Speckle with a finite number of steps

The statistical properties of classical, fully developed speckle must be modified when the speckle is generated by a random walk with a finite number of steps. It is shown that for such speckle, the standard negative-exponential probability density function for speckle intensity often overestimates the probability that the intensity exceeds a given threshold. In addition, while any linear transformation of the fields in a classical speckle pattern does not change the intensity statistics, the same is not true for finite-step speckle. The implications of these facts in certain applications are discussed.