Quasi-periodic Green Function Research Articles

Analyzing diffraction gratings by a boundary integral equation Neumann-to-Dirichlet map method

For analyzing diffraction gratings, a new method is developed based on dividing one period of the grating into homogeneous subdomains and computing the Neumann-to-Dirichlet (NtD) maps for these subdomains by boundary integral equations. For a subdomain, the NtD operator maps the normal derivative of the wave field to the wave field on its boundary. The integral operators used in this method are simple to approximate, since they involve only the standard Green's function of the Helmholtz equation in homogeneous media. The method retains the advantages of existing boundary integral equation methods for diffraction gratings but avoids the quasi-periodic Green's functions that are expensive to evaluate.

Stable and efficient evaluation of periodized Green’s functions for the Helmholtz equation at high frequencies

A difficulty that arises in the context of infinite d-periodic rough-surface scattering relates to the effective numerical evaluation of the corresponding “quasi-periodic Green function” G qp. Due to its relevance in a variety of applications, this problem has generated significant interest over the last 40 years, and a variety of numerical methods have been devised for this purpose. None of these methods to evaluate G qp however, were designed for high-frequency calculations. As a result, in this regime, these methods become prohibitively expensive and/or unstable. Here we present a novel scheme that can be shown to outperform every alternative numerical evaluation procedure and is especially effective for high-frequency calculations. Our new algorithm is based on the use of some exact integrals that arise on judicious manipulation of the integral representation of G qp and which reduce the overall problem to that of evaluation of a sequence of simpler integrals that can be effectively handled by standard quadrature formulas. We include a variety of numerical results that confirm that, indeed, our algorithm compares favorably with alternative methods.

Fast algorithms for Helmholtz Green's functions

The formal representation of the quasi-periodic Helmholtz Green's function obtained by the method of images is only conditionally convergent and, thus, requires an appropriate summation convention for its evaluation. Instead of using this formal sum, we derive a candidate Green's function as a sum of two rapidly convergent series, one to be applied in the spatial domain and the other in the Fourier domain (as in Ewald's method). We prove that this representation of Green's function satisfies the Helmholtz equation with the quasi-periodic condition and, furthermore, leads to a fast algorithm for its application as an operator.We approximate the spatial series by a short sum of separable functions given by Gaussians in each variable. For the series in the Fourier domain, we exploit the exponential decay of its terms to truncate it. We use fast and accurate algorithms for convolving functions with this approximation of the quasi-periodic Green's function. The resulting method yields a fast solver for the Helmholtz equation with the quasi-periodic boundary condition. The algorithm is adaptive in the spatial domain and its performance does not significantly deteriorate when Green's function is applied to discontinuous functions or potentials with singularities. We also construct Helmholtz Green's functions with Dirichlet, Neumann or mixed boundary conditions on simple domains and use a modification of the fast algorithm for the quasi-periodic Green's function to apply them.The complexity, in dimensiond≥2, of these algorithms is(κd log κ+C(log ϵ−1)d), whereϵis the desired accuracy,κis proportional to the number of wavelengths contained in the computational domain andCis a constant. We illustrate our approach with examples.

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

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 Schlomilch 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.