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

Abstract

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.

1. Introduction

We consider the problem of scattering of a monochromatic plane wave by a periodic array of cylinders of general cross section. We approach this problem by means of the methodology introduced in [1], which, based on the use of a certain shifted Green function, provides a solver for problems of scattering by periodic surfaces that is valid and accurate throughout the spectrum—including Wood frequencies [2,3], at which the classical quasi-periodic Green function ceases to exist

A variety of approaches have been used to tackle this important problem, including, notably, methods based on the use of integral equations; see [1,4–8] and references therein. The success of the integral-equation approach results from its inherent dimensionality reduction (only the boundary of the domain needs to be discretized) and associated automatic enforcement of radiation conditions.

For the sake of simplicity the methodology presented in this article assumes perfectly conducting obstacles under transverse electric polarization. The method can be easily extended to transverse magnetic polarization and dielectric cylinders: the dielectric case does not give rise to additional difficulties, and the hyper-singular operators that arise in the transverse magnetic case of polarization can be handled by means of existing regularization techniques (e.g. [9] and references therein). The main strategy developed in this paper can thus be applied in those contexts without significant modifications.

As is well known, classical expansions for quasi-periodic Green functions converge extremely slowly, and they of course completely fail to converge at Wood anomalies. A number of methods have been introduced to tackle the slow-convergence difficulty, including the well-known Ewald summation method for two- and three-dimensional problems [10–13] and many other contributions [14–20]. Unfortunately, however, none of these methods resolve the difficulties posed by Wood anomalies. Recently, a new quasi-periodic Green function was introduced [1] for the problem of scattering by periodic surfaces which, relying on the use of certain linear combinations of shifted free-space Green functions (which amount to discrete finite-differencing of the Green functions), can be used to produce arbitrary (user-prescribed) algebraic convergence order for frequencies throughout the spectrum, including Wood frequencies [2,3].

A straightforward application of this procedure leads to an operator equation that contains denominators which tend to zero as a Wood anomaly is approached. To remedy this situation a strategy based on the use of the Woodbury–Sherman–Morrison formulae is introduced which completely regularizes the problem and provides a limiting solution as Wood frequencies are approached. To our knowledge, this is the first approach ever presented that is applicable to problems of scattering by periodic arrays of bounded obstacles at Wood anomalies on the basis of quasi-periodic Green functions. It is worth mentioning that an alternative method, not based on the use of quasi-periodic Green functions and which is also applicable at Wood anomalies, was proposed in [4,5]. In that approach, whose generalization to corresponding three-dimensional problems at Wood anomalies has not been provided, the quasi-periodicity is enforced through the use of auxiliary layer potentials on the boundaries of the periodic cell. As suggested by the treatment [21] for the problem of scattering by bi-periodic surfaces in three dimensions, the shifted Green-function approach can be extended to three-dimensional problems without difficulty.

In order to demonstrate the character of the new approach, we present numerical methods based on the use of a combination of three main elements: the half-space quasi-periodic Green function, the smooth-windowing methodology [1,21,22] (which gives rise to super-algebraically fast convergence away from Wood anomalies) and high-order quadratures for singular integrals [23–26]. As shown by means of a variety of numerical results, highly accurate solvers result from this strategy—even at and around Wood anomalies. As can be seen in §6 the present approach can solve the complete scattering problem for scatterers of arbitrary shape even at anomalous configurations, in fast computing times.

The remainder of this article is organized as follows. Section 2 presents necessary background on the problem of scattering by periodic media. Section 3 summarizes the convergence properties of the shifted quasi-periodic Green-function approximation introduced in [1], and §4 then presents an associated integral-equation formulation for the solution of the problem of wave scattering by a periodic array of bounded obstacles at all frequencies. The actual numerical implementation we propose, which relies, in addition, on the aforementioned smooth-windowing methodology, is presented in §5 (the super-algebraic convergence of the smooth-windowing methodology at non-anomalous configurations has been established in [21] for bi-periodic structures in three-dimensional space; the same ideas can be applied to establish the result for a linear periodic array in the two-dimensional case). Section 6, finally, presents a variety of numerical results demonstrating the properties of the overall approach.

2. Preliminaries

(a). Scattering by a periodic array of bounded obstacles

We consider problems of scattering by a one-dimensional perfectly-conducting diffraction grating of period L (L>0) in two-dimensional space under transverse electric polarization. The scatterer is assumed to equal a union of the form

$$D=\bigcup _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{D}_n$$ 2.1

of mutually disjoint closed sets D_n, where D₀ equals a union of a finite number of non-overlapping connected bounded components, and where $D_n=D_0+nL\hat{x}$ ($\hat{x}=(1,0)$, $n\in \mathbb{Z}$) (figure 1). We assume the grating is illuminated by a plane wave of spatial frequency k>0 and incidence angle θ (measured from the y-axis). The scattered wave u^s_k satisfies the partial differential equation (PDE)

$$\mathrm{\Delta }{u}_k^{\mathrm{s}}+k^2{u}_k^{\mathrm{s}}=0\mathrm{in}\hspace{0.17em}\mathrm{\Omega }={\mathbb{R}}^2\setminus D$$ 2.2

along with the condition of radiation at infinity (see remark 2.1) and a Dirichlet-type boundary condition

$$u_k^{\mathrm{s}}=-u_k^{\mathrm{inc}}\text{on}\hspace{0.17em}\mathrm{\partial }D,$$ 2.3

where, letting $\alpha =k\sin (\theta )$ and $\beta =k\cos (\theta ),$ we have set

$$u_k^{\mathrm{inc}}(x,y)={\mathrm{e}}^{\mathrm{i}\alpha x-\mathrm{i}\beta y}.$$

Figure 1.

The physical problem: a plane wave with angle of incidence θ is scattered by an array of identical cylindrical obstacles distributed periodically in the x-direction. The particular star-shaped cylinder cross section depicted here is given by the parametrization $C(t)=r(t)(\cos (t),\sin (t))$, where $r(t)=1+0.1\cos (5t)+0.01\cos (10t)$. (Online version in colour.)

As is well known, the incoming and scattered waves u^inc_k and u^s_k are α quasi-periodic functions, that is, they satisfy the identities

$$u_k^{\mathrm{inc}}(x+nL,y)={\mathrm{e}}^{\mathrm{i}\alpha nL}{u}_k^{\mathrm{inc}}(x,y)\text{and}{u}_k^{\mathrm{s}}(x+nL,y)={\mathrm{e}}^{\mathrm{i}\alpha nL}{u}_k^{\mathrm{s}}(x,y).$$

It follows that, assuming

$$D\subset \mathbb{R}\times (M^{-},M^{+})$$ 2.4

(figure 1) and letting α_n=α+(2π/L)n, ${\beta }_n^2=k^2-{\alpha }_n^2$ (Im(β_n)≥0), the solution u^s_k admits Rayleigh expansions [27] of the form

$$u_k^{\mathrm{s}}(x,y)=\sum _{n\in \mathbb{Z}}{A}_n^{\pm }\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_{n}x+\mathrm{i}{\beta }_{n}y}+B_n^{\pm }\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_{n}x-\mathrm{i}{\beta }_{n}y}.$$ 2.5

Remark 2.1 —

The scattered field consists of a superposition of plane waves that propagate away from D and which remain bounded in the far field; a scattered solution having this property is called radiating. In the present context, we impose such a radiation condition by requiring that [27]

$$u_k^{\mathrm{s}}(x,y)=\sum _{n\in \mathbb{Z}}{C}_n^{+}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_{n}x+\mathrm{i}{\beta }_{n}y},y>M^{+}$$ 2.6

and

$$u_k^{\mathrm{s}}(x,y)=\sum _{n\in \mathbb{Z}}{C}_n^{-}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_{n}x-\mathrm{i}{\beta }_{n}y},y<M^{-}.$$ 2.7

Remark 2.2 —

Throughout this paper, $\mathcal{N}$ denotes the finite subset of integers for which $k^2-{\alpha }_n^2>0$. For $n\in \mathcal{N}\mathcal{,}$ the functions e^{iα_nx+iβ_ny} and e^{iα_nx−iβ_ny} are outwardly propagating waves (above and below D, respectively). If $k^2-{\alpha }_n^2<0$, the modes e^{iα_nx+iβ_ny} (resp. e^{iα_nx−iβ_ny}) are evanescent waves, i.e. they decrease exponentially as $y\to +\mathrm{\infty }$ (resp. $y\to -\mathrm{\infty }$). A scattering set-up for which $k^2={\alpha }_{n_0}^2$ for some $n_0\in \mathbb{Z}$ is called a Wood-anomaly configuration and the wavenumber k will be referred to as a Wood frequency or Wood-anomaly frequency; note that, in such cases, the plane wave e^{iα_n0x±iβ_n0y} travels in the directions parallel to the obstacle array. For fixed L and θ, we consider the set

$${\mathcal{K}}_{\mathrm{wa}}={\mathcal{K}}_{\mathrm{wa}}(L,\theta )=\{k>0:\hspace{0.17em}\mathrm{\exists }\hspace{0.17em}n\in \mathbb{Z}\text{ such that }{k}^2={\alpha }_n^2\}$$ 2.8

of all Wood frequencies for a given period L and incidence angle θ. Clearly, for each $k_0\in {\mathcal{K}}_{\mathrm{wa}}$, the set

$${\mathcal{N}}_{\mathrm{wa}}(k_0)=\{n\in \mathbb{Z}:{\beta }_n(k_0)=0\}$$ 2.9

has at least one and at most two elements: either ${\mathcal{N}}_{\mathrm{wa}}=\{n_1\}$ or ${\mathcal{N}}_{\mathrm{wa}}=\{n_1,n_2\}$. For conciseness, throughout this paper, we write ${\mathcal{N}}_{\mathrm{wa}}=\{n_1,\dots ,n_r\}$ with r=1 or r=2, as appropriate, with corresponding expressions such as $\sum _{j=1}^r$, etc.—in spite of the fact that the set and the sum, etc. can only have one or two elements.

The energy-balance relation for the type of geometrical arrangement under consideration, which follows easily from consideration of Green’s identities (see [27]), is given by

$$\mathrm{Re}(C_0^{-})+\sum _{n\in \mathcal{N}}{e}_n^{+}+e_n^{-}=0,$$ 2.10

where

$$e_n^{\pm }={\left|C_n^{\pm }\right|}^2\frac{{\beta }_n}{\beta }n\in \mathcal{N}.$$ 2.11

(b). Classical quasi-periodic Green function

Given $k\notin {\mathcal{K}}_{\mathrm{wa}}$, the classical quasi-periodic Green function is defined by

$$G_{0,k}^q(X,Y)=\sum _{n\in \mathbb{Z}}{\mathrm{e}}^{-\mathrm{i}\alpha nL}{G}_{0,k}(X+nL,Y),$$ 2.12

where G_0,k denotes the free space Green function for the Helmholtz equation

$$G_{0,k}(X,Y)=\frac{\mathrm{i}}{4}{H}_0^{(1)}(k\sqrt{X^2+Y^2})$$ 2.13

and $H_0^{(1)}$ is the zero-order Hankel function of the first kind. (The subindex ‘0’ in G_0,k and $G_{0,k}^q$ indicates that the number of ‘shifts’ used is zero; see §3.) This infinite sum converges at all points (X,Y)≠(md,0), $m\in \mathbb{Z}$ and, moreover, the truncated series

$$\sum _{n\in \mathbb{Z},\left|n\right|\ge 2}{\mathrm{e}}^{-\mathrm{i}\alpha nL}{G}_{0,k}(X+nL,Y)$$

converges uniformly over compact subsets of the plane not containing singularities of $G_{0,k}^q$ [28].

The lack of convergence of the classical quasi-periodic Green function (2.12) at Wood anomalies has historically presented some of the most significant challenges in the solution of periodic scattering problems. Section 3 introduces a modified quasi-periodic Green function, namely the shifted quasi-periodic Green function, which does not suffer from this difficulty.

3. Shifted quasi-periodic Green function

In order to solve scattering problems at all frequencies, we make use of the shifted quasi-periodic Green function introduced in [1] which converges even at and around Wood-anomaly frequencies. In this method, convergence at Wood anomalies (and, indeed, fast convergence even away from Wood anomalies) is achieved via addition of a number j of new Green-function poles positioned outside the physical propagation domain, along a certain given direction (e.g. vertically underneath the true pole) and at distances h,2h,…jh from the true Green-function pole. For example, taking j=1 the respective shifted Green-function is given by

$$G_{1,k}(X,Y)=G_{0,k}(X,Y)-G_{0,k}(X,Y+h).$$ 3.1

In order to appreciate the advantages that result from use of this Green function we use the mean value theorem together with the relation ${(H_0^{(1)})}^{\mathrm{\prime }}=-H_1^{(1)}$ and obtain

$$G_{1,k}(X,Y)=\frac{\mathrm{i}}{4}\frac{hk(Y+\xi )}{\sqrt{X^2+{(Y+\xi )}^2}}{H}_1^{(1)}(k\sqrt{X^2+{(Y+\xi )}^2})$$ 3.2

for some ξ∈(0,h). In view of the asymptotic expression

$$H_{\nu }^{(1)}(t)=\sqrt{\frac{2}{\pi t}}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}(t-\nu (\pi /2)-\pi /4)}\{1+O\left(\frac{1}{t}\right)\}\text{as }t\to \mathrm{\infty }(\nu \in \mathbb{R}),$$ 3.3

it follows that there exists C>0 such that for large values of $\left|X\right|$ and bounded $\left|Y\right|$ we obtain the enhanced decay

$$|G_{1,k}(X,Y)|\le \frac{C}{{\left|X\right|}^{3/2}}.$$ 3.4

As shown in [1], Green functions with arbitrary algebraic decay can be obtained by generalizing this idea: given h>0 and $j\in \mathbb{N}$ the half-space-shifted Green function G_j,k containing j shifts is defined by ‘finite-differencing’ Green functions associated with various poles:

$$G_{j,k}(X,Y)=\frac{\mathrm{i}}{4}\sum _{\ell =0}^j{(-1)}^{\ell }\left(\genfrac{}{}{0ex}{}{j}{\ell }\right)H_0^{(1)}(k\sqrt{X^2+{(Y+\ell h)}^2})$$ 3.5

for $(X,Y)\in {\mathbb{R}}^2$, (X,Y)≠(0,ℓh), ℓ=0⋯j. (Note the jth order finite-difference coefficients (−1)^l(j ℓ) in equation (3.5).) The fast decay of this function is established by lemma 3.1, whose proof can be found in [1].

Lemma 3.1 —

Let $j\in \mathbb{N},$ h>0, ℓ=0 or ℓ=1, m=1 or m=2 and k>0 be given. Then, for each M>0, there exists a positive constant C_M (that also depends on j,k and h) such that, for all y∈(−M,M) and for all real numbers X with $\left|X\right|>1,$ we have

$$|{\mathrm{\partial }}_m^{\ell }{G}_{j,k}(X,Y)|\le \{\begin{array}{ll}{C}_M{\left|X\right|}^{-(j+1)/2}& j\mathit{ is\; even}\\ C_M{\left|X\right|}^{-j/2-1}& j\mathit{ is\; odd,}\end{array}$$ 3.6

where ${\mathrm{\partial }}_m^{\ell }$ denotes differentiation of order ℓ in the mth coordinate direction.

The corresponding shifted quasi-periodic Green function is thus defined by

$$G_{j,k}^q(X,Y)=\sum _{n\in \mathbb{Z}}{\mathrm{e}}^{-\mathrm{i}\alpha nL}{G}_{j,k}(X+nL,Y).$$ 3.7

The fast convergence of such series, which follows from lemma 3.1, is laid out in theorem 3.2.

Theorem 3.2 —

Let $j\in \mathbb{N},$ h>0, ℓ=0 or ℓ=1, m=1 or m=2 and k₀>0 be given. Then, for each M>0 and δ>0, there exists a constant D_M,δ>0 (that also depends on j,k₀ and h) such that, for all X,Y satisfying −L≤X≤L and −M<Y <M, for all frequencies k>0 such that $|k-k_0|<\delta ,$ and for all integers N>1, we have

$$|\sum _{n\in \mathbb{Z}:\left|n\right|\ge N}{\mathrm{e}}^{-\mathrm{i}\alpha nL}{\mathrm{\partial }}_m^{\ell }{G}_{j,k}(X+nL,Y)|\le \{\begin{array}{ll}{D}_{M,\delta }{\left|N\right|}^{-(j-1)/2}& j\mathit{ is\; even}\\ D_{M,\delta }{\left|N\right|}^{-j/2}& j\mathit{ is\; odd}.\end{array}$$ 3.8

It follows that for ℓ=0 and ℓ=1 with m=1 or m=2:

• (1) The finite sums $\sum _{\left|n\right|\le N}{\mathrm{e}}^{-\mathrm{i}\alpha nL}{\mathrm{\partial }}_m^{\ell }{G}_{j,k}(X+nL,Y)$ converge as $N\to \mathrm{\infty }$ to the corresponding quantities ${\mathrm{\partial }}_m^{\ell }{G}_{j,k}^q$.

• (2) The corresponding approximation errors decrease at least as fast as ${\left|N\right|}^{-(j-1)/2}$ if j is even and as fast as ${\left|N\right|}^{-j/2}$ if j is odd.

Proof. —

This result, which is equivalent to theorem 4.4 in [1], emphasizes the fact that the constant D_M,δ can be taken to be independent of the wavenumber k for all k in a neighbourhood of a given wavenumber k₀. The proof of the present version of the result can be obtained by inspection of the corresponding proof in [1]. □

4. Periodic array scattering at and around Wood anomalies

(a). Strategy

As is well known [12,13,28], the classical quasi-periodic Green function displayed in equation (2.12) can alternatively be expressed in the spectral form

$$G_{0,k}^q(X,Y)=\frac{\mathrm{i}}{2L}\sum _{n\in \mathbb{Z}}\frac{{\mathrm{e}}^{\mathrm{i}{\alpha }_{n}X+\mathrm{i}{\beta }_n\left|Y\right|}}{{\beta }_n},$$ 4.1

which, of course, is only valid provided β_n≠0 for all $n\in \mathbb{Z}$. In other words, equation (4.1) is only meaningful away from Wood anomalies (see remark 2.2). Accordingly, the ‘spatial’ series (2.12) fails to converge at Wood-anomaly frequencies, and its convergence slows down significantly as a Wood anomaly is approached. In the context of equation (4.1), this difficulty clearly arises from those terms in equation (4.1) whose denominator tends to zero as a Wood anomaly is approached: if such ‘Wood’ terms are excluded then the infinite sum (4.1) converges and, in fact, it yields an analytic function with respect to k for each (X,Y)≠(mL,0), $m\in \mathbb{Z}$. The solution strategy described in the present §4, which is based on detailed analysis around the individual Wood terms, can be briefly summarized as follows:

• (1) Integral equations for array scattering based on the classical Green function (2.12) are obtained.

• (2) The resulting integral operator is re-expressed as an integral operator defined in terms of the shifted Green function (which is defined for every frequency, including Wood anomalies), plus a certain ‘Rayleigh-series operator’.

• (3) For a given Wood-anomaly frequency $k_0\in {\mathcal{K}}_{\mathrm{wa}}$, the Rayleigh-series operator mentioned in point (2) includes a finite-rank operator which encapsulates the singular character of the Wood-anomaly problem. Solutions for the resulting linear operator equation can then be obtained at and around k₀ by resorting to ideas closely related to the Woodbury–Sherman–Morrison formulae.

Points (1) and (2) are considered in §4b while point (3) is addressed in §4c.

(b). Hybrid ‘Rayleigh-expansion/integral-equation’ formulation

Given $k\notin {\mathcal{K}}_{\mathrm{wa}}$ the potential

$$u_k^{\mathrm{s}}(r)={\int }_{\mathrm{\partial }{D}_0}(\frac{\mathrm{\partial }{G}_{0,k}^q}{\mathrm{\partial }\nu (r^{\mathrm{\prime }})}-\mathrm{i}\gamma G_{0,k}^q)(r-r^{\mathrm{\prime }})\psi (r^{\mathrm{\prime }})\hspace{0.17em}\mathrm{d}S(r^{\mathrm{\prime }})$$ 4.2

is a quasi-periodic radiating solution of the Helmholtz equation in $\mathrm{\Omega }={\mathbb{R}}^2\setminus D$. Note that, in view of the quasi-periodicity of u^s_k and u^inc_k, u^s_k satisfies the boundary conditions (2.3) if and only if it satisfies the boundary conditions

$$u_k^{\mathrm{s}}=-u_k^{\mathrm{inc}}\text{on}\hspace{0.17em}\mathrm{\partial }{D}_0,$$ 4.3

where ∂D₀ is the portion of the scattering boundary contained in the unit-cell. Using the well-known jump relations of the single- and double-layer potentials [29,26], it follows that u^s_k is a solution to equations (2.2) and (2.3) if and only if ψ∈C(∂D₀) satisfies the integral equation

$$\frac{1}{2}\psi (r)+\underset{\mathrm{\partial }{D}_0}{\int }(\frac{\mathrm{\partial }{G}_{0,k}^q}{\mathrm{\partial }\nu (r^{\mathrm{\prime }})}-\mathrm{i}\gamma G_{0,k}^q)(r-r^{\mathrm{\prime }})\psi (r^{\mathrm{\prime }})\hspace{0.17em}\mathrm{d}S(r^{\mathrm{\prime }})=-u_k^{\mathrm{inc}}(r),r\in \mathrm{\partial }{D}_0.$$ 4.4

Here γ is a non-negative real number, and the linear combination of single- and double-layer potentials in equation (4.2) is used to ensure invertibility of the integral-equation formulation at wavenumbers that equal eigenvalues of the Laplace operator within D₀ (e.g. [29] as well as the related literature [26,30–32]).

An alternative operator equation can be obtained by noting that

$$G_{0,k}^q(X,Y)=G_{j,k}^q(X,Y)-\sum _{\ell =1}^j{(-1)}^{\ell }\left(\genfrac{}{}{0ex}{}{j}{\ell }\right)G_{0,k}^q(X,Y+\ell h)$$ 4.5

and that, using (4.1), for Y >−h the rightmost sum in the above equation can be expressed in the form

$$\frac{\mathrm{i}}{2L}\sum _{\ell =1}^j{(-1)}^{\ell }\left(\genfrac{}{}{0ex}{}{j}{\ell }\right)G_{0,k}^q(X,Y+\ell h)=\frac{\mathrm{i}}{2L}\sum _{n\in \mathbb{Z}}\frac{{\sigma }_n(k)}{{\beta }_n(k)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)X+\mathrm{i}{\beta }_n(k)Y},$$ 4.6

where, in view of Newton’s binomial formula, we have σ_n(k)=(1−e^iβ_n(k)h)^j−1. Thus, selecting a sufficiently large shift spacing,

$$h>max\{|y-y^{\mathrm{\prime }}|:(x,y),(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\in \mathrm{\partial }{D}_0\text{for some }x,x^{\mathrm{\prime }}\},$$ 4.7

the integral operator in equation (4.4) equals

$${\int }_{\mathrm{\partial }{D}_0}(\frac{\mathrm{\partial }{G}_{j,k}^q}{\mathrm{\partial }\nu (r^{\mathrm{\prime }})}-\mathrm{i}\gamma G_{j,k}^q)(r-r^{\mathrm{\prime }})\psi (r^{\mathrm{\prime }})\hspace{0.17em}\mathrm{d}S(r^{\mathrm{\prime }})-\frac{\mathrm{i}}{2L}\sum _{n\in \mathbb{Z}}{\sigma }_n(k)\frac{I_{n,k}^{+}[\psi ]}{{\beta }_n(k)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)x+\mathrm{i}{\beta }_n(k)y},$$ 4.8

where

$$I_{n,k}^{+}[\psi ]={\int }_{\mathrm{\partial }{D}_0}(\frac{\mathrm{\partial }}{\mathrm{\partial }\nu (r^{\mathrm{\prime }})}({\mathrm{e}}^{-\mathrm{i}{\alpha }_n(k)x^{\mathrm{\prime }}-\mathrm{i}{\beta }_n(k)y^{\mathrm{\prime }}})-\mathrm{i}\gamma \hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}{\alpha }_n(k)x^{\mathrm{\prime }}-\mathrm{i}{\beta }_n(k)y^{\mathrm{\prime }}})\psi (r^{\mathrm{\prime }})\hspace{0.17em}\mathrm{d}S(r^{\mathrm{\prime }}).$$ 4.9

It follows that, letting (x,y)=r∈∂D₀ denote the Cartesian coordinates of r, equation (4.4) is equivalent to

$$\frac{1}{2}\psi (r)+{\int }_{\mathrm{\partial }{D}_0}(\frac{\mathrm{\partial }{G}_{j,k}^q}{\mathrm{\partial }\nu (r^{\mathrm{\prime }})}-\mathrm{i}\gamma G_{j,k}^q)(r-r^{\mathrm{\prime }})\psi (r^{\mathrm{\prime }})\hspace{0.17em}\mathrm{d}S(r^{\mathrm{\prime }})-\frac{\mathrm{i}}{2L}\sum _{n\in \mathbb{Z}}{\sigma }_n(k)\frac{I_{n,k}^{+}[\psi ]}{{\beta }_n(k)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)x+\mathrm{i}{\beta }_n(k)y}=-u_k^{\mathrm{inc}}(r).$$ 4.10

The corresponding expression for the potential (4.2) in terms of the shifted Green function $G_{j,k}^q$ is given by the expression

$$u_k^{\mathrm{s}}(r)={\int }_{\mathrm{\partial }{D}_0}(\frac{\mathrm{\partial }{G}_{j,k}^q}{\mathrm{\partial }\nu (r^{\mathrm{\prime }})}-i\gamma G_{j,k}^q)(r-r^{\mathrm{\prime }})\psi (r^{\mathrm{\prime }})\hspace{0.17em}\mathrm{d}S(r^{\mathrm{\prime }})-\frac{\mathrm{i}}{2L}\sum _{n\in \mathbb{Z}}{\sigma }_n(k)\frac{I_{n,k}^{+}[\psi ]}{{\beta }_n(k)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)x+\mathrm{i}{\beta }_n(k)y},$$ 4.11

which is valid in the region Ω⁺={y≥M⁻−h} (figure 1). For points in Ω⁻={y<M⁻−h}, the potential is represented by the Rayleigh series

$$u_k^{\mathrm{s}}(r)=\frac{\mathrm{i}}{2L}\sum _{n\in \mathbb{Z}}\frac{I_{n,k}^{-}[\psi ]}{{\beta }_n(k)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)x-\mathrm{i}{\beta }_n(k)y},$$ 4.12

where we have set

$$I_{n,k}^{-}[\psi ]={\int }_{\mathrm{\partial }{D}_0}(\frac{\mathrm{\partial }}{\mathrm{\partial }\nu (r^{\mathrm{\prime }})}({\mathrm{e}}^{-\mathrm{i}{\alpha }_n(k)x^{\mathrm{\prime }}+\mathrm{i}{\beta }_n(k)y^{\mathrm{\prime }}})-\mathrm{i}\gamma \hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}{\alpha }_n(k)x^{\mathrm{\prime }}+\mathrm{i}{\beta }_n(k)y^{\mathrm{\prime }}})\psi (r^{\mathrm{\prime }})\hspace{0.17em}\mathrm{d}S(r^{\mathrm{\prime }}).$$ 4.13

Away from Wood anomalies, the integral-equation formulation (4.4) or, equivalently, the hybrid integral-equation/Rayleigh-series operator formulation (4.10) are well posed: the same arguments used in the proof of the invertibility of the combined field formulation for a single bounded obstacle can be applied to the periodic problem provided that we assume uniqueness of solutions for the periodic PDE problem. Once these operator equations are solved, the representation formulae (4.2) or (4.11) and (4.12) can be used to obtain scattering solutions.

However, at and around Wood anomalies, further work is needed. As mentioned previously, the classical quasi-periodic Green function ceases to exist at Wood frequencies and therefore the integral equation (4.4) cannot be used. Even though the shifted Green function does exist at Wood anomalies (and, therefore, integral operators that have it as kernel are well defined) the hybrid formulation (4.10) cannot be used directly at this singular case. Indeed, at Wood anomalies the infinite sum in equation (4.10) contains some vanishing denominators. The merit of equation (4.10), however, is that it makes explicit the singular behaviour at Wood anomalies in the form of a finite-rank operator (the finite sum of those terms whose denominators are close to zero), whereas the remainder of the hybrid integral/Rayleigh-series operator on the left-hand side of equation (4.10) is well defined. Thus, the strategy to solve the problem relies on being able to invert equations of the general form

$$(A+R)\psi =f,$$ 4.14

where A is assumed to be an invertible operator and R a finite-rank operator containing vanishing denominators. Section 4c introduces a general linear algebra result (which can be seen as a slight re-interpretation of the Woodbury–Sherman–Morrison formulae) and then applies it to equation (4.10).

(c). Solution at and around Wood-anomaly frequencies

Let $k_0\in {\mathcal{K}}_{\mathrm{wa}}$ be a Wood frequency and consider the set of integers ${\mathcal{N}}_{\mathrm{wa}}=\{n\in \mathbb{Z}:{\beta }_n(k_0)=0\}$, which, in view of remark 2.2, has at most two elements. For a frequency k in a neighbourhood of k₀, we define the finite-rank operators

$$\begin{array}{rl}{R}_k(\psi )(x,y)& =-\frac{\mathrm{i}}{2L}\sum _{n\in {\mathcal{N}}_{\mathrm{wa}}}{\sigma }_n(k)\frac{I_{n,k}^{+}[\psi ]}{{\beta }_n(k)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)x+\mathrm{i}{\beta }_n(k)y}\end{array}$$ 4.15

$$\begin{array}{rl}{\tilde{R}}_k(\psi )(x,y)& =-\frac{\mathrm{i}}{2L}\sum _{n\in {\mathcal{N}}_{\mathrm{wa}}}{\sigma }_n(k)I_{n,k}^{+}[\psi ]\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)x+\mathrm{i}{\beta }_n(k)y},\end{array}$$ 4.16

where (x,y) are the Cartesian coordinates of a point in ∂D₀. The operator R_k isolates the finitely many terms in the infinite sum in equation (4.10) that contain denominators that vanish as k₀ is approached. The operator ${\tilde{R}}_k$ is a re-scaled version of R_k that does not contain vanishing denominators, and that will play a major role in the application of the inversion formula (4.22) to the scattering problem under consideration. Equation (4.10) can then be re-expressed in the form

$$(A_k+R_k)\psi =-u_k^{\mathrm{inc}},$$ 4.17

where

$$A_k\psi (r)=\frac{1}{2}\psi (r)+{\int }_{\mathrm{\partial }{D}_0}(\frac{\mathrm{\partial }{G}_{j,k}^q}{\mathrm{\partial }\nu (r^{\mathrm{\prime }})}-\mathrm{i}\gamma G_{j,k}^q)(r-r^{\mathrm{\prime }})\psi (r^{\mathrm{\prime }})\hspace{0.17em}\mathrm{d}S(r^{\mathrm{\prime }})-\frac{\mathrm{i}}{2L}\sum _{n\notin {\mathcal{N}}_{\mathrm{wa}}}{\sigma }_n(k)\frac{I_{n,k}^{+}[\psi ]}{{\beta }_n(k)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)x+\mathrm{i}{\beta }_n(k)y}.$$ 4.18

The proposed strategy to solve equation (4.10) (or, equivalently, (4.17)) at and around Wood anomalies relies strongly on an operator formula, related to the Woodbury and Sherman–Morrison relations [33], sections 2.7.1 and 2.7.3, which is presented in lemma 4.2.

Remark 4.1. —

The notation and hypotheses used in the lemma are as follows: X denotes a general vector space over $\mathbb{C}$, $\mathcal{L}(X)$ denotes the set of linear (not necessarily continuous) operators from X to X, $A\in \mathcal{L}(X)$ is an invertible operator and $R_{\mathit{b}}\in \mathcal{L}(X)$ denotes a finite-rank operator which for f∈X takes the value

$$R_{\mathit{b}}[f]=\sum _{j=1}^r\frac{{\ell }_j[f]}{b_j}{w}_j,$$ 4.19

where b_j denote non-zero complex numbers, ℓ_j denote linear functionals defined on X, and where w₁,…,w_r form a linearly independent set of elements of X. Additionally, we consider the re-scaled finite-rank operator

$$R_{\mathbf{\text{1}}}[f]=\sum _{j=1}^r{\ell }_j[f]w_j.$$ 4.20

Setting $\mathcal{W}=\{w_1,\dots ,w_r\}$ and $S=\mathrm{span}(\mathcal{W})\subseteq X$ we see that both R_bA⁻¹ and R₁A⁻¹ map all of X into S and, in particular, they map S into itself. We thus consider the restrictions of R_bA⁻¹ and R₁A⁻¹ to the subspace S and denote them by T_b and T₁, respectively. Letting, further, D_b:S→S denote the diagonal operator defined by

$$D_{\mathit{b}}{w}_j=b_j{w}_j,$$ 4.21

we obtain $R_{\mathit{b}}=D_{\mathit{b}}^{-1}{R}_{\mathbf{\text{1}}}$ throughout X and thus, in particular, $T_{\mathit{b}}=D_{\mathit{b}}^{-1}{T}_{\mathbf{\text{1}}}$.

Lemma 4.2. —

Using the notations and hypotheses as in remark 4.1, assume that the operator (D_b+T₁):S→S is invertible. Then the operator (A+R_b) is also invertible, and its inverse is given by

$${(A+R_{\mathit{b}})}^{-1}=A^{-1}(\mathbb{I}-{(D_{\mathit{b}}+T_{\mathbf{1}})}^{-1}{R}_{\mathbf{1}}{A}^{-1}),$$ 4.22

where $\mathbb{I}$ denotes the identity operator.

Proof. —

In view of the hypotheses of the lemma, the operator on the right-hand side of equation (4.22) is well defined. The lemma follows by direct verification (via multiplication) that the right-hand side operator in (4.22) is both a left and right inverse of A+R_b. (The expression (4.22) for an operator (A+R_b) in a normed space can be derived by writing ${(A+z{R}_{\mathit{b}})}^{-1}=A^{-1}{(\mathbb{I}+z{R}_{\mathit{b}}{A}^{-1})}^{-1}$, expressing the inverse operator ${(\mathbb{I}+z{R}_{\mathit{b}}{A}^{-1})}^{-1}$ by means of a Neumann series which is convergent for z sufficiently small, and performing some simple manipulations. The resulting formula holds for any value of z, and in particular for z=1.) ▪

The solutions of equation (4.10), for frequencies around Wood anomalies, are obtained by inverting the operator (4.18); lemma 4.3 establishes a few regularity properties of this operator which are necessary to establish theorem 4.4. In what follows, C(∂D₀) denotes the Banach space of complex-valued continuous functions along ∂D₀, endowed with the maximum norm.

Lemma 4.3. —

Let $k_0\in {\mathcal{K}}_{\mathrm{wa}}$. Then there exists δ>0 such that, for k∈(k₀−δ,k₀+δ), the operator A_k maps C(∂D₀) into C(∂D₀) and the mapping k→A_k is continuous.

Proof. —

Let δ>0, ρ>0 be such that $|{\beta }_n(k)|>\rho$ for all $n\notin {\mathcal{N}}_{\mathrm{wa}}(k_0)$ and all k satisfying $|k-k_0|<\delta$. (The existence of such δ and ρ values is easily checked from the definition of β_n(k) in §2a.) Thus, for $|k-k_0|<\delta ,$ all the denominators in the series in equation (4.18) are bounded away from zero. To study the mapping properties of A_k, we consider the integral operator and the ‘Rayleigh-series’ operator in equation (4.18) separately.

The kernel of the integral operator can be viewed as the sum of a weakly singular kernel and a continuous kernel. The weakly singular kernel is the portion of the term n=0 in the infinite sum (3.7), which results by using ℓ=0 in the corresponding finite sum G_j,k given in equation (3.5); the continuous kernel, in turn, equals the sum of all of the remaining terms in (3.7). Clearly, the weakly singular kernel equals a combination of the Hankel function $H_0^{(1)}(k|r|)$ ($|r|=\sqrt{X^2+Y^2}$) and its normal derivative.

In view of [29], theorem 2.6, it follows that the integral operator in equation (4.18) maps continuous densities ψ into continuous functions on ∂D₀. Further, because the singular kernel can be expressed in the form $F_1(k|r|)\log (k|r|)+F_2(k|r|)$ with smooth functions F₁ and F₂, it is easy to check that the singular-term portion of the integral operator varies continuously with respect to k (because $\log (k|r|)=\log (k)+\log (|r|)$ and because all other integrands in the singular-term operator vary smoothly with k). The continuity of the remaining terms of the integral operator in (4.18) as a function of k follows easily from the uniform convergence, established in theorem 3.2, of the series that defines the shifted quasi-periodic Green function. Therefore, the whole integral operator is a continuous function of k with values in the space C(∂D₀).

The mapping properties of the ‘Rayleigh-series’ operator, in turn, follow from the uniform convergence of the series (4.17)—whose terms, as shown in what follows, decay at a uniform exponential rate as soon as n is sufficiently large that ${\beta }_n=i\left|{\beta }_n\right|$. In detail, given ψ∈C(∂D₀) and n sufficiently large, for an arbitrary point (x,y)∈∂D₀, we obtain the following estimate:

$$|{\sigma }_n(k)\frac{I_{n,k}^{+}[\psi ]}{{\beta }_n(k)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)x+\mathrm{i}{\beta }_n(k)y}|\le \frac{(k+\gamma )}{\rho }\parallel \psi \parallel \sum _{\ell =1}^j\left(\genfrac{}{}{0ex}{}{j}{\ell }\right){\int }_{\mathrm{\partial }{D}_0}\left|{\mathrm{e}}^{-|{\beta }_n(k)|(y-y^{\mathrm{\prime }}+\ell h)}\right|\hspace{0.17em}\mathrm{d}S(r^{\mathrm{\prime }}),$$ 4.23

which, in view of the assumption (4.7) on the shift parameter h, clearly exhibits the claimed uniform exponential decay. Because each one of the terms in the infinite series is a continuous function of both the spatial variable along ∂D₀ as well as the frequency k, it follows that the infinite series operator in (4.18) defines a continuous function along ∂D₀ which varies continuously with k in the space C(∂D₀). This completes the proof of the lemma. ▪

Theorem 4.4, whose proof relies on lemmas 4.2 and 4.3, produces (unique) solutions of equation (4.10) for non-Wood frequencies arbitrarily close to a Wood frequency, it shows that those solutions admit a limit as the frequency tends to the Wood anomaly, and it provides an explicit expression for the limit solution which does not require use of a limiting process.

Theorem 4.4. —

Let $k_0\in {\mathcal{K}}_{\mathrm{wa}}$ and let S=span{w₁,…,w_r}(r=1 or r=2) be the (finite-dimensional) subspace of C(∂D₀) spanned by the elements $w_j=\exp (\mathrm{i}{\alpha }_{n_j}(k)x+\mathrm{i}{\beta }_{n_j}(k)y),$ with n_j (j=1,…,r) defined as in remark 2.2. If the operator A_k0:C(∂D₀)→C(∂D₀) (equation (4.18) with k=k₀) is invertible, then there exists δ>0 such that for $|k-k_0|<\delta$ we have

• (1) The operator A_k is invertible and the inverse $A_k^{-1}$ is a continuous function of k for $|k-k_0|<\delta$.

• (2) The restriction ${\tilde{T}}_k={\tilde{R}}_k{A}_k^{-1}{|}_S$ of the composite operator ${\tilde{R}}_k{A}_k^{-1}$ to the subspace S maps S into itself bijectively and bicontinuously, and the inverse ${\tilde{T}}_k^{-1}$ is a continuous function of k for $|k-k_0|<\delta$.

• (3) Let β(k)=(β_n1(k),…,β_nr(k)) and let $0<|k-k_0|<\delta$. Then, the solution ψ of equation (4.10) (or, equivalently, (4.17)) is given by ψ=ψ_k, where

  $${\psi }_k=-A_k^{-1}(\mathbb{I}-{(D_{\mathit{\beta }\mathbf{(}\mathit{k}\mathbf{)}}+{\tilde{T}}_k)}^{-1}{\tilde{R}}_k{A}_k^{-1})u_k^{\mathrm{inc}}.$$ 4.24

  Here, the operator D_β(k):S→S is defined by

  $$D_{\mathit{\beta }(\mathit{\text{k}})}{w}_j={\beta }_{n_j}{w}_j.$$ 4.25

Finally, the operator on the right-hand side of equation (4.24) and therefore ψ_k itself are well defined for $|k-k_0|<\delta ,$ and, in particular, at k=k₀. The correspondence k→ψ_k maps k∈(k₀−δ,k₀+δ) continuously into C(∂D₀). In particular, the solutions (4.24) tend uniformly to ψ_k0 as k→k₀.

Proof. —

The existence of a certain δ₁>0 such that A_k admits an inverse $A_k^{-1}$ for |k−k₀|<δ₁, as well as the continuity of $A_k^{-1}$ for 0<|k−k₀|<δ₁, follows easily from the invertibility of A_k0—by means of a simple perturbative argument based on the use of a Neumann series and the continuity of A_k with respect to k (lemma 4.3). Noting that the finite-rank operators ${\tilde{R}}_k$ vary continuously, further, a similar perturbative argument allows one to deduce the invertibility of ${\tilde{R}}_k{A}_k^{-1}$ and the continuity of its inverse for |k−k₀|<δ₁ (perhaps reducing the value of δ₁, if necessary) from the invertibility of ${\tilde{R}}_{k_0}{A}_{k_0}^{-1}$—which is itself established in appendix B under the hypothesis of the present theorem. We have thus established that, for some δ₁>0, points (1) and (2) hold for all k satisfying |k−k₀|<δ₁.

Point (3), in turn, follows by relating the inversion formula (4.22) to equation (4.17). Indeed, the invertibility of ${\tilde{T}}_k={\tilde{R}}_k{A}_k^{-1}$ (point (2)) implies the invertibility of $D_{\mathit{\beta }(k)}+{\tilde{T}}_k$ provided $\left|{\beta }_{n_j}\right|$ is sufficiently small for all j—which is certainly guaranteed provided $|k-k_0|<{\delta }_2$ for a sufficiently small value of δ₂>0. Thus, identifying D_b and T₁ (in lemma 4.2) with D_β(k) and ${\tilde{T}}_k$, respectively, the (D_b+T₁) invertibility assumption in lemma 4.2 is satisfied for $|k-k_0|<{\delta }_2$, and, therefore, equation (4.24) is obtained from equation (4.22), as desired.

To complete the proof of the theorem we note that because ${\tilde{T}}_{k_0}={\tilde{R}}_{k_0}{A}_{k_0}^{-1}:S\to S$ is invertible (appendix B), because the image of ${\tilde{R}}_{k_0}{A}_{k_0}^{-1}$ is contained in S, and because for k=k₀ we have D_β(k)=0 (and thus $D_{\mathit{\beta }(k_0)}+{\tilde{T}}_{k_0}={\tilde{T}}_{k_0}$), it follows that the right-hand side of equation (4.24) is also defined for k=k₀. The uniform convergence of ψ_k to ψ_k0 as k→k₀ is established by relying on the k-continuity at k=k₀ (established in points (1) and (2)) of the operators in equation (4.24) together with the smoothness of the incident field as a function of k. ▪

Theorem 4.4 relies on the invertibility of the operator A_k0. Unfortunately, the study of solution uniqueness for the operator A_k0 for $k_0\in {\mathcal{K}}_{\mathrm{wa}}$ presents difficulties: as pointed out in remark A.1 (appendix A), the potential

$$v_{k_0}(r)={\int }_{\mathrm{\partial }{D}_0}(\frac{\mathrm{\partial }{G}_{j,k_0}^q}{\mathrm{\partial }\nu (r^{\mathrm{\prime }})}-\mathrm{i}\gamma G_{j,k_0}^q)(r-r^{\mathrm{\prime }})\psi (r^{\mathrm{\prime }})\hspace{0.17em}\mathrm{d}S(r^{\mathrm{\prime }})-\frac{\mathrm{i}}{2L}\sum _{n\notin {\mathcal{N}}_{\mathrm{wa}}}{\sigma }_n(k_0)\frac{I_{n,k_0}^{+}[\psi ]}{{\beta }_n(k_0)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k_0)x+i{\beta }_n(k_0)y}$$ 4.26

associated with A_k0 is not necessarily a radiating solution and, therefore, classical arguments based on the uniqueness of a radiating solution for the associated PDE problem are not immediately applicable in this context. A detailed consideration of this problem is out of the scope of the present paper and is left for future work. Throughout this article, however, it is assumed that, as verified numerically (§6a), the operator A_k0 is indeed invertible and, therefore, the conclusions of theorem 4.4 hold.

(d). Solutions to the PDE problem (eqn2.2) and (eqn2.3) at and around Wood frequencies

This section provides a brief discussion of equation (4.24), and it uses the solutions to that equation (which, from theorem 4.4, are well defined for k in the neighbourhood $U_{\mathrm{wa}}=\{k\in \mathbb{R}:|k-k_0|<\delta \}$ of a given Wood frequency $k_0\in {\mathcal{K}}_{\mathrm{wa}}$) to construct solutions of the PDE (2.2) for frequencies k∈U_wa—including, in particular, the Wood frequency k=k₀.

To do this, we first note that, for k∈U_wa, the quantity ψ_k in equation (4.24) is the sum of the two terms

$${\psi }_k^{(1)}=-A_k^{-1}{u}_k^{\mathrm{inc}}\text{and}{\psi }_k^{(2)}=A_k^{-1}{(D_{\mathit{\beta }(k)}+{\tilde{T}}_k)}^{-1}{\tilde{R}}_k{A}_k^{-1}{u}_k^{\mathrm{inc}},$$ 4.27

both of which involve the inverse $A_k^{-1}$ of the operator A_k. (A theoretical discussion concerning the invertibility of the operator A_k is given at the end of §4c, whereas practical matters concerning actual inversion of A_k are discussed in §5.)

The term ${\psi }_k^{(1)}$ is obtained by a direct application of the inverse operator $A_k^{-1}$. The evaluation of ${\psi }_k^{(2)}$ can be viewed as a three-step process: (i) evaluation of $R_k{A}_k^{-1}{u}_k^{\mathrm{inc}}$, (ii) application of the inverse ${(D_{\mathit{\beta }(k)}+{\tilde{T}}_k)}^{-1}$ and, finally, (iii) application of $A_k^{-1}$. In view of equation (4.16) for point (i) we have

$${\tilde{R}}_k{A}_k^{-1}{u}_k^{\mathrm{inc}}=\sum _{j=1}^r{c}_j{w}_j=\sum _{j=1}^r{c}_j\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_{n_j}(k)x+\mathrm{i}{\beta }_{n_j}(k)y},$$ 4.28

where $c_j=c_j(k)=-(\mathrm{i}/2L){\sigma }_{n_j}(k)I_{n_j,k}^{+}[A_k^{-1}{u}_k^{\mathrm{inc}}]$. The inverse of the finite-dimensional operator $D_{\mathit{\beta }(k)}+\tilde{T}$ mentioned in point (ii), on the other hand, can easily be applied to ${\tilde{R}}_k{A}_k^{-1}{u}_k^{\mathrm{inc}}$ by first obtaining the matrices ${\mathcal{D}}_{\mathit{\beta }(k)}$ and $\mathcal{T}$ of the operators ${\mathcal{D}}_{i,j}$ and ${\mathcal{T}}_{i,j}$ in the basis {w_j=e^{iα_nj(k)x+iβ_nj(k)y}:j=1,…,r} of the space S. We obtain

$${\mathcal{D}}_{i,j}={\delta }_i^j{\beta }_{n_j}\text{and}{\mathcal{T}}_{i,j}=-\frac{\mathrm{i}}{2L}{\sigma }_{n_i}(k)I_{n_i,k}^{+}[A_k^{-1}{w}_j]i,j=1,\dots ,r.$$ 4.29

Thus, letting $c(k)={(c_1(k),\dots ,c_r(k))}^t\in {\mathbb{C}}^r$ and calling $d(k)={(d_1(k),\dots ,d_r(k))}^t\in {\mathbb{C}}^r$ the solution of the linear system

$$({\mathcal{D}}_{\mathit{\beta }(k)}+\mathcal{T})d(k)=c(k),$$ 4.30

a straightforward linear algebra argument yields

$${(D_{\mathit{\beta }(k)}+{\tilde{T}}_k)}^{-1}{R}_k{A}_k^{-1}{u}_k^{\mathrm{inc}}=\sum _{j=1}^r{d}_j(k)w_j=\sum _{j=1}^r{d}_j\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_{n_j}(k)x+\mathrm{i}{\beta }_{n_j}(k)y}.$$ 4.31

(Note that for each k a unique solution d(k) of (4.30) indeed exists and varies continuously with k for |k−k₀| sufficiently small, because, as indicated in theorem 4.4, $D_{\mathit{\beta }(k)}+{\tilde{T}}_k$ is invertible in a neighbourhood of the Wood frequency k₀.) Upon application of $A_k^{-1}$ (point (iii)) we thus obtain the relation

$${\psi }_k=-A_k^{-1}(u_k^{\mathrm{inc}})+\sum _{j=1}^r{d}_j(k)A_k^{-1}{w}_j.$$ 4.32

In other words, for k∈U_wa, the density ψ_k can be viewed as the solution of an operator equation involving the operator A_k which is then corrected by addition of finitely many terms which involve the Rayleigh modes w_j=e^{iα_nj(k)x+iβ_nj(k)y}.

Once ψ_k has been obtained as indicated above, the corresponding radiating solution u^s_k of the PDE problem (2.2) and (2.3) can be produced for k∈U_wa. While for k∈U_wa∖{k₀} (and, indeed, for all $k\notin {\mathcal{K}}_{\mathrm{wa}}$) a direct substitution ψ=ψ_k in equations (4.11) and (4.12) yields the desired PDE solution u^s_k, we proceed differently—so as to ensure uniformly fast convergence near the Wood anomaly k₀. Indeed, to obtain the PDE solution for all k∈U_wa we exploit the fact that, as shown in what follows, the singular (or nearly singular) ‘plus/minus’ quotients

$$\frac{I_{n_j,k}^{\pm }[{\psi }_k]}{{\beta }_{n_j}(k)}(n_j\in {\mathcal{N}}_{\mathrm{wa}}(k_0))$$ 4.33

in equations (4.11) and (4.12) can be re-expressed, for all k∈U_wa, in terms of quantities that are well behaved at and around k₀; namely, the coefficients d_j(k) in equation (4.30) and the functionals ${\tilde{I}}_{n,k}$ defined in equation (4.38) below.

To do this let us consider the ‘plus’ quotients first. Noting that ψ_k is a solution of (4.17) we obtain the expression

$$R_k{\psi }_k=-u_k^{\mathrm{inc}}-A_k{\psi }_k,$$ 4.34

which, using equation (4.32), becomes

$$R_k{\psi }_k=\sum _{j=1}^r{d}_j(k)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_{n_j}(k)x+\mathrm{i}{\beta }_{n_j}(k)y}.$$ 4.35

In view of (4.15), it follows that for all k∈U_wa the ‘plus’ singular quotient in (4.33) can be expressed in the form

$$-\frac{\mathrm{i}}{2L}{\sigma }_n(k)\frac{I_{n_j,k}^{+}[{\psi }_k]}{{\beta }_{n_j}(k)}=d_j(k),j=1,\dots ,r$$ 4.36

in terms of the ‘regular’ quantities d_j=d_j(k).

The ‘minus’ quotients, in turn, can be expressed in terms of the ‘plus’ quotients. Indeed, subtracting (4.13) from (4.9), we obtain

$$I_{n,k}^{+}[\psi ]-I_{n,k}^{-}[\psi ]=-2\mathrm{i}{\beta }_n(k){\tilde{I}}_{n,k}[\psi ],$$ 4.37

where

$${\tilde{I}}_{n,k}[\psi ]={\int }_{\mathrm{\partial }{D}_0}(\frac{\mathrm{\partial }}{\mathrm{\partial }\nu (r^{\mathrm{\prime }})}({\mathrm{e}}^{-\mathrm{i}{\alpha }_n(k)x^{\mathrm{\prime }}}{y}^{\mathrm{\prime }}\text{sinc}({\beta }_n(k)y^{\mathrm{\prime }}))-\mathrm{i}\gamma \hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}{\alpha }_n(k)x^{\mathrm{\prime }}}{y}^{\mathrm{\prime }}\text{sinc}({\beta }_n(k)y^{\mathrm{\prime }}))\psi (r^{\mathrm{\prime }})\hspace{0.17em}\mathrm{d}S(r^{\mathrm{\prime }}),$$ 4.38

and where $\text{sinc}(t)=\sin (t)/t$. Thus, in view of (4.36), we obtain the expression

$$\frac{\mathrm{i}}{2L}\frac{I_{n_j,k}^{-}[{\psi }_k]}{{\beta }_{n_j}(k)}=-\frac{d_j(k)}{{\sigma }_{n_j}(k)}-\frac{1}{L}{\tilde{I}}_{n_j,k_0}[{\psi }_{k_0}],j=1,\dots ,r$$ 4.39

for the ‘minus’ quotients in terms of regular quantities.

The scattered-field functions (4.11) and (4.12) can now be evaluated in terms of the regular expressions (4.36) and (4.39). Indeed, using the solution d_j(k) of equation (4.30) and the integers n_j (j=1,…,r) introduced in remark 2.2, we obtain

$$\begin{array}{rl}{u}_k^{\mathrm{s}}(r)& ={\int }_{\mathrm{\partial }{D}_0}(\frac{\mathrm{\partial }{G}_{j,k}^q}{\mathrm{\partial }\nu (r^{\mathrm{\prime }})}-i\gamma G_{j,k}^q)(r-r^{\mathrm{\prime }}){\psi }_k(r^{\mathrm{\prime }})\hspace{0.17em}\mathrm{d}S(r^{\mathrm{\prime }})-\frac{\mathrm{i}}{2L}\sum _{n\notin {\mathcal{N}}_{\mathrm{wa}}}{\sigma }_n(k)\frac{I_{n,k}^{+}[{\psi }_k]}{{\beta }_n(k)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)x+\mathrm{i}{\beta }_n(k)y}\\ & +\sum _{j=1}^r{d}_j(k)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_{n_j}(k)x+\mathrm{i}{\beta }_{n_j}(k)y}\end{array}$$ 4.40

in Ω⁺={y≥M⁻−h}∖D (figure 1) and

$$u_k^{\mathrm{s}}(r)=\frac{\mathrm{i}}{2L}\sum _{n\notin {\mathcal{N}}_{\mathrm{wa}}}\frac{I_{n,k}^{-}[{\psi }_k]}{{\beta }_n(k)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)x-\mathrm{i}{\beta }_n(k)y}+\sum _{j=1}^r(d_j(k)-\frac{1}{L}{\tilde{I}}_{n_j,k}[{\psi }_k]){\mathrm{e}}^{\mathrm{i}{\alpha }_{n_j}(k)x-\mathrm{i}{\beta }_{n_j}(k)y}$$ 4.41

in Ω⁻={y<M⁻−h}. Note that (4.40) and (4.41) are defined for all values of k∈U_wa—including k=k₀.

To conclude this section, we establish that, as claimed, the Wood-anomaly potential u^s_k0 given by equations (4.40) and (4.41) is a radiating solution of the problem (2.2) and (2.3). To do this it suffices to show that

• (1) u^s_k0 is a solution of the Helmholtz equation in the interior of the regions Ω⁺ and Ω⁻.

• (2) The right-hand expressions in (4.40) and (4.41) agree in the region M⁻−h<y<M⁻.

• (3) u^s_k0 satisfies the condition of radiation at infinity in both Ω⁺ and Ω⁻ (see remark 2.1).

• (4) u^s_k0 verifies the boundary condition (2.3) or, equivalently, the boundary condition (4.3).

The validity of point (1) follows directly by inspection of equations (4.40) and (4.41). To establish point (2), in turn, we show that, in the region M⁻−h<y<M⁻, the Rayleigh expansion of the right-hand side in (4.40) coincides with the Rayleigh series (4.41). To produce the Rayleigh series of (4.40), we use equation (4.26) with ψ=ψ_k0, which results in an expression of the form

$$u_{k_0}^{\mathrm{s}}(x,y)=v_{k_0}(x,y)+\sum _{j=1}^r{d}_j(k_0)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_{n_j}(k_0)x}.$$ 4.42

Point (2) now follows directly by considering the Rayleigh expansion of v_k0 (equation (A 6) in appendix A). Point (3) follows directly by inspection of equations (4.41) and (4.42), using, for the latter, the appendix equation (A 5). Point (4), finally, results by evaluation of the potential u^s_k0 in (4.42) for points (x,y)∈∂D₀ (see equation (4.3) and associated text). The boundary values of the potential v_k0 in equation (4.42) can be obtained by evaluating (4.26) (with ψ=ψ_k0) on ∂D₀; the result is (4.18) (with ψ=ψ_k0) or, in other words, v_k0|_∂D0=A_k0ψ_k0. But this quantity can be obtained from theorem 4.4 applied at k=k₀ or, equivalently, from equation (4.32). But, then, equation (4.42) tells us that u^s_k0 verifies the boundary condition (2.3), as desired. Having established that points (1)–(4) hold, it follows that u^s_k0 is a radiating solution of the PDE problem posed by equations (2.2) and (2.3) at the Wood frequency k₀, as desired.

5. Numerical algorithm

Section 4 presents an integral-equation framework for evaluation of scattering solutions of the PDE problem (2.2) and (2.3) for wavenumbers k at and around a given Wood wavenumber k₀. For values of k away from all Wood wavenumbers, in turn, the proposed algorithm resorts to use of the windowed Green-function approach introduced in [1,21,22]. In fact, following these references, our algorithm applies the windowing approach in conjunction with the shifted Green-function method at all wavenumbers. This section presents a numerical discretization of the resulting continuous formulations. We consider the near-Wood-anomaly case first and we then succinctly describe the modifications that are necessary to obtain an algorithm valid for all frequencies. For values of k at and around a given Wood frequency $k_0\in {\mathcal{K}}_{\mathrm{wa}}$, the proposed algorithm for evaluation of the density ψ_k (that is used in equations (4.40) and (4.41)) results as a discrete analogue of the strategy embodied in equation (4.24)—or, equivalently (but in a form more closely related to the actual implementation), equation (4.32). The numerical integrations mentioned in what follows can be effected by means of any numerical integration method applicable to the kinds of smooth and logarithmic-singular integrands associated with the problems under consideration. The integration algorithms used in our implementation were derived in a direct manner from the high-order pointwise-discretization (Nyström) methods described in [26], section 3.5; in particular, this presentation assumes a discretization of the boundary of the scatterer D₀ by means of a given Nyström mesh such as those considered in [26]. Naturally, only one period of the scattering boundaries needs to be discretized; we assume a number n_i of Nyström nodes are used to discretize the boundaries contained in the reference unit period. Finally, the exponentially convergent infinite sum in equation (4.18) is truncated by including all propagating modes as well as a number N_ev of evanescent modes to the right (resp. to the left) of the largest (resp. smallest) element of the set $\mathcal{N}$, excepting all Wood frequency modes.

Away from Wood frequencies the evaluation of the shifted quasi-periodic Green function $G_{j,k}^q$ needed to produce the operator A_k and its inverse, which is required in (4.32) (see equation (4.18)), can be additionally accelerated by means of the windowing methodology introduced in [1,21]—which smoothly truncates the infinite sum (3.7) in such a way that $G_{j,k}^q$ is approximated by the finite sum

$$G_{j,k}^A(x,y)=\sum _{n\in \mathbb{Z}}{\mathrm{e}}^{-\mathrm{i}\alpha nL}{G}_{j,k}(ku(x+nL,y))W\left(\frac{x+nL}{A}\right),$$ 5.1

where A is a positive real number, and where $W:\mathbb{R}\to \mathbb{R}$ (figure 2) is the smooth cut-off function given by

$$W(t)=\{\begin{array}{ll}1& \mathrm{if}\hspace{0.17em}\left|t\right|\le c\\ \exp \left(\frac{2{\mathrm{e}}^{-1/\left|t\right|}}{\left|t\right|-1}\right)& \mathrm{if}\hspace{0.17em}c\le \left|t\right|\le 1\\ 0& \mathrm{if}\hspace{0.17em}\left|t\right|\ge 1\end{array}$$ 5.2

with 0<c<1. As shown in [21] the smooth-windowing methodology converges super-algebraically fast away from Wood anomalies as $A\to \mathrm{\infty }$, and thus this approach improves upon the shifted Green-function convergence—at least away from Wood frequencies. Bruno et al. [21] established super-algebraic convergence to the corresponding quasi-periodic Green function in the context of the problem of scattering by bi-periodic structures in three-dimensional space. Similar ideas can be applied in the two-dimensional problem under consideration. In fact, the proof is simpler in the present case—which does not require summation of doubly infinite sums.

Figure 2.

Windowing function W for c=0.5. (Online version in colour.)

We can now consider the problem of evaluation of the operator A_k, whose inverse appears on the right-hand side of (4.32). Clearly, using (4.18) and the values of a given continuous function ψ on the Nyström mesh, we can obtain the numerical values of A_kψ on the same Nyström mesh by numerical integration of the integral terms containing the Green function as well as those associated with the functionals I⁺_n,k, followed by summation of the infinite sum over $n\notin {\mathcal{N}}_{\mathrm{wa}}$ (whose general term decays exponentially, in accordance with (4.23)). Clearly, such an algorithm can be implemented in terms of a matrix–vector product for the vector which contains the discrete values of the function ψ. The inverse of the corresponding matrix provides the necessary discrete approximation of the operator $A_k^{-1}$. (In practice, the action of the numerical inverse on a given vector can be obtained either by means of an iterative linear-algebra solver or, as in the approach followed in the present two-dimensional context, directly by means of Gaussian elimination.)

In order to complete the evaluation of the solution ψ_k in (4.32) it is necessary to produce the coefficients d_j(k). But these values can be obtained by solving the r×r linear system (4.30) (see remark 2.2). The necessary elements in this matrix equation can be produced as follows: the diagonal matrix ${\mathcal{D}}_{\mathit{\beta }(k)}$ results from simple algebra, and the matrix $\mathcal{T}$ (equation (4.29)) and right-hand side c(k) (right below equation (4.28)) can be produced through respective applications of the aforementioned matrix form of the operator $A_k^{-1}$ followed by numerical integration and simple algebra.

The case in which k is away from Wood anomalies, finally, can be treated by means of a slightly modified version of the ‘Wood and near-Wood’ strategy described above—because, as it is readily checked, the operator A_k in equation (4.18) and the mixed-integral/Rayleigh-series operator in equation (4.10) differ only by the finite sum of terms with $n\in {\mathcal{N}}_{\mathrm{wa}}$. Thus, for a configuration away from Wood anomalies, it is only necessary to incorporate those terms which were excluded to obtain A_k in the near-Wood-anomaly case. (Note, however, that, away from Wood frequencies, one may select j=0 in (4.10), in which case the sum on the left-hand side of this equation vanishes, and a classical formulation is recovered.) In any case, the resulting discrete formulations can be inverted by means of an iterative solver or, for sufficiently small discretizations, by means of Gaussian elimination.

6. Numerical results

This section presents numerical tests and examples that demonstrate the character of the scattering solvers introduced in section 5, with an emphasis on the impact of the Wood-anomaly methods described in §4c. Detailed numerical results presented in what follows concern, in particular, the invertibility of the operator A_k around Wood frequencies (see theorem 4.4 and discussion immediately thereafter), as well as the conditioning, accuracy and computing costs associated with the proposed methodology for ranges of frequencies which, once again, include Wood anomalies. All solutions presented in this section were produced by means of a Fortran implementation of the algorithms described in §5, together with the LAPACK Gaussian elimination routines, on a single core of an Intel i7-4600M processor.

(a). Invertibility and conditioning at and around Wood frequencies

In order to provide numerical evidence of the invertibility of the operator A_k (equation (4.18)) for k in a neighbourhood of $k_0\in {\mathcal{K}}_{\mathrm{wa}}$ we consider the maximum and minimum singular values for a discretization of the operator A_k based on n_i=64 Nyström nodes (see §5), with varying values of both the window-size parameter A and the number j of shifts, and with N_ev=20 (see §5). We have found that, in all cases, the maximum singular values of A_k do not grow as k approaches k₀, and that the minimum singular values are all bounded away from zero. For definiteness, we present examples for the fairly generic test geometry depicted in figure 1 with L=4 and θ=0; similar results were obtained for periodic arrays of cylinders of various cross sections and for other numbers n_i of discretization points. In what follows, we express the window radius A in terms of the number N_per of periods L it spans: A=N_perL.

Figure 3 displays the smallest singular values of the discrete approximation ${\tilde{A}}_k$ of the operator A_k in equation (4.18) obtained using N_per=1600 together with a 64-point discretization of the scatterer D₀ depicted in figure 1. In order to provide close refinements near several Wood anomalies we use the parameters δ defined by k=k₀+sign(δ)⋅δ⁸ for $k_0=p(2\pi /L(1+\sin (\theta )))\in {\mathcal{K}}_{\mathrm{wa}}$ with p=2,4,6. (The δ parametrization of the frequency domain is used to achieve a fine resolution near the Wood-anomaly values corresponding to δ=0 for each one of the integers p=2,4,6.) Visually indistinguishable curves for both the smallest and largest singular values are obtained using a 128-point discretization of D₀ (while fixing the value A=N_perL with N_per=1600), which suggests that the approximate singular values provide close approximations of the corresponding smallest and largest singular values of the continuous operator A_k. Clearly, the smallest singular value σ_min remains bounded in the limit as δ→0.

Figure 3.

Smallest singular values of the discrete approximation ${\tilde{A}}_k$ of the operator A_k, for the scattering structure described in the text, as a function of δ (k=k₀+δ⁸), around the Wood frequencies $k_0=p(2\pi /L(1+\sin (\theta )))$ with p=2,4,6. (For reference, the maximum condition numbers κ=σ_max/σ_min for the cases p=2, 4 and 6 are approximately 280, 1508 and 2401, respectively.) Note that, in particular, the smallest singular values for δ⁸=10⁻¹⁶ and δ=0 $(k_0\in {\mathcal{K}}_{\mathrm{wa}})$ are included in this graph. (Online version in colour.)

Figure 4 presents the energy-balance errors (given by the left-hand side of equation (2.10) for efficiency values $e_n^{\pm }$ computed numerically with n_i=64) as a function of δ, for various values of N_per and for numbers j=0 and j=5 of shifts with shift-spacing h=3.5 (equations (3.5) and (3.7)). Studies based on the use of finer integration meshes and larger window sizes suggest that the actual solution errors are well described by the energy-balance curves presented in figure 4. This figure demonstrates clearly the beneficial effect induced by the presence of the Green-function shifts.

Figure 4.

Energy-balance error, as a function of δ (k=k₀+δ⁸ with $k_0=2\pi /L(1+\sin (\theta ))$), resulting from the use of the methodology introduced in §4c with j=0 (un-shifted windowed Green function) and j=5 (five-shifted windowed Green function) for a configuration with L=4 and θ=0 (normal incidence). The beneficial effect induced by the use of the shifted Green-function and Sherman–Morrison inversion (cf. theorem 4.4) can be clearly appreciated. (Online version in colour.)

(b). Computing cost

The Green-function representations and solvers presented in this paper give rise to fast numerical algorithms: like the rough-surface solvers [1], the present methods for periodic arrays of scatterers enable evaluation of highly accurate scattering solutions, with frequencies away from, at and around Wood anomalies, in computing times of the order of hundreds of milliseconds. In particular, the approach is highly competitive with other available solvers even for configurations away from Wood anomalies [4,6]. The method can be additionally accelerated by means of fast Fourier transform (FFT)-based approaches [21,34]; use of such acceleration methods in conjunction with shifted Green functions will be described elsewhere.

Sections 6b(i)–6b(iii) present the computing times required by the present solver for problems of scattering by periodic arrays (period L=2π) of circular cylinders. Examples for arrays of cylinders of radii R=0.05L, R=0.1L and R=0.25L, and configurations far from Wood anomalies, at Wood anomalies and around Wood anomalies are considered. Incidence angles in Littrow mount of order ℓ=−1 (for which the scattered plane wave of order ℓ=−1 propagates in the backscattering direction) were used; such a configuration is obtained provided the triplet (k,L,θ) verifies the relation $kL\sin (\theta )=\pi$. As noted in §5, Wood frequencies can be obtained by enforcing the relation $k_0=(2\pi /L(1+\sin (\theta )))n$ for some positive integer n; it can be easily checked that, under the Littrow-mount assumption, the Wood-frequency condition reduces to k₀L=(2n−1)π for some positive integer n. In what follows, we consider the particular case n=2, which yields the Wood frequency k₀=1.5⋅2π/L (for which ${\mathcal{N}}_{\mathrm{wa}}=\{1,-2\}$; see remark 2.2). In particular, k=1⋅2π/L is not a Wood frequency while the frequency k=1.49⋅2π/L is close to the Wood frequency k₀.

Throughout this section, the computing times cited sufficed to produce full scattering solutions with an energy-balance error of the order of 10⁻⁸.

(i). Computing cost I: wavenumbers away from Wood anomalies

Tables 1–3 present the computing times required by the proposed algorithm to evaluate the scattered field with an energy–balance error of the order of 10⁻⁸ for the wavenumber k=1⋅L (away from Wood anomalies). The best computing times in each table are displayed in italicized characters. Note that the best computing times are obtained for a shifted Green function of relatively low order (j=1). The incremental gains in convergence speed resulting from the use of higher-order methods does give rise to reductions in the numbers N_per of periods necessary to achieve convergence within the prescribed tolerance, but these gains are outweighed by the additional computing cost required for evaluation of the larger numbers of shifted free-space Green-function values required in the higher-order Green function.

Table 2.

Cylinders of radius R=0.1L at the frequency k=1⋅2π/L.

j	0	1	2	3	4
ni	18	16	16	16	18
Nper	125	38	38	37	36
Nev	0	20	20	20	20
time (s)	0.1	0.07	0.07	0.08	0.09

Table 1.

Cylinders of radius R=0.05L at the frequency k=1⋅2π/L.

j	0	1	2	3	4
ni	18	16	18	18	18
Nper	100	30	45	24	22
Nev	0	18	18	16	16
time (s)	0.06	0.04	0.08	0.08	0.07

Table 3.

Cylinders of radius R=0.25L at the frequency k=1⋅2π/L.

j	0	1	2	3	4
ni	20	14	14	14	16
Nper	125	66	75	73	58
Nev	0	10	10	10	10
time (s)	0.13	0.06	0.09	0.11	0.13

(ii). Computing cost II: a Wood-anomaly frequency

Tables 4–6 present results for the structures considered in the previous section except for the frequency, which is here taken to equal the Wood frequency k=1.5 L. The best computing times are now 0.14, 0.6 and 2.6 s. These times, which are now obtained for the higher-order Green function (j=6), are somewhat higher than the corresponding 0.04, 0.07 and 0.06 times required for the non-Wood frequency k=1 L. The windowing procedure fails to accelerate convergence at Wood anomalies, and use of the higher-order shifted Green functions is therefore highly advantageous in such cases.

Table 5.

Same as table 2 but with k=1.5⋅2π/L.

j	1	2	3	4	5	6
ni	18	18	16	18	18	18
Nper	3500	5500	500	400	200	200
Nev	20	20	20	20	20	20
time (s)	2.35	3.96	0.7	0.6	0.7	0.8

Table 4.

Same as table 1 but with k=1.5⋅2π/L.

j	1	2	3	4	5	6
ni	18	20	16	18	18	18
Nper	450	600	100	100	50	30
Nev	20	20	20	20	20	20
time (s)	0.62	0.8	0.25	0.31	0.19	0.14

Table 6.

Same as table 3 but with k=1.5⋅2π/L.

j	1	2	3	4	5	6
ni	20	20	20	20	20	20
Nper	38 500	42 000	2500	2000	1000	750
Nev	20	20	20	20	20	20
time (s)	32.1	35.5	4.5	4.5	2.6	2.6

(iii). Computing cost III: frequencies near Wood anomalies

Tables 7–9 present results for the structures considered in the previous two sections, but now for the near-Wood frequency k=1.49 L. The computing times are comparable to Wood-frequency times presented in the previous section.

Table 8.

Same as table 2 but with k=1.49(2π/L).

j	0	1	2	3	4	5	6
ni	18	18	18	18	18	18	18
Nper	3700	900	1000	350	320	175	100
Nev	0	20	20	20	20	20	20
time (s)	3.2	0.76	0.91	0.46	0.49	0.47	0.4

Table 7.

Same as table 1 but with k=1.49(2π/L).

j	0	1	2	3	4	5	6
ni	18	16	16	16	16	18	18
Nper	3700	850	650	200	200	100	75
Nev	0	20	20	20	20	20	20
time (s)	3.2	0.59	0.5	0.36	0.4	0.37	0.32

Table 9.

Same as table 3 but with k=1.49(2π/L).

j	0	1	2	3	4	5	6
ni	20	20	20	20	20	20	20
Nper	3700	1200	1000	600	600	400	380
Nev	0	10	10	20	20	20	20
time (s)	3.2	1.21	1.14	1.05	0.99	1.07	1.56

7. Conclusion

This paper introduced a new methodology for solutions of problems of scattering by periodic arrays of cylinders with applicability throughout the spectrum—even at and around Wood frequencies. To the best of our knowledge, this is the first particle-array periodic-Green-function method that remains applicable around Wood frequencies. The algorithm yields fast solutions for frequencies in the resonance region, where wavelengths are comparable to the structural period. High-frequency problems are also amenable to efficient treatment by these methods in conjunction with FFT-based acceleration approaches [21,34]; the development of such accelerated methods, however, is left for future work.

Appendix A. Rayleigh expansion of v_k0 for $k_0\in {\mathcal{K}}_{\mathrm{wa}}$

The Rayleigh expansion of v_k0 can be obtained by substituting the Rayleigh series of the shifted Green function $G_{j,k_0}^q$ for $k_0\in {\mathcal{K}}_{\mathrm{wa}}$ in equation (4.26). In the case j≥2, we have [1], eqs. (54), (56)

$$G_{j,k_0}^q(X,Y)=\frac{\mathrm{i}}{2L}\sum _{n\notin {\mathcal{N}}_{\mathrm{wa}}}\frac{{(1-{\mathrm{e}}^{\mathrm{i}{\beta }_n(k_0)})}^j}{{\beta }_n(k_0)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)X+\mathrm{i}{\beta }_n(k)Y}$$ A 1

for Y >0, while, for −h<Y <0, the corresponding modified version of the equation [1], equation (54) yields

$$G_{j,k_0}^q(X,Y)=\frac{\mathrm{i}}{2L}\sum _{n\notin {\mathcal{N}}_{\mathrm{wa}}}\frac{{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)X-\mathrm{i}{\beta }_n(k)Y}}{{\beta }_n(k_0)}+\frac{{(1-{\mathrm{e}}^{\mathrm{i}{\beta }_n(k_0)})}^j-1}{{\beta }_n(k_0)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)X+\mathrm{i}{\beta }_n(k)Y}+\frac{1}{L}\sum _{n\in {\mathcal{N}}_{\mathrm{wa}}}Y\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k_0)X}.$$ A 2

The analogous expressions for j=1 are

$$G_{j,k_0}^q(X,Y)=\frac{\mathrm{i}}{2L}\sum _{n\notin {\mathcal{N}}_{\mathrm{wa}}}\frac{{(1-{\mathrm{e}}^{\mathrm{i}{\beta }_n(k_0)})}^j}{{\beta }_n(k_0)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)X+\mathrm{i}{\beta }_n(k)Y}+\frac{h}{2L}\sum _{n\in {\mathcal{N}}_{\mathrm{wa}}}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k_0)X}$$ A 3

for Y >0 and

$$\begin{array}{rl}{G}_{j,k_0}^q(X,Y)& =\frac{\mathrm{i}}{2L}\sum _{n\notin {\mathcal{N}}_{\mathrm{wa}}}\frac{{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)X-i{\beta }_n(k)Y}}{{\beta }_n(k_0)}+\frac{{(1-{\mathrm{e}}^{\mathrm{i}{\beta }_n(k_0)})}^j-1}{{\beta }_n(k_0)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)X+\mathrm{i}{\beta }_n(k)Y}\\ & +\frac{1}{L}\sum _{n\in {\mathcal{N}}_{\mathrm{wa}}}(\frac{h}{2}+Y)e^{i{\alpha }_n(k_0)X}\end{array}$$ A 4

for −h<Y <0. For j≥2, we thus obtain

$$v_{k_0}(x,y)=\frac{\mathrm{i}}{2L}\sum _{n\notin {\mathcal{N}}_{\mathrm{wa}}}\frac{I_{n,k_0}^{+}[\psi ]}{{\beta }_n(k)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)x+\mathrm{i}{\beta }_n(k)y}$$ A 5

in the region {y>M⁺} and

$$v_{k_0}(x,y)=\frac{\mathrm{i}}{2L}\sum _{n\notin {\mathcal{N}}_{\mathrm{wa}}}\frac{I_{n,k_0}^{-}[\psi ]}{{\beta }_n(k_0)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k_0)x-\mathrm{i}{\beta }_n(k_0)y}+\frac{1}{L}\sum _{n\in {\mathcal{N}}_{\mathrm{wa}}}{I}_{n,k_0}^{+}[\psi ]y\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k_0)x}-{\tilde{I}}_{n,k_0}[\psi ]\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k_0)x}$$ A 6

in the region {y<M⁻} (figure 1). Here, the functional ${\tilde{I}}_{n,k_0}$ is defined in equation (4.38); note, further, that for $n\in {\mathcal{N}}_{\mathrm{wa}}$ we have I⁺_n,k0=I⁻_n,k0.

In the case j=1, finally, we obtain

$$v_{k_0}(x,y)=\frac{\mathrm{i}}{2L}\sum _{n\notin {\mathcal{N}}_{\mathrm{wa}}}\frac{I_{n,k_0}^{+}[\psi ]}{{\beta }_n(k)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)x+\mathrm{i}{\beta }_n(k)y}+\frac{h}{2L}\sum _{n\in {\mathcal{N}}_{\mathrm{wa}}}{I}_{n,k_0}^{+}[\psi ]\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k_0)x}$$ A 7

in the region {y>M⁺} and

$$\begin{array}{rl}{v}_{k_0}(x,y)& =\frac{\mathrm{i}}{2L}\sum _{n\notin {\mathcal{N}}_{\mathrm{wa}}}\frac{I_{n,k_0}^{-}[\psi ]}{{\beta }_n(k_0)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k_0)x-\mathrm{i}{\beta }_n(k_0)y}+\frac{1}{L}\sum _{n\in {\mathcal{N}}_{\mathrm{wa}}}{I}_{n,k_0}^{+}[\psi ]y\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k_0)x}\\ & +(\frac{h}{2}{I}_{n,k_0}^{+}[\psi ]-{\tilde{I}}_{n,k_0}[\psi ]){\mathrm{e}}^{\mathrm{i}{\alpha }_n(k_0)x}\end{array}$$ A 8

in the region {y<M⁻} (figure 1).

Remark A.1. —

From equations (A 6) and (A 8), we see that the potential v_k0 could in principle contain unbounded modes (that grow linearly with y) in the region {y<M⁻}—and, thus, the solution v_k0 could, in principle, fail to satisfy the radiation condition put forth in remark 2.1. The unbounded modes are not present in the expansions (A 6) and (A 8) provided the density ψ satisfies the conditions $I_{n,k_0}^{+}[\psi ]=0$ for all $n\in {\mathcal{N}}_{\mathrm{wa}}$. But, in view of equation (4.30) and the parenthetical comment that follows (4.31), equation (4.36) tells us that $I_{n_j,k}^{+}$ (j=1,…,r) vanishes for k=k₀—because so does the denominator on the left-hand side of that equation. Because n_j (j=1,…,r) is an enumeration of ${\mathcal{N}}_{\mathrm{wa}}$, we see that the density ψ_k0 obtained at a Wood anomaly in theorem 4.4 indeed satisfies the condition $I_{n,k_0}^{+}[\psi ]=0$ for all $n\in {\mathcal{N}}_{\mathrm{wa}}$. It follows that the corresponding potential v_k0 (defined in terms of ψ_k0) is a radiating potential both above and below the obstacles.

Remark A.2. —

In connection with the previous remark, we note that the functionals $I_{n,k_0}^{+}[\psi ]$ ($n\in {\mathcal{N}}_{\mathrm{wa}}$) do not identically vanish: e.g. setting ψ=e^iα_nx in (4.9) yields a non-zero value, because the integral involving the normal derivative does vanish. It follows that, for a given (arbitrary) density ψ, the potential v_k0 does not generally satisfy the radiation condition and, therefore, as indicated in §4c, classical arguments based on the uniqueness of the radiating solution for the associated PDE problem are not immediately applicable to the study of uniqueness of the operator A_k0 with $k_0\in {\mathcal{K}}_{\mathrm{wa}}$.

Appendix B. Invertibility of ${\tilde{R}}_{k_0}{A}_{k_0}^{-1}$

Using the representations (A 5) and (A 6) (in the case j≥2) or (A 7) and (A 8) (in the case j=1), we can now deduce the invertibility of the restriction of the operator ${\tilde{R}}_{k_0}{A}_{k_0}^{-1}$ to the finite-dimensional space S (see theorem 4.4), provided uniqueness of the solution of the problem (2.2) and (2.3) holds for k=k₀. Indeed, for f∈S such that ${\tilde{R}}_{k_0}{A}_{k_0}^{-1}f=0$, in view of equation (4.16), we must have $I_{n,k_0}^{+}[A_{k_0}^{-1}f]=0$ for all $n\in {\mathcal{N}}_{\mathrm{wa}}$. In view of remark A.1, then, the potential v_k0 defined in (4.26) with $\psi =A_{k_0}^{-1}f$ is a radiating solution of the Helmholtz equation. Moreover, using the jump relations of the single- and double-layer potentials it follows that v_k0 equals f along the boundary of D₀. But, because f∈S,

$$f=\sum _{n\in {\mathcal{N}}_{\mathrm{wa}}}{C}_n\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)x},$$ B 1

it follows (assuming the uniqueness of radiating solutions for the PDE problem for k=k₀) that

$$v_k(x,y)=\sum _{n\in {\mathcal{N}}_{\mathrm{wa}}}{C}_n\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\alpha }_n(k)x}$$ B 2

in the complete exterior domain $\mathrm{\Omega }={\mathbb{R}}^2\setminus D$. Because $I_{n,k_0}^{+}[\psi ]=0$, $n\in {\mathcal{N}}_{\mathrm{wa}}$, then v_k does not contain the Wood modes in the region y>M⁺ (see equations (A 5) or (A 7) in the respective cases j≥2 and j=1) and therefore C_n=0 for all $n\in {\mathcal{N}}_{\mathrm{wa}}$. In other words, the operator ${\tilde{R}}_{k_0}{A}_{k_0}^{-1}:S\to S$ in finite-dimensional vector space S has trivial null space, and it must therefore be invertible, as desired.

Authors' contributions

O.P.B. and A.G.F.-L. contributed to all aspects of this article.

Competing interests

No competing interests exist.

Funding

This work was supported by NSF and AFOSR through contracts DMS-1411876 and FA9550-15-1-0043, and by the NSSEFF Vannevar Bush Fellowship under contract no. N00014-16-1-2808.