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Autocorrelation function of the sum of two spectrally decorrelated laser speckle fields: comparison of Fresnel and Fraunhofer approximation

Laser speckle field can be generated as the result of interaction of a laser wave with randomly rough surface. Information on statistical properties of the surface is encoded in statistical properties of the light intensity distribution of the scattered wave. We can get this information from the degree of correlation of two laser speckle fields obtained step by step with two laser waves of different wavelengths scattered from the same area of the surface. The angle of incidence is the same for those two waves. As a quantitative parameter for evaluation of the rate of correlation of those two laser speckle fields we use the autocorrelation function of their sum. We calculate this function in the framework of the scalar Kirrchoff theory of wave scattering from random surfaces. We compare results obtained in the Fresnel approximation and in the Fraunhofer approximation of the scattered wave. We discuss the disparity of solutions from the point of view of surface roughness measurement.