Open AccessEllipsometric characterization of highly non-uniform thin films with the shape of thickness non-uniformity modeled by polynomials
Author(s) -
Jiří Vohánka,
Daniel Franta,
Martin Čermák,
Vojtěch Homola,
Vilma Buršı̂ková,
Ivan Ohlı́dal
Publication year - 2020
Publication title -
optics express
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.394
H-Index - 271
ISSN - 1094-4087
DOI - 10.1364/oe.380657
Subject(s) - ellipsometry , gaussian quadrature , gaussian , quadrature (astronomy) , optics , polynomial , parameterized complexity , distribution (mathematics) , materials science , characterization (materials science) , physics , thin film , mathematics , mathematical analysis , nyström method , integral equation , quantum mechanics , algorithm
A common approach to non-uniformity is to assume that the local thicknesses inside the light spot are distributed according to a certain distribution, such as the uniform distribution or the Wigner semicircle distribution. A model considered in this work uses a different approach in which the local thicknesses are given by a polynomial in the coordinates x and y along the surface of the film. An approach using the Gaussian quadrature is very efficient for including the influence of the non-uniformity on the measured ellipsometric quantities. However, the nodes and weights for the Gaussian quadrature must be calculated numerically if the non-uniformity is parameterized by the second or higher degree polynomial. A method for calculating these nodes and weights which is both efficient and numerically stable is presented. The presented method with a model using a second-degree polynomial is demonstrated on the sample of highly non-uniform polymer-like thin film characterized using variable-angle spectroscopic ellipsometry. The results are compared with those obtained using a model assuming the Wigner semicircle distribution.