Intensity fluctuations due to a deep phase screen with a power-law spectrum

The spatial frequency spectrum of intensity fluctuations arising in a plane wave after it traverses a one-dimensional screen which imposes large random fluctuations of phase is investigated in the case when the random phase of the screen has a modified power-law spectrum of the form (1 + L2q2)–½m, m = 2, 4, where q is the wavenumber and L determines the outer scale of the fluctuations. The case of a screen with an unmodified power-law phase spectrum, corresponding to a linear structure function of phase, is also investigated by letting L become very large for m = 2. It is shown that the unmodified power-law spectrum can only be used if the observer is so close to the screen that L, the outer scale of the irregular phase, cannot be seen. The intensity spectrum corresponding to m = 4 is considered and it is shown that its behaviour differs fundamentally from the previous case with m = 2. It is shown that for a more exact evaluation of the intensity spectrum the integral formulation must be extended to the complex plane and that an important contribution, partly responsible for a peak in the spectrum, comes from an hitherto unnoticed saddle point. Analytic forms for the spectrum are derived and curves for some typical spectra are given.