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

The vector problem of electromagnetic wave diffraction by a system of bodies and infinitely thin screens is considered in a quasiclassical formulation. The solution is sought in the classical sense but is defined not in the entire space R3 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.


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 2 Advances in Mathematical Physics 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.

The Statement of the Diffraction Problem
Let be the union of a finite number of connected oriented unclosed and nonintersecting bounded surfaces of class  ∞ in R 3 .The edge Ω  := Ω  \ Ω  of the surface Ω  consists of a finite number of curves of the class  ∞ without selfintersection points that intersect at nonzero angles and Ω := ∪  Ω  .We consider all screens to be infinitely conducting.Define tubular neighborhoods Ω  of the screen edge as Assume that   are bounded regions whose boundaries   =   \   are piecewise smooth closed oriented surfaces consisting of a finite number of surfaces of the class  2 .Let  := ∪    .Assume that ∩Ω = ⌀.The bodies are anisotropic and inhomogeneous.The inhomogeneity of the problem is described by the tensor function where ε () ∈  2 (  ) ∩  1, (  ).Hereinafter, Î is the unit 3 × 3 tensor;   := R 3 \ ; the free space is isotropic and homogeneous with constant   and   > 0. In the inhomogeneity area , the condition μ ≡   Î holds.We assume that ε () are symmetric complex tensors with nonnegative imaginary part: The conditions Re   > 0, Im   ≥ 0, Im   ≥ 0, and   =  √     are satisfied in the free space R 3 \ (Ω ∪ ).
We consider the problem of diffraction of electromagnetic wave E 0 , H 0 depending on time harmonically as  − by a system of bodies and infinitely thin screens.
The incident field (E 0 , H 0 ) is the solution to Maxwell's equations in homogeneous entire space: Here current j 0, has a compact support such that supp(j 0, )∩ (Ω ∪ ) = ⌀.
We also define  + ,  − as arbitrary domains exterior and interior with respect to the screen Ω assuming that Ω ⊂  ± .
The diffraction problem consists in finding a total electromagnetic field (E, H): satisfying Maxwell's equations outside the screens and the boundaries of the bodies, the transmission conditions on the boundary  of the inhomogeneous bodies, the Dirichlet boundary conditions in the interior points of the screen (i.e., everywhere on Ω except for the edge points), the energy finiteness condition in any bounded space volume, and the the Silver-Mueller radiation conditions at infinity for the scattered field E  := E − E 0 , H  := H − H 0 ( = || and e  = /||).
It suffices to show that the homogeneous boundary value problem for the scattered field has only the trivial solution.
The boundary value problem for E  , H  is formulated as follows: rot The Silver-Mueller conditions should also be satisfied.
The screen Ω is extended to an arbitrary piecewise smooth closed simply connected orientable surface  1 surrounding a bounded region  1 ⊂ R 3 such that ∩ 1 = ⌀.
Henceforward,  n := /n denotes the derivative in the direction of the unit outward normal vector n to the domain of  and E  , H  denote the scattered field as considered in the region   .
The boundary value problem for the scattered field is reduced to the following transmission problem in the regions We will apply the integral Lorentz lemma in the bounded regions  1 ,  2 , and  3 .
In the region  2 = , consider Maxwell's equations for the scattered field as well as for the conjugate field E 2 , H 2 : rot Replacing E 2 by − Ẽ2 , we obtain rot Applying the Lorentz lemma to the fields , and H  = H 2 and currents j   = (ε − ε) Ẽ2 , j   = (  −   )H 2 , we derive the following equality Replacing Ẽ2 with −E 2 and taking into account the media properties we obtain

Advances in Mathematical Physics
In the regions  1 and  3 , the similar integral relations hold Since the Silver-Mueller conditions hold, we obtain Sum equalities (20), (21), and (22): Consider several cases.Suppose that Im ε > 0 everywhere in (Ω)  .Then radiation conditions imply E, H ≡ 0 in the entire space from (25).
If Im ε() > 0 in  and Im   = 0 in the free space, then we have As both terms in the upper relation are nonnegative, we obtain By the Rellich lemma, we conclude that E  , H  ≡ 0 outside the scatterers; the second equation implies E 2 , H 2 ≡ 0 in .Finally, if permittivity ε() =   Î is everywhere in R 3 \ Ω real and positive we derive by the Rellich lemma that E  , H  ≡ 0 outside the scatterers.In paper [6] it is proved that E  , H  ≡ 0 holds in .

Integrodifferential Equations
Represent the total field as the sum where E 0 is the incident field and E 1 is the field scattered by Ω in the absence of .Hence, we have As mentioned above, (E 0 , H 0 ) satisfies Maxwell's equations as well as transmission conditions The field satisfying radiation conditions (11) as well as transmission conditions The field E 1 is sought in the following form: where (, ) = 1/4 ⋅    |−| /| − |, u is unknown density of the surface current on Ω, and (grad div +  2  )k ≡ grad div k +  2   k for all vector functions k.We assume that u is a tangential vector field: u ⋅ ] = 0 on Ω (] denotes unit normal vector field on Ω).
Define "new" incident field (E  0 , H  0 ) = (E 0 , H 0 ) + (E 1 , H 1 ) and write system (7) as follows: Here current j  is defined as where j  0, are the currents corresponding to the field (E  0 , H  0 ) and j , is the polarization current in the inhomogeneity domain : Advances in Mathematical Physics 5 In  we define electric field E via vector potential according to well-known ( [11], page 61) formulas Thus definition of the fields E 0 and E 1 as well as equalities (28), (32), (36), and (37) implies the following integrodifferential equation: In a similar manner, the field outside the bodies and the screens is represented as To obtain the second equation we use condition (9).Placing the point  on Ω in (39) and considering the tangential components of all terms we have ( The space  = () of vector bundles over  was introduced in [7] as the closure of  ∞ 0 () with respect to the norm ‖ ⋅ ‖  : Here ‖u‖ −1/2 denotes the norm in Sobolev space  −1/2 (Ω) and   =   () is the antidual space to  (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  and u is the density of the surface current on Ω.