The purpose of this chapter is to lay the foundation for describing phase and amplitude fluctuations on an equal footing. That foundation rests on the Rytov approximation or method of smooth perturbations that was introduced in Chapter 1. This method provides a powerful technique with which one can solve the random-wave equation (1.3) subject to the condition (1.7). For weak scattering of waves in random media, the Rytov approximation provides a significant improvement both on geometrical optics and on the Born approximation.

There are two steps in this approach. The first and most important step is to completely change the structure of the problem. Rytov discovered a transformation of the electric-field strength that changes the random-wave equation to one that can be solved in many interesting situations [1]. The second step is to expand the transformed field strength into a series of terms proportional to successively higher powers of the dielectric variations [2].

The same approach is often applied to the parabolic-wave equation, which represents a special case of (1.3) that describes small-angle forward scattering [3]. We shall use the more general Helmholtz equation here so as to include microwave propagation which often includes wide-angle scattering. We specialize these results using the paraxial approximation for optical and infrared applications – where the scattering angle is usually small. This chapter is necessarily analytical and the reader who is primarily interested in applications may wish to proceed directly to those topics which begin with the next chapter.