# On Energy Conditions for Electromagnetic Diffraction by Apertures

Authors: Norbert Gorenflo; Matthias Kunik
Journal: ISRN Mathematical Physics
 (2012)

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## 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.


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## Body

## 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 Theorem4.1. 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 wavenumberk>0 satisfy the first-order system of Maxwell-Helmholtz equations(2.1)ikE̲*(x̲)+∇×B̲*(x̲)=0̲,-ikB̲*(x̲)+∇×E̲*(x̲)=0̲.
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 ej,bj, j=1,2,3,(2.2)(e1(x,x3)e2(x,x3)e3(x,x3))=E̲*(x,x3),(b1(x,x3)b2(x,x3)b3(x,x3))=B̲*(x,x3),
consists of functions defined in the upper half-space(2.3)H:={x̲=(x,x3)∈R3∣x∈R2,x3>0}.The diffraction problem is considered for an open bounded aperture(2.4)Ω⊂{(x,0)∣x∈R2}
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 screenΩc:=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(2.5)(e1,0(x)e2,0(x)e3,0(x))=lim⁡x3↓0E̲*(x,x3),(b1,0(x)b2,0(x)b3,0(x))=lim⁡x3↓0B̲*(x,x3).The screenΩc is assumed to be an ideal conducting wall. This implies the physical boundary conditions(2.6)e1,0(x)=e2,0(x)=0∀x∈Ωc.
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 againx̲=(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 transform(2.7)û(ξ1,ξ2,x3)=12π∬R2u(x1,x2,x3)e-i(ξ1x1+ξ2x2)dx1dx2,ξ1,ξ2∈R.
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 have(2.8)ik(ê1ê2ê3)+(iξ2b̂3-ddx3b̂2ddx3b̂1-iξ1b̂3iξ1b̂2-iξ2b̂1)=(000),-ik(b̂1b̂2b̂3)+(iξ2ê3-ddx3ê2ddx3ê1-iξ1ê3iξ1ê2-iξ2ê1)=(000).
Here we have replaced the partial derivative with respect to x3 by the ordinary derivative d/dx3.We define(2.9)C:=1k(∂2∂x1∂x2-(k2+∂2∂x12)k2+∂2∂x22-∂2∂x1∂x2).
With the two-dimensional Laplace operator Δ we have(2.10)C2=-(k2+Δ)(1001).
For the action of C in the Fourier domain we obtain multiplication by the matrix(2.11)Ĉ(ξ):=1k(-ξ1ξ2-(k2-ξ12)k2-ξ22ξ1ξ2),ξ=(ξ1,ξ2)∈R2.
We replace (2.8) with the ordinary differential equations(2.12)ddx3(ê1ê2)=-iĈ(b̂1b̂2),ddx3(b̂1b̂2)=iĈ(ê1ê2),
and the two algebraic conditions(2.13)ê3=1k(ξ2b̂1-ξ1b̂2),b̂3=1k(ξ1ê2-ξ2ê1).
For all fixed ξ∈ℝ2 we supplement the system of differential equations (2.12) by the initial conditions(2.14)êj,0(ξ):=êj(ξ,0),b̂j,0(ξ):=b̂j(ξ,0),j=1,2,
and put
(2.15)m(ξ):={k2-|ξ|2,|ξ|≤k,i|ξ|2-k2,|ξ|>k.
Then the general solution of the homogeneous linear system (2.12) is(2.16)(ê1(ξ,x3)ê2(ξ,x3)b̂1(ξ,x3)b̂2(ξ,x3))=(cos⁡(m(ξ)x3)E-isin⁡(m(ξ)x3)m(ξ)Ĉ(ξ)isin⁡(m(ξ)x3)m(ξ)Ĉ(ξ)cos⁡(m(ξ)x3)E)(ê1,0(ξ)ê2,0(ξ)b̂1,0(ξ)b̂2,0(ξ))
with the 2×2 unit matrix E.But the termscos(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:(2.17)m⋅(b̂1,0b̂2,0)=Ĉ(ê1,0ê2,0).
By using (2.17), for x3↓0 and |ξ|>k we can replace (2.13) with(2.18)ê3,0=-1m(ξ1ê1,0+ξ2ê2,0),b̂3,0=-1k(ξ2ê1,0-ξ1ê2,0).
From the general solution and (2.17), (2.18) we obtain the following decay conditions for x3>0:(2.19)êj(ξ,x3)=eix3m(ξ)êj,0(ξ),b̂j(ξ,x3)=eix3m(ξ)b̂j,0(ξ),j=1,2,3.
With x=(x1,x2)∈ℝ2, x̲=(x,x3)∈ℝ3 and x3>0 there holds the important and well-known Sommerfeld-Weyl integral representation(2.20)Fk(x,x3):=eik|x̲||x̲|=i2π∬R2eix3m(ξ)+i〈x,ξ〉m(ξ)dξ.
The left-hand side in (2.20) is the singular solution of the three-dimensional Helmholtz equation (Δ+k2)Fk=-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.Distributionsu∈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 êj(·,x3) and b̂j(·,x3).Theorem 2.1.
Let there be givene1,0,e2,0∈𝒮′(ℝ2) with support in the bounded region Ω¯. Then the following functions ej,bj:ℋ→ℂ constitute a C∞-solution of the Maxwell-Helmholtz system (2.1) in the upper half-space ℋ, j=1,2,3;
(2.21)e1(x,x3)=12π∬R2ê1,0(ξ)eix3m(ξ)+i〈x,ξ〉dξ,e2(x,x3)=12π∬R2ê2,0(ξ)eix3m(ξ)+i〈x,ξ〉dξ,b1(x,x3)=-12πk∬R2ξ1ξ2ê1,0(ξ)+(k2-ξ12)ê2,0(ξ)m(ξ)eix3m(ξ)+i〈x,ξ〉dξ,b2(x,x3)=12πk∬R2(k2-ξ22)ê1,0(ξ)+ξ1ξ2ê2,0(ξ)m(ξ)eix3m(ξ)+i〈x,ξ〉dξ,e3(x,x3)=-12π∬R2ξ1ê1,0(ξ)+ξ2ê2,0(ξ)m(ξ)eix3m(ξ)+i〈x,ξ〉dξ,b3(x,x3)=-12πk∬R2[ξ2ê1,0(ξ)-ξ1ê2,0(ξ)]eix3m(ξ)+i〈x,ξ〉dξ.Proof.
The calculation of the partial derivatives ofej and bj can be interchanged with integration. This can be used to check the Maxwell-Helmholtz equations independent from the previous representations of the Fourier-transforms êj and b̂j in terms of ê1,0 and ê2,0.Definition 2.2.
The electromagnetic field in the half-spacex3>0 behind the screen is completely determined by the electric boundary components e1,0, e2,0. We call ej,bj:ℋ→ℂ, j=1,2,3, the half-space solution determined by the boundary distributions e1,0,e2,0∈𝒮′(ℝ2) with compact support in Ω¯.Remark 2.3.
Assume thate1,0,e2,0 are smooth functions with compact support in Ω¯ and that E̲*, B̲* 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∈ℝ2(2.22)lim⁡x3↓0⁡ej(x,x3)=ej,0(x),j=1,2.Remark 2.4.
Assume again thate1,0,e2,0 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
(2.23)b1(x,x3)=-i2πk∂2∂x1∂x2∬Ωe1,0(y)Fk(x-y,x3)dy+i2πk(k2+∂2∂x12)∬Ωe2,0(y)Fk(x-y,x3)dy,b2(x,x3)=-i2πk(k2+∂2∂x22)∬Ωe1,0(y)Fk(x-y,x3)dy+i2πk∂2∂x1∂x2∬Ωe2,0(y)Fk(x-y,x3)dy.
Equations (2.23) involve the boundary conditions (2.6) for the electric field components on the ideal conducting plane screen Ωc. In order to obtain a coupled system of boundary integro-differential equations we pass for x∈ℝ2 to the limit x3↓0 and obtain
(2.24)b1,0(x)=-i2πk∂2∂x1∂x2∬Ωe1,0(y)eik|x-y||x-y|dy+i2πk(k2+∂2∂x12)∬Ωe2,0(y)eik|x-y||x-y|dy,b2,0(x)=-i2πk(k2+∂2∂x22)∬Ωe1,0(y)eik|x-y||x-y|dy+i2πk∂2∂x1∂x2∬Ωe2,0(y)eik|x-y||x-y|dy.
In general the electric boundary fields e1,0 and e2,0 are unknown distributions with compact support in Ω¯, whereas b1,0 and b2,0 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.
Letej,bj:ℋ→ℂ, j=1,2,3, be the half-space solution determined by the boundary distributions e1,0,e2,0∈𝒮′(ℝ2) with compact support in Ω¯.(a)
The solution is calledphysical admissible if and only if it satisfies the local energy condition(2.25)12∑j=13∭G(|ej(x̲)|2+|bj(x̲)|2)dx̲<∞for every bounded domain G⊂ℋ.(b)
The solution satisfies the strongerglobal energy condition if and only if
(2.26)12∑j=13∭Lh(|ej(x̲)|2+|bj(x̲)|2)dx̲<∞for some h>0, and therewith for all h>0, with the layerℒh:={(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 fieldse1,0 and e2,0.


## 3. The Global Energy Condition

Throughout the rest of this paper let the open apertureΩ:=⋃j=1κΩj⊂ℝ2 be a finite union of nonempty bounded Lipschitz domains Ωj in the screen plane, such that the compact sets Ω¯j are pairwise disjoint.ByHs(ℝ2), s∈ℝ, we denote the Sobolev space of tempered distributions h, for which the Fourier transform ĥ is locally integrable with(3.1)‖h‖Hs(R2):=(∬R2|ĥ(ξ)|2(1+|ξ|2)sdξ)1/2<∞
(cf. [3], Chapter  8.8). ∥·∥Hs(ℝ2) is the norm on the Banach space Hs(ℝ2).Fors∈ℝ the Sobolev space H̃s(Ω) is given by(3.2)H̃s(Ω)={h∈Hs(R2)∣supp⁡h⊂Ω¯}.
Here supph denotes the support of the distribution h in the compact set Ω¯. The space H̃s(Ω) is equipped with the norm of Hs(ℝ2), which makes it into a Banach space (cf. [4, Chapter  3]). The Lipschitz property of ∂Ω guarantees that H̃s(Ω) is the closure of 𝒟(Ω) in Hs(ℝ2). For a more general result see [4, Theorem  3.29].Theorem 3.1.
Letej,bj:ℋ→ℂ, j=1,2,3, be the half-space solution determined by the boundary distributions e1,0,e2,0∈𝒮′(ℝ2) with compact support in Ω¯. Then the diffraction solution has global finite energy if and only if(3.3)e1,0,e2,0,∇⋅(-e2,0e1,0)∈H̃-1/2(Ω)
and ê1,0(ξ)ξ1+ê2,0(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 fieldsêj,0,b̂j,0, j=1,2,3, the quantity
(3.4)W0(ξ):=12∑j=13(|êj,0(ξ)|2+|b̂j,0(ξ)|2).
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
(3.5)W0(ξ)={|ê1,0(ξ)|2+|ê2,0(ξ)|2+|ê3,0(ξ)|2,|ξ|<k,|ê3,0(ξ)|2+|b̂3,0(ξ)|2,|ξ|>k.
WithV1:={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>0(3.6)E(h):=12∑j=13∫0h∬R2(|ej(x,x3)|2+|bj(x,x3)|2)dxdx3=h∬V1W0(ξ)dξ+∬V21-e-2h|ξ|2-k22|ξ|2-k2W0(ξ)dξ.
Using for all ξ∈ℝ2 with |ξ|≠k the estimate
(3.7)12(|ê1,0(ξ)|2+|ê2,0(ξ)|2+|ê3,0(ξ)|2+|b̂3,0(ξ)|2)≤W0(ξ)≤|ê1,0(ξ)|2+|ê2,0(ξ)|2+|ê3,0(ξ)|2+|b̂3,0(ξ)|2,
and observing that we have uniformly in r>k for appropriate constants α,β>0(3.8)α1+r2≤1-e-2hr2-k22r2-k2≤β1+r2,
we conclude from (3.6) that ℰ(h) is finite if and only if
(3.9)e1,0,e2,0,e3,0,b3,0=ik∇⋅(-e2,0e1,0)∈H-1/2(R2).
Note that with e1,0 and e2,0 also ∇·(-e2,0,e1,0)T has support in Ω¯.
Assume first thatℰ(h)<∞ and that ê1,0(ξ)ξ1+ê2,0(ξ)ξ2≠0 for a certain ξ=(ξ1,ξ2)∈ℝ2 with |ξ|=k. Since e1,0,e2,0 have compact support, we conclude from the Paley-Wiener theorem that ê1,0,ê2,0:ℝ2→ℂ are smooth and especially continuous. Thus |ê1,0(ξ)ξ1+ê2,0(ξ)ξ2|2≥δ>0 in a bounded domain
(3.10)(ξ1,ξ2)=r(cos⁡φ,sin⁡φ)withφ1<φ<φ2,k<r<k+ɛ,
and hence |ê3,0|2 because of (2.18) is not integrable. Due to (3.1) this violates the necessary condition e3,0∈H-1/2(ℝ2) in (3.9).
For showing the other direction of the equivalence stated in the theorem we assume that(3.11)e1,0,e2,0,∇⋅(-e2,0e1,0)∈H̃-1/2(Ω),(3.12)ê1,0(ξ1,ξ2)ξ1+ê2,0(ξ1,ξ2)ξ2=0,
for all ξ1,ξ2∈ℝ with ξ12+ξ22=k2. In order to prove (3.9) it remains to show that e3,0∈H-1/2(ℝ2). Since e1,0, e2,0 have compact support in Ω¯, we obtain from the Paley-Wiener theorem that the Fourier transforms ê1,0, ê2,0 can be continuated to entire functions. We also denote these entire functions by ê1,0,ê2,0. We define the entire function f by
(3.13)f(z1,z2):=ê1,0(z1,z2)z1+ê2,0(z1,z2)z2.
Using (3.12) and (2.18), it follows for example from Theorem A.1 in the appendix that |ê3,0|2 is a locally integrable function.
From the first equation in (2.18) and the Cauchy-Schwarz inequality we obtain the estimate
(3.14)|ê3,0(ξ)|2≤|ξ|2|ξ|2-k2⋅(|ê1,0(ξ)|2+|ê2,0(ξ)|2),|ξ|>k.
Now e3,0∈H-1/2(ℝ2) follows from the assumption e1,0,e2,0∈H̃-1/2(Ω), because |e3,0|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 functionsu1,u2∈H̃1/2(Ω), in the following called boundary potential functions, satisfying the regularity condition
(3.15)∇⋅(-u2u1)∈H̃1/2(Ω).Define
(3.16)(e1,0e2,0):=C(u1u2).Then the corresponding electromagnetic field ej,bj, j=1,2,3, determined by e1,0 and e2,0 according to Theorem 2.1, has global finite energy.(b)
Lete1,0, e2,0 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 e1,0 and e2,0 can be represented by two boundary potential functions u1,u2∈H̃1/2(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(3.17)b1,0(x)=i2π(k2+Δ)∬Ωu1(y)eik|x-y||x-y|dy,b2,0(x)=i2π(k2+Δ)∬Ωu2(y)eik|x-y||x-y|dyon the whole screen plane ℝ2 with
(3.18)b1,0,b2,0,∇⋅(-b2,0b1,0)∈H-1/2(R2).Proof.
Part (a) follows in the Fourier domain by representingê1,0, ê2,0, ê3,0 and b̂3,0 in terms of û1 and û2 as
(3.19)ê1,0=-kû2-ξ1k(ξ2û1-ξ1û2),ê2,0=+kû1-ξ2k(ξ2û1-ξ1û2),ê3,0=-mk(ξ2û1-ξ1û2),b̂3,0=ξ1û1+ξ2û2.
These equations and the assumptions imply that e1,0,e2,0,b3,0∈H-1/2(ℝ2), and regarding |m(ξ)|≤α1+|ξ|2 uniformly in ξ∈ℝ2 for a certain constant α>0 also e3,0∈H-1/2(ℝ2). Since e1,0, e2,0 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 toe1,0,e2,0∈H̃-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
(3.20)v1(ξ1,ξ2)=+1kê2,0(ξ1,ξ2)-ξ2k⋅ê1,0(ξ)ξ1+ê2,0(ξ)ξ2ξ12+ξ22-k2,v2(ξ1,ξ2)=-1kê1,0(ξ1,ξ2)+ξ1k⋅ê1,0(ξ)ξ1+ê2,0(ξ)ξ2ξ12+ξ22-k2.
From (3.20) we get for j=1,2 that
(3.21)∬R2|vj(ξ)|2(1+|ξ|2)1/2dξ<∞.
Thus for j=1,2 the inverse Fourier transform uj of vj lies in H1/2(ℝ2).
Now we show thatu1 and u2 have their support in Ω¯ and hence, by reason of (3.21), u1,u2∈H̃1/2(Ω). To this aim we choose some R>0 such that
(3.22)Ω⊂BR:={x∈R2∣|x|<R}.
Since e1,0,e2,0 are supported in Ω¯, we have suppe1,0,suppe2,0⊂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 ê1,0, ê2,0, are of exponential type not larger than R.
Equations (3.20) can be resolved with respect to ê1,0, ê2,0. This gives (3.16). Together with (2.10) it follows that in the complement of Ω¯ there holds
(3.23)-(k2+Δ)(u1u2)=C2(u1u2)=C(e1,0e2,0)=(00).
Now we make use of the fact that the complement of Ω¯ is connected. Since suppuj⊂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, sincevj=ûj for j=1,2, from (3.20) we obtain forξ∈ℝ2(3.24)-ξ1û2+ξ2û1=-kξ1ê1,0+ξ2ê2,0ξ12+ξ22-k2.
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 û1, û2, b̂1,0, and b̂2,0.Remark 3.3.
Consider the Sobolev spacesHs(Ω) given for s∈ℝ by
(3.25)Hs(Ω)={F|Ω∣F∈Hs(R2)},
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.
Hs(Ω) is a Banach space with respect to the norm ∥·∥Hs(Ω) given by(3.26)‖f‖Hs(Ω)=inf⁡{‖F‖Hs(R2)∣F∈Hs(R2)is a continuation off}.
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
(3.27)b1,0,b2,0,∇⋅(-b2,0b1,0)∈H-1/2(Ω).
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 b1,0 and b2,0 are considered only in the aperture Ω, reflects the physical fact that b1,0 and b2,0 are only prescribed in Ω. Namely, b1,0 and b2,0 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 bee1,0,e2,0∈𝒮′(ℝ2) and suppe1,0,suppe2,0⊂Ω¯. Let ej,bj:ℋ→ℂ, j=1,2,3, be defined as in Theorem 2.1. Then the diffraction solution ej,bj, j=1,2,3, has local finite energy if and only if
(4.1)e1,0,e2,0,∇⋅(-e2,0e1,0)∈H̃-1/2(Ω).For proving Theorem4.1 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. Forr>0 we define the open ball Br by(4.2)Br:={x∈R2∣|x|<r}.
For r>0 and h>0 the open cylinder Zr,h is defined by(4.3)Zr,h:={x̲=(x,x3)∈R3∣x∈Br,0<x3<h}.Lemma 4.2.
Let bee1,0,e2,0∈𝒮′(ℝ2) and suppe1,0,suppe2,0⊂Ω¯. Let ej,bj:ℋ→ℂ, j=1,2,3, be the diffraction solution given in Theorem 2.1. Let be R′>0 with Ω⊂BR′ and let be R>R′ and H>0. Then one has the following equivalences and implications.
(a)
ej∈L2(ZR,H)⇔ej∈L2(ℝ2×(0,H))forj∈{1,2}.(b)
b3∈L2(ZR,H)⇔b3∈L2(ℝ2×(0,H)).(c)
ej∈L2(ℝ2×(0,H))⇔ej,0∈H̃-1/2(Ω)forj∈{1,2}.(d)
b3∈L2(ℝ2×(0,H))⇔b3,0∈H̃-1/2(Ω).(e)
e1,0,e2,0∈H̃-1/2(Ω)⇒e3∈L2(ZR,H).(f)
e1,0,e2,0,∇·(-e2,0e1,0)∈H̃-1/2(Ω)⇒b1,b2∈L2(ZR,H).Proof of Theorem4.1.
Let there be givene1,0,e2,0∈𝒮′(ℝ2) with support in Ω¯and let ej,bj, j=1,2,3, be defined as in Theorem 2.1.
Firstly, we assume that the diffraction solutionej,bj, 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(ZR,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 e1,0,e2,0,b3,0∈H̃-1/2(Ω). Now the validity of (4.1) follows from(4.4)b3,0=ik∇⋅(-e2,0e1,0)
(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(ZR,H). Since R>R′ and H>0 can be chosen arbitrarily large, the diffraction solution ej,bj, j=1,2,3, has local finite energy.We have yet to prove Lemma4.2.Proof of Lemma4.2.
Let the assumptions of Lemma4.2 be fulfilled.
Proof of Part (a)
Let bej∈{1,2}. Obviously we have only to show the validity of the implication(4.5)ej∈L2(ZR,H)⟹ej∈L2(R2×(0,H)).
We setɛ=(R-R′)/2 and R′′=R′+ɛ. For φ∈[0,2π) we consider the rotation matrix
(4.6)Aφ=(cos⁡φ-sin⁡φsin⁡φcos⁡φ)
and define the function ej(φ):ℋ→ℂby ej(φ)(x,x3)=ej(Aφx,x3)for x∈ℝ2 and x3>0. We show that
(4.7)ej(φ)∈L2((R′′,∞)×R×(0,H))forφ∈[0,2π),
regardless of whether ej∈L2(ZR,H) or not.
From (4.7) we have
(4.8)ej∈L2(Aφ((R′′,∞)×R)×(0,H))forφ∈[0,2π),
where Aφ((R′′,∞)×ℝ) is the image of the set (R′′,∞)×ℝunder the linear mapping Aφ, that is, the set (R′′,∞)×ℝrotated by the angle φ.
SinceR′′<R, by considering a sufficient number of angles φ, from (4.8) we find that there is a polygon P⊂BR with ej∈L2((ℝ2∖P)×(0,H)). Thus
(4.9)ej∈L2((R2×(0,H))∖ZR,H),
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 ofej given in Theorem 2.1 we conclude that
(4.10)ej(φ)(x1,x2,x3)=12π∬Rγφ(ξ)eix3m(ξ)+i〈x,ξ〉dξ1dξ2,(x1,x2,x3)∈H,
where the function γφ is given by
(4.11)γφ(ξ)=êj,0(Aφξ).
By assumption, it holds that suppej,0⊂Ω¯⊂B¯R′. Thus, from the Paley-Wiener theorem and the fact that Aφ is an orthogonal matrix it follows that
(4.12)|γφ(ξ)|≤c(1+|ξ|2)N/2eR′|Im⁡ξ|,ξ∈C2,
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).
Forr>0 we define the curve ηr by
(4.13)ηr={reiα∣π2≤α≤π}.
Let Γr be the closed contour consisting of the parts {it∣0≤t<r}, ηr and (-r,0). By Cauchy’s theorem we have
(4.14)∫Γrγφ(ξ)eix3m(ξ)+i〈x,ξ〉dξ1=0,
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
(4.15)lim⁡r→∞⁡∫ηrγφ(ξ)eix3m(ξ)+i〈x,ξ〉dξ1=0.
From the last two equations we obtain
(4.16)∫-∞0γφ(ξ)eix3m(ξ)+i〈x,ξ〉dξ1=-∫0i∞γφ(ξ)eix3m(ξ)+i〈x,ξ〉dξ1,x1>R′.
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-ξ12-ξ22*. Regarding the integration variable ξ2 we distinguish the two cases |ξ2|<k and |ξ2|>k.
Firstly we assume that|ξ2|<k. We have
(4.17)∫0∞γφ(ξ)eix3m(ξ)+i〈x,ξ〉dξ1=∫0k2-ξ22γφ(ξ)eix3m*(ξ)+i〈x,ξ〉dξ1+∫k2-ξ22∞γφ(ξ)e-ix3m*(ξ)+i〈x,ξ〉dξ1=∫0∞γφ(ξ)e-ix3m*(ξ)+i〈x,ξ〉dξ1+2i∫0k2-ξ22γφ(ξ)sin⁡(x3m*(ξ))ei〈x,ξ〉dξ1.
By contour integration analogous as in the derivation of (4.16), applied to the next-to-last integral, eventually we get
(4.18)∫0∞γφ(ξ)eix3m(ξ)+i〈x,ξ〉dξ1=∫0i∞γφ(ξ)e-ix3m*(ξ)+i〈x,ξ〉dξ1+2i∫0k2-ξ22γφ(ξ)sin⁡(x3m*(ξ))ei〈x,ξ〉dξ1for|ξ2|<k,x1>R′.
In the case|ξ2|>k it holds that m(ξ)=-m*(ξ)for ξ1>0, and contour integration yields
(4.19)∫0∞γφ(ξ)eix3m(ξ)+i〈x,ξ〉dξ1=∫0i∞γφ(ξ)e-ix3m*(ξ)+i〈x,ξ〉dξ1for|ξ2|>k,x1>R′.
In the following, forξ1∈ℝor ξ1∈{it|0<t<∞} we set m(ξ)=k2-ξ12-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
(4.20)∫R∫0∞γφ(ξ)eix3m(ξ)+i〈x,ξ〉dξ1dξ2=∫-kk∫0i∞γφ(ξ)e-ix3m(ξ)+i〈x,ξ〉dξ1dξ2+2i∫-kk∫0k2-ξ22γφ(ξ)sin⁡(x3m(ξ))ei〈x,ξ〉dξ1dξ2+∫R∖[-k,k]∫0iξ22-k2γφ(ξ)eix3m(ξ)+i〈x,ξ〉dξ1dξ2+∫R∖[-k,k]∫iξ22-k2i∞γφ(ξ)e-ix3m(ξ)+i〈x,ξ〉dξ1dξ2forx1>R′.
Together with (4.16) we thus find
(4.21)∬Rγφ(ξ)eix3m(ξ)+i〈x,ξ〉dξ1dξ2=2i∫-kk∫0k2-ξ22γφ(ξ)sin⁡(x3m(ξ))ei〈x,ξ〉dξ1dξ2-2i∫-kk∫0i∞γφ(ξ)sin⁡(x3m(ξ))ei〈x,ξ〉dξ1dξ2-2i∫R∖[-k,k]∫iξ22-k2i∞γφ(ξ)sin⁡(x3m(ξ))ei〈x,ξ〉dξ1dξ2forx1>R′.
Now we show that each of the three addends in the right-hand side of (4.21), considered as a function of x̲=(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. LetA:={ξ∈ℝ2||ξ|≤k,ξ1≥0}, let χA 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
(4.22)‖[χAγφsin⁡(x3m(⋅))]̌‖L2(R2)=‖χAγφsin⁡(x3m(⋅))‖L2(R2)≤c.
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 defineR̃ by R̃=(R′+R′′)/2=R′+(ɛ/2). From (4.12) we find that there is a constant c1>0 with
(4.23)|γφ(it,ξ2)|≤c1eR̃tforξ2∈[-k,k],t≥0.
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
(4.24)|∫0∞γφ(it,ξ2)sin⁡(x3m(it,ξ2))e-tx1dt|≤c1∫0∞e(R̃-x1)tdt=c1x1-R̃forξ2∈[-k,k],x3∈(0,H),x1>R̃.
Thus we have
(4.25)∫-kk|∫0∞γφ(it,ξ2)sin⁡(x3m(it,ξ2))e-tx1dt|2dξ2≤2kc12(x1-R̃)2forx3∈(0,H),x1>R̃.
Because the inverse Fourier transform (here with regard to the variable ξ2) is isometric on L2, we see that
(4.26)∫R|∫-kk∫0∞γφ(it,ξ2)sin⁡(x3m(it,ξ2))e-tx1dteiξ2x2dξ2|2dx2≤4πkc12(x1-R̃)2forx3∈(0,H),x1>R̃.
It follows that
(4.27)∫0H∫R′′∞∫R|∫-kk∫0∞γφ(it,ξ2)sin⁡(x3m(it,ξ2))e-tx1dteiξ2x2dξ2|2dx2dx1dx3<∞.
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
(4.28)|∫ξ22-k2∞γφ(it,ξ2)sin⁡(x3m(it,ξ2))e-tx1dt|≤c∫ξ22-k2∞(1+t2+ξ22)N/2e(R′-x1)tdtforξ2∈R∖[-k,k],x3∈(0,H),x1>R′.
Now let be R̃=R′+(ɛ/2) as above. By reason of
(4.29)(1+t2+ξ22)N/2≤(1+t2)N/2(1+ξ22)N/2,
we conclude that there are constants c2,c3>0such that
(4.30)|∫ξ22-k2∞γφ(it,ξ2)sin⁡(x3m(it,ξ2))e-tx1dt|≤c2(1+ξ22)N/2∫ξ22-k2∞e(R̃-x1)tdt=c2x1-R̃(1+ξ22)N/2e(R̃-x1)ξ22-k2≤c3x1-R̃e-(ɛ/4)ξ22-k2forξ2∈R∖[-k,k],x3∈(0,H),x1>R′′.
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 forj∈{1,2} it holds that
(4.31)êj(ξ,x3)=eix3m(ξ)êj,0(ξ)forξ∈R2,x3>0,(4.32)supp⁡ej,0⊂Ω¯.
The relation (4.31), which is equivalent to the representation of ej given in Theorem 2.1, has led to (4.10).
Since the third magnetic componentb3 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 suppb3,0⊂Ω¯is a direct consequence of
(4.33)b3,0=ik∇⋅(-e2,0e1,0),
following from the second equation in (2.18), and suppe1,0,suppe2,0⊂Ω¯.Proof of Parts  (c) and (d)
These parts of the lemma can be proved along the lines of the proof of Theorem3.1. For example, with respect to ej, j∈{1,2}, one has
(4.34)∫0H∬R2|ej(x,x3)|2dxdx3=∫0H∬R2|êj(ξ,x3)|2dξdx3=∬R2|êj,0(ξ)|2∫0H|eix3m(ξ)|2dx3dξ=H∬|ξ|<k|êj,0(ξ)|2dξ+∬|ξ|>k|êj,0(ξ)|21-e-2H|ξ|2-k22|ξ|2-k2dξ.
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 obtain(4.35)ê3(ξ,x3)=-1m(ξ)[ξ1ê1(ξ,x3)+ξ2ê2(ξ,x3)],ξ∈R2,x3≥0.
Thus for (x,x3)∈ℋ we have
(4.36)e3(x,x3)=-12π∬R21m(ξ)[ξ1ê1(ξ,x3)+ξ2ê2(ξ,x3)]ei〈x,ξ〉dξ.
Now we define the function m0:ℝ2→ℂby
(4.37)m0(ξ)=ik2+|ξ|2
and the function e3(0):ℋ→ℂby
(4.38)e3(0)(x,x3)=-12π∬R21m0(ξ)[ξ1ê1(ξ,x3)+ξ2ê2(ξ,x3)]ei〈x,ξ〉dξ.
Using (4.36), we find
(4.39)|e3(0)(x,x3)-e3(x,x3)|≤12π∑j=12(Ij(1)(x3)+Ij(2)(x3)),
with
(4.40)Ij(1)(x3)=∬|ξ|<2k|1m0(ξ)-1m(ξ)||ξj||êj(ξ,x3)|dξ,j=1,2,Ij(2)(x3)=∬|ξ|>2k|1m0(ξ)-1m(ξ)||ξj||êj(ξ,x3)|dξ,j=1,2.
From (4.31) we see that there is a constant c1>0 such that
(4.41)|Ij(1)(x3)|≤c1forj∈{1,2},x3>0.
For |ξ|>2k it holds that
(4.42)|1m0(ξ)-1m(ξ)|=O(1|ξ|3).
Therefore with some constant c2>0 we have for j∈{1,2} and x3>0(4.43)|Ij(2)(x3)|≤c2∬|ξ|>2k1|ξ|3/2|1|ξ|1/2êj(ξ,x3)|dξ.
From this we get from the Cauchy-Schwarz inequality that
(4.44)|Ij(2)(x3)|≤c2(∬|ξ|>2k1|ξ|3dξ)1/2(∬|ξ|>2k1|ξ||êj(ξ,x3)|2dξ)1/2.
Using the fact that the first integral in the right-hand side of the last expression is finite and using (4.31), we conclude that there is a c3>0 with
(4.45)|Ij(2)(x3)|≤c3(∬|ξ|>2k1|ξ||eix3m(ξ)|2|êj,0(ξ)|2dξ)1/2.
Since |eix3m(ξ)|≤1 for |ξ|>2k and x3>0, from the assumption e1,0,e2,0∈H̃-1/2(Ω) of part (e) of the lemma it now follows that there is a c4>0 with
(4.46)|Ij(2)(x3)|≤c4forj∈{1,2},x3>0.
From (4.39), (4.41), and (4.46) we obtain that there is a constant c5>0 such that
(4.47)|e3(0)(x,x3)-e3(x,x3)|≤c5for(x,x3)∈H.
Thus we have
(4.48)e3(0)-e3∈L2(ZR,H),
since ZR,H is a bounded set.
Now we show that alsoe3(0)∈L2(ZR,H). From this, together with (4.48), we get the assertion e3∈L2(ZR,H) of part (e) of the lemma.
We show thate3(0) actually is square integrable over the whole of ℝ2×(0,H). By definition of e3(0), it suffices to prove that
(4.49)∫0H∬R2|∬R2ξjm0(ξ)êj(ξ,x3)ei〈x,ξ〉dξ|2dxdx3<∞forj∈{1,2}.
By applying the Parseval formula with respect to the variables x and ξ and using Formula (4.31) we find that the last integral is equal to
(4.50)4π2∫0H∬R2ξj2k2+|ξ|2|eix3m(ξ)|2|êj,0(ξ)|2dξdx3.
Now we split the domain of integration with respect to ξ into the parts |ξ|≤k and |ξ|>k. Due to the continuity of the integrand we have
(4.51)∫0H∬|ξ|≤kξj2k2+|ξ|2|eix3m(ξ)|2|êj,0(ξ)|2dξdx3<∞.
The fact that also
(4.52)∫0H∬|ξ|>kξj2k2+|ξ|2|eix3m(ξ)|2|êj,0(ξ)|2dξdx3<∞
follows from the assumption e1,0,e2,0∈H̃-1/2(Ω) similar to the lines of the parts (c) and (d).Proof of Part  (f)
From the representations ofb1 and b2 in Theorem 2.1 together with (2.19) we see that for ξ∈ℝ2 and x3≥0 there holds(4.53)b̂1(ξ,x3)=-km(ξ)ê2(ξ,x3)+ξ1km(ξ)[ξ1ê2(ξ,x3)-ξ2ê1(ξ,x3)],b̂2(ξ,x3)=km(ξ)ê1(ξ,x3)+ξ2km(ξ)[ξ1ê2(ξ,x3)-ξ2ê1(ξ,x3)].
Now the proof of part (f) proceeds along the same lines as the proof of part (e) if one takes the last two equations as starting point, as we have used (4.35) as starting point for the proof of part (e). The assumption ∇·(-e2,0,e1,0)T∈H̃-1/2(Ω) of part (f) is needed because in the formulas (4.53) there occur the terms
(4.54)ξjkm(ξ)[ξ1ê2(ξ,x3)-ξ2ê1(ξ,x3)]=iξjkm(ξ)eix3m(ξ)[∇⋅(-e2,0e1,0)]̂(ξ).


## 5. Conclusions and Outlook

If we interpret the system (2.24) in the distributional sense, then we obtain a vectorial pseudodifferential operator A, acting on the electric boundary fields e1,0, e2,0 and resulting in the magnetic boundary fields b1,0, b2,0:(5.1)A(e1,0e2,0)=(b1,0b2,0).For the electromagnetic diffraction problemb1,0 and b2,0 are distributions which are only prescribed in the aperture Ω whereas e1,0 and e2,0 are unknown. Then the diffracted electromagnetic field is completely determined for x3>0, that is, behind the screen, by Theorem 2.1.We define the vectorial Sobolev-Hilbert spacesV and W by(5.2)V:={(v1v2)∣v1,v2,∇⋅(-v2v1)∈H̃-1/2(Ω)},‖(v1v2)‖V:=(‖v1‖H̃-1/2(Ω)2+‖v2‖H̃-1/2(Ω)2+‖∇⋅(-v2v1)‖H̃-1/2(Ω)2)1/2,W:={(w1w2)∣w1,w2,∇⋅(-w2w1)∈H-1/2(Ω)},‖(w1w2)‖W:=(‖w1‖H-1/2(Ω)2+‖w2‖H-1/2(Ω)2+‖∇⋅(-w2w1)‖H-1/2(Ω)2)1/2.
Now it is an interesting future task to prove that A:V→W is a homeomorphism. For this purpose we want to develop a purely “intrinsic” proof, related only to the boundary equations in the aperture and without making use of the time harmonic Maxwell equations (2.1). For the slit diffraction problem this approach has been realised by the authors in the paper [2].Note that(e1,0,e2,0)∈V is a necessary and sufficient condition for a diffraction solution with local finite energy, see part (a) of Definition 2.5 and Theorem 4.1.In the case of global finite energy in Definition2.5, part (b), we have decoupled the vectorial boundary equations (cf. Theorem 3.2). In the future it should also be examined how this result can be used to derive representations of the physical admissible solutions satisfying only the condition of local finite energy.



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