Statistics of speckle focused through a hard-edged pupil

We examine the statistical properties of speckle patterns in and near the far-field, produced by an ordinary optical system having a hard-edged pupil. By applying Fresnel diffraction theory, expressions for the contrast in terms of the statistical parameters of rough surfaces and the geometry of the optical system are derived and discussed. It is shown that the contrast of speckle has maxima and minima in the focal plane and also along the axis of the optical system.