Publisher Summary This chapter discusses the statistics of speckle patterns. The statistical properties of a speckle pattern depend, in general, on both the coherence of the incident light and the statistics of the scattering surface or medium. In a large number of practical situations normal speckle patterns are produced; the statistics of these speckle patterns are well-established both theoretically and experimentally, and in particular, they do not depend on the detailed scattering properties of the scattering medium. When examining the effects of polychromatic illumination on speckle statistics, it was found that the value of the r.m.s. height variation of a scattering surface strongly influenced the statistical properties of the scattered intensity. In addition, the chapter shows that the general relationship between the scattered intensity and the scattering surface may be very complicated, even for surfaces with a Gaussian distribution of surface heights.

I The Statistics of Speckle Patterns

A unified, analysis is presented for the spatial and the spectral sensitivity of speckle (the rapid spatial variations which occur in an image when illumination of narrow spectral width is used) in a space-invariant linear system. In prior work considering speckle size, others have shown that its spatial variation is functionally dependent primarily on the autocorrelation function of the system's impulse response, but effects of varying the wavelength were largely ignored. In the present paper we treat the general problem in which a diffuse object, illuminated by a collimated, monochromatic beam, is imaged by a system whose amplitude impulse response isz(x, η), wherex and η are space and normalized (temporal) frequency coordinates, respectively. An expression is derived for the multidimensional autocorrelation functionRu(Δx,η1,η2) of the intensityu(x,η) in the image plane. Functionally, it depends upon a convolution of the system autocorrelation functionRu(Δx,η1,η2) with the characteristic function of the distribution function for heights, which is used to model the input object's surface. Examples are presented; and, it is shown that one can infer valuable information about the variation of heights for points on the surface of the input diffuse object, which are separated by much less than the classical resolution limit.

Space and wavelength dependence of speckle intensity

Measurement of surface roughness properties by means of laser speckle techniques

Chapter 3 – surface roughness measurement

Photomultiplier tubes which perform virtually ideal digital detection of electromagnetic radiation at visible frequencies are now widely available. The only sources of noise are a generally low dark count, typically a few counts per second, and the unavoidable shot noise due to the random (Poisson) nature of the detection process. The output of such detectors consists of a series of discrete pulses of charge, each corresponding to a single photodetection, which constitutes a Poisson process rate-modulated by variations in the incident intensity. Coincident with the development of these detectors has been the development of hardware based on integrated circuits which can perform errorless analysis of digital signals at frequencies up to about 108 Hz. Combination of these two technologies provides the techniques of photon correlation.1 (We will include the study of single-interval photon statistics in the generic term “photon correlation”.) With speed and accuracy close to the theoretical limit these techniques can be used to analyze fluctuating optical signals of any origin (provided the fluctuation time is ≳10−7 sec).

Statistical Properties of Scattered Radiation

(1972). Speckle Patterns: A Historical Retrospect. Optica Acta: International Journal of Optics: Vol. 19, No. 9, pp. 791-793.

P. Hariharan

Papers

Speckle Patterns: A Historical Retrospect