Study of the evolutions of speckle intensities and their statistical properties in the extremely deep Fresnel diffraction region

From the Kirchhoff approximation theory, we have derived the formulas for speckle fields and numerically simulated speckle fields in the extremely deep Fresnel diffraction region generated by three samples with different roughness. It is found that the speckle characteristics are obviously different from that in a far field. First, the properties of speckle patterns are closely related to the morphology of the corresponding samples. It is interesting that for a sample the speckle pattern at some scattering distance is similar to the morphology of the sample to some degree. Second, for a given scattering distance, the probability density and the contrast vary depending on the roughness of the sample. Similarly, these quantities change for a given sample if the scattering distance is changed within the deep Fresnel region. The calculated results are roughly consistent with experimental ones. Qualitative analyses for these results are also given. The studies in this paper are significant for the realization of the evolutions of speckle properties and their statistical characteristics in the extremely deep Fresnel diffraction region.