Quasi-periodic Green Function Research Articles

Robust fast direct integral equation solver for three-dimensional doubly periodic scattering problems with a large number of layers

Frequency-domain wave scattering problems that arise in acoustics and electromagnetism can be often described by the Helmholtz equation. This work presents a boundary integral equation method for the Helmholtz equation in 3-D multilayered media with many doubly periodic smooth layers. Compared with conventional quasi-periodic Green's function method, the new method is robust at all scattering parameters. A periodizing scheme is used to decompose the solution into near- and far-field contributions. The near-field contribution uses the free-space Green's function in an integral equation on the interface in the unit cell and its immediate eight neighbors; the far-field contribution uses proxy point sources that enclose the unit cell. A specialized high-order quadrature is developed to discretize the underlying surface integral operators to keep the number of unknowns per layer small. We achieve overall linear computational complexity in the number of layers by reducing the linear system into block tridiagonal form and then solving the system directly via block LU decomposition. The new solver is capable of handling a 100-interface structure with 961.3k unknowns to 10−5 accuracy in less than 2 hours on a desktop workstation.

On the evaluation of quasi-periodic Green functions and wave-scattering at and around Rayleigh-Wood anomalies

This article presents full-spectrum, well-conditioned, Green-function methodologies for evaluation of scattering by general periodic structures, which remains applicable on a set of challenging singular configurations, usually called Rayleigh-Wood (RW) anomalies (at which the quasi-periodic Green function ceases to exist), where most existing quasi-periodic solvers break down. After reviewing a variety of existing fast-converging numerical procedures commonly used to compute the classical quasi-periodic Green-function, the present work explores the difficulties they present around RW-anomalies and introduces the concept of hybrid “spatial/spectral” representations. Such expressions allow both the modification of existing methods to obtain convergence at RW-anomalies as well as the application of a slight generalization of the Woodbury-Sherman-Morrison formulae together with a limiting procedure to bypass the singularities. (Although, for definiteness, the overall approach is applied to the scalar (acoustic) wave-scattering problem in the frequency domain, the approach can be extended in a straightforward manner to the harmonic Maxwell's and elasticity equations.) Ultimately, this thorough study of RW-anomalies yields fast and highly-accurate solvers, which are demonstrated with a variety of simulations of wave-scattering phenomena by arrays of particles, crossed impenetrable and penetrable diffraction gratings and other related structures. In particular, the methods developed in this article can be used to “upgrade” classical approaches, resulting in algorithms that are applicable throughout the spectrum, and it provides new methods for cases where previous approaches are either costly or fail altogether. In particular, it is suggested that the proposed shifted Green function approach may provide the only viable alternative for treatment of three-dimensional high-frequency configurations with either one or two directions of periodicity. A variety of computational examples are presented which demonstrate the flexibility of the overall approach.

Domain Decomposition for Quasi-Periodic Scattering by Layered Media via Robust Boundary-Integral Equations at All Frequencies

We develop a non-overlapping domain decomposition method (DDM) for scalar wave scattering by periodic layered media. Our approach relies on robust boundary-integral equation formulations of Robin-to-Robin (RtR) maps throughout the frequency spectrum, including cutoff (or Wood) frequencies. We overcome the obstacle of non-convergent quasi-periodic Green functions at these frequencies by incorporating newly introduced shifted Green functions. Using the latter in the definition of quasi-periodic boundary-integral operators leads to rigorously stable computations of RtR operators. We develop Nystr\"om discretizations of the RtR maps that rely on trigonometric interpolation, singularity resolution, and fast convergent windowed quasi-periodic Green functions. We solve the tridiagonal DDM system via recursive Schur complements and establish rigorously that this procedure is always completed successfully. We present a variety of numerical results concerning Wood frequencies in two and three dimensions as well as large numbers of layers.

Spectrally-accurate numerical method for acoustic scattering from doubly-periodic 3D multilayered media

A periodizing scheme and the method of fundamental solutions are used to solve acoustic wave scattering from doubly-periodic three-dimensional multilayered media. A scattered wave in a unit cell is represented by the sum of the near and distant contribution. The near contribution uses the free-space Green's function and its eight immediate neighbors. The contribution from the distant sources is expressed using proxy source points over a sphere surrounding the unit cell and its neighbors. The Rayleigh-Bloch radiation condition is applied to the top and bottom layers. Extra unknowns produced by the periodizing scheme in the linear system are eliminated using a Schur complement. The proposed numerical method avoids using singular quadratures and the quasi-periodic Green's function or complicated lattice sum techniques. Therefore, the proposed scheme is robust at all scattering parameters including Wood anomalies. The algorithm is also applicable to electromagnetic problems by using the dyadic Green's function. Numerical examples with 10-digit accuracy are provided. Finally, reflection and transmission spectra are computed over a wide range of incident angles for device characterization.

Shifted equivalent sources and FFT acceleration for periodic scattering problems, including Wood anomalies

This paper introduces a fast algorithm, applicable throughout the electromagnetic spectrum, for the numerical solution of problems of scattering by periodic surfaces in two-dimensional space. The proposed algorithm remains highly accurate and efficient for challenging configurations including randomly rough surfaces, deep corrugations, large periods, near grazing incidences, and, importantly, Wood-anomaly resonant frequencies. The proposed approach is based on use of certain “shifted equivalent sources” which enable FFT acceleration of a Wood-anomaly-capable quasi-periodic Green function introduced recently (Bruno and Delourme (2014) [4]). The Green-function strategy additionally incorporates an exponentially convergent shifted version of the classical spectral series for the Green function. While the computing-cost asymptotics depend on the asymptotic configuration assumed, the computing costs rise at most linearly with the size of the problem for a number of important rough-surface cases we consider. In practice, single-core runs in computing times ranging from a fraction of a second to a few seconds suffice for the proposed algorithm to produce highly-accurate solutions in some of the most challenging contexts arising in applications.

An FFT-based Algorithm for Efficient Computation of Green's Functions for the Helmholtz and Maxwell's Equations in Periodic Domains

The integral equation method is widely used in numerical simulations of 2D/3D acoustic and electromagnetic scattering problems, which needs a large number of values of the Green's functions. A significant topic is the scattering problems in periodic domains, where the corresponding Green's functions are quasi-periodic. The quasi-periodic Green's functions are defined by series that converge too slowly to be used for calculations. Many mathematicians have developed several efficient numerical methods to calculate quasi-periodic Green's functions. In this paper, we will propose a new FFT-based fast algorithm to compute the 2D/3D quasi-periodic Green's functions for both the Helmholtz equations and Maxwell's equations. The convergence results and error estimates are also investigated in this paper. Further, the numerical examples are given to show that, when a large number of values are needed, the new algorithm is very competitive.

Three-dimensional quasi-periodic shifted Green function throughout the spectrum, including Wood anomalies.

This work, part II in a series, presents an efficient method for evaluation of wave scattering by doubly periodic diffraction gratings at or near what are commonly called 'Wood anomaly frequencies'. At these frequencies, there is a grazing Rayleigh wave, and the quasi-periodic Green function ceases to exist. We present a modification of the Green function by adding two types of terms to its lattice sum. The first type are transversely shifted Green functions with coefficients that annihilate the growth in the original lattice sum and yield algebraic convergence. The second type are quasi-periodic plane wave solutions of the Helmholtz equation which reinstate certain necessary grazing modes without leading to blow-up at Wood anomalies. Using the new quasi-periodic Green function, we establish, for the first time, that the Dirichlet problem of scattering by a smooth doubly periodic scattering surface at a Wood frequency is uniquely solvable. We also present an efficient high-order numerical method based on this new Green function for scattering by doubly periodic surfaces at and around Wood frequencies. We believe this is the first solver able to handle Wood frequencies for doubly periodic scattering problems in three dimensions. We demonstrate the method by applying it to acoustic scattering.

Controlling Flexural Waves in Semi-Infinite Platonic Crystals with Resonator-Type Scatterers

We address the scattering and transmission of a plane flexural wave through a semi-infinite array of point scatterers/resonators, which take a variety of physically interesting forms. The mathematical model accounts for several classes of point defects, including mass-spring resonators attached to the top surface of the flexural plate and their limiting case of concentrated point masses. We also analyse the special case of resonators attached to opposite faces of the plate. The problem is reduced to a functional equation of the Wiener–Hopf type, whose kernel varies with the type of scatterer considered. A novel approach, which stems from the direct connection between the kernel function of the semi-infinite system and the quasi-periodic Green's functions for corresponding infinite systems, is used to identify special frequency regimes. We thereby demonstrate dynamically anisotropic wave effects in semi-infinite platonic crystals, with particular attention paid to designing systems that exhibit dynamic neutrality (perfect transmission) and localisation close to the structured interface.

Rapidly convergent quasi-periodic Green functions for scattering by arrays of cylinders-including Wood anomalies.

This paper presents a full-spectrum Green-function methodology (which is valid, in particular, at and around Wood-anomaly frequencies) for evaluation of scattering by periodic arrays of cylinders of arbitrary cross section-with application to wire gratings, particle arrays and reflectarrays and, indeed, general arrays of conducting or dielectric bounded obstacles under both transverse electric and transverse magnetic polarized illumination. The proposed method, which, for definiteness, is demonstrated here for arrays of perfectly conducting particles under transverse electric polarization, is based on the use of the shifted Green-function method introduced in a recent contribution (Bruno & Delourme 2014 J. Computat. Phys.262, 262-290 (doi:10.1016/j.jcp.2013.12.047)). A certain infinite term arises at Wood anomalies for the cylinder-array problems considered here that is not present in the previous rough-surface case. As shown in this paper, these infinite terms can be treated via an application of ideas related to the Woodbury-Sherman-Morrison formulae. The resulting approach, which is applicable to general arrays of obstacles even at and around Wood-anomaly frequencies, exhibits fast convergence and high accuracies. For example, a few hundreds of milliseconds suffice for the proposed approach to evaluate solutions throughout the resonance region (wavelengths comparable to the period and cylinder sizes) with full single-precision accuracy.

Superalgebraically convergent smoothly windowed lattice sums for doubly periodic Green functions in three-dimensional space.

This work, part I in a two-part series, presents: (i) a simple and highly efficient algorithm for evaluation of quasi-periodic Green functions, as well as (ii) an associated boundary-integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three-dimensional space. Except for certain 'Wood frequencies' at which the quasi-periodic Green function ceases to exist, the proposed approach, which is based on smooth windowing functions, gives rise to tapered lattice sums which converge superalgebraically fast to the Green function-that is, faster than any power of the number of terms used. This is in sharp contrast to the extremely slow convergence exhibited by the lattice sums in the absence of smooth windowing. (The Wood-frequency problem is treated in part II.) This paper establishes rigorously the superalgebraic convergence of the windowed lattice sums. A variety of numerical results demonstrate the practical efficiency of the proposed approach.

A volume integral equation method for periodic scattering problems for anisotropic Maxwell's equations

This paper presents a volume integral equation method for an electromagnetic scattering problem for three-dimensional Maxwell's equations in the presence of a biperiodic, anisotropic, and possibly discontinuous dielectric scatterer. Such scattering problem can be reformulated as a strongly singular volume integral equation (i.e., integral operators that fail to be weakly singular). In this paper, we firstly prove that the strongly singular volume integral equation satisfies a Gårding-type estimate in standard Sobolev spaces. Secondly, we rigorously analyze a spectral Galerkin method for solving the scattering problem. This method relies on the periodization technique of Gennadi Vainikko that allows us to efficiently evaluate the periodized integral operators on trigonometric polynomials using the fast Fourier transform (FFT). The main advantage of the method is its simple implementation that avoids for instance the need to compute quasiperiodic Green's functions. We prove that the numerical solution of the spectral Galerkin method applied to the periodized integral equation converges quasioptimally to the solution of the scattering problem. Some numerical examples are provided for examining the performance of the method.

A fast and robust solver for the scattering from a layered periodic structure containing multi-particle inclusions

We present a solver for plane wave scattering from a periodic dielectric grating with a large number M of inclusions lying in each period of its middle layer. Such composite material geometries have a growing role in modern photonic devices and solar cells. The high-order scheme is based on boundary integral equations, and achieves many digits of accuracy with ease. The usual way to periodize the integral equation—via the quasi-periodic Green's function—fails at Wood's anomalies. We instead use the free-space Green's kernel for the near field, add auxiliary basis functions for the far field, and enforce periodicity in an expanded linear system; this is robust for all parameters. Inverting the periodic and layer unknowns, we are left with a square linear system involving only the inclusion scattering coefficients. Preconditioning by the single-inclusion scattering matrix, this is solved iteratively in O(M) time using a fast matrix-vector product. Numerical experiments show that a diffraction grating containing M=1000 inclusions per period can be solved to 9-digit accuracy in under 5 minutes on a laptop.

Robust fast direct integral equation solver for quasi-periodic scattering problems with a large number of layers.

We present a new boundary integral formulation for time-harmonic wave diffraction from two-dimensional structures with many layers of arbitrary periodic shape, such as multilayer dielectric gratings in TM polarization. Our scheme is robust at all scattering parameters, unlike the conventional quasi-periodic Green's function method which fails whenever any of the layers approaches a Wood anomaly. We achieve this by a decomposition into near- and far-field contributions. The former uses the free-space Green's function in a second-kind integral equation on one period of the material interfaces and their immediate left and right neighbors; the latter uses proxy point sources and small least-squares solves (Schur complements) to represent the remaining contribution from distant copies. By using high-order discretization on interfaces (including those with corners), the number of unknowns per layer is kept small. We achieve overall linear complexity in the number of layers, by direct solution of the resulting block tridiagonal system. For device characterization we present an efficient method to sweep over multiple incident angles, and show a 25× speedup over solving each angle independently. We solve the scattering from a 1000-layer structure with 3 × 105 unknowns to 9-digit accuracy in 2.5 minutes on a desktop workstation.

Range of Validity of the Rayleigh Hypothesis for Sound Wave Scattering From a Periodic Seafloor

In this paper, sound scattering at a water/sediment periodic interface is modeled by an integral equation method to investigate the range of validity of the Rayleigh hypothesis. Due to the periodicity of surface, coupled integral equations are simplified by introducing the quasi-periodic Green's function, which can be evaluated with a relatively small number of terms of the spectral series. By comparing the scattering amplitudes obtained from the integral equation method and the method based on the Rayleigh hypothesis, the range of validity of the Rayleigh hypothesis is determined. For a given wavenumber κ of the periodic surface, the maximum surface amplitude hmax beyond which the Rayleigh hypothesis is invalid decreases with frequency of incident waves. For the grazing angle θ = 90 ° , h max is larger than h c =0.448/κ (the upper limit of surface amplitude determined by the criterion of the Rayleigh hypothesis) in the frequency range 2.5-40 kHz. However, for a small grazing angle, such as, θ = 35 ° , h max is observed to be smaller than h c for the large surface period Λ at frequency 40 kHz.

Rapidly convergent two-dimensional quasi-periodic Green function throughout the spectrum—including Wood anomalies

We introduce a new methodology, based on new quasi-periodic Green functions which converge rapidly even at and around Wood-anomaly configurations, for the numerical solution of problems of scattering by periodic rough surfaces in two-dimensional space. As is well known the classical quasi-periodic Green function ceases to exist at Wood anomalies. The approach introduced in this text produces fast Green function convergence throughout the spectrum on the basis of a certain “finite-differencing” approach and smooth windowing of the classical Green function lattice sum. The resulting Green-function convergence is super-algebraically fast away from Wood anomalies, and it reduces to an arbitrarily-high (user-prescribed) algebraic order of convergence at Wood anomalies.

A trigonometric Galerkin method for volume integral equations arising in TM grating scattering

Transverse magnetic (TM) scattering of an electromagnetic wave from a periodic dielectric diffraction grating can mathematically be described by a volume integral equation.This volume integral equation, however, in general fails to feature a weakly singular integral operator. Nevertheless, after a suitable periodization, the involved integral operator can be efficiently evaluated on trigonometric polynomials using the fast Fourier transform (FFT) and iterative methods can be used to solve the integral equation. Using Fredholm theory, we prove that a trigonometric Galerkin discretization applied to the periodized integral equation converges with optimal order to the solution of the scattering problem. The main advantage of this FFT-based discretization scheme is that the resulting numerical method is particularly easy to implement, avoiding for instance the need to evaluate quasiperiodic Green's functions.

Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction

Boundary integral equation methods for diffraction gratings are particularly suitable for gratings with complicated material interfaces but are difficult to implement due to the quasi-periodic Green's function and the singular integrals at the corners. In this paper, the boundary integral equation Neumann-to-Dirichlet map method for in-plane diffraction problems of gratings [Y. Wu and Y. Y. Lu, J. Opt. Soc. Am. A26, 2444 (2009)] is extended to conical diffraction problems. The method uses boundary integral equations to calculate the so-called Neumann-to-Dirichlet maps for homogeneous subdomains of the grating, so that the quasi-periodic Green's functions can be avoided. Since wave field components are coupled on material interfaces with the involvement of tangential derivatives, a least squares polynomial approximation technique is developed to evaluate tangential derivatives along these interfaces for conical diffraction problems. Numerical examples indicate that the method performs equally well for dielectric or metallic gratings.

An integral representation of the Green function for a linear array of acoustic point sources

We present a new algorithm for the evaluation of the quasi-periodic Green function for a linear array of acoustic point sources such as those arising in the analysis of line array loudspeakers. A variety of classical algorithms (based on spatial and spectral representations, Ewald transformation, etc.) have been implemented in the past to evaluate these acoustic fields. However as we show, these methods become unstable and/or impractically expensive as the frequency of use of the sources increases. Here we introduce a new numerical scheme that overcomes some of these limitations allowing for simulations at unprecedentedly large frequencies. The method is based on a new integral representation derived from the classic spatial form, and on suitable further manipulations of the relevant integrands to render the integrals amenable to efficient and accurate approximations through standard quadrature formulas. We include a variety of numerical results that demonstrate that our algorithm compares favorably with several classical methods both for points close to the line where the poles are located and at high-frequencies while remaining competitive with them in every other instance.

Lattice Sums for the Helmholtz Equation

A survey of different representations for lattice sums for the Helmholtz equation is made. These sums arise naturally when dealing with wave scattering by periodic structures. One of the main objectives is to show how the various forms depend on the dimension d of the underlying space and the lattice dimension $d_\Lambda$. Lattice sums are related to, and can be calculated from, the quasi-periodic Green's function and this object serves as the starting point of the analysis.

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.