Abstract
The vector electromagnetic problem for eigenwaves of optical fibers, originally formulated on the whole plane, is equivalently reduced to a linear parametric eigenvalue problem posed in a circle, convenient for numerical solution. The study of the solvability of this problem is based on the spectral theory of compact self-adjoint operators. Asymptotic properties of the dispersion curves and their smoothness are investigated for the new formulation of the problem. A numerical method based on finite element approximations combined with an exact non-reflecting boundary condition is developed. Error estimates for approximating eigenvalues and eigenfunctions are derived.
Funding statement: This paper has been supported by the Kazan Federal University Strategic Academic Leadership Program.
References
[1] A. Bamberger and A. S. Bonnet, Mathematical analysis of the guided modes of an optical fiber, SIAM J. Math. Anal. 21 (1990), no. 6, 1487–1510. 10.1137/0521082Search in Google Scholar
[2] Y. Chai, W. Li, T. Li, Z. Gong and X. You, Analysis of underwater acoustic scattering problems using stable node-based smoothed finite element method, Eng. Anal. Bound. Elem. 72 (2016), 27–41. 10.1016/j.enganabound.2016.08.005Search in Google Scholar
[3] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics Appl. Math. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2002. 10.1137/1.9780898719208Search in Google Scholar
[4] R. Z. Dautov and E. M. Karchevskii, Solution of the vector eigenmode problem for cylindrical dielectric waveguides based on a nonlocal boundary a condition, Comput. Math. Math. Phys. 42 (2002), 1012–1027. Search in Google Scholar
[5] R. Z. Dautov and E. M. Karchevskii, Error estimates for a Galerkin method with perturbations for spectral problems of the theory of dielectric waveguides, Lobachevskii J. Math. 37 (2016), no. 5, 610–625. 10.1134/S1995080216050024Search in Google Scholar
[6] R. Z. Dautov and E. M. Karchevskii, Numerical modeling of optical fibers using the finite element method and an exact non-reflecting boundary condition, Comput. Methods Appl. Math. 18 (2018), no. 4, 581–601. 10.1515/cmam-2017-0049Search in Google Scholar
[7] R. Z. Dautov, E. M. Karchevskii and G. P. Kornilov, A numerical method for determining the dispersion curves and natural waves of optical waveguides, Comput. Math. Math. Phys. 45 (2005), 2119–2134. Search in Google Scholar
[8] S. Eriksson and J. Nordström, Exact non-reflecting boundary conditions revisited: Well-posedness and stability, Found. Comput. Math. 17 (2017), no. 4, 957–986. 10.1007/s10208-016-9310-3Search in Google Scholar
[9] S. Falletta and G. Monegato, An exact non reflecting boundary condition for 2D time-dependent wave equation problems, Wave Motion 51 (2014), no. 1, 168–192. 10.1016/j.wavemoti.2013.06.001Search in Google Scholar
[10] Y. He, M. Min and D. P. Nicholls, A spectral element method with transparent boundary condition for periodic layered media scattering, J. Sci. Comput. 68 (2016), no. 2, 772–802. 10.1007/s10915-015-0158-5Search in Google Scholar
[11] P. Joly and C. Poirier, Mathematical analysis of electromagnetic open waveguides, RAIRO Modél. Math. Anal. Numér. 29 (1995), no. 5, 505–575. 10.1051/m2an/1995290505051Search in Google Scholar
[12] P. Joly and C. Poirier, A numerical method for the computation of electromagnetic modes in optical fibres, Math. Methods Appl. Sci. 22 (1999), no. 5, 389–447. 10.1002/(SICI)1099-1476(19990325)22:5<389::AID-MMA31>3.0.CO;2-ESearch in Google Scholar
[13] J. B. Keller and D. Givoli, Exact nonreflecting boundary conditions, J. Comput. Phys. 82 (1989), no. 1, 172–192. 10.1016/0021-9991(89)90041-7Search in Google Scholar
[14] M. A. Krasnosel’skii, G. M. Vaĭnikko, P. P. Zabreĭko, Y. B. Rutitskii and V. Y. Stetsenko, Approximate Solution of Operator Equations, Wolters-Noordhoff, Groningen, 1972. 10.1007/978-94-010-2715-1Search in Google Scholar
[15] M. Zlámal, Curved elements in the finite element method. I, SIAM J. Numer. Anal. 10 (1973), 229–240. 10.1137/0710022Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Robust Iterative Solvers for Gao Type Nonlinear Beam Models in Elasticity
- Using Complete Monotonicity to Deduce Local Error Estimates for Discretisations of a Multi-Term Time-Fractional Diffusion Equation
- Accurate Full-Vectorial Finite Element Method Combined with Exact Non-Reflecting Boundary Condition for Computing Guided Waves in Optical Fibers
- Robust Hybrid High-Order Method on Polytopal Meshes with Small Faces
- A Totally Relaxed, Self-Adaptive Subgradient Extragradient Method for Variational Inequality and Fixed Point Problems in a Banach Space
- Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization
- Dual System Least-Squares Finite Element Method for a Hyperbolic Problem
- Artificial Compressibility Methods for the Incompressible Navier–Stokes Equations Using Lowest-Order Face-Based Schemes on Polytopal Meshes
- Numerical Analysis of a Finite Element Approximation to a Level Set Model for Free-Surface Flows
- Inexact Inverse Subspace Iteration with Preconditioning Applied to Quadratic Matrix Polynomials
- Convergence Theory for IETI-DP Solvers for Discontinuous Galerkin Isogeometric Analysis that is Explicit in ℎ and 𝑝
- Poincaré Inequality for a Mesh-Dependent 2-Norm on Piecewise Linear Surfaces with Boundary