Speckle contrast in the diffraction field of weak and strong diffusers, in the presence of spherical aberration

We study theoretically the speckle contrast at the center of the diffraction field of phase diffusers illuminated by a Gaussian beam focused by a lens which introduces primary spherical aberration. Using a Gaussian approximation for the speckle complex amplitude, and assuming that the diffusers introduce Gaussian-distributed random phase fluctuations with an autocorrelation function of Gaussian form, we derive analytical expressions for the mean speckle complex amplitude or direct component, the mean speckle intensity and the speckle contrast, which are valid for arbitrary values of the standard deviation of the phase fluctuations and arbitrary strength of the aberration; the results are valid in both, the Fresnel and Fraunhofer zones of the diffuser. For certain combinations of the aberration coefficient, beam parameters and focal length of the focusing lens, the speckle contrast shows an oscillatory behavior as the strength of the aberration is increased, which has a significant effect on the rate at which the speckle contrast increases as the phase standard deviation increases. We mention the relevance of our results for the measurement of surface roughness.