Intensity correlations between reflected and transmitted speckle patterns

We study theoretically the spatial correlations between the intensities measured at the input and output planes of a disordered scattering medium. We show that at large optical thicknesses, a long-range spatial correlation persists and takes negative values. For small optical thicknesses, short-range and long-range correlations coexist, with relative weights that depend on the optical thickness. These results may have direct implications for the control of wave transmission through complex media by wavefront shaping, thus finding applications in sensing, imaging and information transfer.