Solving the inverse problem for determining the optical characteristics of materials

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.В работе изложена методология определения оптических и физических свойств анизотропных тонкоплёночных материалов. Такой подход позволяет в дальнейшем проектировать многослойные тонкоплёночные покрытия с заданными свойствами. Сформулирована обратная задача определения тензора диэлектрической проницаемости и толщины тонкой плёнки, нанесённой на стеклянную подложку, с известными оптическими свойствами и толщиной. Предварительная информация о принадлежности тонкоплёночного покрытия к определённому классу позволяет значительно сократить время расчёта и увеличить точность определения тензора диэлектрической проницаемости на всём исследуемом интервале длин волн и толщины плёнки в точке измерения отражения и пропускания. В зависимости от поставленных целей возможна постановка и, следовательно, решение различных обратных задач: o определение тензора диэлектрической проницаемости и уточнение толщины толстой (до 1 см) подложки, часто изотропной; o определение тензора диэлектрической проницаемости тонкой изотропной или анизотропной плёнки, нанесённой на подложку, с известными оптическими свойствами. Сложность решения каждой из задач весьма различна и каждая требует своего определённого набора измеренных входных данных. Окончательные результаты решения обратной задачи верифицируются с помощью сравнения вычисленных коэффициентов пропускания и отражения с измеренными для произвольных углов падения и отражения

Symbolic-numeric implementation of the four potential method for calculating normal modes of square electromagnetic waveguide with rectangular insert

In this paper, the Maple computer algebra system is used to construct a symbolic-numeric implementation of the method for calculating normal modes of square closed waveguides in a vector formulation. The method earlier proposed by Malykh et al. [M.D. Malykh, L.A. Sevastianov, A.A. Tiutiunnik, N.E. Nikolaev. On the representation of electromagnetic fields in closed waveguides using four scalar potentials // Journal of Electromagnetic Waves and Applications, 32 (7), 886-898 (2018)] will be referred to as the method of four potentials. The Maple system is used at all stages of treating the system of differential equations for four potentials: the generation of the Galerkin basis, the substitution of approximate solution into the system under study, the formulation of a computational problem, and its approximate solution. The paper presents the results of the verification method. © 2019 Author(s)

Symbolic Investigation of Eigenvectors for General Solution of a System of ODEs with a Symbolic Coefficient Matrix

Abstract: This paper investigates the problem of symbolic representation for the general solution of a system of ordinary differential equations (ODEs) with symbolically defined constant coefficients in the case where some symbolic constants can vanish. In addition, the symbolic representation of eigenvectors for the system’s coefficient matrix is not unique. It is shown that standard procedures of computer algebra systems search for specific symbolic representations of eigenvectors while ignoring the other symbolic representations. In turn, the eigenvectors found by a computer algebra system can be inadequate for constructing numerical algorithms based on them, which is demonstrated by an example. We propose an algorithm for finding various symbolic representations of eigenvectors for symbolically defined matrices. This paper considers a particular system of ODEs obtained by investigating some solutions of Maxwell’s equations; however, the proposed algorithm can be applied to an arbitrary system with a normal matrix of coefficients. © 2021, Pleiades Publishing, Ltd

Calculation of normal modes of the closed waveguides in general vector case

The article is devoted to the calculation of normal modes of the closed waveguides with an arbitrary filling , μ in the system of computer algebra Sage. Maxwell equations in the cylinder are reduced to the system of two bounded Helmholtz equations, the notion of weak solution of this system is given and then this system is investigated as a system of ordinary differential equations. The normal modes of this system are an eigenvectors of a matrix pencil. We suggest to calculate the matrix elements approximately and to truncate the matrix by usual way but further to solve the truncated eigenvalue problem exactly in the field of algebraic numbers. This approach allows to keep the symmetry of the initial problem and in particular the multiplicity of the eigenvalues. In the work would be presented some results of calculations. © 2018 SPIE

Calculation of normal modes of the closed waveguides in general vector case

The article is devoted to the calculation of normal modes of the closed waveguides with an arbitrary filling , μ in the system of computer algebra Sage. Maxwell equations in the cylinder are reduced to the system of two bounded Helmholtz equations, the notion of weak solution of this system is given and then this system is investigated as a system of ordinary differential equations. The normal modes of this system are an eigenvectors of a matrix pencil. We suggest to calculate the matrix elements approximately and to truncate the matrix by usual way but further to solve the truncated eigenvalue problem exactly in the field of algebraic numbers. This approach allows to keep the symmetry of the initial problem and in particular the multiplicity of the eigenvalues. In the work would be presented some results of calculations. © 2018 SPIE

Symbolic-Numeric Study of Geometric Properties of Adiabatic Waveguide Modes

The eikonal equation links wave optics to ray optics. In the present work, we show that the eikonal equation is also valid for an approximate description of the phase of vector fields describing guided-wave propagation in inhomogeneous waveguide structures in the adiabatic approximation. The main result of the work was obtained using the model of adiabatic waveguide modes. Highly analytical solution procedure makes it possible to obtain symbolic or symbolic-numerical expressions for vector fields of guided modes. Making use of advanced computer algebra systems, we describe fundamental properties of adiabatic modes in symbolic form. Numerical results are also obtained by means of computer algebra systems. © 2020, Springer Nature Switzerland AG

On the representation of electromagnetic fields in closed waveguides using four scalar potentials

The investigation of the electromagnetic field in a regular waveguide filled with a homogeneous substance reduces to the study of two independent boundary value problems for the Helmholtz equation. In the case of a waveguide filled with an inhomogeneous substance, a relationship arises between the modes of these two problems, which in numerical experiments can not always be fully taken into account. In this paper, we will show how to rewrite the Helmholtz equations in the “Hamiltonian form” to express this connection explicitly. In this case, the problem of finding monochromatic waves in a waveguide with arbitrary filling will be reduced to an infinite system of ordinary differential equations in a properly constructed Hilbert space. The results of numerical experiments on finding normal waves, realized in the computer algebra system Sage, are presented. © 2017 Informa UK Limited, trading as Taylor & Francis Group

Scalar Product in the Space of Waveguide Modes of an Open Planar Waveguide

To implement the method of adiabatic waveguide modes for modeling the propagation of polarized monochromatic electromagnetic radiation in irregular integrated optics structures it is necessary to expand the desired solution in basic adiabatic waveguide modes. This expansion requires the use of the scalar product in the space of waveguide vector fields of integrated optics waveguide. This work solves the first stage of this problem - the construction of the scalar product in the space of vector solutions of the eigenmode problem (classical and generalized) waveguide modes of an open planar waveguide. In constructing the mentioned sesquilinear form, we used the Lorentz reciprocity principle of waveguide modes and tensor form of the Ostrogradsky-Gauss theorem. © Owned by the authors

The application of Helmholtz decomposition method to investigation of multicore fibers and their application in next-generation communications systems

New optical multicore fibers use their spatial properties in the designs of next-generation systems. To investigate light propagation in such fiber waveguides we use Helmholtz decomposition method. We consider a waveguide having the constant cross-section S with ideally conducting walls. We assume that the filling of waveguide does not change along its axis and is described by the piecewise continuous functions ɛ and μ defined on the waveguide cross section. We show that it is possible to make a substitution, which allows dealing only with continuous functions. Instead of discontinuous cross components of the electromagnetic field E and H we propose to use four potentials ue, uh and ve, vh. Generalizing the Thikhonov-Samarskii theorem, we have proved that any field in the waveguide allows such representation, if we consider the potentials ue, uh as elements of the Sobolev space (Formula presented) and the potentials ue, uh as elements of the Sobolev space W1/2(S). If ε and μ are piecewise constant functions, then in terms of four potentials the Maxwell equations reduce to a pair of Helmholtz equations. This fact means that a few dielectric waveguides placed between ideally conducting walls can be described by a scalar boundary problem. This statement offers a new approach to the investigation of spectral properties of waveguides. First, we can prove the completeness of the system of the normal waves in closed waveguides using standard functional spaces. Second, we can propose a new technique for calculating the normal waves using standard finite elements. © Springer Nature Switzerland AG 2018

Scalar Product in the Space of Waveguide Modes of an Open Planar Waveguide

To implement the method of adiabatic waveguide modes for modeling the propagation of polarized monochromatic electromagnetic radiation in irregular integrated optics structures it is necessary to expand the desired solution in basic adiabatic waveguide modes. This expansion requires the use of the scalar product in the space of waveguide vector fields of integrated optics waveguide. This work solves the first stage of this problem – the construction of the scalar product in the space of vector solutions of the eigenmode problem (classical and generalized) waveguide modes of an open planar waveguide. In constructing the mentioned sesquilinear form, we used the Lorentz reciprocity principle of waveguide modes and tensor form of the Ostrogradsky-Gauss theorem