Simultaneous determination of dispersion model parameters and local thickness of thin films by imaging spectrophotometry

• Published in: Applied surface science Vol. 350; pp. 149 - 155
• Main Authors: Nečas, David, Vodák, Jiří, Ohlídal, Ivan, Ohlídal, Miloslav, Majumdar, Abhijit, Zajíčková, Lenka
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 30-09-2015
• Subjects: Algorithms; Dispersions; Fittings; Imaging; Imaging spectrophotometry; Least squares method; Levenberg–Marquardt; Mathematical models; Non-linear least-squares; Optical properties; Reflectometry; Thickness mapping; Thin films; Thin films; Thickness mapping; Non-linear least-squares; Levenberg–Marquardt; Optical properties; Imaging spectrophotometry
• ISSN: 0169-4332; 1873-5584
• DOI: 10.1016/j.apsusc.2015.01.093

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•Imaging spectroscopic reflectometry data processing method is developed.•Local film thicknesses in each pixel and shared optical constants are determined.•The method permits efficient analysis of very large experimental data sets.•It is built on top of Levenberg–Marquardt algorithm, permitting easy implementation.•Efficiency is demonstrated on SiOxCyHz and CNx:H thin films and by numerical means.
A least-squares data fitting procedure is developed for the analysis of measurements of thin films non-uniform in thickness by imaging spectroscopic reflectometry. It solves the problem of simultaneous least-squares fitting of film thicknesses in all image pixels together with shared dispersion model parameters. Since the huge number of mutually correlated fitting parameters prevents a straightforward application of the standard Levenberg–Marquardt algorithm, the presented procedure exploits the special structure of the specific least-squares problem. The free parameters are split into thicknesses and dispersion model parameters. Both groups of parameters are fitted alternately, utilising an unmodified Levenberg–Marquardt algorithm, correcting however the thicknesses during the dispersion model fitting step to preserve effective optical thicknesses. The behaviour of the algorithm is studied using experimental data of two highly non-uniform thin films of different materials, SiOxCyHz and CNx:H, and by numerical simulations using artificial data. It is found that the optical thickness correction enables the procedure to converge rapidly, permitting the analysis of large imaging spectroscopic reflectometry data sets with reasonable computational resources.