Characterization of Thicknesses of Multi-Layered Thin Films Based on Recursive Calculations of Refraction at Layers' Interfaces*

Since the discovery of Graphene in 2004, a variety of methods have been developed to determine the thickness of thin films of two-dimensional atomic crystals. Optical methods are non-invasive, highly accurate, cost-efficient, and can be used to determine the thickness of large samples. Here, we apply a numerical method based on recursive calculations of the reflectance over a range of incident angles as determined by the numerical aperture in multi-layered thin films to calculate the optical contrast. We show that the calculated optical contrast can be used to characterize the thickness of the thin films. Our results agree with other work in the literature.

Additionally, we develop a method based on Total Color Difference (TCD) that can precisely determine the thicknesses of substrates that achieve optimal visibility of thin films in the visible range. A method of predicting the color of a sample of a fixed thickness through values of reflectance is also developed and agrees with experimental observation.