Statistical Properties of Laser Speckle Patterns

Since speckle plays an important role in many physical phenomena, it is essential to fully understand its statistical properties. Starting from the basic idea of a random walk in the complex plane, we derive the first-order statistics of the complex amplitude, intensity and phase of speckle. Sums of speckle patterns are also considered, the addition being either on an amplitude or on an intensity basis, with partially polarized speckle being a special case. Next we consider the sum of a speckle pattern and a coherent background, deriving the first-order probability density functions of intensity and phase. Attention is then turned to second-order statistics. The autocorrelation function and power spectral density are derived, both for a free-space propagation geometry and for an imaging geometry. In some cases the recorded speckle pattern may be spatially integrated or blurred, and accordingly consideration is given to the statistics of such patterns. Finally, the relationship between detailed surface structure and the resulting speckle pattern is explored, with emphasis on the effects of the surface autocorrelation function and the effects of finite surface roughness.