Planewave expansion methods for photonic crystal fibres
Photonic crystal fibres are novel optical devices that can be designed to guide light of a particular frequency. In this paper the performance of planewave expansion methods for computing spectral gaps and trapped eigenmodes in photonic crystal fibres is carefully analysed. The occurrence of discont...
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| Published in | Applied numerical mathematics Vol. 63; pp. 88 - 104 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Elsevier B.V
01.01.2013
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| Abstract | Photonic crystal fibres are novel optical devices that can be designed to guide light of a particular frequency. In this paper the performance of planewave expansion methods for computing spectral gaps and trapped eigenmodes in photonic crystal fibres is carefully analysed. The occurrence of discontinuous coefficients in the governing equation means that exponential convergence is impossible due to the limited regularity of the eigenfunctions. We show through a numerical convergence study and rigorous analysis on a simplified problem that the convergence of the planewave expansion method is controlled by the regularity of the eigenfunctions and performs thus no better (but also no worse) than we would expect (non-adaptive) finite element methods to perform, both in terms of error convergence and computational efficiency. We also consider the performance of two variants of the planewave expansion method: (a) coupling the planewave expansion method with a regularisation technique where the discontinuous coefficients in the governing equation are approximated by smooth functions, and (b) approximating the Fourier coefficients of the discontinuous coefficients in the governing equation. There is no evidence that regularisation improves the planewave expansion method, but with the correct choice of parameters both variants can be used efficiently without adding significant errors. |
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| AbstractList | Photonic crystal fibres are novel optical devices that can be designed to guide light of a particular frequency. In this paper the performance of planewave expansion methods for computing spectral gaps and trapped eigenmodes in photonic crystal fibres is carefully analysed. The occurrence of discontinuous coefficients in the governing equation means that exponential convergence is impossible due to the limited regularity of the eigenfunctions. We show through a numerical convergence study and rigorous analysis on a simplified problem that the convergence of the planewave expansion method is controlled by the regularity of the eigenfunctions and performs thus no better (but also no worse) than we would expect (non-adaptive) finite element methods to perform, both in terms of error convergence and computational efficiency. We also consider the performance of two variants of the planewave expansion method: (a) coupling the planewave expansion method with a regularisation technique where the discontinuous coefficients in the governing equation are approximated by smooth functions, and (b) approximating the Fourier coefficients of the discontinuous coefficients in the governing equation. There is no evidence that regularisation improves the planewave expansion method, but with the correct choice of parameters both variants can be used efficiently without adding significant errors. |
| Author | Norton, R.A. Scheichl, R. |
| Author_xml | – sequence: 1 givenname: R.A. surname: Norton fullname: Norton, R.A. email: Richard.Norton@latrobe.edu.au organization: Department of Mathematics and Statistics, La Trobe University, Bundoora, Victoria 3086, Australia – sequence: 2 givenname: R. surname: Scheichl fullname: Scheichl, R. email: R.Scheichl@bath.ac.uk organization: Department of Mathematical Sciences, University of Bath, BA2 7AY, United Kingdom |
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| Cites_doi | 10.1016/j.physleta.2004.05.030 10.1364/OE.8.000173 10.1103/PhysRevB.48.8434 10.1364/OPEX.12.000069 10.1137/040616875 10.1103/PhysRevB.45.13962 10.1137/S0036142902403012 10.1016/j.cam.2006.01.042 10.1137/050636887 10.1137/090756545 10.1103/PhysRevB.71.195108 10.1364/OPEX.13.006937 10.1364/OE.11.002854 |
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| Keywords | Spectral approximation Fourier methods Error analysis Sampling Smoothing Planewave expansion method |
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Anal. doi: 10.1137/050636887 contributor: fullname: Buffa – volume: 47 start-page: 4356 year: 2010 ident: 10.1016/j.apnum.2012.09.008_br0160 article-title: Convergence analysis of planewave expansion methods for 2d Schrödinger operators with discontinuous periodic potentials publication-title: SIAM J. Numer. Anal. doi: 10.1137/090756545 contributor: fullname: Norton – volume: 71 year: 2005 ident: 10.1016/j.apnum.2012.09.008_br0180 article-title: Adaptive curvilinear coordinates in a plane-wave solution of Maxwellʼs equations in photonic crystals publication-title: Phys. Rev. B doi: 10.1103/PhysRevB.71.195108 contributor: fullname: Pearce – start-page: 207 year: 2001 ident: 10.1016/j.apnum.2012.09.008_br0110 article-title: The mathematical theory of photonic crystals contributor: fullname: Kuchment – volume: 13 start-page: 6937 year: 2005 ident: 10.1016/j.apnum.2012.09.008_br0190 article-title: Hollow-core PCF for guidance in the mid to far infra-red publication-title: Opt. 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| SubjectTerms | Convergence Crystal fibers Eigenfunctions Error analysis Fibers Fourier methods Mathematical analysis Mathematical models Photonic crystals Planewave expansion method Regularity Sampling Smoothing Spectral approximation |
| Title | Planewave expansion methods for photonic crystal fibres |
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