Statistical properties of laser speckle patterns are discussed, starting from the concept of random walk in the complex plane. The first-order statistics of complex amplitude, intensity, and phase of speckle are derived. The addition of speckle patterns, either on amplitude or intensity basis, is considered, with partially polarized speckle as a special case. The first-order probability density functions of intensity and phase for the sum of a speckle pattern and a coherent background are derived. Second-order statistics, including the autocorrelation function and power spectral density, are analyzed for both free-space propagation and imaging geometries. The statistics of spatially integrated or blurred speckle patterns are also considered. The relationship between surface structure and speckle pattern is explored, emphasizing the effects of surface autocorrelation function and finite surface roughness. The origin of laser speckle was first observed in 1960 when objects viewed in highly coherent light appeared granular. This phenomenon is due to the roughness of most materials on the scale of an optical wavelength. When nearly monochromatic light reflects from such a surface, the optical wave at a distant point consists of many coherent wavelets from different microscopic elements of the surface. The interference of these dephased wavelets results in the granular intensity pattern known as speckle. In an imaging system, diffraction and interference both contribute to speckle formation. Even in an aberration-free system, the intensity at an image point can result from the coherent addition of contributions from many independent surface areas. Speckle can arise from free-space propagation or imaging operations. The basic random interference phenomenon underlying laser speckle has parallels in many other branches of physics and engineering. Early mathematical investigations of speckle-like phenomena were conducted by Verdet.Statistical properties of laser speckle patterns are discussed, starting from the concept of random walk in the complex plane. The first-order statistics of complex amplitude, intensity, and phase of speckle are derived. The addition of speckle patterns, either on amplitude or intensity basis, is considered, with partially polarized speckle as a special case. The first-order probability density functions of intensity and phase for the sum of a speckle pattern and a coherent background are derived. Second-order statistics, including the autocorrelation function and power spectral density, are analyzed for both free-space propagation and imaging geometries. The statistics of spatially integrated or blurred speckle patterns are also considered. The relationship between surface structure and speckle pattern is explored, emphasizing the effects of surface autocorrelation function and finite surface roughness. The origin of laser speckle was first observed in 1960 when objects viewed in highly coherent light appeared granular. This phenomenon is due to the roughness of most materials on the scale of an optical wavelength. When nearly monochromatic light reflects from such a surface, the optical wave at a distant point consists of many coherent wavelets from different microscopic elements of the surface. The interference of these dephased wavelets results in the granular intensity pattern known as speckle. In an imaging system, diffraction and interference both contribute to speckle formation. Even in an aberration-free system, the intensity at an image point can result from the coherent addition of contributions from many independent surface areas. Speckle can arise from free-space propagation or imaging operations. The basic random interference phenomenon underlying laser speckle has parallels in many other branches of physics and engineering. Early mathematical investigations of speckle-like phenomena were conducted by Verdet.