Autocorrelation Function and Power Spectral Density of Laser-Produced Speckle Patterns

When a diffuse surface is illuminated by a coherent monochromatic source such as a laser, the illuminated area appears speckled. Exposing photographic film directly to the backscattered radiation confirms the independent existence of the speckles. The autocorrelation function of the speckle pattern so recorded is shown to be proportional to the diffraction pattern corresponding to the illumination function plus a constant. The power spectral density of the pattern is shown to be the convolution of the illumination function against itself displaced by an amount proportional to the space frequency of interest. The cases of single beam and double beam uniform circular illumination are treated explicitly and experimental verification is offered.