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

Abstract

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.1 periodic Green function is not even defined (see Remark 2.2 for details on nomenclature concerning Wood anomalies). To address this difficulty we propose new quasi-periodic Green functions and associated series representation which converge rapidly even at and around Wood anomalies. More precisely, we present a set of rapidly convergent quasi-periodic Green functions G q j whose N -term truncated series converge, at Wood anomalies, at least as fast as (1/N ) (j−1)/2 for j even (resp. (1/N ) j/2 for j odd) as N → ∞; in view of the fact that this approach also incorporates the smooth windowing function methodology [36], the new Green functions also enjoy super-algebraically fast convergence (faster than any power of N ) away from Wood anomalies.The approach introduced in this text produces fast Green-function convergence at and around Wood anomalies on the basis of a certain order-j "finite-differencing" method (for positive integer values of j). To our knowledge, this is the first approach ever presented that is applicable to problems of scattering by diffraction gratings at Wood anomalies on the basis of quasi-periodic Green functions.It should be noted that a "method-of-images Green-function", which is related to our j = 1 Green function approach, was used in [15,48] to treat problems of scattering by nonlocal, non-periodic perturbations of a line in two dimensions. The j = 1 method, which suffices to yield (slow) convergence in the two dimensional case, does not give rise to convergence in three dimensions: for three dimensional configurations convergence only results for j ≥ 2 [12]. In this context we also mention the recent work [3,4] which, for two-dimensional problems, introduces an alternative integral equation which does not utilize a quasi-periodic Green function, and which is also applicable at Wood anomalies: in that approach quasi-periodicity is enforced through use of auxiliary layer potentials on the periodic cell boundaries. The practical feasibility of an extension of this methodology to three dimensional problems has not as yet been established.In order to demonstrate the character of the new approach we present efficient numerical methods, based on the new Green functions, for the solution of quasi-periodic scattering problems through...

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

“…For wavenumbers that are Wood frequencies, the series in the definition of the Green function G q k in equation (3.8) does not converge. In the case when k is a Wood frequency, we introduce the following shifted Green functions [10]…”

“…On account of the windowed function χ, the summations in equations (4.1) and (4.2) are over a finite range of indices n. The functions G q,A k were shown to converge superalgebraically fast to G q k as A → ∞ when k is not a Wood frequency [16,15,10], whereas the functions G q,j,A k,h were shown to converge algebraically fast to a α-quasi-periodic Green function as A → ∞ (the rate increases as the number of shifts j grows) in the half-plane x 2 > 0 when h > 0 and respectively x 2 < 0 when h < 0 for all frequencies k, including at and around Wood frequencies [10].…”

“…This approach gives rise to efficient direct solvers for transmission problems in two-dimensional periodic layered media, and can yield results even at Wood frequencies [18,36]. Its rigorous analysis appears to be absent in the literature, and we are not aware of evidence that these methods are capable of handling Wood frequencies in three dimensions.For the first time in this arena, a rigorous solution was provided to the problem of boundaryintegral equation formulations of quasi-periodic problems at Wood frequencies in both two and three dimensions through a new method, in which the well-posedness of the formulation and the stability of the numerical scheme were proven, each in its own right [16,10,15]. Smooth windowed truncations of the lattice sums for the Green functions were introduced and analyzed in [16].…”

“…For the first time in this arena, a rigorous solution was provided to the problem of boundaryintegral equation formulations of quasi-periodic problems at Wood frequencies in both two and three dimensions through a new method, in which the well-posedness of the formulation and the stability of the numerical scheme were proven, each in its own right [16,10,15]. Smooth windowed truncations of the lattice sums for the Green functions were introduced and analyzed in [16].…”

High-Performance Computational Electromagnetics in Frequency-Domain and Time-Domain

“…A number of contributions in the area of rough surfaces and periodic Green functions [8][9][10][11] over the span of this contract include use of a new "windowing" approach to greatly accelerate Greenfunction calculations [8], extension to periodic arrays of cylinders [9], introduction of acceleration in the periodic solver [10] and extension to three-dimensional periodic problems [11] as well as the novel construction, produced under sponsorship of this contract, of rapidly convergent periodic Green functions at Wood anomaly frequencies [8]. The latter problem is very well known and had defied solution since the early twentieth century.…”