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Chapter 9:
Finite Beam Characteristics As Examples with a Single Complex Parameter
In Chap. 2 I introduced a method for easily expressing in integral form the second moment of phase and log-amplitude related quantities for infinite collimated and focused waves propagating through turbulence. It was shown how these integrals could be evaluated using Mellin transform techniques. I extend these methods here to solve for the propagation of beam waves with Gaussian shape. Earlier results are a special case of this more general case. To express results as integrals, the approach in Chap. 2 is followed to the point where the propagation parameter γ was assumed to be real. Here γ must be complex to handle beams of finite extent. I derive the filter function for two counterpropagating or copropagating waves. The filter function for the general case has a complex propagation parameter, which results in functions with exponential growth in the integrand. The Mellin convolution integral was generalized to this contingency in the last chapter by applying Parseval's equality. The Taylor series obtained using pole-residue integration are not more difficult to derive in this case. For large parameters I derive asymptotic expressions that have a more complicated dependence than for the infinite-wave case, because the functional form can change depending on the argument of the propagation constant in the complex plane. The method is applied to find the scintillation on a beam wave. Recent results from computer simulations show that these results are not correct for every geometry. The restrictions on the validity of the solution are discussed. In addition, these results only apply to the case in which the beam is tracked so that its center stays on axis. Heuristic formulas that apply in the other cases for both the tracked and untracked cases are given.
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