Dynamical properties of speckled speckles

We consider the dynamical properties of speckles observed through a second static diffuser arising from a linear or angularly displaced first diffuser. Analytical expressions are obtained for general situations where both the space between the displaced and the static diffuser and the space between the static diffuser and the plane of observation consist of an optical system that can be characterized by a complex-valued ABCD-matrix (e.g. simple and complex imaging systems, free space propagation in both the near-and far-field, and Fourier transform systems). The use of the complex ABCD-method means that diffraction due to inherent apertures is included. One of the diffusers is assumed to give rise to fully developed speckle, i.e. the scattered phase is assumed to be delta-correlated, whereas the second and dynamic diffuser has a finite lateral scale. The illumination of the displaced diffuser is assumed to be Gaussian but the derived expressions are not restricted to a plane incident beam. The results are applicable for speckle-based systems for determining mechanical displacements, especially for long-range systems, and for analyzing systems for measuring biological activity beyond a diffuse layer, e.g. blood flow measurements through human skin.