The Second-Order Rytov Approximation

An explicit and useful formulation of the solution of the second-order Rytov approximation for estimating the effect of a turbulent medium on laser propagation. The first-order Rytov approximation has been much used in theoretical analyses but recently there has been a controversy on the ranges for which it gives an adequate estimate. RM-5697-PR showed that the calculations of the EM field must include all the second-order terms of the fluctuating part of the index of refraction, proved that the Born approximation is valid only over a limited range (a few kilometers at optical wavelengths), and assumed that the same validity condition applied to the Rytov approximation; this study explicitly demonstrates that this is the case. The second-order Rytov approximation is obtained by equating the second-order terms in the Rytov and Born expansions and is explicitly solved in quadrature. The field equation that results gives the correct average field, phase and intensity statistics, conserves average energy, and satisfies the optical theorem — within the domain of validity, which is the same for the Rytov and the Born approximations. Analyses that apply the Rytov approximation over longer distances, and therefore predict large dispersion, are incorrect.