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Chapter 6:
Second-Order Statistics: Weak Fluctuation Theory
Overview: The main purpose of this chapter is to introduce the mutual coherence function (MCF), which is a two-point field moment. It is used to determine the additional beam spreading caused by atmospheric turbulence over that due to diffraction alone, and to determine the spatial coherence radius of the wave at the receiver pupil plane. Knowledge of beam spreading is important in a free space optics (FSO) communications link, for example, because it determines the loss of power at the receiver. Also, the spatial coherence radius defines the effective receiver aperture size in a heterodyne detection system. The mean irradiance is obtained from the MCF when the two points in the MCF coincide. For the special case when both points are zero, the resulting expression corresponds to the maximum mean irradiance on the beam optical axis from which we deduce the long-term beam radius. The normalized MCF defines the modulus of the complex degree of coherence (DOC) from which the wave structure function (WSF) is identified (and hence, the log-amplitude and phase structure functions are also identified). The separation distance at which the DOC falls to 1/e defines the spatial coherence radius ρ 0 . The root-mean-square (rms) angle-of-arrival and rms image jitter are both derived from the phase structure function. Movement of the short-term beam instantaneous center (or "€œhot spot"€) is commonly called beam wander. A new model for rms beam wander displacement is developed here by analyzing the refractive effects of turbulent eddies equal to or larger than the beam up to the outer scale of turbulence. An estimate of the short-term beam radius is then obtained by removing beam wander effects from the long-term beam radius. Expressions for the beam spot size and spatial coherence radius derived here are based on weak fluctuation theory using the Rytov method. Many of the results that we develop are based on the Kolmogorov power-law spectrum for reasons of mathematical simplicity. However, in attempting to compare models with measured data taken in outdoor experiments, it may be necessary to use models found in Appendix III based on the modified atmospheric spectrum because it is a better representation of actual atmospheric conditions.
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