  A well-characterised wavefront is important for many X-ray free-electron
laser (XFEL) experiments, especially for single-particle imaging (SPI), where
individual bio-molecules randomly sample a nanometer-region of highly-focused
femtosecond pulses. We demonstrate high-resolution multiple-plane wavefront
imaging of an ensemble of XFEL pulses, focused by Kirkpatrick-Baez (KB)
mirrors, based on mixed-state ptychography, an approach letting us infer and
reduce experimental sources of instability. From the recovered wavefront
profiles, we show that while local photon fluence correction is crucial and
possible for SPI, a small diversity of phase-tilts likely has no impact. Our
detailed characterisation will aid interpretation of data from past and future
SPI experiments, and provides a basis for further improvements to experimental
design and reconstruction algorithms.
