Rational eigenvalue problems and applications to photonic crystals
(2017) In Journal of Mathematical Analysis and Applications 445(1). p.240-279- Abstract
We establish new analytic results for a general class of rational spectral problems. They arise e.g. in modelling photonic crystals whose capability to control the flow of light depends on specific features of the eigenvalues. Our results comprise a complete spectral analysis including variational principles and two-sided bounds for all eigenvalues, as well as numerical implementations. They apply to the eigenvalues between the poles where classical variational principles fail completely. In the application to multi-pole Lorentz models of permittivity functions we show, in particular, that our abstract two-sided eigenvalue estimates are optimal and we derive explicit bounds on the band gap above a Lorentz pole. A high order finite... (More)
We establish new analytic results for a general class of rational spectral problems. They arise e.g. in modelling photonic crystals whose capability to control the flow of light depends on specific features of the eigenvalues. Our results comprise a complete spectral analysis including variational principles and two-sided bounds for all eigenvalues, as well as numerical implementations. They apply to the eigenvalues between the poles where classical variational principles fail completely. In the application to multi-pole Lorentz models of permittivity functions we show, in particular, that our abstract two-sided eigenvalue estimates are optimal and we derive explicit bounds on the band gap above a Lorentz pole. A high order finite element method (FEM) is used to compute the two-sided bounds for a selection of eigenvalues for several concrete Lorentz models, e.g. polaritonic materials and multi-pole models.
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- author
- Engström, Christian LU ; Langer, Heinz and Tretter, Christiane
- publishing date
- 2017-01-01
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Eigenvalue, Finite element method, Non-linear spectral problem, Photonic crystal, Spectral gap, Variational principle
- in
- Journal of Mathematical Analysis and Applications
- volume
- 445
- issue
- 1
- pages
- 40 pages
- publisher
- Elsevier
- external identifiers
-
- scopus:84981736312
- ISSN
- 0022-247X
- DOI
- 10.1016/j.jmaa.2016.07.048
- language
- English
- LU publication?
- no
- additional info
- Funding Information: The authors gratefully acknowledge the support of the Swedish Research Council under Grant No. 621-2012-3863 . H. Langer and C. Tretter thank the Institutionen för matematik och matematisk statistik at Umeå universitet very much for the kind hospitality. C. Tretter also gratefully acknowledges the support of Schweizerischer Nationalfonds (SNF) under Grant No. 200020_146477 , and a guest professorship of the Knut och Alice Wallenbergs Stiftelse at Stockholms universitet where this work was completed. Last, but not least, we thank both referees for the very careful reading of our manuscript and their most valuable comments. Publisher Copyright: © 2016 Elsevier Inc.
- id
- 5f332ab0-5a7d-4c15-9adf-2a15ee868235
- date added to LUP
- 2023-03-24 11:08:57
- date last changed
- 2023-03-24 14:00:08
@article{5f332ab0-5a7d-4c15-9adf-2a15ee868235,
abstract = {{<p>We establish new analytic results for a general class of rational spectral problems. They arise e.g. in modelling photonic crystals whose capability to control the flow of light depends on specific features of the eigenvalues. Our results comprise a complete spectral analysis including variational principles and two-sided bounds for all eigenvalues, as well as numerical implementations. They apply to the eigenvalues between the poles where classical variational principles fail completely. In the application to multi-pole Lorentz models of permittivity functions we show, in particular, that our abstract two-sided eigenvalue estimates are optimal and we derive explicit bounds on the band gap above a Lorentz pole. A high order finite element method (FEM) is used to compute the two-sided bounds for a selection of eigenvalues for several concrete Lorentz models, e.g. polaritonic materials and multi-pole models.</p>}},
author = {{Engström, Christian and Langer, Heinz and Tretter, Christiane}},
issn = {{0022-247X}},
keywords = {{Eigenvalue; Finite element method; Non-linear spectral problem; Photonic crystal; Spectral gap; Variational principle}},
language = {{eng}},
month = {{01}},
number = {{1}},
pages = {{240--279}},
publisher = {{Elsevier}},
series = {{Journal of Mathematical Analysis and Applications}},
title = {{Rational eigenvalue problems and applications to photonic crystals}},
url = {{http://dx.doi.org/10.1016/j.jmaa.2016.07.048}},
doi = {{10.1016/j.jmaa.2016.07.048}},
volume = {{445}},
year = {{2017}},
}