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

Abstract

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.

1. Introduction

This work presents the second part of a two-part contribution. The first part [1], which will be referenced as Part I throughout this paper, introduced a ‘windowed Green function’ method, which, using a smooth cut-off, approximates the quasi-periodic Green function with super-algebraically small errors—that is, errors that admit upper bounds proportional to any negative power of the numbers of terms used—for configurations that are not close to a certain set of ‘Wood frequencies’. As discussed in Part I and references therein, at Wood frequencies the classical quasi-periodic Green function ceases to exist, and, therefore, integral equation methods based on such Green functions are inapplicable. Following upon the two-dimensional work [2], the present Part II introduces an additional element in the method: the ‘shifted’ quasi-periodic Green function. By adding copies of the quasi-periodic Green function that are shifted perpendicular to the plane of periodicity, one obtains an algebraic convergence rate that increases with the number of shifts, even at and around Wood frequencies. Using the new quasi-periodic Green function $G_p^q(\mathbf{\text{x}})$, we establish, for the first time, that the Dirichlet problem of scattering by a smooth doubly periodic surface, or diffraction grating, at a Wood frequency is uniquely solvable. We present an efficient high-order numerical method based on the Green function $G_p^q(\mathbf{\text{x}})$ for the problem of scattering by doubly periodic three-dimensional gratings at and around Wood frequencies. We believe this is the first solver in existence that is applicable to doubly periodic scattering problems at Wood frequencies.

Wood frequencies depend on the wave vector parallel to the grating, and they occur when one of the Rayleigh waves, or diffraction orders, is at the transition from propagating to evanescent in the ambient medium and is thus exactly grazing the surface. Because of this, they are often called ‘cut-off frequencies’. Certain scattering anomalies at or near these frequencies, known as ‘Wood anomalies’, were first observed experimentally by Wood [3] and treated mathematically by Rayleigh [4] and Fano [5]. A brief discussion concerning historical aspects in these regards can be found in [2], Remark 2.2; see also [6–9].

Computation of scattering by three-dimensional doubly periodic structures at or near Wood frequencies is particularly challenging. As mentioned above, in boundary integral methods the classical quasi-periodic Green function ceases to exist at Wood frequencies, even when the scattering problem admits a unique solution. A boundary-integral method that employs the free-space Green function and enforces periodicity through auxiliary layer potentials on the boundary of a period has been developed for two-dimensional problems [10,11], but a three-dimensional version of this method does not as yet exist. Approaches based on finite-element methods [12] often rely on the classical quasi-periodic Green functions in order to enforce the radiation condition at infinity; such approaches must also necessarily fail at Wood anomalies. Finally, finite-element methods exist, such as that presented in the contribution [13], which enforce the radiation condition on the basis of sponge layers such as the perfectly matched-layer technique. The results of that reference indicate that such approaches may become problematic for truly quasi-periodic problems and, we suggest, the difficulties are compounded at Wood frequencies, at which it would be necessary to damp waves which travel in directions parallel to the absorbing layer.

The classical quasi-periodic Green function G^q(x) ($\mathbf{\text{x}}=(x,y,z)\in {\mathbb{R}}^3$) can be constructed as an infinite sum of translated copies of the free-space Helmholtz Green function

$$G(\mathbf{\text{x}})=\frac{{\mathrm{e}}^{\mathrm{i}k|\mathbf{\text{x}}|}}{4\pi |\mathbf{\text{x}}|},$$ 1.1

with doubly periodically distributed monopole singularities. Indeed, let $\widetilde{\mathbf{\text{x}}}=(x,y)$, α:=(α,β) (the Bloch wavevector) and

$$r_{mn}^2=|\mathbf{\text{x}}+m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2{|}^2=|\widetilde{\mathbf{\text{x}}}+m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2{|}^2+z^2,$$ 1.2

in which v₁ and v₂ denote two independent vectors in ${\mathbb{R}}^2$ that characterize the periodicity. The quasi-periodic Green function can be expressed in the form

$$G^q(\mathbf{\text{x}})=\sum _{m,n\in \mathbb{Z}}G(\mathbf{\text{x}}+m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2)\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}\mathit{\alpha }\cdot (m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2)}=\frac{1}{4\pi }\sum _{m,n\in \mathbb{Z}}\frac{{\mathrm{e}}^{\mathrm{i}k{r}_{mn}}}{r_{mn}}\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}\mathit{\alpha }\cdot {\mathbf{\text{v}}}_{mn}},$$ 1.3

in which v_mn=mv₁+nv₂. As is well known, this expansion suffers from notoriously poor convergence properties. Various methods to accelerate its convergence, notably the Ewald method [14–16] among others [17–19], have been proposed. Importantly, however, the sum does not converge at Wood frequencies. This is a difficulty that is not addressed by any of the aforementioned acceleration approaches.

The convergence problem is elucidated by the Fourier representation of the Green function that results from application of the Poisson summation formula to the series (1.3). Letting ${\mathbf{\text{v}}}_1^{\ast }$ and ${\mathbf{\text{v}}}_2^{\ast }$ denote the dual vectors defined by ${\mathbf{\text{v}}}_i^{\ast }\cdot {\mathbf{\text{v}}}_j={\delta }_{ij}$, and setting D=∥v₁×v₂∥, ${\mathbf{\text{v}}}_{j\ell }^{\ast }=(2\pi j\hspace{0.17em}{\mathbf{\text{v}}}_1^{\ast }+2\pi \ell {\mathbf{\text{v}}}_2^{\ast })+\mathit{\alpha }$ and ${\gamma }_{j\ell }={(k^2-\parallel {\mathbf{\text{v}}}_{j\ell }^{\ast }{\parallel }^2)}^{1/2}$, one obtains the alternative expansion

$$G^q(\widetilde{\mathbf{\text{x}}},z)=\frac{\mathrm{i}}{2D}\sum _{j,\ell \in \mathbb{Z}}\frac{1}{{\gamma }_{j\ell }}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }|z|},$$ 1.4

which converges provided γ_jℓ≠0 for all j,ℓ—that is, away from Wood configurations (α,β,k) where one of these exponents vanishes. The branch of the square root that defines γ_jℓ is selected so that $\sqrt{1}=1$, with a branch cut along the negative imaginary semiaxis.

The primary and dual periodicity lattices are defined by

$$\mathrm{\Lambda }=\{{\mathbf{\text{v}}}_{mn}:\hspace{0.17em}m,n\in \mathbb{Z}\}\text{and}{\mathrm{\Lambda }}^{\ast }=\{{\mathbf{\text{v}}}_{j\ell }^{\ast }:\hspace{0.17em}j,\ell \in \mathbb{Z}\}.$$ 1.5

The lattice sum (1.3) is only defined if γ_jℓ≠0 for all integer pairs (j,ℓ). A Wood frequency (for given α) is a value of k for which at least one of the constants γ_jℓ vanishes. In such cases, both the spatial expansion (1.3) and the spectral representation (1.4) cease to exist.

The present contribution presents a Green function method that enables efficient and accurate evaluation of wave scattering by doubly periodic structures throughout the spectrum, including frequencies at and around Wood frequencies. The work additionally incorporates the smooth windowing approach introduced in Part I, which accelerates the Green-function convergence: a windowed version of the series (1.3) converges super-algebraically fast (i.e. faster than any power of the window size) for non-Wood configurations. The convergence of this windowed series deteriorates near Wood frequencies: the constants in the super-algebraic convergence estimates established in Part I grow without bound as Wood configurations are approached, and the windowed version of the lattice sum (1.3) once again fails to converge at Wood frequencies. When applied to the shifting method introduced in this work, the smooth windowing method increases the algebraic convergence rate at Wood anomalies by a factor equal to the truncation size raised to the power −1/2.

To establish convergence at Wood frequencies, the method introduced in this article replaces the free-space Green function term e^ikr_mn/r_mn in (1.3) by a pth-order finite difference for this function with respect to z, with p≥3. The combined effect of the shifting, or finite-differencing, plus the windowing yields a Green function that converges with algebraic order at all frequencies (see (3.1) and (3.2) below). The error is bounded by C_p/(A^{⌈p/2⌉−1}) as the window diameter A increases, as stated in theorem 3.2.

The algebraic convergence is based on the following idea. This new Green function is built of weighted combinations of z-shifted terms of the Fourier sum (1.4) (see (3.38) below), which, for z>0, are $\sum _{p=0}^q{a}_{pq}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }(z-dq)}/{\gamma }_{j\ell }$ with spacing d>0 between shifts. As $\widetilde{\mathbf{\text{x}}}$ becomes large compared to z, this finite difference essentially approaches a pth derivative of ${\mathrm{e}}^{\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }z}/{\gamma }_{j\ell }$ with respect to z, and this eliminates γ_jℓ from the denominator. But in doing so, it also annihilates the grazing mode completely at a Wood configuration where γ_jℓ=0 (for p>1). As the grazing mode is present in wave scattering from the periodic surface, it must be reinstated into the Green function, as described in §3b below. The proof of algebraic convergence is more complex than meets the eye, as indicated by the order of convergence ⌈p/2⌉−1. The proof that the integral equation involving this new Green function does yield a unique solution of the scattering problem is subtle and involves an interesting interplay between the spatial and spectral (Fourier) forms of the Green function.

The approach is demonstrated in §6 through an application to the problems of sound-soft and sound-hard scattering by doubly periodic surfaces throughout the spectrum. Other scattering problems can be treated similarly, as demonstrated in the contributions [2,20] concerning two-dimensional diffraction gratings and periodic arrays of cylinders.

This paper is organized as follows. After some preliminaries in §2, the shifted quasi-periodic Green function is introduced in §3. That section contains our main results: proof of algebraic convergence of the shifted Green function, theorem 3.2; and proof of unique solvability of the problem of reflection from a periodic grating via integral equations employing this new Green function, including at Wood configurations, theorem 4.1 (appendix A contains a lemma used in the aforementioned uniqueness proof, as well as a simplified existence and uniqueness proof which is only valid away from Wood anomalies.) §5 then outlines the numerical implementation. A variety of results presented in §6 demonstrate the character of the proposed solvers, for all configurations, far from, near and at Wood anomalies.

2. Periodic surface

We consider the sound-soft problem of scattering by a doubly periodic scattering surface

$$\mathrm{\Gamma }=\{(\widetilde{\mathbf{\text{x}}},z):\widetilde{\mathbf{\text{x}}}\in {\mathbb{R}}^2\text{ and }z=f(\widetilde{\mathbf{\text{x}}})\},$$

where f is a smooth doubly periodic function of periodicity Λ:

$$f(\widetilde{\mathbf{\text{x}}}+m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2)=f(\widetilde{\mathbf{\text{x}}})\text{for all}\hspace{0.17em}\widetilde{\mathbf{\text{x}}}\in {\mathbb{R}}^2\text{and all}\hspace{0.17em}m,n\in \mathbb{Z}.$$ 2.1

Letting

$${\mathrm{\Omega }}^{+}=\{(\widetilde{\mathbf{\text{x}}},z):\widetilde{\mathbf{\text{x}}}\in {\mathbb{R}}^2\text{ and }z>f(\widetilde{\mathbf{\text{x}}})\}$$ 2.2

and assuming an incident field

$$u^{\mathrm{inc}}(\mathbf{x})=\exp [\mathrm{i}(\mathit{\alpha }\cdot \widetilde{\mathbf{\text{x}}}-\gamma z)]$$ 2.3

impinges upon the surface from above (where (α,−γ) is the wavevector and |α|²+γ²=k²), the scattered field u under sound-soft conditions satisfies the equations

$$\begin{array}{rl}& \mathrm{\Delta }u+k^{2}u=0\mathrm{in}\hspace{0.17em}{\mathrm{\Omega }}^{+},\\ & \text{and}u=-u^{\mathrm{inc}}\mathrm{on}\hspace{0.17em}\mathrm{\Gamma }\end{array}\}$$ 2.4

together with the Sommerfeld radiation condition [21]: letting $z_{+}=maxf,$ we have

$$u(\widetilde{\mathbf{\text{x}}},z)=\sum _{j,\ell \in \mathbb{Z}}{B}_{j\ell }^{+}\{\mathrm{exp\hspace{0.17em}}\mathrm{i}[(2\pi j{\mathbf{\text{v}}}_1^{\ast }+2\pi \ell {\mathbf{\text{v}}}_2^{\ast })+\mathit{\alpha }]\cdot \widetilde{\mathbf{\text{x}}}\}\exp [\mathrm{i}{\gamma }_{j\ell }z],z>z_{+}.$$ 2.5

Although not physically relevant for the grating problems considered in this paper, the set

$${\mathrm{\Omega }}^{-}=\{(\widetilde{\mathbf{\text{x}}},z):\widetilde{\mathbf{\text{x}}}\in {\mathbb{R}}^2\text{ and }z<f(\widetilde{\mathbf{\text{x}}})\}$$ 2.6

below Γ and the associated radiation condition

$$u(\widetilde{\mathbf{\text{x}}},z)=\sum _{j,\ell \in \mathbb{Z}}{B}_{j\ell }^{-}\exp \{\mathrm{i}[(2\pi j{\mathbf{\text{v}}}_1^{\ast }+2\pi \ell {\mathbf{\text{v}}}_2^{\ast })+\mathit{\alpha }]\cdot \widetilde{\mathbf{\text{x}}}\}\exp [-\mathrm{i}{\gamma }_{j\ell }z],z<z_{-}.$$ 2.7

$(z_{-}=minf)$ will be used in the existence and uniqueness proofs in §3d and appendix A.

3. Shifted quasi-periodic Green function

The finite-difference half-space-shifted Green function we use can be viewed as a generalization of the Dirichlet half-space Green that results from the method of images. The method-of-images Green function decays more rapidly at infinity than the free-space Green function itself. In the three-dimensional case under consideration, such decay does not suffice to induce fast convergence, or even absolute convergence, in a corresponding series of the form (1.3). (By contrast, absolute convergence does result from use of the method-of-images Green function in the two-dimensional case [2], lemma 4.3 and theorem 4.4). But viewing the method-of-images Green function as a finite difference of the lowest order, a generalization of this idea emerges: as shown in what follows, a higher-order finite difference of the free-space Green function tends to zero more rapidly at infinity, and thus gives rise to an absolutely convergent quasi-periodic Green function series. In fact, the resulting quasi-periodic Green function can be made to converge with a prescribed order of accuracy provided finite differences of sufficiently high order are used.

To pursue this idea, fix a ‘shift’ value d>0 and define the shifted half-space Green function

$${\tilde{G}}_p(\mathbf{\text{x}})=\sum _{q=0}^p{a}_{pq}G(\mathbf{\text{x}}+(0,0,qd))=\sum _{q=0}^p{a}_{pq}G(\widetilde{\mathbf{\text{x}}}+(0,0,z+qd)),$$ 3.1

where a_pq denote the finite-difference coefficients [2,22]:

$$a_{pq}={(-1)}^q\left(\begin{array}{c}p\\ q\end{array}\right),0\le q\le p.$$

Clearly, the function ${\tilde{G}}_p(\mathbf{\text{x}})$ has poles at the points (0,0,−qd) for 0≤q≤p. As shown in lemma 3.1 below, the shifted half-space Green function ${\tilde{G}}_p(\mathbf{\text{x}})$ tends to zero algebraically fast as $\widetilde{\mathbf{\text{x}}}$ tends to infinity while z remains bounded. The shifted quasi-periodic Green function is then defined by

$${\tilde{G}}_p^q(\mathbf{\text{x}})=\sum _{m,n\in \mathbb{Z}}{\tilde{G}}_p(\mathbf{\text{x}}+m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2)\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}\mathit{\alpha }\cdot (m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2)}.$$ 3.2

The function ${\tilde{G}}_p^q(\mathbf{\text{x}})$ has poles at the points

$${\mathbf{\text{x}}}_{mnq}^{\prime }=(m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2,-qd),m,n,q\in \mathbb{Z},\hspace{0.17em}0\le q\le p.$$ 3.3

Theorem 3.2 below shows that the sum on the right-hand side of equation (3.2) converges algebraically fast: a truncation of this series to |mv₁+nv₂|≤A results in errors of order ⌈p/2⌉−1 (where ⌈r⌉ denotes the smallest integer greater than or equal to r) for all wavevectors k=(α,−γ) and all periodicity lattices Λ, including Wood-anomaly configurations. That theorem also establishes that somewhat improved accuracies result when a smooth windowed truncation, as introduced in part I, is additionally employed.

When evaluated at $(\widetilde{\mathbf{\text{x}}}-{\widetilde{\mathbf{\text{x}}}}^{\prime },z-z^{\prime })$ with $(\widetilde{\mathbf{\text{x}}},z)\in \mathrm{\Gamma }$ and $({\widetilde{\mathbf{\text{x}}}}^{\prime },z^{\prime })\in \mathrm{\Gamma }$, the terms with q=1,…,p in the series (3.1)–(3.2) are weighted copies of the free-space Green function with sources at the points $({\widetilde{\mathbf{\text{x}}}}^{\prime },z^{\prime }-qd)$ below the grating surface Γ, while the terms corresponding to q=0 produce the necessary sources on Γ. It follows that the layer potentials associated with the shifted Green function (cf. equation (4.1) below) are well defined for all points $\mathbf{\text{x}}=(\widetilde{\mathbf{\text{x}}},z)$ on and above the grating surface, and satisfy the Helmholtz equation for x above Γ.

It is important to note that, as shown in §3b, the addition of the shifted terms effectively suppresses all contributions in equation (1.4) that contain vanishing denominators at Wood frequencies, and thereby reinstates convergence of the quasi-periodic Green function even at such frequencies. But such modes are required in the complete solution of the grating scattering problem. Therefore, corresponding quasi-periodic plane-wave terms need to be added to the Green function to incorporate (now with controlled coefficients) all the necessary grazing modes, as detailed in §3b. The corresponding treatments for simply periodic scattering surfaces and arrays of cylinders in two dimensions is presented in [2,20], respectively.

(a). Convergence of the modified Green function at Wood frequencies

Lemma 3.1 below states that, as desired, the shifted half-space Green functions (3.1) enjoy enhanced degrees of decay as $\widetilde{\mathbf{\text{x}}}\to \mathrm{\infty }$. On the basis of this result, theorem 3.2 then establishes the fast convergence of two different truncations (using discontinuous and smooth window functions) for the spatial lattice sum (3.2). In preparation for the proofs, we examine the individual terms ${\tilde{G}}_p^q(\mathbf{\text{x}}+m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2)$. Considering the relations (1.2), (1.3) and (3.1), and using the notations

$$g(\rho ,\epsilon ):=\frac{{\mathrm{e}}^{\mathrm{i}k\rho \sqrt{1+{\epsilon }^2}}}{\rho \sqrt{1+{\epsilon }^2}},{\rho }_{mn}=|\widetilde{\mathbf{\text{x}}}+m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2|,{\epsilon }_{mn}=\frac{z}{{\rho }_{mn}}\text{and}{\hat{\epsilon }}_{mn}=\frac{d}{{\rho }_{mn}},$$ 3.4

the translated Green-function terms in the sum (3.2) can be expressed in the form

$${\tilde{G}}_p(\mathbf{\text{x}}+m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2)=\sum _{q=0}^p{a}_{pq}g({\rho }_{mn},{\epsilon }_{mn}+q{\hat{\epsilon }}_{mn}).$$ 3.5

Equivalently, letting

$$h(\rho ,\epsilon ,\hat{\epsilon }):=\sum _{q=0}^p{a}_{pq}g(\rho ,\epsilon +q\hat{\epsilon }),$$ 3.6

we have

$${\tilde{G}}_p(\mathbf{\text{x}}+m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2)=h({\rho }_{mn},{\epsilon }_{mn},{\hat{\epsilon }}_{mn}).$$ 3.7

To estimate the asymptotics of the function h as $\rho \to \mathrm{\infty },$ we use the finite-difference relation [23], p. 262, equation 7

$$\sum _{q=0}^p{a}_{pq}f(\epsilon +q\hat{\epsilon }))={(-1)}^p{\hat{\epsilon }}^p{f}^{(p)}(\epsilon +\xi ),\xi \in [0,p\hat{\epsilon }],$$ 3.8

which is valid for every p-times continuously differentiable function f. Thus, for each pair of values of ρ and $\hat{\epsilon }$ there exists ${\xi }_{\rho ,\hat{\epsilon }}\in [0,p\hat{\epsilon }]$ such that

$$h(\rho ,\epsilon ,\hat{\epsilon })={(-1)}^p{\hat{\epsilon }}^p\frac{{\mathrm{\partial }}^{p}g}{\mathrm{\partial }{\epsilon }^p}(\rho ,\epsilon +{\xi }_{\rho ,\hat{\epsilon }})$$

and, therefore,

$$h(\rho ,\frac{z}{\rho },\frac{d}{\rho })={(-1)}^p{\left(\frac{d}{\rho }\right)}^p\hspace{0.17em}\frac{{\mathrm{\partial }}^{p}g}{\mathrm{\partial }{\epsilon }^p}(\rho ,\frac{z}{\rho }+{\xi }_{\rho ,d/\rho }),{\xi }_{\rho ,d/\rho }\in [0,\frac{pd}{\rho }].$$ 3.9

In view of the asymptotic bounds on ∂^pg/∂ε^p in lemma 3.1 below, one obtains $h_p(\rho ,z/\rho ,d/\rho )=\mathcal{O}(1/{\rho }^{\lceil p/2\rceil +1})$ as $(\rho \to \mathrm{\infty })$ where the constant in the $\mathcal{O}$-term, which depends on z, d and p, can be taken to be fixed if d and p are given and z is contained in a bounded subset of $\mathbb{R}$.

Lemma 3.1 —

The pth-order derivative of the function g with respect to ε satisfies

$$|\frac{1}{{\rho }^p}\frac{{\mathrm{\partial }}^{p}g}{\mathrm{\partial }{\epsilon }^p}(\rho ,\epsilon )|\le \frac{C}{{\rho }^{\lceil p/2\rceil +1}}(\rho \ge 1,\hspace{0.17em}|\epsilon |<1),$$ 3.10

where the constant C is independent of ρ and ε for all ρ≥1 and all ε satisfying |ε|<1. Here, for real x, ⌈x⌉ denotes the smallest integer larger than or equal to x. The estimate (3.10) is sharp: the left-hand side in that equation does not decay like 1/ρ^t for any t>⌈p/2⌉+1. Furthermore,

$$h(\rho ,\frac{z}{\rho },\frac{d}{\rho })=\mathcal{O}\left(\frac{1}{{\rho }^{\lceil p/2\rceil +1}}\right)(\rho \to \mathrm{\infty }).$$ 3.11

Proof. —

It suffices to establish that (3.10) holds and is sharp; equation (3.11) then follows directly from (3.9) and (3.10). To obtain the relation (3.10), we first note that the pth derivative ∂^pg/∂ε^p of g with respect to ε can be expressed as a finite linear combination of the form

$$\frac{{\mathrm{\partial }}^{p}g}{\mathrm{\partial }{\epsilon }^p}(\rho ,\epsilon )=\sum _{(m,n,\ell )\in S_p}{C}_{m,n,\ell }^p\frac{{\epsilon }^m{(\mathrm{i}k\rho )}^n}{{\sqrt{1+{\epsilon }^2}}^{\ell }}\hspace{0.17em}\frac{{\mathrm{e}}^{\mathrm{i}k\rho \sqrt{1+{\epsilon }^2}}}{\rho },$$ 3.12

in which S_p is a certain set of triples of non-negative integer indices and $C_{m,n,\ell }^p$ denote real-valued coefficients. Defining

$$T_{m,n,\ell }(\rho ,\epsilon )=\frac{{\epsilon }^m{(\mathrm{i}k\rho )}^n}{{\sqrt{1+{\epsilon }^2}}^{\ell }}\text{for }m,n,\ell \ge 0,$$ 3.13

we may thus write

$$\frac{{\mathrm{\partial }}^{p}g}{\mathrm{\partial }{\epsilon }^p}(\rho ,\epsilon )=\sum _{(m,n,\ell )\in S_p}{C}_{m,n,\ell }^p{T}_{m,n,\ell }(\rho ,\epsilon )\frac{{\mathrm{e}}^{\mathrm{i}k\rho \sqrt{1+{\epsilon }^2}}}{\rho }.$$ 3.14

But, it is easy to check that

$$g(\rho ,\epsilon )=T_{0,0,1}(\rho ,\epsilon )\frac{{\mathrm{e}}^{\mathrm{i}k\rho \sqrt{1+{\epsilon }^2}}}{\rho }$$ 3.15

and that, defining the unary operators

$$\begin{array}{rl}{f}^{-}{T}_{m,n,\ell }(\rho ,\epsilon )& =T_{m+1,n,\ell +2}(\rho ,\epsilon ),\end{array}$$ 3.16

$$\begin{array}{rl}{f}^0{T}_{m,n,\ell }(\rho ,\epsilon )& =T_{m+1,n+1,\ell +1}(\rho ,\epsilon )\end{array}$$ 3.17

$$\begin{array}{rl}\text{and}{f}^{+}{T}_{m,n,\ell }(\rho ,\epsilon )& =\{\begin{array}{ll}{T}_{m-1,n,\ell }(\rho ,\epsilon )& m\ge 1\\ 0m=0\end{array},\end{array}$$ 3.18

we have

$$\frac{\mathrm{\partial }}{\mathrm{\partial }\epsilon }(T_{m,n,\ell }(\rho ,\epsilon )\frac{{\mathrm{e}}^{\mathrm{i}k\rho \sqrt{1+{\epsilon }^2}}}{\rho })=(-\ell f^{-}{T}_{m,n,\ell }+f^0{T}_{m,n,\ell }+m{f}^{+}{T}_{m,n,\ell })\frac{{\mathrm{e}}^{\mathrm{i}k\rho \sqrt{1+{\epsilon }^2}}}{\rho }(m\ge 0).$$ 3.19

We now may (and do) redefine the set S_p to ensure that it contains exactly the triples (m,n,ℓ) corresponding to a sequence of p applications of the unary operators (3.16). Thus calling F={f⁻,f⁰,f⁺}, we let

$$S_p=\{(m,n,\ell ):T_{m,n,\ell }=f_1{f}_2\dots f_p{T}_{0,0,1}\text{ where }{f}_j\in F\text{ for }j=1,\dots ,p\}.$$ 3.20

Application of the operator f⁺ results in a decrease by one in the power of ε in the expression (3.13) for T_m,n,ℓ(ρ,ε), while application of either f⁰ or f⁻ results in an increase by one in that power. Susbstituting ε=z/ρ in (3.13), on the other hand, we obtain

$$T_{m,n,\ell }(\rho ,\frac{z}{\rho })\sim z^m{(\mathrm{i}k)}^n{\rho }^{n-m}(\rho \to \mathrm{\infty })\text{for }m\ge 0$$

and we thus define the ρ-order Q of T_m,n,ℓ by

$$Q(T_{m,n,\ell })=n-m.$$

Note that an application of the operator f⁺ (respectively, f⁰, f⁻) to T_m,n,ℓ results in an increase (respectively, no change, decrease) in the ρ-order Q.

Let (m₀,n₀,ℓ₀) be such that Q₊=Q(T_m0,n0,ℓ0) is greater than or equal to Q(T_m,n,ℓ) for all (m,n,ℓ)∈S_p. We claim that $C_{m_0,n_0,{\ell }_0}^p\ne 0$. To establish this fact, first note that, in view of equations (3.12) and (3.19), the coefficient $C_{m,n,\ell }^p$ in the sum (3.14) equals a sum of several contributions, each one of which results from a sequence of p operations from the set {f⁰,f⁺} applied to the root expression T_0,0,1(ρ,ε). Indeed, while in principle all three elements in the set {f⁻,f⁰,f⁺} appear as a contribution to a coefficient $C_{m,n,\ell }^p$, it is easy to check that f⁻ cannot appear as a contribution towards the maximum-order coefficient C_m0,n0,ℓ0, because, as pointed out above, the f⁻ operator decreases the ρ-order Q.

As none of the f⁰ and f⁺ contributions are negative, it only remains to check that there is at least one positive contribution to the coefficient of C(m₀,n₀,ℓ₀) of T(m₀,n₀,ℓ₀) of ρ-order Q₊. But, in view of equation (3.16), a non-zero contribution of the form f₁f₂…f_pT_0,0,1 to the coefficient C(m₀,n₀,ℓ₀) can only result provided no more than half of p operators used equals f⁺—which implies, in particular, that Q₊≤p/2. In fact, we have Q₊=p/2 (respectively, Q₊=(p−1)/2) for p even (respectively, for p odd), and a positive contribution to C(m₀,n₀,ℓ₀) is provided by T(m₀,n₀,ℓ₀)=(f⁺)^p/2(f⁰)^p/2T(0,0,1) for p even and by T(m₀,n₀,ℓ₀)=(f⁺)^(p−1)/2(f⁰)^(p+1)/2T(0,0,1) for p odd. Taking into account the factor of 1/ρ in each one of the terms in equation (3.14), equation (3.10) follows and, thus, in view of (3.9), so does (3.11). The proof is now complete. □

Now, we are able to prove the algebraic convergence of the lattice sum for ${\tilde{G}}_p^q(\mathbf{\text{x}})$.

Theorem 3.2 Modified Green function for all frequencies; algebraic convergence —

Let χ(r) be a smooth truncation function equal to 1 for r<r₁ and equal to 0 for r>r₂ (0<r₁<r₂), and let p denote an integer such that p≥3. Then, for all real triples (k,α,β)=(k,α) (k≠0), the sums

$$G^{p,A}(\widetilde{\mathbf{\text{x}}},z)=\frac{1}{4\pi }\sum _{\stackrel{m,n\in \mathbb{Z}}{|m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2|\le A}}\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}\mathit{\alpha }\cdot (m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2)}\sum _{q=0}^p{a}_{pq}\frac{{\mathrm{e}}^{\mathrm{i}k{r}_{mn}^q}}{r_{mn}^q}$$ 3.21

and

$${\hat{G}}^{p,A}(\widetilde{\mathbf{\text{x}}},z)=\frac{1}{4\pi }\hspace{0.17em}\sum _{m,n\in \mathbb{Z}}\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}\mathit{\alpha }\cdot (m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2)}\sum _{q=0}^p{a}_{pq}\frac{{\mathrm{e}}^{\mathrm{i}k{r}_{mn}^q}}{r_{mn}^q}\chi \left(\frac{{\tilde{r}}_{mn}}{A}\right),$$ 3.22

where ${(r_{mn}^q)}^2=|\widetilde{\mathbf{\text{x}}}+m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2{|}^2+{(z+qd)}^2$ and ${\tilde{r}}_{mn}=|\widetilde{\mathbf{\text{x}}}+m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2|$, converge to a radiating quasi-periodic modified Green function ${\tilde{G}}_p^q(\mathbf{\text{x}}),$ which satisfies the partial differential equation

$${\mathrm{\nabla }}^2{\tilde{G}}_p^q(\mathbf{\text{x}})+k^2{\tilde{G}}_p^q(\mathbf{\text{x}})=-\sum _{m,n\in \mathbb{Z}}\sum _{q=0}^p\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}\mathit{\alpha }\cdot (m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2)}\delta (\mathbf{\text{x}}-{\mathbf{\text{x}}}_{mnq}^{\prime }),$$ 3.23

as well as the quasi-periodicity condition ${\tilde{G}}_p^q(\mathbf{\text{x}}+(m{\mathbf{\text{v}}}_2+n{\mathbf{\text{v}}}_2,0))={\tilde{G}}_p^q(\mathbf{\text{x}})\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}\mathit{\alpha }\cdot (m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2)}$. Further, there exists a constant C_p=C_p(k,α,β) for which

$$|G^{p,A}(\mathbf{\text{x}})-{\tilde{G}}_p^q(\mathbf{\text{x}})|<\frac{C_p}{A^{\lceil p/2\rceil -1}}$$ 3.24

and

$$|{\hat{G}}^{p,A}(\mathbf{\text{x}})-{\tilde{G}}_p^q(\mathbf{\text{x}})|<\frac{C_p}{A^{\lceil p/2\rceil -1/2}}$$ 3.25

for all sufficiently large values of A.

Proof. —

The sum (3.21) can be re-expressed in the form

$$G^{p,A}(\mathbf{\text{x}})=\frac{1}{4\pi }\sum _{\stackrel{m,n\in \mathbb{Z}}{|m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2|\le A}}h({\rho }_{mn},\frac{z}{{\rho }_{mn}},\frac{d}{{\rho }_{mn}}){\mathrm{e}}^{-\mathrm{i}(\alpha m+\beta n)}.$$ 3.26

But, in view of (3.11) and letting ν=⌈p/2⌉+1, we see that

$$\frac{1}{4\pi }h({\rho }_{mn},\frac{z}{{\rho }_{mn}},\frac{d}{{\rho }_{mn}})=\frac{1}{|{\mathbf{\text{v}}}_{mn}{|}^{\nu }}H(\mathbf{\text{x}};m,n)$$

for some function H which, for certain constants C and M satisfies |H(x;m,n)|<C as long as |v_mn|>M. Thus, the sum in (3.26) converges to a limit ${\tilde{G}}_p^q(\mathbf{\text{x}})$ as $A\to +\mathrm{\infty }$, and for A>M we have

$$|G^{p,A}(\mathbf{\text{x}})-{\tilde{G}}_p^q(\mathbf{\text{x}})|\le \sum _{\stackrel{m,n\in \mathbb{Z}}{|m{\mathbf{\text{v}}}_1+n{\mathbf{\text{v}}}_2|>A}}\frac{C}{|{\mathbf{\text{v}}}_{mn}{|}^{\nu }}<\frac{C_p}{A^{\nu -2}},$$ 3.27

a relation which establishes the desired result (3.24).

The proof of the bound (3.25) follows in part the proof of theorem 2.1 in Part I [1]. For simplicity we assume $\widetilde{\mathbf{\text{x}}}=(x,y)=0$ and α=0; the extension of the proof to non-zero values of these quantities is handled easily via consideration of standard properties of the Fourier transform, as described explicitly at the end of the proof in [1]. Let U denote the finite set

$$U=\{j{\mathbf{\text{v}}}_1^{\ast }+\ell {\mathbf{\text{v}}}_2^{\ast }:{\gamma }_{j\ell }=0\}\subseteq {\mathrm{\Lambda }}^{\ast }.$$ 3.28

This set is non-empty exactly when k is a Wood frequency. As k≠0, one has (0,0)∉U. The assumption p≥3 implies ν≥3.

Let z_q=z+dq. We may re-express (3.22) in the form

$$4\pi {\hat{G}}^{p,A}(0,0,z)=\sum _{\mathit{\text{r}}\in \mathrm{\Lambda }}\chi \left(\frac{\mathit{\text{r}}}{A}\right)\sum _{q=0}^p\frac{\exp (\mathrm{i}k\sqrt{|\mathit{\text{r}}{|}^2+z_q^2})}{\sqrt{|\mathit{\text{r}}{|}^2+z_q^2}}.$$ 3.29

The Poisson Summation Formula transforms this sum into a lattice sum in the Fourier variable:

$$4\pi {\hat{G}}^{p,A}(0,0,z)=\sum _{\mathit{\xi }\in {\mathrm{\Lambda }}^{\ast }}\mathcal{F}\left[\chi \left(\frac{\mathit{\text{r}}}{A}\right)\sum _{q=0}^p\frac{\exp (\mathrm{i}k\sqrt{|\mathit{\text{r}}{|}^2+z_q^2})}{\sqrt{|\mathit{\text{r}}{|}^2+z_q^2}}\right](\mathit{\xi })=S_1+S_2$$ 3.30

in which

$$S_1=\sum _{\mathit{\xi }\in {\mathrm{\Lambda }}^{\ast }\setminus U}\sum _{q=0}^p{\int }_{{\mathbb{R}}^2}\chi \left(\frac{\mathit{\text{r}}}{A}\right)\frac{\exp (\mathrm{i}k\sqrt{|\mathit{\text{r}}{|}^2+z_q^2})}{\sqrt{|\mathit{\text{r}}{|}^2+z_q^2}}\hspace{0.17em}{\mathrm{e}}^{2\pi \mathrm{i}\mathit{\xi }\cdot \mathit{\text{r}}}\hspace{0.17em}\mathrm{d}\mathit{\text{r}}$$ 3.31

and

$$S_2=\sum _{\mathit{\xi }\in U}{\int }_{{\mathbb{R}}^2}\chi (\mathit{\text{r}}/A)\hspace{0.17em}{\mathrm{e}}^{2\pi \mathrm{i}\mathit{\xi }\cdot \mathit{\text{r}}}\left(\sum _{q=0}^p\frac{\exp (\mathrm{i}k\sqrt{|\mathit{\text{r}}{|}^2+z_q^2})}{\sqrt{|\mathit{\text{r}}{|}^2+z_q^2}}\right)\mathrm{d}\mathit{\text{r}}.$$ 3.32

The proof of [1], theorem 2.1 establishes that the sum S₁ (U is empty in that work) converges superalgebraically to a limit L, that is, it is equal to $L+\mathcal{O}(A^{-n})$ for each positive integer n. Thus, it only remains to show that the sum S₂ converges, with the difference from its limit being of order $\mathcal{O}(A^{-(\nu -3/2)})$ (or, equivalently, $\mathcal{O}(A^{-(\lceil p/2\rceil -1/2)})$) as $A\to +\mathrm{\infty }$. Each fraction in S₂ can be expressed as an exponential in r=|r|, multiplied by a Laurent expansion,

$$\frac{\exp (\mathrm{i}k\sqrt{r^2+z_q^2})}{\sqrt{r^2+z_q^2}}=\frac{{\mathrm{e}}^{\mathrm{i}kr}}{r}{g}_q(r),g_q(r)=1+\sum _{j=1}^{\mathrm{\infty }}{a}_j^q{r}^{-j}.$$ 3.33

The coefficients $a_j^q$ depend on z, and the expansion is convergent when r>|z_q|. In view of (3.6) and (3.11), we see that

$$\sum _{q=0}^p{g}_q(r)=\frac{g(r)}{r^{\nu -1}}\text{where}g(r)=\sum _{j=0}^{\mathrm{\infty }}{a}_j{r}^{-j}\text{for}r>z_q(\text{with }{a}_0\ne 0)$$ 3.34

and clearly

$$g(Ar)\to a_0\text{as }A\to \mathrm{\infty }$$ 3.35

with uniform convergence over the set r≥r₁.

To study the contribution by the sum S₂, we define the polar coordinates r=(r,θ) and we note that, because ξ∈U, we must necessarily have ξ=(k/2π,γ). Then, using the rescaling ρ=r/A and the notation ψ(ρ)=1−χ(ρ), and in view of the fact that ν≥3, we obtain

$$\begin{array}{rl}& {\int }_{{\mathbb{R}}^2}\chi (\mathit{\text{r}}/A)\hspace{0.17em}{\mathrm{e}}^{2\pi \mathrm{i}\mathit{\xi }\cdot \mathit{\text{r}}}\left(\sum _{q=0}^p\frac{\exp (\mathrm{i}k\sqrt{|\mathit{\text{r}}{|}^2+z_q^2})}{\sqrt{|\mathit{\text{r}}{|}^2+z_q^2}}\right)\mathrm{d}\mathit{\text{r}}\\ & ={\int }_{{\mathbb{R}}^2}{\mathrm{e}}^{2\pi \mathrm{i}\mathit{\xi }\cdot \mathit{\text{r}}}\left(\sum _{q=0}^p\frac{\exp (\mathrm{i}k\sqrt{|\mathit{\text{r}}{|}^2+z_q^2})}{\sqrt{|\mathit{\text{r}}{|}^2+z_q^2}}\right)\mathrm{d}\mathit{\text{r}}\\ & -{\int }_0^{2\pi }\hspace{0.17em}{\int }_0^{\mathrm{\infty }}\psi (A^{-1}r)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}kr(\cos \theta +1)}\frac{g(r)}{r^{\nu -1}}\hspace{0.17em}\mathrm{d}r\hspace{0.17em}\mathrm{d}\theta \\ & =\tilde{L}-A^{2-\nu }{\int }_{r_1}^{\mathrm{\infty }}\psi (\rho )\frac{g(A\rho )}{{\rho }^{\nu -1}}({\int }_0^{2\pi }{\mathrm{e}}^{\mathrm{i}Ak(\cos \theta +1)\rho }\hspace{0.17em}\mathrm{d}\theta )\mathrm{d}\rho \\ & =\tilde{L}-2\pi A^{2-\nu }{\int }_{r_1}^{\mathrm{\infty }}\psi (\rho )\frac{g(A\rho )}{{\rho }^{\nu -1}}{\mathrm{e}}^{\mathrm{i}Ak\rho }{J}_0(Ak\rho )\hspace{0.17em}\mathrm{d}\rho ,\end{array}$$ 3.36

where J₀ denotes the Bessel function of order 0. Clearly, the number $\tilde{L}$ does not depend on A. In view of the well-known Bessel function asymptotics $J_0(x)\sim {(2/x\pi )}^{1/2}\cos (x-\pi /4),\hspace{0.17em}x\to +\mathrm{\infty }$, and because k≠0, we obtain the relation

$${\int }_{{\mathbb{R}}^2}\chi \left(\frac{\mathit{\text{r}}}{A}\right){\mathrm{e}}^{2\pi \mathrm{i}\mathit{\xi }\cdot \mathit{\text{r}}}\left(\sum _{q=0}^p\frac{\exp (\mathrm{i}k\sqrt{|\mathit{\text{r}}{|}^2+z_q^2})}{\sqrt{|\mathit{\text{r}}{|}^2+z_q^2}}\right)\mathrm{d}\mathit{\text{r}}=\tilde{L}+\mathcal{O}(A^{3/2-\nu }).$$ 3.37

It follows that the sum S₂ converges with an error of $\mathcal{O}(1/A^{\lceil p/2\rceil -1/2})$ as $A\to \mathrm{\infty }$ (because ν=⌈p/2⌉+1). Together with the superalgebraic convergence of S₁ as $A\to \mathrm{\infty }$, this fact establishes (3.25). The proof is now complete. □

(b). Complete Green function in Fourier space

In view of the Fourier expression (1.4) for the Green function away from Wood anomalies, we obtain the corresponding expression

$${\tilde{G}}_p^q(\widetilde{\mathbf{\text{x}}},z)=\frac{\mathrm{i}}{2D}\sum _{j,\ell \in \mathbb{Z}}\frac{1}{{\gamma }_{j\ell }}{\mathrm{e}}^{\mathrm{i}\hspace{0.17em}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}}\hspace{0.17em}\sum _{q=0}^p{a}_{pq}{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }|z+qd|},$$ 3.38

for the shifted Green function. For z>0, this expression can be made to read

$${\tilde{G}}_p^q(\widetilde{\mathbf{\text{x}}},z)=\frac{\mathrm{i}}{2D}\sum _{j,\ell \in \mathbb{Z}}\frac{1}{{\gamma }_{j\ell }}{\mathrm{e}}^{\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }z}{(1-{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }d})}^p.$$ 3.39

In view of the limit $\underset{{\gamma }_{j\ell }\to 0}{lim}({(1-{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }d})}^p/{\gamma }_{j\ell })=0$ for p≥2, we see that (3.39) can be evaluated even at Wood anomalies: letting

$$U=\{(j,\ell )\in {\mathbb{Z}}^2\hspace{0.17em}|\hspace{0.17em}{\gamma }_{j\ell }=0\},$$ 3.40

we may continuously extend the function ${\tilde{G}}_p^q(\widetilde{\mathbf{\text{x}}},z)$ to all frequencies, including Wood configurations, by means of the expression

$${\tilde{G}}_p^q(\widetilde{\mathbf{\text{x}}},z)=\frac{\mathrm{i}}{2D}\sum _{(j,\ell )\notin U}{\mathrm{e}}^{\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }z}\frac{{(1-{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }d})}^p}{{\gamma }_{j\ell }}.$$ 3.41

Unfortunately, however, if γ_jℓ=0 for some (j,ℓ), the corresponding Fourier component is not present in the Green function (3.41) and, therefore, use of this Green function cannot give rise to a uniquely solvable system of integral equations. To tackle this difficulty, we follow [2] and introduce a modified version $G_p^q$ of ${\tilde{G}}_p^q$, that is outgoing for $z\to \mathrm{\infty }$ (but not for $z\to -\mathrm{\infty }$) and which contains all necessary Fourier harmonics, even at Wood frequencies. The modified Green function is given by

$$G_p^q(\mathbf{\text{x}})={\tilde{G}}_p^q(\mathbf{\text{x}})+v(\mathbf{\text{x}}),$$

where $v(\widetilde{\mathbf{\text{x}}},z)$ denotes a solution of the homogeneous Helmholtz equation of the form

$$v(\widetilde{\mathbf{\text{x}}},z)=\frac{\mathrm{i}}{2D}\sum _{(j,\ell )\in U}{b}_{j\ell }\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}}{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }z},$$ 3.42

where b_jℓ≠0 are arbitrary non-zero complex constants. The function $G_p^q(\widetilde{\mathbf{\text{x}}},z)$ is α-quasi-periodic in $\widetilde{\mathbf{\text{x}}}$ with periods v₁ and v₂, it satisfies (3.23) and, crucially, it contains all Fourier harmonics.

As shown in §3d and demonstrated numerically in §6, the complete Green function $G_p^q$ can be used to obtain uniquely solvable integral-equation formulations around Wood frequencies. The analysis presented in §3d relies, in part, on use of yet another Green function, namely, a non-radiating Green function defined by a slowly convergent series which, however, (i) is well defined at Wood frequencies; and (ii) unlike the shifted Green function $G_p^q$, is a Helmholtz solution for z<0, and is therefore well suited for use as part of a proof that concerns the PDE domain and its complement at a Wood frequency. This rather peculiar Green function is introduced in the following section, and it is then used in the uniqueness proof presented in §3d.

(c). All space (non-radiating) Green function at and around Wood frequencies

In view of the relation

$$\frac{{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }|z|}}{{\gamma }_{j\ell }}=\frac{\cos ({\gamma }_{j\ell }z)}{{\gamma }_{j\ell }}+\mathrm{i}\hspace{0.17em}\frac{\sin ({\gamma }_{j\ell }|z|)}{{\gamma }_{j\ell }},$$ 3.43

each term in the classical quasi-periodic Green function (1.4) equals the sum of two quantities, the first of which diverges and the second of which tends to i|z| as γ_jℓ→0. In view of the relation (3.41), both terms can be made to vanish by using a pth order finite difference of shifted Green functions. As an alternative, a direct removal of the diverging term (which amounts to addition of a solution of Helmholtz’ equation) does produce a Helmholtz Green function for (k,α,β) in a neighbourhood of a given Wood anomaly triple (k₀,α₀,β₀) at which γ_jℓ=0. The resulting Green function at the Wood configuration (k₀,α₀,β₀) is thus given by

$$B^q(\widetilde{\mathbf{\text{x}}},z):=\frac{\mathrm{i}}{2D}\sum _{(j,\ell )\notin U}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}}\frac{1}{{\gamma }_{j\ell }}{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }|z|}+\frac{\mathrm{i}}{2D}\sum _{(j,\ell )\in U}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}}i|z|.$$ 3.44

This is not an outgoing Green function, on account of the terms that contain |z| as a factor. But, as indicated in the previous section, the function $B^q(\widetilde{\mathbf{\text{x}}},z)$ is a solution of the Helmholtz equation except at the periodically distributed singular points (mv₁+nv₂,0), and, unlike the shifted Green function $G_p^q$, it does not contain any additional singularities.

4. Uniquely solvable integral equations at and around Wood frequencies

We seek a scattered field above Γ in the form of a combined single- and double-layer potential

$$u(\mathbf{\text{x}})={\int }_{{\mathrm{\Gamma }}^{\mathrm{per}}}(\mathrm{i}\eta G_p^q(\mathbf{x}-{\mathbf{\text{x}}}^{\prime })+\xi \frac{\mathrm{\partial }{G}_p^q(\mathbf{x}-{\mathbf{\text{x}}}^{\prime })}{\mathrm{\partial }n({\mathbf{x}}^{\prime })})\varphi ({\mathbf{\text{x}}}^{\prime })\hspace{0.17em}\mathrm{d}s({\mathbf{\text{x}}}^{\prime })$$ 4.1

in terms of a quasi-periodic density ϕ defined on Γ^per, where

$${\mathrm{\Gamma }}^{\mathrm{per}}=\{(\widetilde{\mathbf{\text{x}}},z):\widetilde{\mathbf{\text{x}}}=a{\mathbf{\text{v}}}_1+b{\mathbf{\text{v}}}_2\text{ with }0\le \{a,b\}\le 1,z=f(\widetilde{\mathbf{\text{x}}})\}.$$ 4.2

The domain Γ^per is that part of Γ that lies above the unit cell

$$Q=\{\widetilde{\mathbf{\text{x}}}=a{\mathbf{\text{v}}}_1+b{\mathbf{\text{v}}}_2:0\le \{a,b\}<1\}$$ 4.3

of the periodic lattice Λ.

The function u defined in (4.1) is quasi-periodic and outgoing as $z\to \mathrm{\infty }$, and it satisfies the Helmholtz equation in ${\mathbb{R}}^3\setminus \bigcup _{q=0}^p({\mathrm{\Gamma }}^{\mathrm{per}}-(0,0,qd))$. This function is a solution of (2.4) if and only if ϕ solves the integral equation

$$\frac{\xi \varphi (\mathbf{x})}{2}+{\int }_{{\mathrm{\Gamma }}^{\mathrm{per}}}(\mathrm{i}\eta G_p^q(\mathbf{x}-{\mathbf{\text{x}}}^{\prime })+\xi \frac{\mathrm{\partial }{G}_p^q(\mathbf{x}-{\mathbf{\text{x}}}^{\prime })}{\mathrm{\partial }n({\mathbf{x}}^{\prime })})\varphi ({\mathbf{\text{x}}}^{\prime })\mathrm{d}s({\mathbf{\text{x}}}^{\prime })=-{\mathrm{e}}^{\mathrm{i}(\mathit{\alpha }\cdot \widetilde{\mathbf{\text{x}}}-\gamma z)},\mathbf{\text{x}}\in {\mathrm{\Gamma }}^{\mathrm{per}}.$$ 4.4

The following theorem establishes that this integral equation is uniquely solvable as long as the shift distance d>0 is selected in such a way that the restrictions

$$(1-{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }d})\ne 0\text{for all}\hspace{0.17em}(j,\ell )\in {\mathbb{Z}}^2$$ 4.5

are satisfied. It is important to note that, as γ_jℓ is an imaginary quantity for j,ℓ large enough, equation (4.5) amounts to a finite number of constraints.

Theorem 4.1 —

Let p≥0, let ξ≠0 and η≠0 denote real numbers satisfying η/ξ<0, and let d>0 be such that (4.5) holds. Then equation (4.4) admits a unique solution for all triples (k,α,β) of wavenumbers, including Wood configurations.

Proof. —

Given that the surface Γ is smooth and that $G_p^q$ has the same singularity as G, it follows that the integral operators on the left-hand side of equation (4.4) are compact operators in the space L²(Γ^per). Thus, by the Fredholm theory, the unique solvability of equation (4.4) is equivalent to the injectivity of the operator on the left-hand side of that equation. To establish injectivity, and thereby complete the proof of the theorem, in what follows we show that any solution ϕ∈L²(Γ^per) of the homogeneous equation

$$\frac{\xi \varphi (\mathbf{x})}{2}+{\int }_{{\mathrm{\Gamma }}^{\mathrm{per}}}(\mathrm{i}\eta G_p^q(\mathbf{x}-{\mathbf{\text{x}}}^{\prime })+\xi \frac{\mathrm{\partial }{G}_p^q(\mathbf{x}-{\mathbf{\text{x}}}^{\prime })}{\mathrm{\partial }n({\mathbf{x}}^{\prime })})\varphi ({\mathbf{\text{x}}}^{\prime })\mathrm{d}s({\mathbf{\text{x}}}^{\prime })=0,\mathbf{\text{x}}\in {\mathrm{\Gamma }}^{\mathrm{per}}$$ 4.6

must necessarily vanish.

Let ϕ satisfy (4.6) and let u denote the corresponding Helmholtz solution (4.1). It follows that u vanishes on Γ^per. As u satisfies the outgoing radiation condition (2.5) at $+\mathrm{\infty }$ (because $G_p^q(\mathbf{\text{x}})$ does), by uniqueness of the solution of the Dirichlet problem (2.4), which holds even at Wood frequencies [21], it follows that u(x)=0 for all x∈Ω⁺. In particular, letting u_jℓ(z) denote the Fourier coefficients of the doubly periodic function $u(\widetilde{\mathbf{\text{x}}},z)\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}\mathit{\alpha }\cdot \widetilde{\mathbf{\text{x}}}}$ with respect to $\widetilde{\mathbf{\text{x}}}$ for z>z₊, we have u_jℓ(z)=0 for all $j,\ell \in \mathbb{Z}$.

Given that, for z>0,

$$G_p^q(\widetilde{\mathbf{\text{x}}},z)=\frac{\mathrm{i}}{2D}\sum _{(j,\ell )\notin U}{\mathrm{e}}^{\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}}\frac{{(1-{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }d})}^p}{{\gamma }_{j\ell }}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }z}+\frac{\mathrm{i}}{2D}\sum _{(j,\ell )\in U}{b}_{j\ell }\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }z}$$

(see §3b), it follows that

$$u_{j\ell }(z)=\{\begin{array}{ll}{c}_{j\ell }^{+}\frac{{(1-{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }d})}^p}{{\gamma }_{j\ell }}{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }z}& \text{for}\hspace{0.17em}(j,\ell )\notin U,\\ c_{j\ell }^{+}{b}_{j\ell }& \text{for}\hspace{0.17em}(j,\ell )\in U,\end{array}$$ 4.7

in which

$$c_{j\ell }^{+}=\frac{1}{2D}{\int }_{\mathrm{\Gamma }}\varphi ({\mathbf{\text{x}}}^{\prime })\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot {\widetilde{\mathbf{\text{x}}}}^{\prime }}\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}{\gamma }_{j\ell }{z}^{\prime }}[\xi ({\mathbf{\text{v}}}_{j\ell }^{\ast },{\gamma }_{j\ell })\cdot \mathbf{\text{n}}({\mathbf{\text{x}}}^{\prime })-\eta ]\hspace{0.17em}\mathrm{d}s({\mathbf{\text{x}}}^{\prime }).$$ 4.8

Under the assumptions of this theorem, u may vanish only if $c_{j\ell }^{+}=0$ for all $j,\ell \in \mathbb{Z}$.

Using the non-radiating Green function introduced in §3c, we then set

$$v(\mathbf{\text{x}})={\int }_{\mathrm{\Gamma }}(\mathrm{i}\eta B^q(\mathbf{\text{x}}-{\mathbf{\text{x}}}^{\prime })+\xi \frac{\mathrm{\partial }{B}^q(\mathbf{\text{x}}-{\mathbf{\text{x}}}^{\prime })}{\mathrm{\partial }n({\mathbf{x}}^{\prime })})\varphi ({\mathbf{\text{x}}}^{\prime })\hspace{0.17em}\mathrm{d}s({\mathbf{\text{x}}}^{\prime }),\mathbf{\text{x}}\notin \mathrm{\Gamma }.$$ 4.9

In view of (3.44), for z>z₊ the function v admits the Fourier expansion

$$v(\widetilde{\mathbf{\text{x}}},z)=\sum _{j,\ell \in \mathbb{Z}}{v}_{j\ell }^{+}(z)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}}(z>z_{+}),$$

in which

$$v_{j\ell }^{+}(z)=c_{j\ell }^{+}\frac{1}{{\gamma }_{j\ell }}{\mathrm{e}}^{\mathrm{i}{\gamma }_{j\ell }z}\text{for }(j,\ell )\notin U$$ 4.10

and

$$v_{j\ell }^{+}(z)=\mathrm{i}(c_{j\ell }^{+}z-c_{j\ell }^{\prime })+c_{j\ell }^{″}\text{for }(j,\ell )\in U,$$ 4.11

in which (recalling that |z|=z−z′ when x is above Γ, or $z>f(\widetilde{\mathbf{\text{x}}})$)

$$c_{j\ell }^{\prime }=\frac{1}{2D}[-\eta {\int }_{\mathrm{\Gamma }}\varphi ({\mathbf{\text{x}}}^{\prime })z^{\prime }\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot {\widetilde{\mathbf{\text{x}}}}^{\prime }}\hspace{0.17em}\mathrm{d}s({\mathbf{\text{x}}}^{\prime })+\xi {\int }_{\mathrm{\Gamma }}\varphi ({\mathbf{\text{x}}}^{\prime })({\mathbf{\text{v}}}_{j\ell }^{\ast },0)\cdot \mathbf{\text{n}}({\mathbf{\text{x}}}^{\prime })z^{\prime }\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot {\widetilde{\mathbf{\text{x}}}}^{\prime }}\hspace{0.17em}\mathrm{d}s({\mathbf{\text{x}}}^{\prime })]$$ 4.12

and

$$c_{j\ell }^{″}=\frac{\xi }{2D}{\int }_{\mathrm{\Gamma }}\varphi ({\mathbf{\text{x}}}^{\prime })(0,0,1)\cdot \mathbf{\text{n}}({\mathbf{\text{x}}}^{\prime })\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot {\widetilde{\mathbf{\text{x}}}}^{\prime }}\hspace{0.17em}\mathrm{d}s({\mathbf{\text{x}}}^{\prime }).$$ 4.13

Given that $c_{j\ell }^{+}=0$ for all $j,\ell \in \mathbb{Z}$, it follows that

$$v_{j\ell }^{+}(z)=\{\begin{array}{ll}0,& (j,\ell )\notin U,\\ -\mathrm{i}{c}_{j\ell }^{\prime }+c_{j\ell }^{″},& (j,\ell )\in U.\end{array}$$

Therefore, $v(\widetilde{\mathbf{\text{x}}},z)=\sum _{(j,\ell )\in U}(-\mathrm{i}{c}_{j\ell }^{\prime }+c_{j\ell }^{″})\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}}$ for z>z₊. The field v(x) satisfies the Helmholtz equation for x∉Γ, and v(x) is independent of z for z>z₊. In view of the real analyticity of v(x) for x∉Γ and the uniqueness of analytic continuation, it follows that v(x) is independent of z everywhere above Γ and we have

$$v(\widetilde{\mathbf{\text{x}}},z)=\sum _{(j,\ell )\in U}(-\mathrm{i}{c}_{j\ell }^{\prime }+c_{j\ell }^{″})\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}}\text{for }z\ge f(\widetilde{\mathbf{\text{x}}}).$$ 4.14

Now we turn our attention to the Fourier expansion

$$v(\widetilde{\mathbf{\text{x}}},z)=\sum _{j,\ell \in \mathbb{Z}}{v}_{j\ell }^{-}(z)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}},z<z_{-}$$

of the function v in the region z<z₋. For (j,ℓ)∈U, the Fourier coefficients in this expansion satisfy $v_{j\ell }^{-}(z)=-v_{j\ell }^{+}$, because the term |z−z′| in B^q(x−x′) equals −(z−z′) for z<z′. (There is no such relation for (j,ℓ)∉U.) For (j,ℓ)∈U we thus have

$$v_{j\ell }^{-}(z)=\mathrm{i}{c}_{j\ell }^{\prime }-c_{j\ell }^{″}.$$

The function v(x) satisfies the radiation condition (2.7) for z<z₋ because, in spite of linear terms that are part of the Green function B^q, v itself does not contain such linear growth for z<z₋.

The right-hand side expression in (4.14), considered as a function defined on all of ${\mathbb{R}}^3$, defines a Helmholtz field that satisfies the outgoing condition for $z\to \mathrm{\infty }$ and $z\to -\mathrm{\infty }$. Thus, by subtracting it from v(x), one obtains the field

$$\tilde{v}(\widetilde{\mathbf{\text{x}}},z):=v(\widetilde{\mathbf{\text{x}}},z)-\sum _{(j,\ell )\in U}(-\mathrm{i}{c}_{j\ell }^{\prime }+c_{j\ell }^{″})\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\mathbf{\text{v}}}_{j\ell }^{\ast }\cdot \widetilde{\mathbf{\text{x}}}}\text{for }(\widetilde{\mathbf{\text{x}}},z)\in {\mathbb{R}}^3$$

that satisfies the radiation conditions at both $+\mathrm{\infty }$ and $-\mathrm{\infty }$, and that vanishes for $z>f(\widetilde{\mathbf{\text{x}}})$. The jump conditions of the single- and double-layer potentials that define v(x) in (4.9) imply that the limits of $\tilde{v}(\mathbf{\text{x}})$ and its normal derivative from below Γ are

$$\tilde{v}(\widetilde{\mathbf{\text{x}}},f(\widetilde{\mathbf{\text{x}}})-0)=-\xi \varphi (\widetilde{\mathbf{\text{x}}},f(\widetilde{\mathbf{\text{x}}}))$$ 4.15

and

$$\frac{\mathrm{\partial }\tilde{v}}{\mathrm{\partial }n(\mathbf{\text{x}})}(\widetilde{\mathbf{\text{x}}},f(\widetilde{\mathbf{\text{x}}})-0)=\mathrm{i}\eta \varphi (\widetilde{\mathbf{\text{x}}},f(\widetilde{\mathbf{\text{x}}})).$$ 4.16

It follows that the function $\tilde{v}(\mathbf{\text{x}})$ restricted to the domain $\{z\le f(\widetilde{\mathbf{\text{x}}})\}$ satisfies the homogeneous impedance boundary condition

$$\frac{\mathrm{\partial }\tilde{v}}{\mathrm{\partial }n(\mathbf{\text{x}})}(\widetilde{\mathbf{\text{x}}},f(\widetilde{\mathbf{\text{x}}}))+\frac{\mathrm{i}\eta }{\xi }\tilde{v}(\widetilde{\mathbf{\text{x}}},f(\widetilde{\mathbf{\text{x}}}))=0\text{on }\mathrm{\Gamma }.$$

As, per the discussion above, $\tilde{v}$ additionally satisfies the radiation condition (2.7), lemma .1 tells us that we must have $\tilde{v}(\mathbf{\text{x}})=0$ for all x below Γ. In view of (4.15) and/or (4.16) and the assumption η/ξ<0, it follows that ϕ(x)=0 for all x∈Γ. □

5. Numerical evaluation of quasi-periodic layer potentials

For our numerical treatment, we reformulate the quasi-periodic scattering integral equation (4.4) in terms of only periodic functions. We use the fact that the solution ϕ=ϕ^qper of the integral equation (4.4) is α-quasi-periodic with respect to the lattice Λ, and that, therefore, the quantity

$${\varphi }^{\mathrm{per}}(\widetilde{\mathbf{\text{x}}})={\mathrm{e}}^{-\mathrm{i}\mathit{\alpha }\cdot \widetilde{\mathbf{\text{x}}}}\hspace{0.17em}{\varphi }^{\mathrm{qper}}(\widetilde{\mathbf{\text{x}}})$$

is periodic with respect to Λ. This allows us to express the integral equation (4.4) in the form

$$\frac{\xi {\varphi }^{\mathrm{per}}(\mathbf{\text{x}})}{2}+{\int }_{{\mathrm{\Gamma }}^{\mathrm{per}}}(\xi \frac{\mathrm{\partial }{G}_p^{\mathrm{per}}(\mathbf{x}-{\mathbf{\text{x}}}^{\prime })}{\mathrm{\partial }n({\mathbf{x}}^{\prime })}+\mathrm{i}\eta G_p^{\mathrm{per}}(\mathbf{x}-{\mathbf{\text{x}}}^{\prime })){\varphi }^{\mathrm{per}}({\mathbf{\text{x}}}^{\prime })\hspace{0.17em}\mathrm{d}s({\mathbf{\text{x}}}^{\prime })=-{\mathrm{e}}^{-\mathrm{i}\gamma f(\widetilde{\mathbf{\text{x}}})},\widetilde{\mathbf{\text{x}}}\in {\mathrm{\Gamma }}^{\mathrm{per}},$$ 5.1

where the Λ-periodic Green function G^per_p is defined by

$$G_p^{\mathrm{per}}(\mathbf{\text{x}},{\mathbf{\text{x}}}^{\prime })=G_p^q(\mathbf{\text{x}},{\mathbf{\text{x}}}^{\prime }){\mathrm{e}}^{\mathrm{i}\mathit{\alpha }\cdot ({\widetilde{\mathbf{\text{x}}}}^{\prime }-\widetilde{\mathbf{\text{x}}})}.$$ 5.2

In practice, for any given p=0,1,… the quantity ${\hat{G}}^{p,A}$ (equation (3.22)) is used as an approximation for $G_p^q$ which, when substituted in (5.2), results in the needed numerical approximation of G^per_p. We solve equation (5.1) by means of the unaccelerated high-order Nyström procedure introduced in [24] and Part I. For all the numerical experiments presented in §6, a single patch was used to represent the biperiodic surfaces under consideration.

6. Numerical results

We present numerical computations of scattering by the doubly periodic scattering surface $f(x,y)=\frac{1}{2}\cos (2\pi x)\cos (2\pi y)$ with periodicity lattice vectors v₁=(1,0,0) and v₂=(0,1,0) and under Dirichlet and Neumann boundary conditions. For non-Wood configurations, we utilize the p=0 (unshifted) version of the algorithm described in § 5. At and around Wood configurations, on the other hand, we use the p=3 version of that algorithm. A fully three-dimensional single-core Matlab implementation of these methods was used, which was neither accelerated nor optimized; accordingly, our numerical error studies at Wood anomalies do not go beyond relative errors of the order 10⁻⁴. Clear high-order convergence is observed in all cases.

We report the quality of the solutions on the basis of two error indicators. The first of these indicators is the energy-conservation defect

$$\epsilon =|\sum _{(j,\ell )\in P}\frac{{\gamma }_{j\ell }}{{\gamma }_{00}}|B_{j\ell }{|}^2-1|$$ 6.1

which we have verified (by means of numerical resolution studies) to be an excellent error predictor for these solvers. An additional error estimator we present, ε₁, on the other hand, equals the absolute error in the Rayleigh coefficient B⁺_0,0 (as estimated by comparison with a reference solution obtained by means of a highly refined discretization, a large value of the window parameter A and a sufficiently small GMRES tolerance). The word ‘ref’ on a table entry indicates that the parameter values on that row were used to produce the reference solution necessary for evaluation of the errors ε₁ for the corresponding frequency k on the that table. The numbers of iterations required by the GMRES solvers to reach specified tolerances are provided in each case.

 Table 1 demonstrates the high-order character, as the window-size parameter A grows, for the proposed p=0 (unshifted) Dirichlet solvers at frequencies k away from Wood anomalies. Table 2 demonstrates the high-order character of the p=3 (shifted) solver, as A grows, also under Dirichlet boundary conditions and for values of k away from Wood anomalies. Tables 3 and 4, in turn, concern configurations at and near Wood anomalies; Dirichlet (respectively, Neumann) boundary conditions are considered in the first (respectively, second) of these tables. In the normal-incidence case considered in those tables, the first three Wood anomalies occur at k=2π, $k=2\sqrt{2}\pi$ and k=4π. Once again, fast convergence is observed as A grows, even at and around Wood anomalies. We note that the number of iterations required by the GMRES solvers based on Combined Field Integral Equations remains small even for Wood and near-Wood parameters for both Dirichlet and Neumann problems.

Table 1.

Convergence of the p=0 Dirichlet solver (unshifted) as A grows, for a configuration away from Wood anomalies. Normal incidence (α=0) was assumed for this example. The reference solution was produced using A=A_ref=320 and 16×16 unknowns, for which ε=1.0×10⁻⁶.

k	unknowns	A	$max|G^A-G^{\mathrm{ref}}|$	iter	ε1	ε
1	16×16	30	4.4×10−2	13	1.0×10−1	1.8×10−1
1	16×16	60	3.1×10−3	13	6.3×10−3	6.6×10−3
1	16×16	120	4.3×10−4	13	4.8×10−4	4.3×10−4
1	16×16	160	3.7×10−5	13	1.3×10−4	2.3×10−4
1	16×16	240	2.4×10−6	13	4.9×10−6	6.7×10−6

Table 2.

Convergence of the p=3 Dirichlet solver with shift parameter d=2.4, away from Wood frequencies, as A grows. Normal incidence. The reference solution was produced using corresponds to A=A_ref=120, 32×32 unknowns, for which ε=4.0×10⁻⁷.

			GA			GA,p,p=3
k	unknowns	A	iter	ε1	ε	iter	ε1	ε
6	16×16	30	16	4.9×10−3	1.6×10−3	12	6.5×10−3	1.2×10−2
6	16×16	60	16	1.5×10−3	4.1×10−4	12	3.8×10−4	1.5×10−5
6	16×16	80	16	2.8×10−5	1.1×10−5	12	4.7×10−6	2.3×10−6

Table 3.

Convergence, as A grows, of the p=3 Dirichlet solver at and around Wood frequencies. Shift parameter d=1.4, GMRES residual tolerance equal to 10⁻⁶. ‘Ref’ refers to finely resolved solutions against which the error of the coarser solutions is evaluated.

k	unknowns	A	iter	ε1	ε
2π	24×24	20	19	3.2×10−2	1.7×10−2
2π	24×24	30	19	2.7×10−3	4.7×10−3
2π	24×24	40	19	6.9×10−4	4.0×10−4
2π	24×24	60	19	ref	2.4×10−6
2π±10−6	24×24	40	19	6.8×10−4	4.3×10−4
$2\sqrt{2}\pi$	24×24	30	25	1.3×10−2	1.5×10−2
$2\sqrt{2}\pi$	24×24	40	25	6.8×10−3	5.5×10−3
$2\sqrt{2}\pi$	24×24	80	25	8.9×10−5	2.1×10−4
$2\sqrt{2}\pi$	24×24	120	25	ref	3.8×10−5
$2\sqrt{2}\pi \pm {10}^{-6}$	24×24	30	25	1.6×10−2	2.5×10−2
$2\sqrt{2}\pi \pm {10}^{-6}$	24×24	80	25	8.9×10−5	1.5×10−4
4π	32×32	30	28	4.6×10−2	4.9×10−2
4π	32×32	60	28	2.4×10−3	1.1×10−3
4π	32×32	180	28	ref	2.2×10−4

Table 4.

Convergence, as A grows, of the p=3 Neumann solver at and around Wood frequencies. Shift parameter d=1.4; GMRES residual tolerance equal to 10⁻⁶. ‘Ref’ refers to finely resolved solutions against which the error of the coarser solutions is evaluated.

k	unknowns	A	iter	ε1	ε
2π	24×24	20	19	5.3×10−3	7.4×10−3
2π	24×24	30	19	5.7×10−4	1.7×10−3
2π	24×24	40	19	1.1×10−4	3.7×10−4
2π	24×24	80	19	ref	4.5×10−6
2π±10−6	24×24	40	19	1.1×10−4	3.6×10−4
$2\sqrt{2}\pi$	24×24	30	25	7.3×10−2	5.1×10−2
$2\sqrt{2}\pi$	24×24	40	25	4.1×10−3	2.8×10−3
$2\sqrt{2}\pi$	24×24	80	25	2.6×10−4	3.4×10−4
$2\sqrt{2}\pi$	24×24	160	25	ref	4.2×10−5
$2\sqrt{2}\pi \pm {10}^{-6}$	24×24	80	25	2.5×10−4	3.5×10−4
4π	32×32	30	28	1.2×10−1	4.5×10−2
4π	32×32	60	28	2.7×10−3	1.6×10−3
4π	32×32	180	28	ref	1.1×10−4

Appendix A. Integral equations away from Wood frequencies

To establish the unique solvability of the integral equations (4.4), the proof of theorem 4.1 relies on the following classical result on solutions to the homogeneous surface-impedance problem.

Lemma A.1 —

Let v(x) is a quasi-periodic field that satisfies the Helmholtz equation Δv+k²v=0 for $z<f(\widetilde{\mathbf{\text{x}}})$ (respectively, $z>f(\widetilde{\mathbf{\text{x}}})),$ the outgoing condition (2.7) (respectively, (2.5)) and the impedance condition

$$\frac{\mathrm{\partial }v}{\mathrm{\partial }n}(\mathbf{\text{x}})-\mathrm{i}\zeta v(\mathbf{\text{x}})=0(\mathbf{\text{x}}\in \mathrm{\Gamma })$$

with ζ>0 (respectively, ζ<0). Then v(x)=0 for z<f(x) (respectively, z>f(x)).

Proof. —

We establish the result in the case z<f(x) and ζ>0; the complementary case is handled analogously. Consider the truncated period

$$\mathrm{\Omega }=\{(\widetilde{\mathbf{\text{x}}},z)\hspace{0.17em}:\hspace{0.17em}\widetilde{\mathbf{\text{x}}}\in Q,\hspace{0.17em}{z}_{-}<z<f(\widetilde{\mathbf{\text{x}}})\}$$

with lower boundary $S=\{(\widetilde{\mathbf{\text{x}}},z):\widetilde{\mathbf{\text{x}}}\in Q,\hspace{0.17em}z=z_{-}\}$ oriented downwards. Integration by parts yields

$$0={\int }_{\mathrm{\Omega }}(\mathrm{\Delta }v+k^{2}v)\overline{v}={\int }_{\mathrm{\Omega }}(-|\mathrm{\nabla }v{|}^2+k^2|v{|}^2)\hspace{0.17em}\mathrm{d}x-{\int }_{{\mathrm{\Gamma }}^{\mathrm{per}}}\frac{\mathrm{\partial }v}{\mathrm{\partial }n}\overline{v}\hspace{0.17em}\mathrm{d}s-{\int }_S\frac{\mathrm{\partial }v}{\mathrm{\partial }n}\hspace{0.17em}\overline{v}\hspace{0.17em}\mathrm{d}s.$$ A 2

(The integrals over the lateral sides of Ω add up to zero as a result of the assumed quasi-periodicity of v.) In view of the impedance condition on Γ and the outgoing condition (2.7), we obtain

$${\int }_{\mathrm{\Gamma }}\frac{\mathrm{\partial }v}{\mathrm{\partial }n}\overline{v}\hspace{0.17em}\mathrm{d}s+{\int }_S\frac{\mathrm{\partial }v}{\mathrm{\partial }n}\hspace{0.17em}\overline{v}\hspace{0.17em}\mathrm{d}s=\mathrm{i}\zeta {\int }_{\mathrm{\Gamma }}|v{|}^2\hspace{0.17em}\mathrm{d}s+\frac{\mathrm{i}}{2D}\sum _{(j,\ell ),{\gamma }_{j\ell }>0}{\gamma }_{j\ell }|c_{j\ell }^{-}{|}^2.$$

The imaginary part of this quantity equals the imaginary part of (A2) and it must therefore vanish. It follows that v=0 on Γ. The impedance condition then also yields ∂v/∂n=0 on Γ. By Green’s identity it follows that v=0 in Ω and, therefore, for all x with $z<f(\widetilde{\mathbf{\text{x}}})$ ($\widetilde{\mathbf{\text{x}}}\in {\mathbb{R}}^2$). □

For completeness we present a simpler alternative proof of theorem 4.1 that is restricted to the case p=0 (unshifted Green function) and to non-Wood configurations.

Theorem A.2 —

Let η/ξ<0, let p=0 and let us assume that k is a wavenumber for which the quasi-periodic Green function $G_0^q$ exists, that is, γ_jℓ≠0 for all pairs $(j,\ell )\in \mathbb{Z}$. Then the integral equation (4.4) is uniquely solvable in L²(Γ^per).

Proof. —

Given that the surface Γ is smooth and that $G_0^q$ has the same singularity as G, it follows that the integral operators on the left-hand side of equation (4.4) are compact operators in the space L²(Γ^per). Thus, by Fredholm theory, the unique solvability of equation (4.4) is equivalent to the injectivity of the operator on the left-hand side of that equation. To establish injectivity, let ϕ₀∈L²(Γ^per) denote a solution of equation (4.4) with zero right-hand side, and let u^± denote the restrictions to Ω^± of the potentials u defined by equation (4.1) above and below Γ with density ϕ=ϕ₀. It follows that u⁺ is a radiating solution of the Helmholtz equation in Ω⁺ with zero Dirichlet boundary conditions, and hence u⁺=0 in Ω⁺ [21]. Using the jump relations satisfied by the layer potentials in equation (4.1), we obtain u⁻|_Γ=−ξϕ₀ and (∂u⁻/∂n)|_Γ=iηϕ₀. Thus u⁻ is a quasi-periodic radiating solution of the Helmholtz equation in Ω⁻ with zero impedance boundary conditions (∂v/∂n)(x)−iζv(x)=0, ζ=−η/ξ>0 on Γ. By lemma .1 it follows that v⁻=0 in Ω⁻, and thus ϕ₀=0 in L²(Γ^per), as desired. The proof of the theorem is now complete. □

No uniqueness results exist for the Helmholtz scattering problem (2.4) under Neumann boundary-value conditions and the radiation condition (2.5), even away from Wood anomalies—although it has been repeatedly conjectured (cf. [25], p. 147), [2]) that such a uniqueness result does hold. Assuming that the wavenumber k is such that this scattering problem does admit a unique solution, however, we may seek the scattered field in the form

$$u(\mathbf{x})={\int }_{\mathrm{\Gamma }}{G}^q(\mathbf{x}-{\mathbf{x}}^{\prime })\psi ({\mathbf{x}}^{\prime })\hspace{0.17em}\mathrm{d}s({\mathbf{x}}^{\prime }),\mathbf{\text{x}}\in {\mathbb{R}}^3\setminus \mathrm{\Gamma }$$ A 3

in terms of the unknown surface density ψ. Using the jump condition for the normal derivatives of single-layer potentials and the sound-hard (Neumann) boundary condition, the unknown density ψ is seen to be a solution of the integral equation

$$-\frac{\psi (\mathbf{x})}{2}+{\int }_{\mathrm{\Gamma }}\frac{\mathrm{\partial }{G}^q(\mathbf{x}-{\mathbf{x}}^{\prime })}{\mathrm{\partial }n(\mathbf{\text{x}})}\psi ({\mathbf{x}}^{\prime })\hspace{0.17em}\mathrm{d}s({\mathbf{x}}^{\prime })=-\mathrm{i}(\mathit{\alpha },-\gamma )\cdot \mathbf{\text{n}}(\mathbf{\text{x}})\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}(\mathit{\alpha }\cdot \widetilde{\mathbf{\text{x}}}-\gamma z)},\mathbf{\text{x}}=(\widetilde{\mathbf{\text{x}}},z)\in \mathrm{\Gamma }.$$ A 4

Given that, additionally, the scattering problems from doubly periodic surfaces and Dirichlet boundary conditions admit unique solutions for all wavenumbers, it follows that the integral equations (A4) themselves have unique solutions.

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Authors' contributions

All authors are equally considered co-contributors in this article.

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There are no competing interests relevant to this article.

Funding

This effort was supported by AFOSR, NSF and a NSSEFF Vannevar Bush Fellowship under contracts FA9550-15-1-0043, DMS-1411876 and N00014-16-1-2808 (O.P.B.); NSF DMS-0807325 (S.P.S.); NSF DMS-1312169 and DMS-1614270 (C.T.) and NSF DMS-1211638 (S.V.).