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

Alex Barnett, Leslie Greengard

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)6898-6914
Number of pages17
JournalJournal of Computational Physics
Volume229
Issue number19
DOIs
StatePublished - Sep 2010

Fingerprint

Green's function
Band structure
Green's functions
integral equations
cells
periodic functions
Helmholtz equation
Helmholtz equations
Boundary integral equations
bypasses
Maxwell equations
Maxwell equation
multipoles
Integral equations
Cells
inclusions
Geometry
geometry

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.

In: Journal of Computational Physics, Vol. 229, No. 19, 09.2010, p. 6898-6914.

Research output: Contribution to journalArticle

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