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Determination of Optical Characteristics of Nanocoatings by Lorentz Transformation Approach

Lorentz transformation formalism from special relativity theory was applied to light interference in films with thicknesses of a few hundreds and a few tens of nanometers. In this analogy between two theories, the speed of light is an imaginary number multiplied to the film refractive index. The interference in the layers of carbon nanotubes on a copper film was proposed to be described as a trajectory in (1+1) or (3+1) spacetimes. By using spacetimes of different dimensions, characteristic trajectories were possible to connect with experimental spectral ellipsometric data. The shape of a trajectory was determined by the real and imaginary parts of the film refractive index and therefore was sensitive to the presence of absorption bands in thin film dispersive curves. The coatings in this approach were characterized by a circular trajectory with singular points for absorption bands. If the optical conductivity of a film had a nonzero slope, the resulted trajectory in (1+1) spacetime formed a spiral. Both spectral ellipsometric and scanning tunneling microscopy experiments have been performed for a few tens of nanometers carbon nanotube thin film on a copper film with two hundred nanometer thickness. The copper film functioned as a Fabry-Perot interferometer with carbon nanotube layer as an interferometer’s mirror. Multiray interference in the copper layer significantly improved the accuracy of the carbon nanotube film optical conductivity curve reconstruction. The thickness of the nanotube coating and its dispersive curves have been found by performing analysis of spectral ellipsometric curves by inverse gradient descent method. The approach based on classical optical experimental techniques allowed to determine parameters of a few tens nanometers layer though its thickness was one order of magnitude less than visible light wavelengths. Scanning tunneling microscopy measurements with high spatial resolution up to 1 nm registered nanotube bundles on the surface of the coating.