The mutual coherence function and the backscatter amplification effect for a reflected Gaussian-beam wave in atmospheric turbulence

The influence of a modified spectrum of refractive-index fluctuations (that includes a high wavenumber rise as well as inner- and outer-scale parameters) on the backscatter amplification effect, arising from double passage of an optical wave through statistically dependent inhomogeneities of a random medium, is studied for the case of a Gaussian-beam wave reflected by a mirror of finite size. A formal expression is first developed for the mutual coherence function, which subsequently leads to tractable analytic models for the mean irradiance in the strictly backward direction. When the inner scale and Fresnel zone are of comparable size, the modified spectrum predicts significantly larger values of the enhancement factor than predicted by the Kolmogorov power-law spectrum. It is also shown in this analysis that by varying the focal length of the mirror the enhancement effects can be greater or less than those of a plane mirror, depending on focus adjustment. All calculations are based on weak irradiance fluctuations using complex ABCD ray-matrix representations for the propagation channel and a generalized spectral representation theory for the complex phase perturbations.