Mathematical Analysis of Nanostructured Surfaces: The Period-Scale Transform

This work has been motivated by the urgent need for accurate and complete characterization of patterns consisting of almost periodic arrangements of specific features (trenches, bumps, holes, spikes, and so on) amply used in the industries of nanotechnology, microelectronics, and photonics. The quantitative characterization of such surface structures demands mathematical methods able to reveal both period- and feature-scale aspects. Given that the conventional approaches (Fourier or wavelet transform) are limited to either periodicity or feature-scale characterization, our work contributes with the proposal of a transformation which combines Fourier and wavelet merits to quantify simultaneously the period and feature scale of a periodic or almost periodic surface pattern. The output of our study has been (a) a detailed investigation of the mathematical properties of the proposed period-scale transform (PST) along with its relationship with other well-known transforms, (b) a presentation of some examples of PST of model 1D periodic surfaces to identify its benefits, and (c) first applications of PST in real profiles extracted from experimental polymer surfaces after plasma treatment.