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Chapter 8:
Fourth-Order Statistics: Weak Fluctuation Theory
Overview: In this chapter we examine various fourth-order statistical quantities like the scintillation index and the irradiance covariance function. Knowledge of the scintillation index is crucial for determining system performance in a laser communication system or laser radar link (Chaps. 11 and 13). In particular, beam wander may be an important factor for scintillation, depending on whether or not the beam is tracked (i.e., whether beam wander is removed), and whether it is collimated or focused. All expressions developed in this chapter are restricted to weak irradiance fluctuation regimes for which the Rytov method can be used. For this case the scintillation index is directly related to the log-amplitude variance studied in most early publications. In Chap. 9 we will expand these results into the moderate-to-strong fluctuation regimes by employing the extended Rytov method (Chap. 5). Many results developed here are based on the Kolmogorov power-law spectrum for reasons of mathematical tractability. However, in attempting to compare scintillation models with measured data taken in outdoor experiments, for example, it may be necessary to use the more general models found in Appendix III based on the modified atmospheric spectrum. This spectrum model features both inner scale and outer scale parameters, and the high wave number "€œbump."€ This so-called "€œbump" near the start of the dissipation regime in the spatial power spectrum can have a profound effect on irradiance fluctuations (leading to potentially large scintillation values). By invoking the frozen-turbulence hypothesis, we can infer the temporal covariance function from which we calculate the temporal spectrum of irradiance fluctuations. In weak fluctuations, we find the spectral width is determined by the transverse wind velocity scaled by the first Fresnel zone. Last, we examine phase fluctuations and the phase covariance function in a manner that parallels our treatment of irradiance fluctuations. Knowledge of atmospheric phase fluctuations is important in the use of coherent heterodyne receivers and for phase modulation techniques applied to optical communications. We also briefly discuss the phase temporal spectrum.
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