The second-order term in the Rytov series plays an influential role in determining moments of the field strength and in demonstrating that energy is conserved. The importance of this term was recognized in early studies in which the Rytov approximation was developed to describe propagation in random media [1][2]. Some treatments have used the expected equivalence of the mean irradiance and its free-space values for plane and spherical waves to relate the second-order Rytov term to the phase and amplitude variances estimated to first order [3]. This is often characterized as a consequence of conservation of energy and we will examine that proposition in Chapter 9.

It soon became clear that the preferred approach was to calculate the second-order solution from first principles and then use the result to predict properties of the signal. This was done first for plane waves [1][4][5] and provided reasonable estimates for important properties of signals. That agreement finally established the Rytov approximation as a trustworthy description for the weak-scattering regime [6]. This work was later extended to describe spherical waves [7]. The development of optical lasers stimulated many applications involving beam waves and the corresponding second-order solution was calculated with improving accuracy over the next two decades [8][9][10][11].

The material presented in this chapter is highly analytical and is intended primarily for specialists in the field. The reader who is more interested in applications may wish to proceed directly to Chapters 9 and 10.