# Integrodifferential Equations of the Vector Problem of Electromagnetic Wave Diffraction by a System of Nonintersecting Screens and Inhomogeneous Bodies

Authors: Y. G. Smirnov; A. A. Tsupak
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

The vector problem of electromagnetic wave diffraction by a system of bodies and infinitely thin screens is considered in a quasi-classical formulation. The solution is sought in the classical sense but is defined not in the entire spaceR3 but rather everywhere except for the screen edges. The original boundary value problem for Maxwell’s equations system is reduced to a system of integrodifferential equations in the regions occupied by the bodies and on the screen surfaces. The integrodifferential operator is treated as a pseudodifferential operator in Sobolev spaces and is shown to be zero-index Fredholm operator.


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## Body

## 1. Introduction

In this work, we obtain new integrodifferential equations of the vector problem of electromagnetic wave diffraction by a set of two- and three-dimensional scatterers of complex geometry.Acoustic and electromagnetic problems of diffraction by screens or inhomogeneous bodies are classical ones, which have been thoroughly investigated by many researchers. In [1], a wide class of acoustic scattering problems in bounded and unbounded regions is described. In [2, 3], boundary value problems (or transmission problems) are reduced to surface integral equations over the boundary of a region, followed by their analysis and numerical solution by applying projection methods. Scalar problems of diffraction by unclosed screens and their numerical solution are described, for example, in [4].Methods of volume integral equations (as well as integrodifferential equations) are also applied for theoretical study and numerical solving of electromagnetic problems (see [5, 6] and the bibliography inside these works).Diffraction problems (mostly, vector ones) are the subject of a number of works by the authors of this paper. In [7], vector problems of diffraction by thin conducting screens are studied by applying methods of the theory of pseudodifferential operators. In [8], diffraction problems were investigated both theoretically (the existence and uniqueness of boundary value problems, their equivalence to integral equations, and the invertibility of integral operators were considered) and numerically (Galerkin method was formulated and its convergence was proved). In [9], methods of pseudodifferential operators were applied to prove ellipticity and Fredholm property of volume integrodifferential operator.In the present work, we consider a more complicated problem: the scattering structure is formed of three-dimensional bodies and infinitely thin screens. Such a problem (namely, the scalar problem of diffraction by bodies and surfaces) was treated first in [10]. Both in [10] and in this paper, the solution is understood as a function defined everywhere, except for the edge of the screens; it satisfies Maxwell’s equations in the classical sense, continuity conditions on the interface between the media, Dirichlet condition in the interior points of the screens, and radiation conditions at infinity. For this problem, we prove a uniqueness theorem.Below, the original boundary value problem is reduced to a system of integrodifferential equations. According to well-known results on the behavior of the field near a screen edge, it is necessary to consider the equations obtained as pseudodifferential equations in Sobolev spaces. Using this approach we prove that the solution is smooth at points other than the screen edges and establish the Fredholm property of the matrix integrodifferential operator of the considered diffraction problem.


## 2. The Statement of the Diffraction Problem

Let(1)Ω=⋃jΩj,Ω¯i⋂Ω¯j=⌀i≠jbe the union of a finite number of connected oriented unclosed and nonintersecting bounded surfaces of class C∞ in R3. The edge ∂Ωj≔Ω¯j∖Ωj of the surface Ωj consists of a finite number of curves of the class C∞ without self-intersection points that intersect at nonzero angles and ∂Ω≔∪j∂Ωj. We consider all screens to be infinitely conducting.Define tubular neighborhoods∂Ωδ of the screen edge as (2)∂Ωδ≔x∈R3:dist⁡x,∂Ω<δ.Assume thatQi are bounded regions whose boundaries ∂Qi=Q¯i∖Qi are piecewise smooth closed oriented surfaces consisting of a finite number of surfaces of the class C2. Let Q≔∪iQi. Assume that Q¯∩Ω¯=⌀. The bodies are anisotropic and inhomogeneous. The inhomogeneity of the problem is described by the tensor function (3)ε^x=εeI^,x∈Q¯∪Ω¯cε^ix,x∈Q¯i,where ε^i(x)∈C2(Qi)∩C1,α(Q¯i). Hereinafter, I^ is the unit 3×3 tensor; Mc≔R3∖M; the free space is isotropic and homogeneous with constant εe and μe>0. In the inhomogeneity area Q¯, the condition μ^≡μeI^ holds. We assume that ε^i(x) are symmetric complex tensors with nonnegative imaginary part: (4)ε^i=ε^iT,Im⁡ε^i≥0∀i.The conditionsReεe>0, Imεe≥0, Imke≥0, and ke=ωεeμe are satisfied in the free space R3∖(Ω¯∪Q¯).We consider the problem of diffraction of electromagnetic waveE0, H0 depending on time harmonically as e-iωt by a system of bodies and infinitely thin screens.The incident field(E0,H0) is the solution to Maxwell’s equations in homogeneous entire space:(5)rot⁡H0=-iωεeE0+j0,E,rot⁡E0=iωμeH0.Here current j0,E has a compact support such that supp(j0,E)∩(Ω¯∪Q¯)=⌀.We also defineP+, P- as arbitrary domains exterior and interior with respect to the screen Ω assuming that Ω⊂∂P±.The diffraction problem consists in finding a total electromagnetic field(E,H):(6)E,H∈C2∂Q∪Ω¯c⋂C1Q¯c∖Ω¯⋂C1Q¯·⋂δ>0CP+¯∖∂Ωδ⋂δ>0CP-¯∖∂Ωδ,satisfying Maxwell’s equations(7)rot⁡H=-iωε^E+j0,E,rot⁡E=iωμeH,in∂Q∪Ω¯coutside the screens and the boundaries of the bodies, the transmission conditions(8)Eτ∂Q=Hτ∂Q=0,on the boundary ∂Q of the inhomogeneous bodies, the Dirichlet boundary conditions(9)EτΩ=0,in the interior points of the screen (i.e., everywhere on Ω except for the edge points), the energy finiteness condition(10)E,H∈L2,loc⁡R3=Hloc⁡0R3in any bounded space volume, and the the Silver-Mueller radiation conditions at infinity(11)Es,Hs=o1r,Im⁡ke>0,Hs×er-Es=o1r,Es×er-Hs=o1rr⟶∞,Es,Hs=O1r,Im⁡ke=0,for the scattered field Es≔E-E0, Hs≔H-H0 (r=|x| and er=x/|x|).Definition 1.
The solutionE,H of problem (7)–(11) satisfying condition (6) is called quasi-classical.Theorem 2.
The inhomogeneous boundary value problem (7)–(11) has at most one quasi-classical solution.It suffices to show that the homogeneous boundary value problem for the scattered field has only the trivial solution.The boundary value problem forEs,Hs is formulated as follows:(12)rot⁡Hs=-iωε^Esrot⁡Es=iωμeHs,Es,τ∂Q=Hs,τ∂Q=0,Es,τΩ=0.The Silver-Mueller conditions should also be satisfied.The screenΩ is extended to an arbitrary piecewise smooth closed simply connected orientable surface ∂V1 surrounding a bounded region V1⊂R3 such that Q¯∩V¯1=⌀.LetB≔BR(0) be a ball of a sufficiently large radius R such that (13)Q¯,V¯1⊂B.LetV2≔Q. Define the region V3≔B∖(V¯1∪V¯2) with the boundary ∂V3=∂B∪∂V1∪∂V2, and consider V4≔B¯c.Henceforward,un≔∂u/∂n denotes the derivative in the direction of the unit outward normal vector n to the domain of u and El,Hl denote the scattered field as considered in the region Vl.The boundary value problem for the scattered field is reduced to the following transmission problem in the regionsVl: (14)rot⁡Hl=-iωεeElrot⁡El=iωμeHl,l=1,3,4;rot⁡H2=-iωε^E2rot⁡E2=iωμeH2E2,τ∂Q=E3,τ∂Q;H2,τ∂Q=H3,τ∂Q;E3,τ∂B=E4,τ∂B;H3,τ∂B=H4,τ∂B;E1,τ∂V1∖Ω¯=E3,τ∂V1∖Ω¯;H1,τ∂V1∖Ω¯=H3,τ∂V1∖Ω¯;E1,τΩ=E3,τΩ=0;E4,H4=o1r,Im⁡ke>0H4×er-E4=o1r,E4×er-H4=o1rr⟶∞.E4,H4=O1r,Im⁡ke=0.We will apply the integral Lorentz lemma(15)∫∂VE′×H′′-E′′×H′,nds=∫VjH′′,H′¯+jE′,E′′¯-jH′,H′′¯-jE′′,E′¯dVin the bounded regions V1, V2, and V3.In the regionV2=Q, consider Maxwell’s equations(16)rot⁡H2=-iωε^E2,rot⁡E2=iωμeH2for the scattered field as well as for the conjugate field E¯2,H¯2:(17)rot⁡H¯2=iωε^¯E¯2,rot⁡E¯2=-iωμ¯eH¯2.Replacing E¯2 by -E~2, we obtain(18)rot⁡H¯2=-iωε^¯E~2=-iωε^E~2+iωε^-ε^¯E~2,rot⁡E~2=iωμ¯eH¯2=iωμeH¯2-iωμe-μ¯eH¯2,inV2.Applying the Lorentz lemma to the fields E′=E2, H′=H2, E′′=E~2, and H′′=H¯2 and currents jE′′=iω(ε^-ε^¯)E~2, jH′′=iω(μe-μ¯e)H¯2, we derive the following equality(19)∫∂QE2×H¯2-E~2×H2,nds=-∫Qiωε^-ε^¯E~2,E¯2dv+∫Qiωμe-μ¯eH¯2,H¯2dv.Replacing E~2 with -E¯2 and taking into account the media properties we obtain(20)∫∂QE2×H¯2+E¯2×H2,nds=∫Qiωε^-ε^¯E¯2,E¯2dv.In the regionsV1 and V3, the similar integral relations hold(21)∫∂V1∖Ω¯E1×H¯1+E¯1×H1,nds=∫V1iωεe-ε¯eE¯1,E¯1dv.(22)∫∂Q∪∂V1∖Ω¯∪∂BE3×H¯3+E¯3×H3,nds=∫V3iωεe-ε¯eE¯3,E¯3dv.Define(23)Ψ≔∫∂BE3×H¯3+E¯3×H3,nds.Since the Silver-Mueller conditions hold, we obtain(24)Ψ=∫∂BE3×H¯3+E3×H¯3¯,nds=2Re∫∂BE3×H¯3,nds=2Re∫∂BH3×n+o1r×H¯3,nds=2Re∫∂BH3×n×H¯3,nds+o1=2Re∫∂BH3×n,H¯3×nds+o1=∫∂BH3×n2ds+o1=∫∂BH3,τ2ds+o1.Sum equalities (20), (21), and (22):(25)∫∂BH3,τ2ds+∫QIm⁡ε^E¯2,E¯2dv+∫QIm⁡μeH¯2,H¯2dv+∑l=1,3∫VlIm⁡εeE¯l,E¯l+Im⁡μeH¯1,H¯1dv=o1.Consider several cases.Suppose thatImε^>0 everywhere in (Ω¯)c. Then radiation conditions imply E,H≡0 in the entire space from (25).IfImε^(x)>0 in Q and Imεe=0 in the free space, then we have (26)∫∂BH3,τ2ds+∫QIm⁡ε^E¯2,E¯2dv=o1.As both terms in the upper relation are nonnegative, we obtain (27)∫∂BH3,τ2ds=o1,R⟶∞;∫QIm⁡ε^E¯2,E¯2dv=0.By the Rellich lemma, we conclude that Es,Hs≡0 outside the scatterers; the second equation implies E2,H2≡0 in Q.Finally, if permittivityε^(x)=εeI^ is everywhere in R3∖Ω¯ real and positive we derive by the Rellich lemma that Es,Hs≡0 outside the scatterers. In paper [6] it is proved that Es,Hs≡0 holds in Q.


## 3. Integrodifferential Equations

Represent the total field as the sum(28)E=E0+E1+E2,where E0 is the incident field and E1 is the field scattered by Ω in the absence of Q. Hence, we have Es=E1+E2.As mentioned above,(E0,H0) satisfies Maxwell’s equations as well as transmission conditions(29)E0,τ∂Q=H0,τ∂Q=0.The field(E1,H1) is a solution to Maxwell’s equations in R3∖Ω¯:(30)rot⁡H1=-iωεeE1,rot⁡E1=iωμeH1satisfying radiation conditions (11) as well as transmission conditions(31)E1,τ∂Q=H1,τ∂Q=0,E1,τ+E0,τΩ=0.The fieldE1 is sought in the following form:(32)E1x=grad⁡div+ke2∫ΩGx,yuydsy,where G(x,y)=1/4π·eike|x-y|/|x-y|,  u is unknown density of the surface current on Ω, and (grad⁡div+ke2)v≡grad⁡div⁡v+ke2v for all vector functions v. We assume that u is atangential vector field: u·ν=0 on Ω (ν denotes unit normal vector field on Ω).Define “new” incident field(E0′,H0′)=(E0,H0)+(E1,H1) and write system (7) as follows:(33)rot⁡H=-iωεeE+jE,rot⁡E=iωμeH.Here current jE is defined as (34)jE=j0,E′+jp,E,where j0,E′ are the currents corresponding to the field (E0′,H0′) and jp,E is the polarization current in the inhomogeneity domain Q: (35)jp,E=-iωε^-εeI^E.InQ we define electric field E via vector potential(36)AEx=∫QGx,yjEydyaccording to well-known ([11], page 61) formulas(37)E=iωμeAE-1iωεegrad⁡div⁡AE.Thus definition of the fieldsE0 and E1 as well as equalities (28), (32), (36), and (37) implies the following integrodifferential equation:(38)Ex-ke2+grad⁡div∫QGx,yε^yεe-I^Eydy-ke2+grad⁡div∫ΩGx,yuydsy=E0x,x∈Q.In a similar manner, the field outside the bodies and the screens is represented as(39)Ex=ke2+grad⁡div∫QGx,yε^yεe-I^Eydy+ke2+grad⁡div∫ΩGx,yuydsy+E0x,x∈Q¯∪Ω¯c.To obtain the second equation we use condition (9). Placing the point x on Ω in (39) and considering the tangential components of all terms we have(40)-ke2+grad⁡div∫QGx,yε^yεe-I^Eydy-ke2+grad⁡div∫ΩGx,yuydsyτ=E0,τx,x∈Ω.Defineξ^≔[ε^(y)/εe-I^]-1 and J≔[ε^(y)/εe-I^]E and write system (39), (40) for currents:(41)ξ^Jx-ke2+grad⁡div∫QGx,yJydy-ke2+grad⁡div∫ΩGx,yuydsy=E0x,x∈Q;-ke2+grad⁡div∫QGx,yJydy-ke2+grad⁡div∫ΩGx,yuydsyτ=E0,τx,x∈Ω.Define the matrix operatorL^ of system (41):(42)L^≔L^1+L^2≔A00S+0K1K20.Here operators A, S, and Ki are defined according to (41):(43)AJ≔ξJx-ke2+grad⁡div∫QGx,yJydy,Su≔-ke2+grad⁡div∫ΩGx,yuydsyτ,K1u≔-ke2+grad⁡div∫ΩGx,yuydsy,K2J≔-ke2+grad⁡div∫QGx,yJydyτ,and they are regarded as mappings in the spaces:(44)AJ≔L2Q⟶L2Q,Su≔WΩ⟶W′Ω,K1u≔WΩ⟶L2Q,K2J≔L2Q⟶W′Ω.The spaceW=W(S¯) of vector bundles over S was introduced in [7] as the closure of C0∞(S) with respect to the norm ∥·∥W: (45)uW2=u-1/22+div⁡u-1/22.Here u-1/2 denotes the norm in Sobolev space H-1/2(Ω) and W′=W′(S) is the antidual space to W (see [7], page 88).The solution to the system of integrodifferential equations is understood as a pair of functions(J,u), where J is the polarization current corresponding to the total electric field in Q and u is the density of the surface current on Ω.Theorem 3.
In the spaces chosen,L^ is a Fredholm operator.Throughout the following, a Fredholm operator is treated as a zero-index Noether operator.As the permittivity is a positive function inQ then the operator A is a Fredholm operator in L2(Q) (see [9]). The operator S is a Fredholm operator from W(Q) to W′(Q) because ke≠0 ([7], page 109). Therefore, L^1 is a Fredholm operator.The operatorL^2 is compact because the kernels of the operators K1 and K2 are infinitely smooth.Finally,L^=L^1+L^2 is a Fredholm operator in the above-mentioned spaces.Statement 1.
If solution(J,u)∈L2(Q)×W(Ω) exists, then J and u are infinitely differentiable in the interior points of Q and Ω, respectively.Suppose that(J,u) is a solution to system (41) with E0∈C∞(R3). The ellipticity of the operator A:Hcomps(Q)→Hloc⁡s(Q) (see [9]) and smoothness in Q¯ of E0-K1u (since Ω¯∩Q¯=⌀) imply J∈C∞(Q).Similarly, coercivity ofS ([7], page 69) implies infinite smoothness of u in Ω.



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*Source: 945965-2015-05-06.xml*