Theory and simulation of far field diffraction patterns in Talbot-based transient grating spectroscopy at X-ray free electron lasers.

The recent extension of transient grating (TG) spectroscopy to the X-ray regime at X-ray free electron lasers (XFEL) facilities has opened new possibilities for studying ultrafast dynamics and nanoscale transport. Recent experiments have employed the Talbot effect to generate excitation gratings in the hard X-rays, e.g., at 7 keV, using X-ray phase masks, enabling simplified, collinear TG setups with only two beams. Despite promising experimental progress, a comprehensive theoretical framework for understanding far-field diffraction patterns in Talbot-based X-ray TG is still lacking. In this work, we present a detailed theoretical study of stationary far-field diffraction patterns in TG spectroscopy using the Talbot effect, providing essential insights for interpreting and optimizing recent XFEL experiments. We systematically investigate: (1) the influence of the sample material properties such as, index of refraction and thickness, and beam intensity, included in the phase effectivity parameter causing broadening of the spatial spectrum; (2) the role of wavefront curvature in modulating diffraction regimes; (3) the effect of sample position on far-field patterns; (4) the emergence of heterodyne effects due to phase shifts across diffraction orders; and (5) the impact of using two different wavelengths for pump and probe beams and analyzing the effect of their relative intensities. Our analysis focuses on static spatial properties rather than dynamical transients, offering a foundation for precise material characterization and nonlinear spectroscopy. Simulations are performed using parameter values representative of XFEL conditions, including hard X-ray wavelengths and sub-micrometer grating periods. The results highlight how phase effectivity, beam parameters, and sample placement govern the formation of non-trivial diffraction patterns, providing critical guidance for future X-ray TG experiments. The model focuses on nonlinear phase effects at hard X-ray energies, which are generally very weak and become relevant only at high intensities. This study delivers the first systematic theoretical framework for Talbot-based TG spectroscopy, bridging a vital gap in understanding far-field diffraction effects in cutting-edge XFEL applications.