On Energy Conditions for Electromagnetic Diffraction by Apertures

Abstract

The diffraction of light is considered for a plane screen with an open bounded aperture. The corresponding solution behind the screen is given explicitly in terms of the Fourier transforms of the tangential components of the electric boundary field in the aperture. All components of the electric as well as the magnetic field vector are considered. We introduce solutions with global finite energy behind the screen and describe them in terms of two boundary potential functions. This new approach leads to a decoupling of the vectorial boundary equations in the aperture in the case of global finite energy. For the physically admissible solutions, that is, the solutions with local finite energy, we derive a characterisation in terms of the electric boundary fields.

1. Introduction

This paper deals with the classical diffraction problem for electromagnetic waves passing a bounded aperture in an ideally conducting plane screen. We treat the problem within the exact theory, that is, we consider the corresponding solutions of the time harmonic Maxwell equations that fulfil the correct boundary conditions on the screen.

The problem of diffraction of electromagnetic waves by an infinite slit has been treated by the Fourier method in the papers [1, 2]. In [1] especially representations of the solutions that fulfil a certain energy condition have been given in terms of distributional electric boundary fields satisfying special regularity properties. In [2] mapping properties of the corresponding boundary operators between Sobolev spaces have been studied. These Sobolev spaces have been chosen such that the corresponding diffraction solutions satisfy the correct physical energy condition.

While the slit problem treated in [1, 2] can be decoupled into two scalar problems by considering two kinds of polarisations of the electromagnetic field, in the case of a bounded aperture such a decoupling is not possible in general. However, for the latter case we derive a new kind of decoupling of the vectorial system which can be performed if and only if the condition of global finite energy in part (b) of Definition 2.5 is fulfilled, see Theorem 3.2.

Finally we study the condition of local finite energy which covers all physically admissible solutions. Here we give a characterisation of solutions with local finite energy in terms of a regularity property of the electric boundary fields, see Theorem 4.1. This is done in a self-contained way by using the Paley-Wiener theorem for distributions defined on the bounded aperture as well as a special contour integration method in the Fourier domain.

2. Electromagnetic Diffraction by an Aperture in a Plane Screen

We start with an informal physical description of the electromagnetic diffraction problem and fix some notations which will be used in the sequel. Then we will develop a more general mathematical frame with boundary distributions in Sobolev spaces in order to obtain diffraction solutions satisfying physical energy conditions.

Monochromatic light waves with a fixed wavenumber 𝑘>0 satisfy the first-order system of Maxwell-Helmholtz equations−𝑖𝑘𝑖𝑘𝐸𝐵_∗∗𝑥+∇×𝐵_∗𝑥=0,𝑥+∇×𝐸_∗𝑥=0.(2.1) In the whole paper we consider a real wavenumber 𝑘>0, although the results can be generalised to the case of a complex wavenumber 𝑘≠0 with Re𝑘≥0 and Im𝑘≥0. We assume that the electromagnetic field with components 𝑒_𝑗,𝑏_𝑗, 𝑗=1,2,3,⎛⎜⎜⎜⎜⎝𝑒𝑒₁₂𝑥,𝑥₃𝑒₃𝑥,𝑥₃𝑥,𝑥₃⎞⎟⎟⎟⎟⎠=𝐸^∗𝑥,𝑥₃,⎛⎜⎜⎜⎜⎝𝑏¹𝑏₂𝑥,𝑥₃𝑏₃𝑥,𝑥₃𝑥,𝑥₃⎞⎟⎟⎟⎟⎠=𝐵^∗𝑥,𝑥₃,(2.2) consists of functions defined in the upper half-spaceℋ∶=𝑥=𝑥,𝑥₃∈ℝ³∣𝑥∈ℝ²,𝑥₃>0.(2.3)

The diffraction problem is considered for an open bounded apertureΩ⊂(𝑥,0)∣𝑥∈ℝ²(2.4) in the screen plane 𝑥₃=0. In the sequel we will suppress the notation of the third component 0 for the points in the screen plane, and interpret Ω as well as the screen Ω^𝑐∶=ℝ²⧵Ω as subsets of ℝ². For describing the whole electromagnetic field in terms of its boundary values, for the moment we assume that these are functions, given for 𝑥∈ℝ² by⎛⎜⎜⎜⎜⎝𝑒𝑒_1,0(𝑥)_2,0𝑒(𝑥)_3,0(𝑥)⎞⎟⎟⎟⎟⎠=lim^𝑥3↓0𝐸_∗𝑥,𝑥₃,⎛⎜⎜⎜⎜⎝𝑏^1,0𝑏(𝑥)_2,0𝑏(𝑥)_3,0(𝑥)⎞⎟⎟⎟⎟⎠=lim^𝑥3↓0𝐵_∗𝑥,𝑥₃.(2.5)

The screen Ω^𝑐 is assumed to be an ideal conducting wall. This implies the physical boundary conditions𝑒_1,0(𝑥)=𝑒_2,0(𝑥)=0∀𝑥∈Ω^𝑐.(2.6) In the general case the boundary fields (2.5) have to be replaced by appropriate distributions, and the limit 𝑥₃↓0 will be performed in the Fourier domain instead of the half-space ℋ.

For this purpose we write again 𝑥=(𝑥,𝑥₃), with 𝑥=(𝑥₁,𝑥₂)∈ℝ² and fixed 𝑥₃>0, and assume that each field component 𝑢(⋅,𝑥₃) represents a tempered distribution in 𝒮′(ℝ²) with Fourier transform𝑢̂𝜉₁,𝜉₂,𝑥₃=12𝜋_ℝ²𝑢𝑥₁,𝑥₂,𝑥₃𝑒^{−𝑖(𝜉1𝑥1+𝜉2𝑥2)}𝑑𝑥₁𝑑𝑥₂,𝜉₁,𝜉₂∈ℝ.(2.7) Then we obtain from the first-order Maxwell-Helmholtz equations the following Fourier transformed Maxwell-Helmholtz system: for all 𝑥₃≥0 and fixed 𝜉₁,𝜉₂∈ℝ we have𝑖𝑘⎛⎜⎜⎜⎜⎝𝑒𝑒¹₂̂̂𝑒₃̂⎞⎟⎟⎟⎟⎠+⎛⎜⎜⎜⎜⎜⎝𝑖𝜉₂𝑏̂₃−𝑑𝑑𝑥₃𝑏̂₂𝑑𝑑𝑥₃𝑏̂₁−𝑖𝜉₁𝑏̂₃𝑖𝜉₁𝑏̂₂−𝑖𝜉₂𝑏̂₁−𝑖𝑘⎛⎜⎜⎜⎜⎝𝑏₁̂⎞⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎝000⎞⎟⎟⎟⎟⎠,𝑏̂₂𝑏̂₃⎞⎟⎟⎟⎟⎠+⎛⎜⎜⎜⎜⎜⎝𝑖𝜉₂𝑒₃̂−𝑑𝑑𝑥₃𝑒₂̂𝑑𝑑𝑥₃𝑒₁̂−𝑖𝜉₁𝑒₃̂𝑖𝜉₁𝑒₂̂−𝑖𝜉₂𝑒₁̂⎞⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎝000⎞⎟⎟⎟⎟⎠.(2.8) Here we have replaced the partial derivative with respect to 𝑥₃ by the ordinary derivative 𝑑/𝑑𝑥₃.

We define𝐶∶=1𝑘⎛⎜⎜⎜⎜⎜⎝𝜕²𝜕𝑥₁𝜕𝑥₂−𝑘²+𝜕²𝑘₂𝜕𝑥²¹+𝜕²𝜕𝑥²²−𝜕²𝜕𝑥₁𝜕𝑥₂⎞⎟⎟⎟⎟⎟⎠.(2.9) With the two-dimensional Laplace operator Δ we have𝐶²=−𝑘²+Δ⎛⎜⎜⎝1001⎞⎟⎟⎠.(2.10) For the action of 𝐶 in the Fourier domain we obtain multiplication by the matrix𝐶(𝜉)∶=1𝑘⎛⎜⎜⎝−𝜉¹𝑘₂𝜉₂−𝑘²−𝜉²¹−𝜉²²𝜉₁𝜉₂⎞⎟⎟⎠,𝜉=𝜉₁,𝜉₂∈ℝ².(2.11) We replace (2.8) with the ordinary differential equations𝑑𝑑𝑥₃⎛⎜⎜⎝𝑒₁̂𝑒₂̂⎞⎟⎟⎠=−𝑖𝐶⎛⎜⎜⎝𝑏₁̂𝑏̂₂⎞⎟⎟⎠,𝑑𝑑𝑥₃⎛⎜⎜⎝𝑏̂₁𝑏̂₂⎞⎟⎟⎠=𝑖𝐶⎛⎜⎜⎝𝑒₁̂𝑒₂̂⎞⎟⎟⎠,(2.12) and the two algebraic conditions𝑒₃̂=1𝑘𝜉²𝑏̂₁−𝜉₁𝑏̂₂,𝑏̂₃=1𝑘𝜉¹𝑒₂̂−𝜉₂𝑒₁̂.(2.13) For all fixed 𝜉∈ℝ² we supplement the system of differential equations (2.12) by the initial conditions𝑒̂_𝑗,0(𝜉)∶=𝑒_𝑗̂(𝜉,0),𝑏̂_𝑗,0(𝜉)∶=𝑏_𝑗̂(𝜉,0),𝑗=1,2,(2.14) and put 𝑚(𝜉)∶=⎧⎪⎨⎪⎩𝑖𝑘²−||𝜉||²,||𝜉||≤𝑘,||𝜉||²−𝑘²,||𝜉||>𝑘.(2.15) Then the general solution of the homogeneous linear system (2.12) is⎛⎜⎜⎜⎜⎜⎜⎝𝑒𝑒₁₂̂̂𝜉,𝑥₃𝑏̂₁𝜉,𝑥₃𝑏̂₂𝜉,𝑥₃𝜉,𝑥₃⎞⎟⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎝cos𝑚(𝜉)𝑥³𝐸−𝑖sin𝑚(𝜉)𝑥³𝑖sin𝑚(𝜉)𝑥₃𝑚(𝜉)𝐶(𝜉)𝑚(𝜉)𝐶(𝜉)cos𝑚(𝜉)𝑥³𝐸⎞⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎝𝑒̂^1,0𝑒̂_2,0(𝜉)𝑏̂(𝜉)_1,0𝑏̂(𝜉)_2,0(𝜉)⎞⎟⎟⎟⎟⎟⎟⎠(2.16) with the 2×2 unit matrix 𝐸.

But the terms cos(𝑚(𝜉)𝑥₃) and sin(𝑚(𝜉)𝑥₃) are exponentially increasing for fixed 𝑥₃>0 and |𝜉|→∞. For avoiding that the Fourier transformed fields are also exponentially increasing we have to require the following algebraic conditions for |𝜉|>𝑘:𝑚⋅⎛⎜⎜⎝𝑏̂^1,0𝑏̂_2,0⎞⎟⎟⎠=𝐶⎛⎜⎜⎝𝑒̂_1,0𝑒̂_2,0⎞⎟⎟⎠.(2.17) By using (2.17), for 𝑥₃↓0 and |𝜉|>𝑘 we can replace (2.13) with𝑒̂_3,0=−1𝑚𝜉¹𝑒̂_1,0+𝜉₂𝑒̂_2,0,𝑏̂_3,0=−1𝑘𝜉²𝑒̂_1,0−𝜉₁𝑒̂_2,0.(2.18) From the general solution and (2.17), (2.18) we obtain the following decay conditions for 𝑥₃>0:𝑒_𝑗̂𝜉,𝑥₃=𝑒^{𝑖𝑥3𝑚(𝜉)}𝑒̂_𝑗,0(𝜉),𝑏̂_𝑗𝜉,𝑥₃=𝑒^{𝑖𝑥3𝑚(𝜉)}𝑏̂_𝑗,0(𝜉),𝑗=1,2,3.(2.19) With 𝑥=(𝑥₁,𝑥₂)∈ℝ², 𝑥=(𝑥,𝑥₃)∈ℝ³ and 𝑥₃>0 there holds the important and well-known Sommerfeld-Weyl integral representation𝐹_𝑘𝑥,𝑥₃∶=𝑒^{𝑖𝑘|𝑥|}||𝑥||=2𝑖𝜋_ℝ²𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑚(𝜉)𝑑𝜉.(2.20) The left-hand side in (2.20) is the singular solution of the three-dimensional Helmholtz equation (Δ+𝑘²)𝐹_𝑘=−4𝜋𝛿. For this reason it is natural to require the algebraic conditions (2.17), (2.18) also in the case |𝜉|<𝑘, such that (2.19) is generally valid for 𝑥₃>0 and 𝜉∈ℝ², |𝜉|≠𝑘.

Distributions 𝑢∈𝒟′(ℝ²) with compact support in the screen plane are tempered, and it follows from the Paley-Wiener theorem that 𝑢̂ is a smooth function which has polynomial growth on ℝ². This is used in the following theorem, which results if we regard (2.17), (2.18), and (2.19) and apply the Fourier inversion formula for 𝑥₃>0 to each component 𝑒_𝑗̂(⋅,𝑥₃) and 𝑏̂_𝑗(⋅,𝑥₃).

Theorem 2.1. Let there be given 𝑒_1,0,𝑒_2,0∈𝒮′(ℝ²) with support in the bounded region Ω. Then the following functions 𝑒_𝑗,𝑏_𝑗∶ℋ→ℂ constitute a 𝐶^∞-solution of the Maxwell-Helmholtz system (2.1) in the upper half-space ℋ, 𝑗=1,2,3; 𝑒₁𝑥,𝑥₃=12𝜋_ℝ²𝑒𝑒̂_1,0(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉,₂𝑏𝑥,𝑥₃=21𝜋_ℝ²𝑒̂_2,0(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉,₁𝑥,𝑥₃=−2𝜋1𝑘_ℝ²𝜉₁𝜉₂𝑒̂_1,0(𝜉)+𝑘²−𝜉²¹𝑒̂_2,0(𝜉)𝑚(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑏𝑑𝜉,₂𝑥,𝑥₃=2𝜋1𝑘_ℝ²𝑘²−𝜉²²𝑒̂_1,0(𝜉)+𝜉₁𝜉₂𝑒̂_2,0(𝜉)𝑚(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑒𝑑𝜉,₃𝑥,𝑥₃=−21𝜋_ℝ²𝜉₁𝑒̂_1,0(𝜉)+𝜉₂𝑒̂_2,0(𝜉)𝑚(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑏𝑑𝜉,₃𝑥,𝑥₃=−2𝜋1𝑘_ℝ²𝜉₂𝑒̂_1,0(𝜉)−𝜉₁𝑒̂_2,0(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉.(2.21)

Proof. The calculation of the partial derivatives of 𝑒_𝑗 and 𝑏_𝑗 can be interchanged with integration. This can be used to check the Maxwell-Helmholtz equations independent from the previous representations of the Fourier-transforms 𝑒_𝑗̂ and 𝑏̂_𝑗 in terms of 𝑒̂_1,0 and 𝑒̂_2,0.

Definition 2.2. The electromagnetic field in the half-space 𝑥₃>0 behind the screen is completely determined by the electric boundary components 𝑒_1,0, 𝑒_2,0. We call 𝑒_𝑗,𝑏_𝑗∶ℋ→ℂ, 𝑗=1,2,3, the half-space solution determined by the boundary distributions 𝑒_1,0,𝑒_2,0∈𝒮′(ℝ²) with compact support in Ω.

Remark 2.3. Assume that 𝑒_1,0,𝑒_2,0 are smooth functions with compact support in Ω and that 𝐸_∗, 𝐵_∗ is the corresponding electromagnetic field with the components given in Theorem 2.1. Then we obtain from Fourier’s inversion formula that for all 𝑥∈ℝ²lim_𝑥3↓0𝑒_𝑗𝑥,𝑥₃=𝑒_𝑗,0(𝑥),𝑗=1,2.(2.22)

Remark 2.4. Assume again that 𝑒_1,0,𝑒_2,0 are smooth functions with compact support in Ω. Then we obtain from Theorem 2.1 and the Sommerfeld-Weyl integral (2.20) for all 𝑥∈ℝ² and all 𝑥₃>0 that 𝑏₁𝑥,𝑥₃=−𝑖2𝜋𝑘𝜕𝑥𝜕₁𝜕²𝑥₂_Ω+𝑖𝑒_1,0(𝑦)𝐹_𝑘𝑥−𝑦,𝑥₃𝑑𝑦2𝜋𝑘𝑘²+𝜕²𝜕𝑥²¹_Ω𝑏𝑒_2,0(𝑦)𝐹_𝑘𝑥−𝑦,𝑥₃𝑑𝑦,₂𝑥,𝑥₃=−2𝜋𝑖𝑘𝑘²+𝜕²𝜕𝑥²²_Ω+𝑖𝑒_1,0(𝑦)𝐹_𝑘𝑥−𝑦,𝑥₃𝑑𝑦2𝜋𝑘𝜕𝑥𝜕₁𝜕²𝑥₂_Ω𝑒_2,0(𝑦)𝐹_𝑘𝑥−𝑦,𝑥₃𝑑𝑦.(2.23) Equations (2.23) involve the boundary conditions (2.6) for the electric field components on the ideal conducting plane screen Ω^𝑐. In order to obtain a coupled system of boundary integro-differential equations we pass for 𝑥∈ℝ² to the limit 𝑥₃↓0 and obtain 𝑏_1,0(𝑥)=−𝑖2𝜋𝑘𝜕𝑥𝜕₁𝜕²𝑥₂_Ω𝑒_1,0(𝑦)𝑒^{𝑖𝑘|𝑥−𝑦|}+𝑖||𝑥−𝑦||𝑑𝑦2𝜋𝑘𝑘²+𝜕²𝜕𝑥²¹_Ω𝑒_2,0(𝑦)𝑒^{𝑖𝑘|𝑥−𝑦|}𝑏||𝑥−𝑦||𝑑𝑦,_2,0(𝑥)=−𝑖2𝜋𝑘𝑘²+𝜕²𝜕𝑥²²_Ω𝑒_1,0(𝑦)𝑒^{𝑖𝑘|𝑥−𝑦|}+𝑖||𝑥−𝑦||𝑑𝑦2𝜋𝑘𝜕𝑥𝜕₁𝜕²𝑥₂_Ω𝑒_2,0(𝑦)𝑒^{𝑖𝑘|𝑥−𝑦|}||𝑥−𝑦||𝑑𝑦.(2.24) In general the electric boundary fields 𝑒_1,0 and 𝑒_2,0 are unknown distributions with compact support in Ω, whereas 𝑏_1,0 and 𝑏_2,0 are given distributions in the aperture Ω. In order to select physical admissible solutions of the diffraction problem we need some conditions for its electromagnetic energy content, especially in local volume elements 𝐺⊂ℋ. Recall that ℋ∶=ℝ²×ℝ⁺.

Definition 2.5. Let 𝑒_𝑗,𝑏_𝑗∶ℋ→ℂ, 𝑗=1,2,3, be the half-space solution determined by the boundary distributions 𝑒_1,0,𝑒_2,0∈𝒮′(ℝ²) with compact support in Ω.(a)The solution is called physical admissible if and only if it satisfies the local energy condition12³_𝑗=1_𝐺||𝑒^𝑗𝑥||²+||𝑏_𝑗𝑥||²𝑑𝑥<∞(2.25)for every bounded domain 𝐺⊂ℋ.(b)The solution satisfies the stronger global energy condition if and only if 12³_𝑗=1_ℒℎ||𝑒^𝑗𝑥||²+||𝑏_𝑗𝑥||²𝑑𝑥<∞(2.26)for some ℎ>0, and therewith for all ℎ>0, with the layerℒ_ℎ∶={(𝑥₁,𝑥₂,𝑥₃)∈ℋ|𝑥₃<ℎ}.
In the following two sections we determine the solutions with global as well as those with local finite energy in terms of an appropriate functional analytical setting for the electric boundary fields 𝑒_1,0 and 𝑒_2,0.

3. The Global Energy Condition

Throughout the rest of this paper let the open aperture Ω∶=⋃^𝜅𝑗=1Ω_𝑗⊂ℝ² be a finite union of nonempty bounded Lipschitz domains Ω_𝑗 in the screen plane, such that the compact sets Ω_𝑗 are pairwise disjoint.

By 𝐻^𝑠(ℝ²), 𝑠∈ℝ, we denote the Sobolev space of tempered distributions ℎ, for which the Fourier transform ℎ is locally integrable with‖ℎ‖_𝐻^𝑠_(ℝ²₎∶=_ℝ2|||ℎ(𝜉)|||²1+||𝜉||²^𝑠𝑑𝜉^1/2<∞(3.1) (cf. [3], Chapter  8.8). ‖⋅‖_𝐻^𝑠_(ℝ²₎ is the norm on the Banach space 𝐻^𝑠(ℝ²).

For 𝑠∈ℝ the Sobolev space 𝐻^𝑠(Ω) is given by𝐻^𝑠(Ω)=ℎ∈𝐻^𝑠ℝ²∣suppℎ⊂Ω.(3.2) Here suppℎ denotes the support of the distribution ℎ in the compact set Ω. The space 𝐻^𝑠(Ω) is equipped with the norm of 𝐻^𝑠(ℝ²), which makes it into a Banach space (cf. [4, Chapter  3]). The Lipschitz property of 𝜕Ω guarantees that 𝐻^𝑠(Ω) is the closure of 𝒟(Ω) in 𝐻^𝑠(ℝ²). For a more general result see [4, Theorem  3.29].

Theorem 3.1. Let 𝑒_𝑗,𝑏_𝑗∶ℋ→ℂ, 𝑗=1,2,3, be the half-space solution determined by the boundary distributions 𝑒_1,0,𝑒_2,0∈𝒮′(ℝ²) with compact support in Ω. Then the diffraction solution has global finite energy if and only if𝑒_1,0,𝑒_2,0,∇⋅⎛⎜⎜⎝−𝑒^2,0𝑒_1,0⎞⎟⎟⎠∈𝐻_−1/2(Ω)(3.3) and 𝑒̂_1,0(𝜉)𝜉₁+𝑒̂_2,0(𝜉)𝜉₂=0 for all 𝜉=(𝜉₁,𝜉₂)∈ℝ² with |𝜉|=𝑘.

Proof. In order to perform the energy evaluation with Parseval’s theorem we define with the Fourier transformed electromagnetic boundary fields 𝑒̂_𝑗,0,𝑏̂_𝑗,0, 𝑗=1,2,3, the quantity 𝑊₀(𝜉)∶=12³_𝑗=1||𝑒̂^𝑗,0(𝜉)||²+||𝑏̂_𝑗,0(𝜉)||².(3.4) It follows from a lengthy calculation with |𝑧|²=𝑧𝑧 for the values 𝑧∈ℂ of the Fourier transformed boundary fields as well as from the algebraic relations (2.17), (2.18), which are valid for |𝜉|≠𝑘, that 𝑊₀(𝜉)=||𝑒̂^1,0||𝑒̂_3,0(𝜉)||²+||𝑒̂_2,0(𝜉)||²+||𝑒̂_3,0(𝜉)||²,||𝜉||<𝑘,(𝜉)||²+||𝑏̂_3,0(𝜉)||²,||𝜉||>𝑘.(3.5)
With 𝑉₁∶={𝜉∈ℝ²||𝜉|<𝑘}, 𝑉₂∶={𝜉∈ℝ²||𝜉|>𝑘} we conclude from Parseval’s theorem, (2.19) and the definition (2.15) of 𝑚(𝜉) for ℎ>0ℰ(ℎ)∶=12³_𝑗=1^ℎ0_ℝ2||𝑒^𝑗𝑥,𝑥₃||²+||𝑏_𝑗𝑥,𝑥₃||²𝑑𝑥𝑑𝑥³=ℎ_𝑉1𝑊₀(𝜉)𝑑𝜉+_𝑉21−𝑒^{√−2ℎ|𝜉|2−𝑘2}2||𝜉||²−𝑘²𝑊₀(𝜉)𝑑𝜉.(3.6) Using for all 𝜉∈ℝ² with |𝜉|≠𝑘 the estimate 12||𝑒̂^1,0≤𝑊(𝜉₀)||²+||𝑒̂_2,0(𝜉)||²+||𝑒̂_3,0(𝜉)||²+||𝑏̂_3,0(𝜉)||²(𝜉)≤||𝑒̂_1,0(𝜉)||²+||𝑒̂_2,0(𝜉)||²+||𝑒̂_3,0(𝜉)||²+||𝑏̂_3,0(𝜉)||²,(3.7) and observing that we have uniformly in 𝑟>𝑘 for appropriate constants 𝛼,𝛽>0𝛼√1+𝑟²≤1−𝑒^{√−2ℎ𝑟2−𝑘2}2√𝑟²−𝑘²≤𝛽√1+𝑟²,(3.8) we conclude from (3.6) that ℰ(ℎ) is finite if and only if 𝑒_1,0,𝑒_2,0,𝑒_3,0,𝑏_3,0=𝑖𝑘∇⋅⎛⎜⎜⎝−𝑒𝑒^2,0_1,0⎞⎟⎟⎠∈𝐻_−1/2ℝ².(3.9) Note that with 𝑒_1,0 and 𝑒_2,0 also ∇⋅(−𝑒_2,0,𝑒_1,0)^𝑇 has support in Ω.
Assume first that ℰ(ℎ)<∞ and that 𝑒̂_1,0(𝜉)𝜉₁+𝑒̂_2,0(𝜉)𝜉₂≠0 for a certain 𝜉=(𝜉₁,𝜉₂)∈ℝ² with |𝜉|=𝑘. Since 𝑒_1,0,𝑒_2,0 have compact support, we conclude from the Paley-Wiener theorem that 𝑒̂_1,0,𝑒̂_2,0∶ℝ²→ℂ are smooth and especially continuous. Thus |𝑒̂_1,0(𝜉)𝜉₁+𝑒̂_2,0(𝜉)𝜉₂|²≥𝛿>0 in a bounded domain 𝜉₁,𝜉₂=𝑟(cos𝜑,sin𝜑)with𝜑₁<𝜑<𝜑₂,𝑘<𝑟<𝑘+𝜀,(3.10) and hence |𝑒̂_3,0|² because of (2.18) is not integrable. Due to (3.1) this violates the necessary condition 𝑒_3,0∈𝐻^−1/2(ℝ²) in (3.9).
For showing the other direction of the equivalence stated in the theorem we assume that 𝑒_1,0,𝑒_2,0,∇⋅⎛⎜⎜⎝−𝑒^2,0𝑒_1,0⎞⎟⎟⎠∈𝐻_−1/2𝑒̂_1,0(Ω),(3.11)𝜉₁,𝜉₂𝜉₁+𝑒̂_2,0𝜉₁,𝜉₂𝜉₂=0,(3.12) for all 𝜉₁,𝜉₂∈ℝ with 𝜉²¹+𝜉²²=𝑘². In order to prove (3.9) it remains to show that 𝑒_3,0∈𝐻^−1/2(ℝ²). Since 𝑒_1,0, 𝑒_2,0 have compact support in Ω, we obtain from the Paley-Wiener theorem that the Fourier transforms 𝑒̂_1,0, 𝑒̂_2,0 can be continuated to entire functions. We also denote these entire functions by 𝑒̂_1,0,𝑒̂_2,0. We define the entire function 𝑓 by 𝑓𝑧₁,𝑧₂∶=𝑒̂_1,0𝑧₁,𝑧₂𝑧₁+𝑒̂_2,0𝑧₁,𝑧₂𝑧₂.(3.13) Using (3.12) and (2.18), it follows for example from Theorem A.1 in the appendix that |𝑒̂_3,0|² is a locally integrable function.
From the first equation in (2.18) and the Cauchy-Schwarz inequality we obtain the estimate ||𝑒̂_3,0(𝜉)||²≤||𝜉||²||𝜉||²−𝑘²⋅||𝑒̂_1,0(𝜉)||²+||𝑒̂_2,0(𝜉)||²,||𝜉||>𝑘.(3.14) Now 𝑒_3,0∈𝐻^−1/2(ℝ²) follows from the assumption 𝑒_1,0,𝑒_2,0∈𝐻^−1/2(Ω), because |𝑒_3,0|² is locally integrable and |𝜉|²/(|𝜉|²−𝑘²) is bounded for sufficiently large |𝜉|.

Next we present a characterisation of the global finite energy solutions in terms of two boundary potential functions. In the following we use the vectorial differential operator (2.9) satisfying (2.10).

Theorem 3.2. Boundary potential functions(a)Given are two functions 𝑢₁,𝑢₂∈𝐻^1/2(Ω), in the following called boundary potential functions, satisfying the regularity condition ∇⋅⎛⎜⎜⎝−𝑢²𝑢₁⎞⎟⎟⎠∈𝐻_1/2(Ω).(3.15)Define ⎛⎜⎜⎝𝑒_1,0𝑒_2,0⎞⎟⎟⎠∶=𝐶⎛⎜⎜⎝𝑢₁𝑢₂⎞⎟⎟⎠.(3.16)Then the corresponding electromagnetic field 𝑒_𝑗,𝑏_𝑗, 𝑗=1,2,3, determined by 𝑒_1,0 and 𝑒_2,0 according to Theorem 2.1, has global finite energy. (b)Let 𝑒_1,0, 𝑒_2,0 be given such that the half-space solution in Theorem 2.1 has global finite energy. If the set ℝ²⧵Ω is connected, then the boundary fields 𝑒_1,0 and 𝑒_2,0 can be represented by two boundary potential functions 𝑢₁,𝑢₂∈𝐻^1/2(Ω) satisfying (3.15) and (3.16) in part (a).(c)From the assumptions of part (a), or alternatively part (b), one obtains in the distributional sense 𝑏_1,0(𝑥)=𝑖2𝜋𝑘²+Δ_Ω𝑢₁(𝑦)𝑒^{𝑖𝑘|𝑥−𝑦|}𝑏||𝑥−𝑦||𝑑𝑦,_2,0(𝑥)=𝑖2𝜋𝑘²+Δ_Ω𝑢₂(𝑦)𝑒^{𝑖𝑘|𝑥−𝑦|}||𝑥−𝑦||𝑑𝑦(3.17)on the whole screen plane ℝ² with 𝑏_1,0,𝑏_2,0,∇⋅⎛⎜⎜⎝−𝑏^2,0𝑏_1,0⎞⎟⎟⎠∈𝐻_−1/2ℝ².(3.18)

Proof. Part (a) follows in the Fourier domain by representing 𝑒̂_1,0, 𝑒̂_2,0, 𝑒̂_3,0 and 𝑏̂_3,0 in terms of 𝑢₁̂ and 𝑢₂̂ as 𝑒̂_1,0=−𝑘𝑢₂̂−𝜉¹𝑘𝜉²𝑒̂_2,0𝑢₁̂−𝜉₁𝑢₂̂,=+𝑘𝑢₁̂−𝜉²𝑘𝜉²𝑒̂_3,0𝑢₁̂−𝜉₁𝑢₂̂,=−𝑚𝑘𝜉²𝑏̂_3,0𝑢₁̂−𝜉₁𝑢₂̂,=𝜉₁𝑢₁̂+𝜉₂𝑢₂̂.(3.19) These equations and the assumptions imply that 𝑒_1,0,𝑒_2,0,𝑏_3,0∈𝐻^−1/2(ℝ²), and regarding |𝑚(𝜉)|≤𝛼√1+|𝜉|² uniformly in 𝜉∈ℝ² for a certain constant 𝛼>0 also 𝑒_3,0∈𝐻^−1/2(ℝ²). Since 𝑒_1,0, 𝑒_2,0 have compact support in Ω like 𝑢₁, 𝑢₂, the proof of part (a) follows from the fact that condition (3.9) is equivalent to ℰ(ℎ)<∞.
For proving part (b) we assume that the diffraction solution corresponding to 𝑒_1,0,𝑒_2,0∈𝐻^−1/2(Ω) has global finite energy. We conclude from Theorem 3.1 and Theorem A.1 in the appendix that we obtain entire functions 𝑣₁,𝑣₂∶ℂ²→ℂ by 𝑣₁𝜉₁,𝜉₂=+1𝑘𝑒̂^2,0𝜉₁,𝜉₂−𝜉²𝑘⋅𝑒̂^1,0(𝜉)𝜉₁+𝑒̂_2,0(𝜉)𝜉₂𝜉²¹+𝜉²²−𝑘²𝑣₂,𝜉₁,𝜉₂=−𝑘1𝑒̂^1,0𝜉₁,𝜉₂+𝜉𝑘¹⋅𝑒̂^1,0(𝜉)𝜉₁+𝑒̂_2,0(𝜉)𝜉₂𝜉²¹+𝜉²²−𝑘².(3.20) From (3.20) we get for 𝑗=1,2 that _ℝ2||𝑣_𝑗(𝜉)||²1+||𝜉||²^1/2𝑑𝜉<∞.(3.21) Thus for 𝑗=1,2 the inverse Fourier transform 𝑢_𝑗 of 𝑣_𝑗 lies in 𝐻^1/2(ℝ²).
Now we show that 𝑢₁ and 𝑢₂ have their support in Ω and hence, by reason of (3.21), 𝑢₁,𝑢₂∈𝐻^1/2(Ω). To this aim we choose some 𝑅>0 such that Ω⊂𝐵_𝑅∶=𝑥∈ℝ²∣|𝑥|<𝑅.(3.22) Since 𝑒_1,0,𝑒_2,0 are supported in Ω, we have supp𝑒_1,0,supp𝑒_2,0⊂𝐵_𝑅. Therefore we obtain from the Paley-Wiener theorem and (3.20) that 𝑢₁ and 𝑢₂ are also supported in 𝐵_𝑅, because 𝑣₁, 𝑣₂, as 𝑒̂_1,0, 𝑒̂_2,0, are of exponential type not larger than 𝑅.
Equations (3.20) can be resolved with respect to 𝑒̂_1,0, 𝑒̂_2,0. This gives (3.16). Together with (2.10) it follows that in the complement of Ω there holds −𝑘²+Δ⎛⎜⎜⎝𝑢¹𝑢₂⎞⎟⎟⎠=𝐶₂⎛⎜⎜⎝𝑢₁𝑢₂⎞⎟⎟⎠=𝐶⎛⎜⎜⎝𝑒_1,0𝑒_2,0⎞⎟⎟⎠=⎛⎜⎜⎝00⎞⎟⎟⎠.(3.23) Now we make use of the fact that the complement of Ω is connected. Since supp𝑢_𝑗⊂𝐵_𝑅 for 𝑗=1,2, Holmgren’s unique continuation principle, applied to the two scalar Helmholtz equations in (3.23), implies that 𝑢₁, 𝑢₂ have their support in Ω.
Finally, since 𝑣_𝑗=𝑢_𝑗̂ for 𝑗=1,2, from (3.20) we obtain for𝜉∈ℝ²−𝜉₁𝑢₂̂+𝜉₂𝑢₁̂=−𝑘𝜉¹𝑒̂_1,0+𝜉₂𝑒̂_2,0𝜉²¹+𝜉²²−𝑘².(3.24) The asymptotic behaviour of this expression for |𝜉|→∞ shows the validity of the regularity condition (3.15).
The convolution integrals in part (c) can be rewritten in the Fourier domain by using the Sommerfeld-Weyl integral representation (2.20) in the limit 𝑥₃↓0. The resulting relations in the Fourier domain follow from the representations of the components 𝑏₁ and 𝑏₂ in Theorem 2.1.
The validity of (3.18) is a consequence of the algebraic relations between 𝑢₁̂, 𝑢₂̂, 𝑏̂_1,0, and 𝑏̂_2,0.

Remark 3.3. Consider the Sobolev spaces 𝐻^𝑠(Ω) given for 𝑠∈ℝ by 𝐻^𝑠(Ω)=𝐹|_Ω∣𝐹∈𝐻^𝑠ℝ²,(3.25) with the restriction 𝐹|_Ω of the tempered distribution 𝐹∶𝒮(ℝ²)→ℂ to the subspace 𝒮(Ω) of the Schwartz space 𝒮(ℝ²), where 𝒮(Ω) is the closure of the set 𝒟(Ω) in 𝒮(ℝ²) with respect to the topology of 𝒮(ℝ²) [5], §1 in Section 5.
𝐻^𝑠(Ω) is a Banach space with respect to the norm ‖⋅‖_𝐻^𝑠_(Ω) given by‖𝑓‖_𝐻^𝑠_(Ω)=inf‖𝐹‖_𝐻^𝑠_(ℝ²₎∣𝐹∈𝐻^𝑠ℝ²isacontinuationof𝑓.(3.26) The magnetic boundary fields in part (c) of Theorem 3.2 corresponding to global finite energy solutions may also be reinterpreted as distributions restricted to Ω. In this case we obtain 𝑏_1,0,𝑏_2,0,∇⋅⎛⎜⎜⎝−𝑏^2,0𝑏_1,0⎞⎟⎟⎠∈𝐻_−1/2(Ω).(3.27) In general the conditions (3.27) are weaker than the conditions (3.18) in Theorem 3.2, and we assume that they are fulfilled for diffraction solutions with local finite energy.
The conditions (3.27), where 𝑏_1,0 and 𝑏_2,0 are considered only in the aperture Ω, reflects the physical fact that 𝑏_1,0 and 𝑏_2,0 are only prescribed in Ω. Namely, 𝑏_1,0 and 𝑏_2,0 are the tangential magnetic components of the incoming electromagnetic wave in the aperture Ω.

4. The Local Energy Condition

In this section we derive the following characterisation for the diffraction solutions of local finite energy.

Theorem 4.1. Let be 𝑒_1,0,𝑒_2,0∈𝒮^(ℝ²) and supp𝑒_1,0,supp𝑒_2,0⊂Ω. Let 𝑒_𝑗,𝑏_𝑗∶ℋ→ℂ, 𝑗=1,2,3, be defined as in Theorem 2.1. Then the diffraction solution 𝑒_𝑗,𝑏_𝑗, 𝑗=1,2,3, has local finite energy if and only if 𝑒_1,0,𝑒_2,0,∇⋅⎛⎜⎜⎝−𝑒^2,0𝑒_1,0⎞⎟⎟⎠∈𝐻_−1/2(Ω).(4.1)

For proving Theorem 4.1 firstly we formulate the subsequent Lemma 4.2. Then, using this lemma, we give the proof of the theorem. Afterwards we prove the lemma.

In the sequel we will make use of the following notations. For 𝑟>0 we define the open ball 𝐵_𝑟 by𝐵_𝑟∶=𝑥∈ℝ²∣|𝑥|<𝑟.(4.2) For 𝑟>0 and ℎ>0 the open cylinder 𝑍_𝑟,ℎ is defined by𝑍_𝑟,ℎ∶=𝑥=𝑥,𝑥₃∈ℝ³∣𝑥∈𝐵_𝑟,0<𝑥₃<ℎ.(4.3)

Lemma 4.2. Let be 𝑒_1,0,𝑒_2,0∈𝒮′(ℝ²) and supp𝑒_1,0,supp𝑒_2,0⊂Ω. Let 𝑒_𝑗,𝑏_𝑗∶ℋ→ℂ, 𝑗=1,2,3, be the diffraction solution given in Theorem 2.1. Let be 𝑅′>0 with Ω⊂𝐵_𝑅 and let be 𝑅>𝑅′ and 𝐻>0. Then one has the following equivalences and implications.
(a)𝑒_𝑗∈𝐿²(𝑍_𝑅,𝐻)⇔𝑒_𝑗∈𝐿²(ℝ²×(0,𝐻))for𝑗∈{1,2}. (b)𝑏₃∈𝐿²(𝑍_𝑅,𝐻)⇔𝑏₃∈𝐿²(ℝ²×(0,𝐻)). (c)𝑒_𝑗∈𝐿²(ℝ²×(0,𝐻))⇔𝑒_𝑗,0∈𝐻^−1/2(Ω)for𝑗∈{1,2}. (d)𝑏₃∈𝐿²(ℝ²×(0,𝐻))⇔𝑏_3,0∈𝐻^−1/2(Ω). (e)𝑒_1,0,𝑒_2,0∈𝐻^−1/2(Ω)⇒𝑒₃∈𝐿²(𝑍_𝑅,𝐻). (f)𝑒_1,0,𝑒_2,0,∇⋅^−𝑒2,0_𝑒1,0∈𝐻^−1/2(Ω)⇒𝑏₁,𝑏₂∈𝐿²(𝑍_𝑅,𝐻).

Proof of Theorem 4.1. Let there be given 𝑒_1,0,𝑒_2,0∈𝒮^(ℝ²) with support in Ωand let 𝑒_𝑗,𝑏_𝑗, 𝑗=1,2,3, be defined as in Theorem 2.1.
Firstly, we assume that the diffraction solution 𝑒_𝑗,𝑏_𝑗, 𝑗=1,2,3, has local finite energy and show the validity of (4.1). Since the diffraction solution has local finite energy especially we have 𝑒₁,𝑒₂,𝑏₃∈𝐿²(𝑍_𝑅,𝐻). From the parts (a) and (b) of Lemma 4.2 we find that 𝑒₁,𝑒₂,𝑏₃∈𝐿²(ℝ²×(0,𝐻)). From the parts (c) and (d) of the lemma we thus obtain 𝑒_1,0,𝑒_2,0,𝑏_3,0∈𝐻^−1/2(Ω). Now the validity of (4.1) follows from𝑏_3,0=𝑖𝑘∇⋅⎛⎜⎜⎝−𝑒𝑒^2,0_1,0⎞⎟⎟⎠(4.4) (cf. (3.9)).
For proving the other direction, we assume that relation (4.1) is valid. Because of (4.4) from the parts (c) and (d) of Lemma 4.2 it follows that 𝑒₁,𝑒₂,𝑏₃∈𝐿²(ℝ²×(0,𝐻)). Regarding the other three field components, from the parts (e) and (f) of the lemma we obtain 𝑒₃,𝑏₁,𝑏₂∈𝐿²(𝑍_𝑅,𝐻). Since 𝑅>𝑅′ and 𝐻>0 can be chosen arbitrarily large, the diffraction solution 𝑒_𝑗,𝑏_𝑗, 𝑗=1,2,3, has local finite energy.

We have yet to prove Lemma 4.2.

Proof of Lemma 4.2. Let the assumptions of Lemma 4.2 be fulfilled.
Proof of Part (a)
Let be 𝑗∈{1,2}. Obviously we have only to show the validity of the implication𝑒_𝑗∈𝐿²𝑍_𝑅,𝐻⟹𝑒_𝑗∈𝐿²ℝ²×(0,𝐻).(4.5)
We set 𝜀=(𝑅−𝑅′)/2 and 𝑅^=𝑅′+𝜀. For 𝜑∈[0,2𝜋) we consider the rotation matrix 𝐴_𝜑=⎛⎜⎜⎝csoins𝜑𝜑−csoisn𝜑𝜑⎞⎟⎟⎠(4.6) and define the function 𝑒^𝑗(𝜑)∶ℋ→ℂby 𝑒^𝑗(𝜑)(𝑥,𝑥₃)=𝑒_𝑗(𝐴_𝜑𝑥,𝑥₃)for 𝑥∈ℝ² and 𝑥₃>0. We show that 𝑒^𝑗(𝜑)∈𝐿²𝑅^,∞×ℝ×(0,𝐻)for𝜑∈[0,2𝜋),(4.7) regardless of whether 𝑒_𝑗∈𝐿²(𝑍_𝑅,𝐻) or not.
From (4.7) we have 𝑒_𝑗∈𝐿²𝐴^𝜑𝑅^′,∞×ℝ×(0,𝐻)for𝜑∈[0,2𝜋),(4.8) where 𝐴_𝜑((𝑅^′,∞)×ℝ) is the image of the set (𝑅^,∞)×ℝunder the linear mapping 𝐴_𝜑, that is, the set (𝑅^,∞)×ℝrotated by the angle 𝜑.
Since 𝑅^<𝑅, by considering a sufficient number of angles 𝜑, from (4.8) we find that there is a polygon 𝑃⊂𝐵_𝑅 with 𝑒_𝑗∈𝐿²((ℝ²⧵𝑃)×(0,𝐻)). Thus 𝑒_𝑗∈𝐿²ℝ²×(0,𝐻)⧵𝑍_𝑅,𝐻,(4.9) and therefore (4.5) holds true.
It remains to prove (4.7). To this end we use a contour integration technique which one of the authors had already used earlier in the treatment of diffraction problems [6, 7].
From the representation of 𝑒_𝑗 given in Theorem 2.1 we conclude that 𝑒^𝑗(𝜑)𝑥₁,𝑥₂,𝑥₃=12𝜋_ℝ𝛾_𝜑(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁𝑑𝜉₂,𝑥₁,𝑥₂,𝑥₃∈ℋ,(4.10) where the function 𝛾_𝜑 is given by 𝛾_𝜑(𝜉)=𝑒̂_𝑗,0𝐴_𝜑𝜉.(4.11) By assumption, it holds that supp𝑒_𝑗,0⊂Ω⊂𝐵_𝑅^′. Thus, from the Paley-Wiener theorem and the fact that 𝐴_𝜑 is an orthogonal matrix it follows that ||𝛾_𝜑(𝜉)||≤𝑐1+||𝜉||²^𝑁/2𝑒^{𝑅′|Im𝜉|},𝜉∈ℂ²,(4.12) for some constants 𝑐>0 and 𝑁∈ℕ₀.
We split the inner integral in (4.10) into one integral over the interval (−∞,0) and one over (0,∞). Firstly we consider the integral over (−∞,0).
For 𝑟>0 we define the curve 𝜂_𝑟 by 𝜂_𝑟=𝑟𝑒^𝑖𝛼∣𝜋2≤𝛼≤𝜋.(4.13) Let Γ_𝑟 be the closed contour consisting of the parts {𝑖𝑡∣0≤𝑡<𝑟}, 𝜂_𝑟 and (−𝑟,0). By Cauchy’s theorem we have _Γ𝑟𝛾_𝜑(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁=0,(4.14) here for nonreal values of 𝜉₁ the function 𝑚 is defined by analytic continuation of the function 𝜉₁↦𝑚(𝜉₁,𝜉₂), where 𝜉₁<0 and |𝜉|≠𝑘, and ⟨𝑥,𝜉⟩=𝑥₁𝜉₁+𝑥₂𝜉₂.
Because of (4.12), for 𝑥₁>𝑅′ it holds that lim_𝑟→∞_𝜂𝑟𝛾_𝜑(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁=0.(4.15) From the last two equations we obtain ^0−∞𝛾_𝜑(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁=−^0𝑖∞𝛾_𝜑(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁,𝑥₁>𝑅′.(4.16)
Now we treat the integral over (0,∞). Let √^∗ be the principal branch of the square root function (branch cut (−∞,0)). For 𝑧<0 we set √𝑧^∗=−𝑖√|𝑧|,thatis,√^∗ is thought to be defined on the negative real half-axis by continuation from the lower complex half-plane. In what follows, the function 𝑚^∗ is defined by 𝑚^∗(𝜉)=𝑘²−𝜉²¹−𝜉^22∗. Regarding the integration variable 𝜉₂ we distinguish the two cases |𝜉₂|<𝑘 and |𝜉₂|>𝑘.
Firstly we assume that |𝜉₂|<𝑘. We have ^∞0𝛾_𝜑(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁=^{𝑘2−𝜉22}₀𝛾_𝜑(𝜉)𝑒^{𝑖𝑥3𝑚∗(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁+^∞_{𝑘2−𝜉22}𝛾_𝜑(𝜉)𝑒^{−𝑖𝑥3𝑚∗(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁=^∞0𝛾_𝜑(𝜉)𝑒^{−𝑖𝑥3𝑚∗(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁+2𝑖^{𝑘2−𝜉22}₀𝛾_𝜑(𝜉)sin𝑥₃𝑚^∗(𝜉)𝑒^{𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁.(4.17) By contour integration analogous as in the derivation of (4.16), applied to the next-to-last integral, eventually we get ^∞0𝛾_𝜑(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁=^0𝑖∞𝛾_𝜑(𝜉)𝑒^{−𝑖𝑥3𝑚∗(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁+2𝑖^{𝑘2−𝜉22}₀𝛾_𝜑(𝜉)sin𝑥₃𝑚^∗(𝜉)𝑒^{𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁for||𝜉₂||<𝑘,𝑥₁>𝑅^.(4.18)
In the case |𝜉₂|>𝑘 it holds that 𝑚(𝜉)=−𝑚^∗(𝜉)for 𝜉₁>0, and contour integration yields ^∞0𝛾_𝜑(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁=^0𝑖∞𝛾_𝜑(𝜉)𝑒^{−𝑖𝑥3𝑚∗(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁for||𝜉₂||>𝑘,𝑥₁>𝑅^.(4.19)
In the following, for 𝜉₁∈ℝor 𝜉₁∈{𝑖𝑡|0<𝑡<∞} we set 𝑚(𝜉)=𝑘²−𝜉²¹−𝜉²², where the square root is chosen in the way that Re𝑚(𝜉)≥0and Im𝑚(𝜉)≥0. This definition is in accordance with (2.15). In this notation, from (4.18) and (4.19) we obtain _ℝ^∞0𝛾_𝜑(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁𝑑𝜉₂=^𝑘−𝑘^0𝑖∞𝛾_𝜑(𝜉)𝑒^{−𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁𝑑𝜉₂+2𝑖^𝑘−𝑘^{𝑘2−𝜉22}₀𝛾_𝜑(𝜉)sin𝑥₃𝑚(𝜉)𝑒^{𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁𝑑𝜉₂+_{[]ℝ⧵−𝑘,𝑘}^{𝑖𝜉22−𝑘2}₀𝛾_𝜑(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁𝑑𝜉₂+_{[]ℝ⧵−𝑘,𝑘}^{𝑖𝑖∞}_{𝜉22−𝑘2}𝛾_𝜑(𝜉)𝑒^{−𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁𝑑𝜉₂for𝑥₁>𝑅^.(4.20) Together with (4.16) we thus find _ℝ𝛾_𝜑(𝜉)𝑒^{𝑖𝑥3𝑚(𝜉)+𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁𝑑𝜉₂=2𝑖^𝑘−𝑘^{𝑘2−𝜉22}₀𝛾_𝜑(𝜉)sin𝑥₃𝑚(𝜉)𝑒^{𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁𝑑𝜉₂−2𝑖^𝑘−𝑘^0𝑖∞𝛾_𝜑(𝜉)sin𝑥₃𝑚(𝜉)𝑒^{𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁𝑑𝜉₂−2𝑖_{[]ℝ⧵−𝑘,𝑘}^{𝑖𝑖∞}_{𝜉22−𝑘2}𝛾_𝜑(𝜉)sin𝑥₃𝑚(𝜉)𝑒^{𝑖⟨𝑥,𝜉⟩}𝑑𝜉₁𝑑𝜉₂for𝑥₁>𝑅^.(4.21)
Now we show that each of the three addends in the right-hand side of (4.21), considered as a function of 𝑥=(𝑥,𝑥₃), lies in 𝐿²((𝑅^,∞)×ℝ×(0,𝐻)). Then, because of the representation (4.10), the stated relation (4.7) is proved.
We begin with the first addend. Let 𝐴∶={𝜉∈ℝ²||𝜉|≤𝑘,𝜉₁≥0}, let 𝜒_𝐴 be the characteristic function of the set 𝐴 and let ̌‌ denote the inverse Fourier transform. There is a constant 𝑐>0, which does not depend on 𝑥₃∈(0,𝐻), such that ‖‖𝜒_𝐴𝛾_𝜑sin𝑥₃𝑚(⋅)‖‖^̌‌_𝐿2(ℝ2)=‖‖𝜒_𝐴𝛾_𝜑sin𝑥₃𝑚(⋅)‖‖_𝐿2(ℝ2)≤𝑐.(4.22) From this it follows that the first addend in the right-hand side of (4.21) is quadratically integrable over the set ℝ²×(0,𝐻) and thus especially quadratically integrable over (𝑅^,∞)×ℝ×(0,𝐻).
Now we come to the second addend. We define 𝑅 by 𝑅=(𝑅′+𝑅^)/2=𝑅′+(𝜀/2). From (4.12) we find that there is a constant 𝑐₁>0 with ||𝛾_𝜑𝑖𝑡,𝜉₂||≤𝑐₁𝑒𝑅^𝑡for𝜉₂∈[−𝑘,𝑘],𝑡≥0.(4.23) Since for 𝜉₂∈[−𝑘,𝑘] and 𝑡≥0 the quantity 𝑚(𝑖𝑡,𝜉₂) is a real number and therefore |sin(𝑥₃𝑚(𝑖𝑡,𝜉₂))|≤1 for 𝑥₃∈(0,𝐻), we conclude that ||||^∞0𝛾≤_𝜑𝑐₁𝑖𝑡,𝜉₂sin𝑥₃𝑚𝑖𝑡,𝜉₂𝑒^−𝑡𝑥1𝑑𝑡||||^∞0𝑒⁽𝑅_{−𝑥1)𝑡}𝑑𝑡=𝑐¹𝑥₁−𝑅for𝜉²∈[−𝑘,𝑘],𝑥₃∈(0,𝐻),𝑥₁>𝑅.(4.24) Thus we have ^𝑘−𝑘||||^∞0𝛾_𝜑𝑖𝑡,𝜉₂sin𝑥₃𝑚𝑖𝑡,𝜉₂𝑒^−𝑡𝑥1𝑑𝑡||||²𝑑𝜉₂≤2𝑘𝑐²¹𝑥¹−𝑅²for𝑥₃∈(0,𝐻),𝑥₁>𝑅.(4.25) Because the inverse Fourier transform (here with regard to the variable 𝜉₂) is isometric on 𝐿², we see that _ℝ||||^𝑘−𝑘^∞0𝛾_𝜑𝑖𝑡,𝜉₂sin𝑥₃𝑚𝑖𝑡,𝜉₂𝑒^−𝑡𝑥1𝑑𝑡𝑒^{𝑖𝜉2𝑥2}𝑑𝜉₂||||²𝑑𝑥₂≤4𝜋𝑘𝑐²¹𝑥¹−𝑅²for𝑥₃∈(0,𝐻),𝑥₁>𝑅.(4.26) It follows that ^𝐻0^{∞𝑅′′}_ℝ||||^𝑘−𝑘^∞0𝛾_𝜑𝑖𝑡,𝜉₂sin𝑥₃𝑚𝑖𝑡,𝜉₂𝑒^−𝑡𝑥1𝑑𝑡𝑒^{𝑖𝜉2𝑥2}𝑑𝜉₂||||²𝑑𝑥₂𝑑𝑥₁𝑑𝑥₃<∞.(4.27) The substitution 𝑡=−𝑖𝜉₁ in the innermost integral now shows that the second addend in the right-hand side of (4.21) indeed lies in 𝐿²((𝑅^,∞)×ℝ×(0,𝐻)).
Regarding the third addend, from (4.12) we find |||||_{∞𝜉22−𝑘2}≤𝑐𝛾_𝜑𝑖𝑡,_{∞𝜉22−𝑘2}𝜉₂sin𝑥₃𝑚𝑖𝑡,𝜉₂𝑒^−𝑡𝑥1𝑑𝑡|||||1+𝑡²+𝜉²²^𝑁/2𝑒^{(𝑅′−𝑥1)𝑡}𝑑𝑡for𝜉₂∈ℝ⧵[−𝑘,𝑘],𝑥₃∈(0,𝐻),𝑥₁>𝑅^.(4.28) Now let be 𝑅=𝑅′+(𝜀/2) as above. By reason of 1+𝑡²+𝜉²²^𝑁/2≤1+𝑡²^𝑁/21+𝜉²²^𝑁/2,(4.29) we conclude that there are constants 𝑐₂,𝑐₃>0such that |||||_{∞𝜉22−𝑘2}≤𝑐₂𝛾_𝜑𝑖𝑡,𝜉₂sin𝑥₃𝑚𝑖𝑡,𝜉₂𝑒^−𝑡𝑥1𝑑𝑡|||||1+𝜉²²^𝑁/2^∞_{𝜉22−𝑘2}𝑒⁽𝑅_{−𝑥1)𝑡}=𝑐𝑑𝑡₂𝑥₁−𝑅1+𝜉²²^𝑁/2𝑒⁽𝑅_{−𝑥1)𝜉}²²_−𝑘²≤𝑐³𝑥₁−𝑅𝑒^{−(𝜀/4)𝜉22−𝑘2}for𝜉₂∈ℝ⧵[−𝑘,𝑘],𝑥₃∈(0,𝐻),𝑥₁>𝑅^′.(4.30) This estimate corresponds to formula (4.24), used in the case of the second addend. Continuing as in this latter case, it is seen that the third addend in the right-hand side of (4.21) lies in 𝐿²((𝑅^,∞)×ℝ×(0,𝐻)) too.Proof of Part (b)
The preceding proof of part (a) is based on the fact that for 𝑗∈{1,2} it holds that 𝑒_𝑗̂𝜉,𝑥₃=𝑒su^{𝑖𝑥3𝑚(𝜉)}𝑒p̂p_𝑗,0𝑒(𝜉_𝑗,0)for𝜉∈ℝ²,𝑥₃>0,(4.31)⊂Ω.(4.32) The relation (4.31), which is equivalent to the representation of 𝑒_𝑗 given in Theorem 2.1, has led to (4.10).
Since the third magnetic component 𝑏₃ fulfills conditions which are analogous to (4.31) and (4.32), the proof of part (b) of the lemma is completely analogous to the proof of part (a). The condition for 𝑏₃ that is analogous to (4.31) is given in (2.19); note that (2.19) also holds for |𝜉|<𝑘. The condition supp𝑏_3,0⊂Ωis a direct consequence of 𝑏_3,0=𝑖𝑘∇⋅⎛⎜⎜⎝−𝑒𝑒^2,0_1,0⎞⎟⎟⎠,(4.33) following from the second equation in (2.18), and supp𝑒_1,0,supp𝑒_2,0⊂Ω.Proof of Parts (c) and (d)
These parts of the lemma can be proved along the lines of the proof of Theorem 3.1. For example, with respect to 𝑒_𝑗, 𝑗∈{1,2}, one has ^𝐻0_ℝ2||𝑒_𝑗𝑥,𝑥₃||²𝑑𝑥𝑑𝑥₃=^𝐻0_ℝ2||𝑒_𝑗̂𝜉,𝑥₃||²𝑑𝜉𝑑𝑥₃=_ℝ2||𝑒̂_𝑗,0(𝜉)||²^𝐻0=𝐻_{||𝜉||<𝑘}||𝑒^{𝑖𝑥3𝑚(𝜉)}||²𝑑𝑥₃𝑑𝜉||𝑒̂_𝑗,0(𝜉)||²𝑑𝜉+_{||𝜉||>𝑘}||𝑒̂_𝑗,0(𝜉)||²1−𝑒^{√−2𝐻|𝜉|2−𝑘2}2||𝜉||²−𝑘²𝑑𝜉.(4.34) The condition (3.12), used in the proof of Theorem 3.1 because the factor 𝑚(𝜉)⁻¹ is not locally square integrable, is not needed for the special field components 𝑒₁, 𝑒₂, and 𝑏₃.Proof of Part (e)
From the first equation in (2.18) and the first equation in (2.19) we obtain𝑒₃̂𝜉,𝑥₃=−1𝑚(𝜉)𝜉¹𝑒₁̂𝜉,𝑥₃+𝜉₂𝑒₂̂𝜉,𝑥₃,𝜉∈ℝ²,𝑥₃≥0.(4.35) Thus for (𝑥,𝑥₃)∈ℋ we have 𝑒₃𝑥,𝑥₃=−12𝜋_ℝ²1𝑚(𝜉)𝜉¹𝑒₁̂𝜉,𝑥₃+𝜉₂𝑒₂̂𝜉,𝑥₃𝑒^{𝑖⟨𝑥,𝜉⟩}𝑑𝜉.(4.36) Now we define the function 𝑚₀∶ℝ²→ℂby 𝑚₀(𝜉)=𝑖𝑘²+||𝜉||²(4.37) and the function 𝑒³⁽⁰⁾∶ℋ→ℂby 𝑒³⁽⁰⁾𝑥,𝑥₃=−12𝜋_ℝ²1𝑚₀(𝜉)𝜉¹𝑒₁̂𝜉,𝑥₃+𝜉₂𝑒₂̂𝜉,𝑥₃𝑒^{𝑖⟨𝑥,𝜉⟩}𝑑𝜉.(4.38) Using (4.36), we find ||𝑒³⁽⁰⁾𝑥,𝑥₃−𝑒₃𝑥,𝑥₃||≤12𝜋²_𝑗=1𝐼^𝑗(1)𝑥₃+𝐼^𝑗(2)𝑥₃,(4.39) with 𝐼^𝑗(1)𝑥₃=_{||𝜉||<2𝑘}||||1𝑚₀(𝜉)−𝑚1(𝜉)||||||𝜉^𝑗𝐼||||𝑒_𝑗̂𝜉,𝑥₃||𝑑𝜉,𝑗=1,2,_𝑗(2)𝑥₃=_|𝜉|>2𝑘||||1𝑚₀(𝜉)−𝑚1(𝜉)||||||𝜉^𝑗||||𝑒_𝑗̂𝜉,𝑥₃||𝑑𝜉,𝑗=1,2.(4.40) From (4.31) we see that there is a constant 𝑐₁>0 such that ||𝐼^𝑗(1)𝑥₃||≤𝑐₁for𝑗∈{1,2},𝑥₃>0.(4.41) For |𝜉|>2𝑘 it holds that ||||1𝑚₀(𝜉)−𝑚1(𝜉)||||=𝑂||𝜉1||³.(4.42) Therefore with some constant 𝑐₂>0 we have for 𝑗∈{1,2} and 𝑥₃>0||𝐼^𝑗(2)𝑥₃||≤𝑐₂_{||𝜉||>2𝑘}1||𝜉||^3/2|||||1||𝜉||^1/2𝑒_𝑗̂𝜉,𝑥₃|||||𝑑𝜉.(4.43) From this we get from the Cauchy-Schwarz inequality that ||𝐼^𝑗(2)𝑥₃||≤𝑐₂_{||𝜉||>2𝑘}1||𝜉||³𝑑𝜉^1/2_{||𝜉||>2𝑘}1||𝜉||||𝑒^𝑗̂(𝜉,𝑥₃)|