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

Abstract

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 $\Lambda$ is the wavelength of the periodic part of the surface, the

extended matched filter method can detect hidden periodic structures for

$(r_{0}/\Lambda )\ge 0.11$, while conventional methods are limited to

much higher values ($(r_{0}/\Lambda) \ge 0.33$). 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 $Q = 2\pi/\Lambda$, their distinct identities

being obfuscated by the fact that the peaks have width $\Delta v_{x} =

2\pi/r_{0} \gg 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 $r_{0}/\Lambda$, which leads to

the conclusion that there is really a periodic structure hidden behind the

random surface