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Chapter 6:
Examples with a Single Positive Parameter
Several interesting problems relating to wave propagation in turbulence are solved in this chapter. The Zernike tilt jitter is calculated for finite outer scale with both von Karman and Greenwood models for outer scale effects. It is shown that outer scale significantly affects tilt even if the diameter is much smaller than the outer scale. It is shown that inner scale limits the maximum tilt that can be measured on an aperture. Tilt is calculated on an annulus, and it is shown that this tilt does not differ much from that of a filled aperture unless most of the aperture is obscured. The effect of diffraction on tilt is calculated. It is shown that tilt variance goes to half of its near-field value in the far field. The amount of tilt difference between two displaced sources is calculated. The power spectral density of tilt is found. An asymptotic series describes the behavior at high frequencies. I show that finite size sources and apertures can significantly reduce scintillation. Characteristic sizes for the source and aperture at which the reduction is significant are found. Inner scale is found to reduce the scintillation for typical inner-scale sizes. Scintillation of a beam corrected for turbulence by the use of a beacon that is displaced from the corrected beam is found. The scintillation difference between two sources in the sky is found. The phase variance of focus is found, and the correlation function of focus is calculated. The solution for anisoplanatism for any Zernike mode is then addressed. The problem of correcting turbulence with artificial beacons is addressed, and the phase variances with a point, distributed, and offset beacon are calculated. The first few problems are solved in detail, and subsequent discussions leave out some intermediate steps that are repeated from problem to problem. This chapter illustrates how readily solutions are obtained with the algorithmic approach developed in this book. Many of these problems are difficult to solve by other means. The solutions to many problems contain a single turbulence moment; however, for those problems in which the integration parameter is a function of the propagation coordinate, the solution is the sum of terms with different turbulence moments. For most problems considered here this does not pose a problem; however, there are cases in which the individual terms are infinite. In that case, an infinite sum of moments cannot be used, and the function in the axial integral must be evaluated at each position and multiplied by the turbulence strength.
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Algorithm development

Correlation function


Near field

Wave propagation

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