Chapter 9:
Fourth-Order Statistics: Strong Fluctuation Theory
Published: 2005
DOI: 10.1117/3.626196.ch9
Overview: In this chapter we extend our examination begun in Chap. 8 of various fourth-order statistical quantities, like the scintillation index and the irradiance covariance function, to the strong fluctuation regime. We develop separate scintillation models for plane waves, spherical waves, and Gaussian-beam waves. These models evolve from the extended Rytov theory (Chap. 5) by taking into account the role of decreasing spatial coherence of the optical wave as it propagates further and further through the random medium. The net result is a modification of the atmospheric spectrum to an “effective spectrum” arising in the form of a multiplicative spatial filter function that eliminates the effects of moderate-sized refractive-index scales (or turbulent “eddies”) under strong fluctuation conditions. This is similar to the use of spatial filters in adaptive optics applications to eliminate piston and tilt effects (among others) in the received wave front. Under the general irradiance fluctuation theory developed here, the covariance function acquires a two-scale behavior in the strong fluctuation regime, consistent with earlier theories. From the frozen-turbulence hypothesis, we can infer the temporal covariance function from which we calculate the temporal spectrum of irradiance fluctuations. As shown in Chap. 8, the spectral width is determined by the transverse wind velocity scaled by the first Fresnel zone under weak irradiance fluctuations, but the power becomes concentrated at higher and higher frequencies as the strength of turbulence increases. Nonetheless, under strong irradiance fluctuations the two-scale behavior in the covariance function is also evident in the power spectrum. In the last two sections, we review probability distribution models proposed for the irradiance fluctuations, including the gamma-gamma distribution that is theoretically valid under all fluctuation conditions. A favorable characteristic of the gamma-gamma distribution is that it has two parameters that are completely determined by atmospheric conditions.
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