thought of as the centre of the x-ray probe pulse. For a Gaussian, it is simply the time where the intensity is at its maximum. For SASE pulses, this is usually not the case, and τ is better described by the expectation value,

$$\begin{align} \tau = \langle t \rangle = \int^\infty_{-\infty} t\ \mathcal{I}\left(t\right)\ \mathrm{d}t.\end{align} \tag{ 10 }$$

To investigate the effect that the stochasticity of SASE pulses has on the CMS signal, scattering signals are calculated according to equation (5). The simulations use previously published electronic potential energy curves, one-electron scattering matrix elements, and nuclear wavepackets [66, 67]. The nuclear wavepackets have been propagated by numerical integration of the time-dependent Schrödinger equation using the WAVEPACKET code [73] in MATLAB [74], with accurate potential energy surfaces, transition dipole moments, and derivative non-adiabatic coupling matrix elements from Wolniewicz et al [75–78]. The one-electron scattering matrix elements, defined in equation (4), are calculated using our own code [39, 79, 80] from electronic wavefunctions obtained using the state-averaged complete active space self-consistent field method with a large active space and basis, SA-CASSCF(2,30)/d-aug-cc-pVQZ, in MOLPRO [81] for the first nine electronic singlet states of molecular hydrogen.

In the published simulation, an extreme ultraviolet pump pulse centred at $t\,=\,0$ excites the molecule in its electronic and vibrational ground state, $\mathrm{X}^1\Sigma_\mathrm{g}^+(\nu=0)$. This pulse has a Gaussian shape with a FWHM duration of 25 fs, a mean photon energy of 14.3 eV, and an electric field amplitude of 53.8 MV $\mathrm{cm}^{-1}$. It leads to a partial excitation from the electronic and vibrational ground state, $\mathrm{B}^1\Sigma_\mathrm{u}^+ \longleftarrow \mathrm{X}^1\Sigma_\mathrm{g}^+(\nu = 0)$, with 10% of the population transferred to the first electronically excited state, $\mathrm{B}^1\Sigma_\mathrm{u}^+$. The electronic coherence between these two states peaks at $t\,=\,61.92\,\text{fs}$ and the resulting reference CMS signal is calculated for a Gaussian-shaped x-ray pulse. The geometry of the calculations are such that the bond vector of the molecule is aligned with the laboratory $\hat{x}$ axis, the incident x-ray pulse propagates along the $\hat{z}$ axis, and the scattering patterns are calculated in the $(q_x,q_y)$ plane. Finally, the weights $W_{fij}$ are modelled by a box function centered at the mean photon frequency ω₀ with a width Δω chosen to justify the truncation of the sum over final states f in equation (5) at the nine electronic states included.

While the previous work of Simmermacher et al approximated the SASE pulse by a simple bandwidth-limited Gaussian function, in this article more realistic temporal intensity profiles are used. Synthetic SASE pulses are generated applying the partial-coherence method discussed in section 3.1, using the following parameters: $d_\mathrm{f} = 100\ \text{as}$, $\tau_\mathrm{f} = 61.92\, \text{fs}$, $t_\mathrm{c} = 15\ \text{as}$, and $\hbar\omega_\mathrm{c} = 8.5\ \text{keV}$. The intensity of the pulse is assumed to be sufficiently low that non-linear processes can be ignored (see SI for further details). The random phases, $\phi(\omega)$, were generated using the built-in random number generator in Matlab, initialised by a seed based on the current time. We note that pulse durations on the order of attoseconds are significantly shorter than what is currently available at XFEL facilities. However, these values are needed in order to resolve the fast electron dynamics in H₂. Experimentally recorded SASE pulse intensity profiles, scaled down to an average duration of 100 as, are also used for the calculations in section 4.2. Strictly speaking, shorter SASE pulses would most likely undergo qualitative changes, but for the current purpose of matching the overall duration of the pulses to the time-scale of the theoretical model, such linear scaling is sufficient. The method is directly applicable to any SASE pulse as long as its structure is measured and characterised. We also note that the XTCAV measures pulse energy and converting this to intensity requires that the focus diameter is estimated.

To compare CMS signals from different SASE pulses with the signal obtained with the idealized Gaussian pulse in previous work, we introduce a scaled signal,

$$\begin{align} \tilde{S}_\text{cm}\left(\boldsymbol{q}\right) = \frac{S_\mathrm{cm}\left(\boldsymbol{q}\right)}{\mathrm{max}\left(S_{\mathcal{G}}\left(\boldsymbol{q}\right)\right)},\end{align} \tag{ 11 }$$

where $S_\mathrm{cm}(\boldsymbol{q})$ is the CMS signal calculated according to equation (5) with a realistic SASE pulse and $S_{\mathcal{G}}(\boldsymbol{q})$ is the reference CMS signal simulated with a Gaussian pulse with the same parameters as the Gaussian filter function in equation (9). The function max(...) in the denominator gives the maximum value across all values of $\boldsymbol{q}$, providing a global normalisation of the CMS signal.

Figures 2(A) and (C) show two examples of scaled CMS signals. The overall shapes of the scattering patterns are identical and do not deviate from the pattern obtained with the Gaussian pulse. They only differ in their magnitudes. We can therefore represent each shot by the maximum value of the respective signal,

$$\begin{align} \kappa = \mathrm{max}\left(\tilde{S}_\text{cm}\left(\boldsymbol{q}\right)\right).\end{align} \tag{ 12 }$$

The CMS pattern shown in figure 2(A) has a maximum intensity of $\kappa \approx 0.746$, which is notably weaker than the corresponding value for the reference Gaussian pulse, $\kappa = 1$. This 25% reduction in intensity is a direct consequence of the intensity profile $\mathcal{I}(t)$ of the corresponding synthetic SASE pulse shown in figure 2(B). The pulse has an effective pulse duration of $d_\text{p} \approx 138.6\ \text{as}$, which is significantly longer than the duration of the idealised Gaussian pulse of $d_\text{p} \approx 100\ \text{as}$. Hence, the temporal resolution is decreased, consequently leading to a weaker signal. The signal in figure 2(C), in contrast, is considerably stronger with $\kappa \approx 1.186$, which is nearly a 20% improvement over the reference. In this latter case, the underlying SASE pulse in figure 2(D) has a shorter effective pulse duration of only $d_\text{p} \approx 65.9\ \text{as}$.

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Figure 3(A) shows the values of $\kappa$ for 25 000 shots simulated with different synthetic SASE pulses and sorted according to the pulse properties that were introduced in section 3.2 The shots are distributed in a circle centred at $d_\mathrm{f}$ and $\tau_\mathrm{f}$, i.e. the FWHM and the centre of the Gaussian filter function. This circular shape is a consequence of the statistical distribution used in the partial-coherence method. As illustrated in figure 3, $\kappa$ depends strongly on the effective pulse duration $d_\mathrm{p}$, with shorter pulses unsurprisingly giving stronger CMS signals. This is further exemplified by two pulses with $d_\mathrm{p}\approx 153.4 \ \text{as}$ and $d_\mathrm{p}\approx 40.8\ \text{as}$, which are shown in figures 3(B) and (C). These two pulses lead to quite weak and quite strong signals with $\kappa \approx 0.623$ and $\kappa \approx 1.352$, respectively, as shown by the corresponding markers in figure 3(A).

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The magnitude of the CMS signal $\kappa$ also depends on the delay time $\tau$, reflected by the curved appearance of the blue, yellow and red distributions in figure 3(A). The SASE pulse in figure 3(D) illustrates the effect. Even though this pulse has a shorter than average duration of $d_\mathrm{p}\approx 83.5\ \text{as}$, it still results in a weak signal with $\kappa \approx 0.748$ because the pulse is shifted to the shorter delay time of $\tau - \tau_\mathrm{f} \approx -26.2\ \text{as}$. Here, the signal is weaker simply due to the phase of the electronic wavepacket. Essentially, the measurement is performed at a time when the CMS signal is weak. This is due to the fact that the centre of the Gaussian filter function is set to $\tau_\mathrm{f} = 61.92\ \text{as}$ because the CMS signal reaches its maximum at that time. Hence, any deviation of $\tau$ from that value of $\tau_\mathrm{f}$ inevitably results in a weaker signal.

Interestingly, the SASE pulse in figure 3(E) has an average effective pulse duration of $d_\mathrm{p}\approx 106.7\ \text{as}$ and is centred at $\tau - \tau_\mathrm{f}\approx -3.3\ \text{as}$, but nevertheless results in a weakly enhanced signal with $\kappa \approx 1.092$. This can be explained by the presence of a sharp peak in $\mathcal{I}(t)$ around $\tau_\mathrm{f}$ and suggests that the manner in which we characterise the pulses cannot fully account for the distribution of the intensity within the boundaries of $d_\mathrm{p}$. We note that the characterisation may be straightforwardly improved by the introduction of additional parameters such as, for instance, a second pulse duration defined by a different width, e.g. the full-width 3/4-maximum (see SI for details). In the latter case, the two integrals $A_\mathrm{L}$ and $A_\mathrm{R}$ used to determine $d_\mathrm{p}$ via $t_\mathrm{L}$ and $t_\mathrm{R}$ (see section 3.2) take the value $A_\mathrm{L} = A_\mathrm{R} \approx 0.2241$.

Next, we consider how one may apply the above pulse characterisation to an actual XFEL-based experiment to enhance the magnitude of ultrafast signals. For that, the synthetic SASE pulses are replaced with real experimental pulse envelopes taken from XTCAV data measured at the CXI instrument of the Linear Coherent Light Source at SLAC in 2021 (proposal number LV04). The intensity profile of the average pulse from this data is shown in figure 4. It displays a largely Gaussian-like shape with two extra shoulders at negative times that together span about 10 fs and has an effective pulse duration of $d_{\text{p}} \approx 10.6\ \text{fs}$.

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To be usable within the currently employed model of attosecond electron dynamics of $\text{H}_2$, the experimentally characterised pulses are compressed in time by two orders of magnitude to an average duration of $d_\mathrm{p}\approx 100\ \text{as}$. Moreover, the centre of each individual pulse is determined according to equation (10) and shifted to a random value within the range $61.89\ \mathrm{fs} \unicode{x2A7D} \tau \unicode{x2A7D} 62.95 \ \mathrm{fs}$ to emulate a temporal jitter [82] of 60 as with an average time delay $\bar{\tau}$ = 62.92 fs. As expected, this random shift leads to a uniform distribution of pulses in $\tau$, which can be seen in figure 5(A). The overall dependence of the signal strength on the effective pulse duration $d_\mathrm{p}$ in figure 5(A) is identical to the one shown in figure 3(A). The distribution is, however, slightly shifted to lower values of $\tau$. This can be explained by the asymmetry of the average intensity profile seen in figure 4. Because the average intensity has a strong peak at positive values of $t-\tau$, individual pulses give stronger CMS signals on average when they are slightly shifted to negative values of $\tau - \bar{\tau}$.

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Now, an upper limit for the pulse duration $d_\mathrm{p}$ is introduced and only CMS signals simulated with pulses shorter than this limit are averaged. Figure 5(B) demonstrates how the magnitude of the average signal increases as the limit for $d_\mathrm{p}$ decreases and more pulses are discarded. Using only 50% of all shots, for example, the strength of the average signal is increased by 10.8% relative to the average with all shots included. By adding a second pulse duration to the characterisation as described earlier, the signal enhancement may be further improved to 11.2% (see SI for details). As a result of this selection, the magnitude of the average CMS signal can be enhanced without prior knowledge about the magnitude of the CMS signal itself. This is particularly useful considering that the signal cannot be determined for individual shots due to its inherent weakness and insufficient signal-to-noise ratio. The CMS signal can only be discerned once averaged over a sufficient amount of shots. This also points to another important aspect of this analysis: the enhancement of the signal comes at the cost of discarding data. Since there is a requirement for a particularly good signal-to-noise ratio in order to detect a significant and discernible signal, a balance must be struck to ensure enough data is kept for the signal to emerge from the background noise.

Even though the intensity of the CMS signal cannot be measured experimentally for individual shots, in the current analysis the theoretical CMS intensities can be used to create a benchmark for the pulse selection based on $d_\mathrm{p}$. Figure 5(B) also shows how the strength of the average signal increases if shots are selected according to their maximum signal $\kappa$. This selection shows a similar trend as the one based on the effective pulse duration $d_\mathrm{p}$ before, but there is a significant difference in magnitude, illustrating that there is still room for improvement. For example, selecting 50% of all shots based on $\kappa$ yields an increase of 13.5% relative to the average with all shots included, compared to the 10.8% increase previously achieved when using $d_\mathrm{p}$ for the selection. The largest contributor to this difference is the dependence on $\tau$ because the temporal jitter of 60 as covers a significant part of the electronic wavepacket's period of $T_\mathrm{elec} \approx 289\ \text{as}$. This illustrates the well-known fact that the temporal jitter and the pulse duration jointly impose a limit on the temporal resolution of a pump-probe experiment. The impact of the temporal jitter becomes more significant the fewer shots are used and the lower the average pulse duration is. Hence, a smaller temporal jitter will increase the effectiveness of the shot selection by $d_\mathrm{p}$ since the jitter will be less of a limiting factor.