X-ray free-electron lasers (XFELs) are revolutionary tools for studying nonlinear x-ray matter interactions and coupled electronic and nuclear dynamics. Core-level x-ray transient absorption (XTAS) using ultrafast XFEL pulses is a powerful technique, but the stochastic nature of self-amplified spontaneous emission (SASE) XFEL pulses, with their shot-to-shot variations in temporal and spectral profiles, presents challenges. Traditional XTAS approaches using monochromatized beams are inefficient and limit time resolution. An alternative is to analyze the incident and transmitted intensity across the entire SASE bandwidth, enabling correlation analysis techniques that leverage the inherent stochasticity of the pulses. Spectral ghost imaging has shown promise in obtaining high-resolution absorption spectra. However, accurate characterization of the incident pulses is crucial for these covariance spectroscopies. Existing diagnostic tools like beamsplitters (using crystal Bragg diffraction or diffraction gratings) and photoionization with photoelectron spectroscopy offer single-shot measurements but may lack sufficient energy resolution for specific applications. This paper introduces a method that uses a photoelectron spectrometer array (PES array) coupled with a ghost-imaging algorithm to overcome these limitations.

Several methods for single-shot spectral measurements of XFEL pulses exist. Beamsplitters using crystal Bragg diffraction for hard x-rays or diffraction gratings for soft x-rays are common, but these methods are often invasive. Photoionization of a dilute target gas and measurement of the kinetic energy of photoelectrons offers a non-invasive approach, as demonstrated using electron time-of-flight spectrometers (eToFs). While eToF arrays have enabled measurements of position, polarization, and central energy, achieving single-shot spectra with high energy resolution comparable to grating spectrometers has been challenging. Previous work has applied ghost imaging to improve the resolution of spectral measurements in other domains, but its application to XFEL spectral characterization with high resolution remains a significant advance.

The researchers employed a noninvasive method using photoionization of dilute neon gas to characterize the incident x-ray beam. A PES array, consisting of 16 eToFs, measured the kinetic energies of Ne 1s photoelectrons. The incident photon energy was derived from the measured kinetic energy and the known Ne 1s binding energy. Simultaneously, a high-resolution grating spectrometer measured the same XFEL pulse. A key innovation was the application of a ghost-imaging algorithm to improve the energy resolution of the PES array measurements. The ghost-imaging technique leverages the shot-to-shot fluctuations of the SASE XFEL pulses. The high-resolution grating spectrometer provided the 'reference' measurement, while the PES array provided the 'object' measurement. Thousands of simultaneous measurements from the PES array and the spectrometer were used to compute a response matrix through a least squares regression. This matrix maps the low-resolution PES array measurements to high-resolution spectrometer measurements, effectively calibrating the PES array. This response matrix was then applied to reconstruct the x-ray spectrum with significantly improved energy resolution. The algorithm's predictive capabilities were also evaluated by testing its ability to reconstruct spectra from shots not included in the initial training dataset. Additionally, the effect of averaging multiple shots on the spectral accuracy was investigated.

The ghost-imaging algorithm successfully improved the energy resolution of the PES array from ~1 eV to ~0.5 eV at a central energy of 910 eV (ΔE/E ~1/2000). This represents a significant improvement, enabling spectral resolution much finer than the average SASE bandwidth. The response matrix derived from the ghost-imaging analysis provided predictive power for the spectral profiles of future XFEL pulses, demonstrating the method's potential for real-time spectral characterization. Analysis showed that using data from multiple eToFs (six in this case) significantly enhanced the signal-to-noise ratio and the accuracy of the reconstruction. Averaging multiple shots further improved the precision of the reconstructed spectrum, reaching a deviation of ~1% with 1000 shots. The method demonstrated spectral dependence in the deviation, with larger deviations observed at low photon energies due to lower signal-to-noise ratios. The time required to compute the response matrix (around 20 minutes using a single core) highlights the feasibility of rapid implementation during an XFEL beamtime, potentially optimized further by using multiple processor cores.

The enhanced spectral resolution achieved by this method addresses a crucial limitation in XFEL-based spectroscopy. The noninvasive nature of the PES array, combined with the high-resolution provided by ghost imaging, allows for accurate spectral characterization without affecting the XFEL beam. This is particularly beneficial for techniques that rely on the shot-to-shot fluctuations of SASE pulses, such as covariance-based spectroscopies. The results demonstrate the potential for applying this methodology to other XFEL diagnostic instruments, improving their calibration and resolution. The ability to obtain high-resolution averaged spectra is also valuable for transient absorption measurements, offering increased precision in studying ultrafast dynamics. This approach could significantly enhance experimental capabilities in the soft x-ray regime and opens new avenues for exploring nonlinear x-ray phenomena.

This study successfully demonstrates the application of a ghost-imaging algorithm to enhance the spectral characterization of stochastic XFEL pulses using a photoelectron spectrometer array as a non-invasive beamsplitter. The method achieves significantly improved energy resolution, both for single-shot and averaged spectra. The results highlight the potential for real-time spectral diagnostics and for advancing various XFEL-based spectroscopic techniques. Future work could focus on further increasing the resolution by enhancing the sampling rate and flight path of the eToF spectrometers, pushing the limits of single-shot spectral characterization to finer details within the SASE pulse structure.

While this method significantly enhances spectral resolution, achieving complete characterization of the SASE pulse structure (down to ~0.1 eV) requires further improvements to the experimental setup. Specifically, the current sampling rate and flight path of the eToF spectrometer array limit the achievable resolution. The spectral deviations observed, particularly at low photon energies, highlight the need for higher signal-to-noise ratios. Furthermore, while the response matrix calculation time is manageable, further optimization may be necessary for implementation at even higher repetition rates.