Scattering of light by a periodic structure in the presence of randomness VII: Application of statistical detection test

Detection of periodic structures, hidden in random surfaces has been addressed by us for some time and the ‘extended matched filter’ method, developed by us, has been shown to be effective in detecting the hidden periodic part from the light scattering data in circumstances where conventional data analysis methods cannot reveal the successive peaks due to scattering by the periodic part of the surface. It has been shown that if r ₀ is the coherence length of light on scattering from the rough part and Λ is the wavelength of the periodic part of the surface, the extended matched filter method can detect hidden periodic structures for (r ₀/Λ) ≥ 0.11, while conventional methods are limited to much higher values ((r ₀/Λ) ≥ 0.33). In the method developed till now, the detection of periodic structures involves the detection of the central peak, first peak and second peak in the scattered intensity of light, located at scattering wave vectors v _x = 0, Q, 2Q, respectively, where Q = 2Gp/Λ, their distinct identities being obfuscated by the fact that the peaks have width Δv _x = 2Gp/r ₀ ≫ Q. The relative magnitudes of these peaks and the consequent problems associated in identifying them is discussed. The Kolmogorov-Smirnov statistical goodness test is used to justify the identification of the peaks. This test is used to ‘reject’ or ‘not reject’ the null hypothesis which states that the successive peaks do exist. This test is repeated for various values of r ₀/Λ, which leads to the conclusion that there is really a periodic structure hidden behind the random surface.