A broadband ML-FMA for 3-D periodic green's function in 2-D lattice using ewald summation

Michael Wei, Weng C. Chew

Research output: Contribution to journalArticlepeer-review


A periodic fast multipole algorithm (P-FMA) is devised for evaluating 3-D periodic Green's function (PGF) for a 2-D lattice which can be used to solve scattering by a structure with 2-D periodicity. The introduction of periodicity in the Green's function formulation produces image sources at each lattice site. Like multilevel FMA (ML-FMA), P-FMA takes advantage of the distance between image sources and observation points to factorize the field using multipoles. By substituting known factorizations of the free-space Green's function into the expression for PGF, one can isolate the summation over the lattice into the translation phase of the FMA. For both plane wave and multipole factorizations, a common term known as lattice constant appears. The lattice constant is an infinite sum over the lattice which does not converge absolutely when expressed as a spatial sum. Using the Ewald summation technique, the lattice constants can be evaluated with exponential convergence and high accuracy. The resulting P-FMA is between O(N) and O(N log N) in memory use and computational complexity, depending on the object size relative to the wavelength.

Original languageEnglish (US)
Article number7891536
Pages (from-to)3134-3145
Number of pages12
JournalIEEE Transactions on Antennas and Propagation
Issue number6
StatePublished - 2017


  • Ewald summation
  • Fast multipole method (ML-FMA)
  • Lattice sum
  • Method of moments (MoM)
  • Multilevel
  • Periodic Green's function (PGF)
  • Periodic scattering
  • Periodic structures

ASJC Scopus subject areas

  • Electrical and Electronic Engineering


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