Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2518310.22363/2658-4670-2020-28-4-378-397Research ArticleSolving the inverse problem for determining the optical characteristics of materialsLovetskiKonstantin P.<p>Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics</p>lovetskiy-kp@rudn.ruZhukovAndrey A.<p>PhD, lead analyst of “ITL Consulting” company</p>a.zhukov@itlc.ruPaukshtoMichael V.<p>- DSc., Physics & Mechanical Engineering, co-founder and CTO of Fibralign Corporation</p>mpaukshto@fibralignbio.comSevastianovLeonid A.<p>Doctor of Physical and Mathematical Sciences, professor of Department of Applied Probability and Informatics</p>sevastianov-la@rudn.ruTiutiunnikAnastasiia A.<p>Candidate of Physical and Mathematical Sciences, lecturer of Department of Applied Probability and Informatics</p>tyutyunnik-aa@rudn.ruPeoples’ Friendship University of Russia (RUDN University)ITL ConsultingFibralign Corporation1512202028437839709122020Copyright © 2020, Lovetski K.P., Zhukov A.A., Paukshto M.V., Sevastianov L.A., Tiutiunnik A.A.2020<p>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.</p>transmittancereflectancerefractive indices determinationthin filmsmultilayersoptical coatingsoptical propertiesопределение коэффициентов пропусканияотраженияпоказателей преломлениятонкие плёнкимногослойные материалыоптические покрытияоптические свойства<p>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 http://creativecommons.org/licenses/by/4.0/ 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</p>[D. A. Yakovlev, V. G. Chigrinov, and H. S. Kwok, Modeling and Optimization of LCD Optical Performance. New York: Wiley, 2015.][J. A. Dobrowolski, F. C. Ho, and A. Waldorf, “Determination of optical constants of thin film coating materials based on inverse synthesis,” Applied Optics, vol. 22, no. 20, pp. 3191-3200, 1983. DOI: 10.1364/AO. 22.003191.][X. Cheng, B. Fan, J. A. Dobrowolski, L. Wang, and Z. Wang, “Gradientindex optical filter synthesis with controllable and predictable refractive index profiles,” Optics Express, vol. 16, no. 4, pp. 2315-3221, 2008. DOI: 10.1364/OE.16.002315.][A. Tejada et al., “Determination of the fundamental absorption and optical bandgap of dielectric thin films from single optical transmittance measurements,” Applied Optics, vol. 58, no. 35, pp. 9585-9594, 2019. DOI: 10.1364/AO.58.009585.][J. B. Bell, A. N. Tikhonov, and V. Y. Arsenin, “Solutions of Ill-Posed Problems,” Mathematics of Computation, vol. 32, no. 144, pp. 1320-1322, 1978.][S. Nevas, F. Manoocheri, E. Ikonen, A. V. Tikhonravov, M. A. Kokarev, and M. K. Trubetskov, “Optical metrology of thin films using highaccuracy spectrophotometric measurements with oblique angles of incidence,” in Advances in Optical Thin Films, International Society for Optics and Photonics, vol. 5250, SPIE, 2004, pp. 234-242. DOI: 10.1117/12.512700.][L. D. Landau and E. M. Lifshitz, Electromagnetic Waves in Anisotropic Media. Oxford: Pergamon Press, 1984.][A. V. Tikhonravov et al., “Effect of systematic errors in spectral photometric data on the accuracy of determination of optical parameters of dielectric thin films,” Applied Optics, vol. 41, no. 13, pp. 2555-2560, 2002. DOI: 10.1364/AO.41.002555.][M. Nur-E-Alam, M. M. Rahman, M. K. Basher, M. Vasiliev, and K. Alameh, “Optical and chromaticity properties of metal-dielectric composite-based multilayer thin-film structures prepared by rf magnetron sputtering,” Coatings, vol. 10, no. 3, p. 251, 2020. DOI: 10.3390/ coatings10030251.][S. A. Furman and A. V. Tikhonravov, Basics of optics of multilayer systems. Singapore: World Scientific Publishing, 1992.][M. Paukshto, K. Lovetskiy, and A. Zhukov, “P-59: Dielectric Constants of Display Optical Components,” SID Symposium Digest of Technical Papers, vol. 38, no. 1, pp. 410-413, 2007. DOI: 10.1889/1.2785320.][M. Paukshto, K. Lovetsky, A. Zhukov, V. Smirnov, D. Kibalov, and G. King, “P-168: Simulation of Sub-100nm Gratings Incorporated in LCD Stack,” SID Symposium Digest of Technical Papers, vol. 37, no. 1, pp. 848-850, 2006. DOI: 10.1889/1.2433649.][A. M. Alsaad et al., “Measurement and ab initio Investigation of Structural, Electronic, Optical, and Mechanical Properties of Sputtered Aluminum Nitride Thin Films,” Frontiers in Physics, vol. 8, p. 115, 2020. DOI: 10.3389/fphy.2020.00115.][R. M. A. Azzam and N. M. Bashara, Ellipsometry and polarized light. Amsterdam: North-Holland Pub. Co., 1977.][Q. M. Al-Bataineh, A. M. Alsaad, A. A. Ahmad, and A. Telfah, “A novel optical model of the experimental transmission spectra of nanocomposite PVC-PS hybrid thin films doped with silica nanoparticles,” Heliyon, vol. 6, no. 6, p. 04 177, 2020. DOI: 10.1016/j.heliyon.2020.e04177.][A. V. Tikhonravov and M. K. Trubetskov. (2020). “OptiChar Software,” [Online]. Available: http://www.optilayer.com.][M. Born and E. Wolf, Principles of Optics. London: Pergamon Press, 1980.][T. L. Watkins and J. Fendley, “Refractive index,” Physics Education, vol. 18, no. 2, p. 56, 1983. DOI: 10.1088/0031-9120/18/2/102.][K. P. Lovetskiy, N. E. Nikolaev, and A. L. Sevastianov, “Optical Characterization of a Thin-Film Material Based on Light Intensity Measurements,” RUDN Journal of Mathematics, Information Sciences and Physics, vol. 26, no. 3, pp. 252-260, 2018. DOI: 10.22363/2312- 9735-2018-26-3-252-260.][J. A. Nelder and R. Mead, “A Simplex Method for Function Minimization,” The Computer Journal, vol. 7, no. 4, pp. 308-313, 1965. DOI: 10.1093/comjnl/7.4.308.][S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” Journal of the Optical Society of America A, vol. 13, no. 5, pp. 1106-1113, 1996. DOI: 10.1364/JOSAA.13.001106.][P. Yeh, “Extended Jones Matrix Method,” Journal of the Optical Society of America, vol. 72, no. 4, pp. 507-513, 1982. DOI: 10.1364/JOSA.72. 000507.][D. A. Yakovlev and V. G. Chigrinov, “A robust polarization-spectral method for determination of twisted liquid crystal layer parameters,” Journal of Applied Physics, vol. 102, no. 2, p. 023 510, 2007. DOI: 10. 1063/1.2756377.][A. Yariv and P. Yeh, Optical Waves in Crystals. New York: John Wiley and Sons, Inc, 2003.][F. I. Fedorov, Optics of Anisotropic Media [Optika anizotropnykh sred]. Minsk: Academy of Sciences of Belarus, 1958, in Russian.][D. W. Berreman, “Optics in Stratified and Anisotropic Media: 4 × 4 - Matrix Formulation,” The Journal of the Optical Society of America, vol. 62, no. 4, pp. 502-510, 1972.][T. F. Isaev, I. V. Kochikov, D. V. Lukyanenko, A. V. Tikhonravov, and A. G. Yagola, “Comparison of Algorithms for Determining the Thickness of Optical Coatings Online,” Computational Mathematics And Mathematical Physics, vol. 59, no. 3, pp. 465-474, 2019. DOI: 10.1134/ S0965542519030102.][J. L. M. Van Mechelen, A. B. Kuzmenko, and H. Merbold, “Stratified dispersive model for material characterization using terahertz timedomain spectroscopy,” Optics Letters, vol. 39, no. 13, pp. 3853-3856, 2014. DOI: 10.1364/OL.39.003853.][A. B. Kuzmenko, “Kramers-Kronig constrained variational analysis of optical spectra,” Review of Scientific Instruments, vol. 76, no. 8, pp. 1-9, 2005. DOI: 10.1063/1.1979470.][G. Ghosh, “Refractive Index of Quartz for Thin Film Thickness Measurement,” Optics Communications, vol. 163, pp. 95-102, 1999.]