Overview: In this chapter we introduce the stochastic Helmholtz equation as the governing partial differential equation for the scalar field of an optical wave propagating through a random medium. However, we provide only the foundational material here for the classical methods of solving the Helmholtz equation. It is interesting that all such methods are based on the same set of simplifying assumptions - backscattering and depolarization effects are negligible, the refractive index is delta correlated in the direction of propagation (Markov approximation), and the paraxial approximation can be invoked.
The Born and Rytov perturbation methods for solving the stochastic Helmholtz equation are introduced first. Whereas the Born approximation has limited utility in optical wave propagation, the Rytov approximation has successfully been used to predict all relevant statistical parameters associated with laser propagation throughout regimes featuring weak irradiance fluctuations. We also illustrate that the Rytov approximation can be generalized to include wave propagation through a train of optical elements that are all characterized by ABCD matrix representations. Methods applicable also under strong irradiance fluctuations are briefly discussed here but formulated in greater detail in Chap. 7. These methods are the parabolic equation method, which is based on the development of parabolic equations for each of the statistical moments of the field, and the extended Huygens-Fresnel principle.
Early probability density function (PDF) models developed for the irradiance of the optical wave include the modified Rician distribution, which follows from the Born approximation, and the lognormal model, which follows directly from the first Rytov approximation. Of these two PDFs, only the lognormal PDF model compares well with the lower-order irradiance moments calculated from experimental data under weak fluctuation conditions. Hence, in this regime it has been the most often-used model for calculating fade statistics associated with a fading communications channel. Nonetheless, more recent investigations of the lognormal PDF suggest that it may be optimistic in predicting fade probabilities, even in weak fluctuation regimes.
We end the chapter with a modification of the Rytov method called the extended Rytov theory that utilizes the two-scale behavior of the propagating wave encountered in regimes of strong irradiance fluctuations. The formalism of the method presented here permits the development of new models for beam wander and scintillation in subsequent chapters that are applicable under strong fluctuations.
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