Solving the inverse problem for determining the optical characteristics of materials

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The paper describes a methodology for determining the optical and physical properties of anisotropic thin film materials. This approach allows in the future designing multilayer thin-film coatings with specified properties. An inverse problem of determining the permittivity tensor and the thickness of a thin film deposited on a glass substrate is formulated. Preliminary information on the belonging of a thin-film coating to a certain class can significantly reduce the computing time and increase the accuracy of determining the permittivity tensor over the entire investigated range of wavelengths and film thickness at the point of reflection and transmission measurement Depending on the goals, it is possible to formulate and, therefore, solve various inverse problems: o determination of the permittivity tensor and specification of the thickness of a thick (up to 1 cm) substrate, often isotropic; o determination of the permittivity tensor of a thin isotropic or anisotropic film deposited on a substrate with known optical properties. The complexity of solving each of the problems is very different and each problem requires its own specific set of measured input data. The ultimate results of solving the inverse problem are verified by comparing the calculated transmission and reflection with those measured for arbitrary angles of incidence and reflection.

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3. Introduction The efficiency of production of existing devices for solid-state microand nanoelectronics and successful creation of new ones largely depend on the level of development of the technology for manufacturing layers of various © Lovetski K. P., Zhukov A. A., Paukshto M. V., Sevastianov L. A., Tiutiunnik A. A., 2020 This work is licensed under a Creative Commons Attribution 4.0 International License materials with a thickness from several nanometers to tens of micrometers [1]. The design and manufacture of multilayer structures with desired properties from dielectric and/or metal films requires an accurate knowledge of the optical parameters of each layer [2], [3]. Methods for evaluating the electrophysical parameters of dielectric and semiconductor thin-film materials [4] based on regularized methods [5] for solving inverse problems allow accurate determination of the electrophysical parameters of thin-film semiconductor materials [6]-[8]. It becomes possible to create multilayer structures with predetermined properties [9]. The advantage of non-contact methods, which include spectrophotometric and polarimetric methods, is the possibility to carry out measurements without destroying the material and without changing its properties. When using these methods, the interaction of electromagnetic waves in the optical range with the sample material is considered and the intensities of the transmitted and reflected waves are measured. The obtained intensities can be then used to calculate both optical and geometric parameters of the samples [10]-[12]. The advantage of spectrophotometric measurements is the possibility to determine several parameters using one measuring device and one sample [13]. To determine the thickness, permittivity, and electrical conductivity of nanometer films in layered structures, one can use the results of measurements of the reflection and transmission spectra of the optical radiation interacting with them, provided that the mathematical model of their interaction is known [14], [15]. Finding the electrophysical parameters of layered structures from the reflection and transmission spectra of electromagnetic waves is associated with the need to solve inverse ill-posed problems of electrodynamics. The developed program “Multilayer” serves both for modeling the transmission of light through multilayer thin-film layered media [16]-[18] and for determining the dielectric (permittivity tensor of anisotropic films) and geometric (film thickness) parameters of various thin-film coatings. The program was created based on many years of experience of collaboration with organizations engaged in the design of thin-film coatings [11] used in the production of liquid crystal displays. 4. Formulation of extended inverse problem Methods described in [19] for description of the transmission of an electromagnetic wave through an optical system are also used in solving the inverse problem for determining the optical characteristics of materials. Let us consider formulation of the inverse problem to determine optical parameters of thin film coating. Within the framework of the inverse problem, it is required to determine, using data on the transmission


About the authors

Konstantin P. Lovetski

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.

Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Andrey A. Zhukov

ITL Consulting


PhD, lead analyst of “ITL Consulting” company

16, Olkhovskaya St., bldg. 5, Moscow, 105066, Russian Federation

Michael V. Paukshto

Fibralign Corporation


- DSc., Physics & Mechanical Engineering, co-founder and CTO of Fibralign Corporation

32930, Alvarado-Niles Rd., Suite 350, Union City, CA 94587, USA

Leonid A. Sevastianov

Peoples’ Friendship University of Russia (RUDN University)


Doctor of Physical and Mathematical Sciences, professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Anastasiia A. Tiutiunnik

Peoples’ Friendship University of Russia (RUDN University)


Candidate of Physical and Mathematical Sciences, lecturer of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation


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Copyright (c) 2020 Lovetski K.P., Zhukov A.A., Paukshto M.V., Sevastianov L.A., Tiutiunnik A.A.

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This work is licensed under a Creative Commons Attribution 4.0 International License.

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