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Reliable hp finite element computations of scattering resonances in nano optics
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.ORCID iD: 0000-0002-0143-5554
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Eigenfrequencies are commonly studied in wave propagation problems, as they are important in the analysis of closed cavities such as a microwave oven. For open systems, energy leaks into infinity and therefore scattering resonances are used instead of eigenfrequencies. An interesting application where resonances take an important place is in whispering gallery mode resonators.

The objective of the thesis is the reliable and accurate approximation of scattering resonances using high order finite element methods. The discussion focuses on the electromagnetic scattering resonances in metal-dielectric nano-structures using a Drude-Lorentz model for the description of the material properties. A scattering resonance pair satisfies a reduced wave equationand an outgoing wave condition. In this thesis, the outgoing wave condition is replaced by a Dirichlet-to-Neumann map, or a Perfectly Matched Layer. For electromagnetic waves and for acoustic waves, the reduced wave equation is discretized with finite elements. As a result, the scattering resonance problem is transformed into a nonlinear eigenvalue problem.

In addition to the correct approximation of the true resonances, a large number of numerical solutions that are unrelated to the physical problem are also computed in the solution process. A new method based on a volume integral equation is developed to remove these false solutions.

The main results of the thesis are a novel method for removing false solutions of the physical problem, efficient solutions of non-linear eigenvalue problems, and a new a-priori based refinement strategy for high order finite element methods. The overall material in the thesis translates into a reliable and accurate method to compute scattering resonances in physics and engineering.

Place, publisher, year, edition, pages
Umeå: Umeå Universitet , 2019. , p. 35
Series
Research report in mathematics, ISSN 1653-0810 ; 67
Keywords [en]
Scattering resonances, Helmholtz problems, pseudospectrum, Lippmann-Schwinger equation, finite element methods, nonlinear eigenvalue problems, spurious solutions
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-159154ISBN: 978-91-7855-076-0 (print)OAI: oai:DiVA.org:umu-159154DiVA, id: diva2:1316711
Public defence
2019-06-13, MA121, MIT-huset, Umeå, 13:00 (English)
Opponent
Supervisors
Available from: 2019-05-23 Created: 2019-05-20 Last updated: 2019-05-21Bibliographically approved
List of papers
1. On spurious solutions in finite element approximations of resonances in open systems
Open this publication in new window or tab >>On spurious solutions in finite element approximations of resonances in open systems
2017 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 74, no 10, p. 2385-2402Article in journal (Refereed) Published
Abstract [en]

In this paper, we discuss problems arising when computing resonances with a finite element method. In the pre-asymptotic regime, we detect for the one dimensional case, spurious solutions in finite element computations of resonances when the computational domain is truncated with a perfectly matched layer (PML) as well as with a Dirichlet-to-Neumann map (DtN). The new test is based on the Lippmann–Schwinger equation and we use computations of the pseudospectrum to show that this is a suitable choice. Numerical simulations indicate that the presented test can distinguish between spurious eigenvalues and true eigenvalues also in difficult cases.

Place, publisher, year, edition, pages
Elsevier, 2017
Keywords
Scattering resonances, Lippmann–Schwinger equation, Nonlinear eigenvalue problems, Acoustic resonator, Dielectric resonator, Bragg resonator
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-138096 (URN)10.1016/j.camwa.2017.07.020 (DOI)000415908400013 ()
Funder
Swedish Research Council, 621-2012-3863
Available from: 2017-08-09 Created: 2017-08-09 Last updated: 2019-05-20Bibliographically approved
2. Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map
Open this publication in new window or tab >>Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map
2018 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 330, p. 177-192Article in journal (Refereed) Published
Abstract [en]

We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains. We consider a variational formulation and show that the spectrum consists of isolated eigenvalues of finite multiplicity that only can accumulate at infinity. The proposed method is based on a high order finite element discretization combined with a specialization of the Tensor Infinite Arnoldi method (TIAR). Using Toeplitz matrices, we show how to specialize this method to our specific structure. In particular we introduce a pole cancellation technique in order to increase the radius of convergence for computation of eigenvalues that lie close to the poles of the matrix-valued function. The solution scheme can be applied to multiple resonators with a varying refractive index that is not necessarily piecewise constant. We present two test cases to show stability, performance and numerical accuracy of the method. In particular the use of a high order finite element discretization together with TIAR results in an efficient and reliable method to compute resonances.

Place, publisher, year, edition, pages
Amsterdam: Elsevier, 2018
Keywords
Nonlinear eigenvalue problems, Helmholtz problem, Scattering resonances, Dirichlet-to-Neumann map, Arnoldi's method, Matrix functions
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-138325 (URN)10.1016/j.cam.2017.08.012 (DOI)000415783000014 ()
Available from: 2017-08-21 Created: 2017-08-21 Last updated: 2019-05-20Bibliographically approved
3. Computation of scattering resonances in absorptive and dispersive media with applications to metal-dielectric nano-structures
Open this publication in new window or tab >>Computation of scattering resonances in absorptive and dispersive media with applications to metal-dielectric nano-structures
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we consider scattering resonance computations in optics when the resonators consist of frequency dependent and lossy materials, such as metals at optical frequencies. The proposed computational approach combines a novel hp-FEM strategy, based on dispersion analysis for complex frequencies, with a fast implementation of the nonlinear eigenvalue solver NLEIGS.Numerical computations illustrate that the pre-asymptotic phase is significantly reduced compared to standard uniform h and p strategies. Moreover, the efficiency grows with the refractive index contrast, which makes the new strategy highly attractive for metal-dielectric structures. The hp-refinement strategy together with the efficient parallel code result in highly accurate approximations and short runtimes on multi processor platforms.

Keywords
Plasmon resonance, Resonance modes, Nonlinear eigenvalue problems, Helmholtz problem, PML, Dispersion analysis, leaky modes, resonant states, quasimodes, quasi-normal modes
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-159151 (URN)
Available from: 2019-05-20 Created: 2019-05-20 Last updated: 2019-05-21
4. Removal of spurious solutions encountered in Helmholtz scattering resonance computations in R^d
Open this publication in new window or tab >>Removal of spurious solutions encountered in Helmholtz scattering resonance computations in R^d
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we consider a sorting scheme for the removal of spurious scattering resonant pairs in two-dimensional electromagnetic problems and in three-dimensional acoustic problems. The novel sorting scheme is based on a Lippmann-Schwinger type of volume integral equation and can therefore be applied to graded material properties as well as piece-wise constant material properties. For TM/TE polarized electromagnetic waves and for acoustic waves, we compute first approximations of scattering resonances with finite elements. Then, we apply the novel sorting scheme to the computed eigenpairs and use it to remove spurious solutions in electromagnetic and acoustic scattering resonances computations at low computational cost. Several test cases with Drude-Lorentz dielectric resonators as well as with graded material properties are considered.

Keywords
plasmon resonance, acoustic scattering resonances, resonance modes, nonlinear eigenvalue problems, Helmholtz problem, pseudospectrum, PML, DtN, leaky modes, resonant states, quasi-normal modes
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-159153 (URN)
Available from: 2019-05-20 Created: 2019-05-20 Last updated: 2019-06-13Bibliographically approved

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