Three-dimensional generalization of the Van Cittert-Zernike theorem to wave and particle scattering

Coherence properties of primary partially coherent radiations (light, X-rays and particles) elastically scattered from a 3D object consisting of a collection of electrons and nuclei are analyzed in the Fresnel diffraction region and in the far field. The behaviour of the cross-spectral density of the scattered radiation transverse and along to the local direction of propagation is shown to be described by respectively the 3D Fourier and Fresnel transform of the generalized radiance function of a scattering secondary source associated with the object. A relativistic correct expression is derived for the mutual coherence function of radiation which takes account of the dispersive propagation of particle beams in vacuum. An effect of the spatial coherence of radiation on the temporal one is found; in the Fresnel diffraction region, in distinction to the field, both the longitudinal spatial coherence and the spectral width of radiation affect the longitudinal coherence. A solution of the 3D inverse scattering problem for partially coherent radiation is presented. It is shown that squared modulus of the scattering potential and its 2D projections can be reconstructed from measurements of the modulus and phase of the degree of transverse spatial coherence of the scattered radiation. The results provide a theoretical basis for new methods of image formation and structure analysis in X-ray, electron, ion, and neutron optics.