Quasi-periodic Green's functions of the Helmholtz and Laplace equations

Abstract

A classical problem of free-space Green's function $G_{0\Lambda }$ representations of the Helmholtz equation is studied in various quasi-periodic cases, i.e., when an underlying periodicity is imposed in less dimensions than is the dimension of an embedding space. Exponentially convergent series for the free-space quasi-periodic $G_{0\Lambda }$ and for the expansion coefficients $D_L$ of $G_{0\Lambda }$ in the basis of regular (cylindrical in two dimensions and spherical in three dimension (3D)) waves, or lattice sums, are reviewed and new results for the case of a one-dimensional (1D) periodicity in 3D are derived. From a mathematical point of view, a derivation of exponentially convergent representations for Schlömilch series of cylindrical and spherical Hankel functions of any integer order is accomplished. The quasi-periodic Green's functions of the Laplace equation are obtained from the corresponding representations of $G_{0\Lambda }$ of the Helmholtz equation by taking the limit of the wave vector magnitude going to zero. The derivation of relevant results in the case of a 1D periodicity in 3D highlights the common part which is universally applicable to any of remaining quasi-periodic cases. The results obtained can be useful for numerical solution of boundary integral equations for potential flows in fluid mechanics, remote sensing of periodic surfaces, periodic gratings, in many contexts of simulating systems of charged particles, in molecular dynamics, for the description of quasi-periodic arrays of point interactions in quantum mechanics, and in various ab-initio first-principle multiple-scattering theories for the analysis of diffraction of classical and quantum waves.

arXiv comment60 pages, 5 figures; no change in the results; introduction rewritten, refined sectioning, Fig. 1 amended, 3 references added

References in corpus (6)

• Propagation of optical excitations by dipolar interactions in metal nanoparticle chains
• Scattering of elastic waves by periodic arrays of spherical bodies
• Surface Plasmon Dispersion Relations in Chains of Metallic Nanoparticles: Exact Quasistatic Calculation
• Metallo-dielectric diamond and zinc-blende photonic crystals
• Photonic Band Gaps of Three-Dimensional Face-Centered Cubic Lattices
• Density-of-States Calculations and Multiple-Scattering Theory for Photons

Cited by in corpus (6)

• Singular statistics revised
• Logarithmic Expansions and the Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems in $\R^2
• Lattice Sums Accommodating Multiple Sublattices for Solutions of the Helmholtz Equation in Two and Three Dimensions
• A Fast Summation Method for Oscillatory Lattice Sums
• An FFT-based algorithm for efficient computation of Green's functions for the Helmholtz and Maxwell's equations in periodic domains
• Multem 3: An updated and revised version of the program for transmission and band calculations of photonic crystals