On Energy Conditions for Electromagnetic Diffraction by Apertures

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.


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.

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 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 Re k ≥ 0 and Im k ≥ 0. We assume that the electromagnetic field with components e j , b j , j 1, 2, 3, consists of functions defined in the upper half-space The diffraction problem is considered for an open bounded aperture in the screen plane x 3 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 : R 2 \Ω as subsets ISRN Mathematical Physics 3 of R 2 . For describing the whole electromagnetic field in terms of its boundary values, for the moment we assume that these are functions, given for x ∈ R 2 by The screen Ω c is assumed to be an ideal conducting wall. This implies the physical boundary conditions In the general case the boundary fields 2.5 have to be replaced by appropriate distributions, and the limit x 3 ↓ 0 will be performed in the Fourier domain instead of the half-space H.
For this purpose we write again x x, x 3 , with x x 1 , x 2 ∈ R 2 and fixed x 3 > 0, and assume that each field component u ·, x 3 represents a tempered distribution in S R 2 with Fourier transform Then we obtain from the first-order Maxwell-Helmholtz equations the following Fourier transformed Maxwell-Helmholtz system: for all x 3 ≥ 0 and fixed ξ 1 ,

2.8
Here we have replaced the partial derivative with respect to x 3 by the ordinary derivative d/dx 3 . We define

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With the two-dimensional Laplace operator Δ we have For the action of C in the Fourier domain we obtain multiplication by the matrix We replace 2.8 with the ordinary differential equations 2.12 and the two algebraic conditions

2.15
Then the general solution of the homogeneous linear system 2.12 is with the 2 × 2 unit matrix E. But the terms cos m ξ x 3 and sin m ξ x 3 are exponentially increasing for fixed x 3 > 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: From the general solution and 2.17 , 2.18 we obtain the following decay conditions for x 3 > 0: e j ξ, x 3 e ix 3 m ξ e j,0 ξ , b j ξ, x 3 e ix 3 m ξ b j,0 ξ , j 1, 2, 3.

2.19
With x x 1 , x 2 ∈ R 2 , x x, x 3 ∈ R 3 and x 3 > 0 there holds the important and well-known Sommerfeld-Weyl integral representation The left-hand side in 2.20 is the singular solution of the three-dimensional Helmholtz equation Δ k 2 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 x 3 > 0 and ξ ∈ R 2 , |ξ| / k.
Distributions u ∈ D R 2 with compact support in the screen plane are tempered, and it follows from the Paley-Wiener theorem that u is a smooth function which has polynomial growth on R 2 . 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 x 3 > 0 to each component e j ·, x 3 and b j ·, x 3 . Theorem 2.1. Let there be given e 1,0 , e 2,0 ∈ S R 2 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;

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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 j and b j in terms of e 1,0 and e 2,0 .
Definition 2.2. The electromagnetic field in the half-space x 3 > 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 R 2 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 e j x, x 3 e j,0 x , j 1, 2.
Then we obtain from Theorem 2.1 and the Sommerfeld-Weyl integral 2.20 for all x ∈ R 2 and all

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 ∈ R 2 to the limit x 3 ↓ 0 and obtain

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 ISRN Mathematical Physics 7 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 : R 2 × 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 R 2 with compact support in Ω.
a The solution is called physical admissible if and only if it satisfies the local energy condition The solution satisfies the stronger global energy condition if and only if for some h > 0, and therewith for all h > 0, with the layer 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 .

The Global Energy Condition
Throughout the rest of this paper let the open aperture Ω : κ j 1 Ω j ⊂ R 2 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 and e 1,0 ξ ξ 1 e 2,0 ξ ξ 2 0 for all ξ ξ 1 , ξ 2 ∈ R 2 with |ξ| k.
Proof. In order to perform the energy evaluation with Parseval's theorem we define with the Fourier transformed electromagnetic boundary fields e j,0 , b j,0 , j 1, 2, 3, the quantity

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 With V 1 : {ξ ∈ R 2 | |ξ| < k}, V 2 : {ξ ∈ R 2 | |ξ| > k} we conclude from Parseval's theorem, 2.19 and the definition 2.15 of m ξ for h > 0

3.6
Using for all ξ ∈ R 2 with |ξ| / k the estimate ISRN Mathematical Physics 9 and observing that we have uniformly in r > k for appropriate constants α, β > 0 we conclude from 3.6 that E h is finite if and only if Note that with e 1,0 and e 2,0 also ∇ · −e 2,0 , e 1,0 T has support in Ω.
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 u 1 , u 2 ∈ H 1/2 Ω , in the following called boundary potential functions, satisfying the regularity condition

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 R 2 \Ω is connected, then the boundary fields e 1,0 and e 2,0 can be represented by two boundary potential functions u 1 , u 2 ∈ 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 on the whole screen plane Proof. Part a follows in the Fourier domain by representing e 1,0 , e 2,0 , e 3,0 and b 3,0 in terms of u 1 and u 2 as
Since e 1,0 , e 2,0 have compact support in Ω like u 1 , u 2 , 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 v 1 , v 2 :

3.20
From 3.20 we get for j 1, 2 that

3.21
Thus for j 1, 2 the inverse Fourier transform u j of v j lies in H 1/2 R 2 . Now we show that u 1 and u 2 have their support in Ω and hence, by reason of 3.21 , u 1 , u 2 ∈ H 1/2 Ω . To this aim we choose some R > 0 such that 3.22 Since e 1,0 , e 2,0 are supported in Ω, we have supp e 1,0 , supp e 2,0 ⊂ B R . Therefore we obtain from the Paley-Wiener theorem and 3.20 that u 1 and u 2 are also supported in B R , because v 1 , v 2 , as e 1,0 , e 2,0 , are of exponential type not larger than R. Equations 3.20 can be resolved with respect to e 1,0 , e 2,0 . This gives 3.16 . Together with 2.10 it follows that in the complement of Ω there holds Now we make use of the fact that the complement of Ω is connected. Since supp u j ⊂ B R for j 1, 2, Holmgren's unique continuation principle, applied to the two scalar Helmholtz equations in 3.23 , implies that u 1 , u 2 have their support in Ω.

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 x 3 ↓ 0. The resulting relations in the Fourier domain follow from the representations of the components b 1 and b 2 in Theorem 2.1.
The validity of 3.18 is a consequence of the algebraic relations between u 1 , u 2 , b 1,0 , and b 2,0 .

Remark 3.3. Consider the Sobolev spaces H s Ω given for s ∈ R by
with the restriction F| Ω of the tempered distribution F : S R 2 → C to the subspace S Ω of the Schwartz space S R 2 , where S Ω is the closure of the set D Ω in S R 2 with respect to the topology of S R 2 5 , §1 in Section 5. H s Ω is a Banach space with respect to the norm · H s Ω given by 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 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.

The Local Energy Condition
In this section we derive the following characterisation for the diffraction solutions of local finite energy. 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 by For r > 0 and h > 0 the open cylinder Z r,h is defined by e e 1,0 , e 2,0 ∈ H −1/2 Ω ⇒ e 3 ∈ L 2 Z R,H .
f e 1,0 , e 2,0 , ∇ · −e 2,0 e 1,0 Proof of Theorem 4.1. Let there be given e 1,0 , e 2,0 ∈ S R 2 with support in Ω and 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 e 1 , e 2 , b 3 ∈ L 2 Z R,H . From the parts a and b of Lemma 4.2 we find that e 1 , e 2 , b 3 ∈ L 2 R 2 × 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 from the other three field components, from the parts e and f of the lemma we obtain e 3 , b 1 , b 2 ∈ L 2 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 implication We set ε R − R /2 and R R ε. For ϕ ∈ 0, 2π we consider the rotation matrix where A ϕ R , ∞ × R is the image of the set R , ∞ × R under the linear mapping A ϕ , that is, the set R , ∞ × R rotated 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 ∈ L 2 R 2 \ P × 0, H . Thus 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 ϕ j x 1 , x 2 , x 3 1 2π R γ ϕ ξ e ix 3 m ξ i x,ξ dξ 1 dξ 2 , where the function γ ϕ is given by γ ϕ ξ e j,0 A ϕ ξ .

4.11
By assumption, it holds that supp e j,0 ⊂ Ω ⊂ B R . Thus, from the Paley-Wiener theorem and the fact that A ϕ is an orthogonal matrix it follows that for some constants c > 0 and N ∈ N 0 . 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 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 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|, that is, √ * 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 k 2 − ξ 2 1 − ξ 2 2 * . Regarding the integration variable ξ 2 we distinguish the two cases |ξ 2 | < k and |ξ 2 | > k.

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Firstly we assume that |ξ 2 | < k. We have

4.17
By contour integration analogous as in the derivation of 4.16 , applied to the next-to-last integral, eventually we get

4.18
In the case |ξ 2 | > k it holds that m ξ −m * ξ for ξ 1 > 0, and contour integration yields In the following, for ξ 1 ∈ R or ξ 1 ∈ {it | 0 < t < ∞} we set m ξ k 2 − ξ 2 1 − ξ 2 2 , where the square root is chosen in the way that Re m ξ ≥ 0 and Im m ξ ≥ 0. This definition is in accordance with 2.15 . In this notation, from 4.18 and 4.19 we obtain

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, x 3 , lies in L 2 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 ∈ R 2 | |ξ| ≤ k, ξ 1 ≥ 0}, let χ A be the characteristic function of the set A and letˇdenote the inverse Fourier transform. There is a constant c > 0, which does not depend on x 3 ∈ 0, H , such that From this it follows that the first addend in the right-hand side of 4.21 is quadratically integrable over the set R 2 × 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 c 1 > 0 with γ ϕ it, ξ 2 ≤ c 1 e Rt for ξ 2 ∈ −k, k , t ≥ 0.

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 L 2 R , ∞ × R × 0, H too.