Formalism to image the dynamics of coherent and incoherent phonons with dark-field X-ray microscopy using kinematic diffraction theory

Dark-field X-ray microscopy (DFXM) is a novel X-ray imaging technique developed at synchrotrons to image along the diffracted beam with a real-space resolution of ∼100 nm and a reciprocal-space resolution of ∼10⁻⁴ radians. Recent implementations of DFXM at X-ray free electron lasers have demonstrated DFXM's ability to visualize the real-time evolution of coherent gigahertz phonons produced by ultrafast laser excitation of metal transducers. Combining this with DFXM's ability to visualize strain fields due to dislocations makes it possible to study the interaction of gigahertz coherent phonons with the strain fields of dislocations and damping of coherent phonons due to interactions with thermal phonons. For advanced analysis of phonon–dislocation interactions and phonon damping, a formalism is required to relate phonon dynamics to the strains measured by DFXM. Here, kinematic diffraction theory is used to simulate DFXM images of the specific coherent phonons in diamond that are generated by the ultrafast laser excitation of a metal transducer. This formalism is also extended to describe imaging of incoherent phonons of sufficiently high frequency to be relevant for thermal transport, offering future opportunities for DFXM to image signals produced by thermal diffuse scattering. For both coherent and incoherent phonons, opportunities are discussed for optimized sampling of reciprocal space and time for deterministic measurements through advances in the optics and excitation geometry.