### Abstract

In this paper, we consider band structure calculations governed by the Helmholtz or Maxwell equations in piecewise homogeneous periodic materials. Methods based on boundary integral equations are natural in this context, since they discretize the interface alone and can achieve high order accuracy in complicated geometries. In order to handle the quasi-periodic conditions which are imposed on the unit cell, the free-space Green's function is typically replaced by its quasi-periodic cousin. Unfortunately, the quasi-periodic Green's function diverges for families of parameter values that correspond to resonances of the empty unit cell. Here, we bypass this problem by means of a new integral representation that relies on the free-space Green's function alone, adding auxiliary layer potentials on the boundary of the unit cell itself. An important aspect of our method is that by carefully including a few neighboring images, the densities may be kept smooth and convergence rapid. This framework results in an integral equation of the second kind, avoids spurious resonances, and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls may be handled easily and automatically. Our approach is compatible with fast-multipole acceleration, generalizes easily to three dimensions, and avoids the complication of divergent lattice sums.

Original language | English (US) |
---|---|

Pages (from-to) | 6898-6914 |

Number of pages | 17 |

Journal | Journal of Computational Physics |

Volume | 229 |

Issue number | 19 |

DOIs | |

State | Published - Sep 2010 |

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### Keywords

- Band structure
- Bloch
- Eigenvalue
- Helmholtz
- Integral equation
- Lattice
- Maxwell
- Periodic
- Photonic crystal

### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy (miscellaneous)

### Cite this

**A new integral representation for quasi-periodic fields and its application to two-dimensional band structure calculations.** / Barnett, Alex; Greengard, Leslie.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 229, no. 19, pp. 6898-6914. https://doi.org/10.1016/j.jcp.2010.05.029

}

TY - JOUR

T1 - A new integral representation for quasi-periodic fields and its application to two-dimensional band structure calculations

AU - Barnett, Alex

AU - Greengard, Leslie

PY - 2010/9

Y1 - 2010/9

N2 - In this paper, we consider band structure calculations governed by the Helmholtz or Maxwell equations in piecewise homogeneous periodic materials. Methods based on boundary integral equations are natural in this context, since they discretize the interface alone and can achieve high order accuracy in complicated geometries. In order to handle the quasi-periodic conditions which are imposed on the unit cell, the free-space Green's function is typically replaced by its quasi-periodic cousin. Unfortunately, the quasi-periodic Green's function diverges for families of parameter values that correspond to resonances of the empty unit cell. Here, we bypass this problem by means of a new integral representation that relies on the free-space Green's function alone, adding auxiliary layer potentials on the boundary of the unit cell itself. An important aspect of our method is that by carefully including a few neighboring images, the densities may be kept smooth and convergence rapid. This framework results in an integral equation of the second kind, avoids spurious resonances, and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls may be handled easily and automatically. Our approach is compatible with fast-multipole acceleration, generalizes easily to three dimensions, and avoids the complication of divergent lattice sums.

AB - In this paper, we consider band structure calculations governed by the Helmholtz or Maxwell equations in piecewise homogeneous periodic materials. Methods based on boundary integral equations are natural in this context, since they discretize the interface alone and can achieve high order accuracy in complicated geometries. In order to handle the quasi-periodic conditions which are imposed on the unit cell, the free-space Green's function is typically replaced by its quasi-periodic cousin. Unfortunately, the quasi-periodic Green's function diverges for families of parameter values that correspond to resonances of the empty unit cell. Here, we bypass this problem by means of a new integral representation that relies on the free-space Green's function alone, adding auxiliary layer potentials on the boundary of the unit cell itself. An important aspect of our method is that by carefully including a few neighboring images, the densities may be kept smooth and convergence rapid. This framework results in an integral equation of the second kind, avoids spurious resonances, and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls may be handled easily and automatically. Our approach is compatible with fast-multipole acceleration, generalizes easily to three dimensions, and avoids the complication of divergent lattice sums.

KW - Band structure

KW - Bloch

KW - Eigenvalue

KW - Helmholtz

KW - Integral equation

KW - Lattice

KW - Maxwell

KW - Periodic

KW - Photonic crystal

UR - http://www.scopus.com/inward/record.url?scp=77955275838&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955275838&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2010.05.029

DO - 10.1016/j.jcp.2010.05.029

M3 - Article

AN - SCOPUS:77955275838

VL - 229

SP - 6898

EP - 6914

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 19

ER -