Photonic band structure calculations using nonlinear eigenvalue techniques

A Spence, C Poulton

Research output: Contribution to journalArticle

28 Citations (Scopus)
64 Downloads (Pure)

Abstract

This paper considers the numerical computation of the photonic band structure of periodic materials such as photonic crystals. This calculation involves the solution of a Hermitian nonlinear eigenvalue problem. Numerical methods for nonlinear eigenvalue problems are usually based on Newton’s method or are extensions of techniques for the standard eigenvalue problem. We present a new variation on existing methods which has its derivation in methods for bifurcation problems, where bordered matrices are used to compute critical points in singular systems. This new approach has several advantages over the current methods. First, in our numerical calculations the new variation is more robust than existing techniques, having a larger domain of convergence. Second, the linear systems remain Hermitian and are nonsingular as the method converges. Third, the approach provides an elegant and efficient way of both thinking about the problem and organising the computer solution so that only one linear system needs to be factorised at each stage in the solution process. Finally, first- and higher-order derivatives are calculated as a natural extension of the basic method, and this has advantages in the electromagnetic problem discussed here, where the band structure is plotted as a set of paths in the (ω,k) plane.
Original languageEnglish
Pages (from-to)65-81
Number of pages17
JournalJournal of Computational Physics
Volume204
Issue number1
DOIs
Publication statusPublished - Mar 2005

Fingerprint

Band structure
Photonics
eigenvalues
photonics
linear systems
Linear systems
Newton methods
organizing
Newton-Raphson method
Photonic crystals
Numerical methods
critical point
derivation
electromagnetism
Derivatives
matrices
crystals

Keywords

  • Photonic band-gap materials
  • Newton’s method
  • Nonlinear eigenvalue problems
  • Photonic crystals

Cite this

Photonic band structure calculations using nonlinear eigenvalue techniques. / Spence, A; Poulton, C.

In: Journal of Computational Physics, Vol. 204, No. 1, 03.2005, p. 65-81.

Research output: Contribution to journalArticle

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