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Chapter 4:
Zero-Parameter Problems
In the second chapter I derived general formulas that enable one to write phase and log-amplitude variances, power spectral densities, and Strehl ratios as integrals. Different cases differ only in the appearance of different filter functions. In the last chapter explicit filter functions were derived for a variety of problems. With the use of these filter functions, integrands contain the following functions: sin(x) , cos(x) , sin 2 (x) , cos 2 (x) , J ν (x) , J ν (x)J μ (x) , J 2 ν (x) , exp(x) , (1+x) −p , (x−1) −p U(x−1) , and K(x) . The Mellin transforms of these functions are given in Table 1.1. (Integrals containing cos 2 (x) are evaluated by substituting 1−sin 2 (x) .) Therefore, allproblems in which the phase or log-amplitude variances are wanted and in which the filter functions in Chap. 3 and the turbulence spectra given in Chap. 2 can be used are solvable by Mellin transform techniques. In this chapter problems of electromagnetic-wave propagation in turbulence that can be solved by table lookup are addressed. The following expressions are found: (1) variances of the Zernike tilt of collimated and focused waves, (2) gradient tilt variance, (3) variance of the difference between Zernike and gradient tilt, (4) beam movement at a target due to tilt, (5) angle-of-arrival jitter of a tracked target, (6) scintillation of collimated and focused waves, (7) phase variance of a system with temporal filtering, and (8) total variance with a phase-only adaptive-optics system with and without a beacon offset. These examples illustrate how easily solutions are obtained once the technique described in this book is mastered.
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Transform theory


Digital filtering

Optical filters

Phase only filters

Radio propagation

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