Scattering of light by a periodic structure in the presence of randomness. V. Detection of successive peaks in a periodic structure

We address the problem of detecting periodic structures hidden behind roughness. We have shown that if $r_o$ is the coherence length of the scattered radiation, due to the random part of the surface, and \wedge is the wavelength of the periodic part of the surface, then with a matched filtering method that we introduce, and by using simple computations with the intensity data, it is possible to detect the hidden first-order peak even when $(r_o/\wedge)\sim 0.11$. Here we advance the method to bring out very weak second-order peaks, which we demonstrate for what we believe is the first time. The unmistakable presence of both the first- and second-order peaks, which have identical shapes as the zeroth-order peak, is strong evidence of the hidden periodicity and serves as a novel method for the detection of weak periodicities hidden behind strong randomness