### Abstract

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 language | English (US) |
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Article number | 7891536 |

Pages (from-to) | 3134-3145 |

Number of pages | 12 |

Journal | IEEE Transactions on Antennas and Propagation |

Volume | 65 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 2017 |

### Keywords

- 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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## Cite this

*IEEE Transactions on Antennas and Propagation*,

*65*(6), 3134-3145. [7891536]. https://doi.org/10.1109/TAP.2017.2690533