Validity of quadratic two-source spherical wave structure functions in analysis of beam propagation through generalized atmospheric turbulence

Two distinct methods based on which two different quadratic-form expressions for the two-source spherical wave structure function (WSF) can be derived are reviewed. The validity of closed-form expressions for the beam-wave cross-spectral density function (CSDF) due to generalized atmospheric turbulence featuring an arbitrary spectral index ranging from 3 to 4, developed based on the quadratic two-source spherical WSFs, is examined in detail. New formulations for the conditions under which the said closed-form expressions for the beam-wave CSDF are strictly valid are developed and several novel interesting findings are elucidated. In particular, the closed-form beam-wave CSDF derived based on the small-separation asymptotic two-source spherical WSF can be considered a rigorous asymptotic solution under the strong-turbulence condition only when the separation distance between the two observation points is much smaller than the inner scale of turbulence; moreover it is also a rigorous asymptotic solution when a certain relation among the initial beam radius, initial transverse coherence width and inner scale holds, regardless of the turbulence strength and spectral index. On the other hand, the accuracy of the closed-form beam-wave CSDF derived based on the large-separation-approximation two-source spherical WSF depends on the spectral index, and a spectral index closer to 4 results in better accuracy.