Anastasiya Tyutyunnik

Bogolyubov

Chiang

D. Podgainy

Egorov

Gh. Adam

J. Buša

Leonid Sevastianov

M. Hnatič

Mikhail Malykh

Nikolai Nikolaev

Samarskiy

EPJ Web of Conferences

In general, the investigation of the electromagnetic field in an inhomogeneous waveguide doesn’t reduce to the study of two independent boundary value problems for the Helmholtz equation. We show how to rewrite the Helmholtz equations in the “Hamiltonian form” to express the connection between these two problems explicitly. The problem of finding monochromatic waves in an arbitrary waveguide is reduced to an infinite system of ordinary differential equations in a properly constructed Hilbert space. The calculations are performed in the computer algebra system Sage.</jats:p

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Diffraction of Electromagnetic Waves on a Waveguide JointMikhail Malykh1, Leonid Sevastianov1, Anastasiya Tyutyunnik1,, and Nikolai Nikolaev21Department of Applied Probability and Informatics, Peoples’ Friendship University of Russia(RUDN University)2Department of Applied Physics, Peoples’ Friendship University of Russia (RUDN University)Abstract. In general, the investigation of the electromagnetic field in an inhomogeneouswaveguide doesn’t reduce to the study of two independent boundary value problems forthe Helmholtz equation. We show how to rewrite the Helmholtz equations in the “Hamil-tonian form” to express the connection between these two problems explicitly. The prob-lem of finding monochromatic waves in an arbitrary waveguide is reduced to an infinitesystem of ordinary differential equations in a properly constructed Hilbert space. Thecalculations are performed in the computer algebra system Sage.1 IntroductionThe mathematical theory of the waveguide systems, based on the works of A.N. Tikhonov,A.A. Samarsky [1] and A.G. Sveshnikov [2], created in the middle of the last century. It has become animportant chapter of the computational electrodynamics. The solution of waveguide diffraction prob-lems is necessary for the design of integrated optical and fiber optic devices. The linear dimensionsof these devices along different directions may differ by several orders of magnitude, for example,the radius of the Lüneburg thin-film waveguide lens exceeds its thickness by more than 104 times.Such a difference in size leads to insurmountable problems in the discretization of continuous models.The aim of our research is to overcome these difficulties. For example, the height of the Lüneburgwaveguide lens is less than its diameter by 104 times, which inevitably leads to specific computa-tional peculiarities [3, 4]. Therefore the development of modern program packages for the calculationof fields in the closed waveguides with extremal parameters remains an important challenge for thecomputational electrodynamics.2 Reduction of the system of Maxwell’s equations to two scalar problemsThe investigation of the electromagnetic field in a regular waveguide filled with a homogeneous sub-stance is reduced to the investigation of two independent boundary value problems for the Helmholtzequation [5]. In the case of a waveguide filled with an inhomogeneous substance, a relationship arisesbetween the modes of these two problems, which in the numerical experiments can not always be fullytaken into account. The first papers on the mathematical theory of the waveguide [6] had in mind thegeneralization of the Fourier method, therefore the question of reducing the system of Maxwell’se-mail: tyutyunnik_aa@pfur.ru© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).EPJ Web of Conferences 173, 02014 (2018) https://doi.org/10.1051/epjconf/201817302014Mathematical Modeling and Computational Physics 2017equations to two scalar equations was in the shadow of the theorem on the completeness of the systemof normal modes. Meanwhile, writing this pair is not at all difficult, if we go over to the “Hamiltonianform” of the Helmholtz equations.Consider a waveguide of a constant simply connected section S with ideally conducting walls.We will not make any starting assumptions with respect to the filling , µ of this waveguide. Theaxis Oz is directed along the cylinder axis, the normal to ∂S will be denoted as n, the tangent vectorperpendicular to ez as τ. For a basis we take the system of Maxwell’s equations, from which weexclude Ez and Hz. For brevity’s sake let us assume ∇ = (∂x, ∂y)T and ∇′ = (−∂y, ∂x)T . By excludingthe longitudinal components from Maxwell’s equations, we getez × ∂z H⊥ + ∇ 1ikµ (curl E⊥)z × ez + ik E⊥ = 0,ez × ∂z E⊥ − ∇ 1ik (curlH⊥)z × ez − ikµ H⊥ = 0.(1)At the boundary, the conditions of ideal conductivity are satisfied toE⊥ × n = 0, H⊥ · n = 0, Ez = 0. (2)A two-dimensional analogue of the Helmholtz decomposition makes it possible, without loss of gen-erality, to seek a solution of the system (1) in the formE⊥ = ∇ue + ∇′ve, H⊥ = ∇vh + ∇′uh.The four scalar functions introduced here will be called potentials, and we’ll always assume that theysatisfy the boundary conditions ue = uh = n · ∇ve = n · ∇vh = 0 at the boundary of the waveguide.Substituting these expressions into (1), we obtain∇′(∂zvh − 1ikµ∆ve)+ ik∇′ve − ∇∂zuh + ik∇ue = 0,∇′(∂zue +1ik∆uh)− ikµ∇′uh − ∇∂zve − ikµ∇vh = 0.(3)The first two of the three boundary conditions (2) are automatically satisfied. With the usual choice ofbasis, the third condition follows from the Dirichlet conditions imposed on the function uh. If  and µare constants, then the system (3) can be rewritten as a pair of uncoupled Helmholtz equations withrespect to ve, uh.The presentation of a whole series of problems in the mathematical theory of waveguides can becarried out uniformly if we consider the field as a function of zwith values in a suitably chosen Hilbertspace [7]. In other words, we will use the weak formulation of the problem of finding the potentialsin the general case.We will consider the solutionw = (vh, ve, ue, uh)of the system (3) as a map of z ∈ R to a Hilbert spaceH ={vh, ve, ue, uh ∈ W22 (S ): ue, uh, n · ∇ve, n · ∇vh = 0 on ∂S}.In this space the equations of the system (3) can be written as follows∂zb(w˜, w) + ika(w˜, w) +1ikc(w˜, w) = 0, ∀w˜ ∈ H.2EPJ Web of Conferences 173, 02014 (2018) https://doi.org/10.1051/epjconf/201817302014Mathematical Modeling and Computational Physics 2017equations to two scalar equations was in the shadow of the theorem on the completeness of the systemof normal modes. Meanwhile, writing this pair is not at all difficult, if we go over to the “Hamiltonianform” of the Helmholtz equations.Consider a waveguide of a constant simply connected section S with ideally conducting walls.We will not make any starting assumptions with respect to the filling , µ of this waveguide. Theaxis Oz is directed along the cylinder axis, the normal to ∂S will be denoted as n, the tangent vectorperpendicular to ez as τ. For a basis we take the system of Maxwell’s equations, from which weexclude Ez and Hz. For brevity’s sake let us assume ∇ = (∂x, ∂y)T and ∇′ = (−∂y, ∂x)T . By excludingthe longitudinal components from Maxwell’s equations, we getez × ∂z H⊥ + ∇ 1ikµ (curl E⊥)z × ez + ik E⊥ = 0,ez × ∂z E⊥ − ∇ 1ik (curlH⊥)z × ez − ikµ H⊥ = 0.(1)At the boundary, the conditions of ideal conductivity are satisfied toE⊥ × n = 0, H⊥ · n = 0, Ez = 0. (2)A two-dimensional analogue of the Helmholtz decomposition makes it possible, without loss of gen-erality, to seek a solution of the system (1) in the formE⊥ = ∇ue + ∇′ve, H⊥ = ∇vh + ∇′uh.The four scalar functions introduced here will be called potentials, and we’ll always assume that theysatisfy the boundary conditions ue = uh = n · ∇ve = n · ∇vh = 0 at the boundary of the waveguide.Substituting these expressions into (1), we obtain∇′(∂zvh − 1ikµ∆ve)+ ik∇′ve − ∇∂zuh + ik∇ue = 0,∇′(∂zue +1ik∆uh)− ikµ∇′uh − ∇∂zve − ikµ∇vh = 0.(3)The first two of the three boundary conditions (2) are automatically satisfied. With the usual choice ofbasis, the third condition follows from the Dirichlet conditions imposed on the function uh. If  and µare constants, then the system (3) can be rewritten as a pair of uncoupled Helmholtz equations withrespect to ve, uh.The presentation of a whole series of problems in the mathematical theory of waveguides can becarried out uniformly if we consider the field as a function of zwith values in a suitably chosen Hilbertspace [7]. In other words, we will use the weak formulation of the problem of finding the potentialsin the general case.We will consider the solutionw = (vh, ve, ue, uh)of the system (3) as a map of z ∈ R to a Hilbert spaceH ={vh, ve, ue, uh ∈ W22 (S ): ue, uh, n · ∇ve, n · ∇vh = 0 on ∂S}.In this space the equations of the system (3) can be written as follows∂zb(w˜, w) + ika(w˜, w) +1ikc(w˜, w) = 0, ∀w˜ ∈ H.Here the forms a, b, c are bounded in H, therefore, by the Riesz theorem, the above condition can berewritten in terms of the operators generated by these bilinear forms. Thus any electromagnetic fieldin the waveguide satisfies the system of ordinary differential equationsBdwdz+ ikAw +1ikCw = 0 (4)in the Hilbert space H. Since the embedding of the Sobolev spaces is compact, the operators A and Bare compact, and B is also self-adjoint.3 Normal modes of a waveguideSimilar to the finite-dimensional case, any solution of the system (4) is a superposition of normalmodes, that is, solutions that depend on z as eikβz. To calculate these modes we obtain an eigenvalueproblem with respect to the spectral parameter β:βBw = Aw − 1k2Cw. (5)The discreteness of the spectrum and the completeness of the system of normal waves are obtained asa corollary of the theorem of M.V. Keldysh about the operator pencil [8].The derivation of the numerical solution of this eigenvalue problem is divided into two steps:• discretization, that is, truncation of the infinite matrices (the Galerkin method),• solving the resulting algebraic eigenvalue problem.In the standard approach to calculate the modes, we need to select some components of the vectorsE and H and write an approximate problem for them using the finite element method or the finitedifference method [9]. Our computing experiments have shown that the system (4) is more convenientfor applying the Galerkin method because the truncation of operators A, B and C in this basis give usat once the algebraic eigenvalues problem in the formβBˆw = Aˆw − 1k2Cˆw. (6)The work proposed by us contains, in our opinion, an elegant and effective generalization of theincomplete Galerkin method [10]. In practice we calculate the matrix elements numerically withsmall errors but preserve all symmetry of the waveguide. As the 2nd step, the standard approachsuggests using the Krylov Subspace Methods [11], but in modern computer algebra system Sage [12]we may work over the field of algebraic numbers without any numerical errors.This approach to computing eigenmodes was tested on several waveguides [13]. Our computingexperiments have shown that in such a way we can calculate all the running modes (that is, modeswith real eigenvalues β) and the small locked modes with correct multiplicities. The last circumstanceis unexpected and indicates new application for symbolical calculations in numerical analysis.4 The joint of two waveguidesLet the filling , µ of the waveguide have a jump at z = 0, so this plane is the joint of two waveg-uides. Any solution of the differential equation (4) at z < 0 and at z > 0 can be represented as thesuperposition of normal modes. The problem about the diffraction of the wave∑β>0Fnw(1)n eikβ(1)n z3EPJ Web of Conferences 173, 02014 (2018) https://doi.org/10.1051/epjconf/201817302014Mathematical Modeling and Computational Physics 2017at the joint z = 0 is reduced to the following: for the given waveguide, the functions 1,2(x, y) andµ1,2(x, y) and the numbers F1, . . . , find numbers Rn, Tn such that∑β>0Fnw(1)n eikβ(1)n z +∑β<0Rnw(1)n eikβ(1)n z =∑β>0Tnw(2)n eikβ(2)n zat z = 0. To solve this problem we can calculate the normal waves as above and solve a linear systemwith respect to R and T using the standard routine in computer algebra system (CAS).5 ConclusionWe list the main results of our work.• We can introduce four potentials, by means of which the system of Maxwell’s equations in thewaveguide reduces to an infinite system of linear ODEs and is convenient for numerical analysis bythe Galerkin method.• To compute the normal waves we can use CAS and save multiplicities and other qualitative proper-ties of eigenvalues.• The problem about diffraction on joint is reduced to a system of linear algebraic equations whichcan also be solved symbolically.AcknowledgmentsThe calculations in this article are performed in the computer algebra system Sage [12]. The publica-tion was prepared with the support of the “RUDN University Program 5-100”. The work was partiallysupported by RFBF grants No 15-07-08795, No 16-07-00556.References[1] A.A. Samarskiy, Izbrannyye trudy (Moscow: MAX Press, 2003) 531 pp.[2] I. E. Mogilevskii, A. G. Sveshnikov , Matematicheskiye zadachi teorii difraktsii (Moscow: Fac-ulty of Physics MSU, 2010) 197 pp.[3] A.A. Egorov, L.A. Sevastyanov, Kvantovaya Elektronika 39 (6), 566–574 (2009)[4] A.A. Egorov et al., Kvantovaya Elektronika 40 (9), 830–836 (2010)[5] W.C. Chew, Lectures on theory of microwave and optical waveguides (http://wcchew.ece.illinois.edu/chew/course/tgwAll20121211.pdf, 2012) 362 pp.[6] A.A. Samarskiy, A.N. Tikhonov, Zhurnal tehnicheskoy fiziki 18 (7), 959–970 (1948)[7] A.N. Bogolyubov, M.D. Malykh, Computations mathematics and mathematical physics 43 (4),560–563 (2003)[8] A.N. Bogolyubov, A.L. Delitsyn, A.G. Sveshnikov, Computational mathematics and mathemat-ical physics 38 (11), 1815–1823 (1998)[9] K.S. Chiang, Optical and Quantum Electronics 26, S113–S134 (1994)[10] A.N. Bogolyubov, A.I. Erokhin, I.E. Mogilevskii, Computational Mathematics and Mathemati-cal Physics 52 (6), 932–936 (2012)[11] Y. Saad, Numerical methods for large eigenvalue problems (Philadelphia, PA : Society for In-dustrial and Applied Mathematics, 2011) 276 pp.[12] SageMath, http://www.sagemath.org[13] M. D. Malykh, L. A. Sevastianov, A. A. Tiutiunnik, N. E. Nikolaev, Journal of ElectromagneticWaves and Applications, 1–13 (2017)4EPJ Web of Conferences 173, 02014 (2018) https://doi.org/10.1051/epjconf/201817302014Mathematical Modeling and Computational Physics 2017

Diffraction of Electromagnetic Waves on a Waveguide Joint

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Diffraction of Electromagnetic Waves on a Waveguide Joint

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