TY - JOUR

T1 - Photonic band structure calculations using nonlinear eigenvalue techniques

AU - Spence, A

AU - Poulton, C

N1 - The original publication is available at www.sciencedirect.com

PY - 2005/3

Y1 - 2005/3

N2 - 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.

AB - 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.

KW - Photonic band-gap materials

KW - Newton’s method

KW - Nonlinear eigenvalue problems

KW - Photonic crystals

UR - http://www.sciencedirect.com/science/journal/00219991

UR - http://dx.doi.org/10.1016/j.jcp.2004.09.016

U2 - 10.1016/j.jcp.2004.09.016

DO - 10.1016/j.jcp.2004.09.016

M3 - Article

VL - 204

SP - 65

EP - 81

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 1

ER -