We study the effect of coherency saturation in spatially or temporally periodical structures with randomization, applicable to a very broad class of systems. We derive a simple analytical formula in the case of uncorrelated deviations of periods with Gaussian probability distribution. Using Monte Carlo simulations, we also demonstrate that many other distributions show statistical properties that closely coincide with the Gaussian, although some of them are drastically different from it. We observed that the characteristic number of elements necessary for the saturation of the coherency (the “coherency range”) depends only on the normalized standard deviation of the size of the elements and not on their probability distribution function. A greatly simplified heuristic formula found by us also fits all of these results with very reasonable precision. In the specific case of x ray transition radiation of low-to-medium relativistic electron beams in multilayer solid-state nanostructures, we show that a structure of a few hundred layers can generate resonantly enhanced radiation in the hard x ray domain with almost unhampered coherency gain.
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