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A Fast Summation Method for Oscillatory Lattice Sums.

Ryan Denlinger1, Zydrunas Gimbutas2, Leslie Greengard1

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This study introduces a rapid summation method for lattice sums crucial in wave scattering. The new approach rigorously analyzes Wood's anomalies, offering a fast and accurate solution for periodic boundary conditions.

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Area of Science:

  • Physics
  • Computational Science

Background:

  • Wave scattering problems with periodic boundary conditions often involve complex lattice sums.
  • Existing algorithms for these calculations can be computationally intensive.
  • Wood's anomalies, singularities in spectral response, occur under specific grating illumination conditions.

Purpose of the Study:

  • To present a novel, fast summation method for lattice sums.
  • To provide a rigorous analysis of Wood's anomalies using this new method.
  • To develop an efficient algorithm for wave scattering calculations.

Main Methods:

  • Utilized the Euler-Maclaurin formula.
  • Employed a steepest descent argument.
  • Developed a new summation algorithm for lattice sums.

Main Results:

  • The new method achieves super-algebraic convergence.
  • The algorithm successfully analyzes Wood's anomalies.
  • The computational time required is on the order of milliseconds.

Conclusions:

  • The presented fast summation method is highly efficient for lattice sums in wave scattering.
  • This approach offers a rigorous framework for understanding Wood's anomalies.
  • The algorithm demonstrates significant improvements in speed and accuracy.