Application Of Integral Equation Method In Numerical Computation Of Direct And Inverse Scattering Problems

Application of integral equation method in numerical computation of direct and inverse scattering problemIn the acoustics and electromagnetics, the direct scattering problems are to de-termine the scattering field (far field) from a knowledge of incident field and the differential equation governing the wave motion, the inverse scattering problems are to reconstruct the differential equation or the domain of definition from a knowledge of behavior of the scattering field (far field).The electromagnetic scattering and inverse problems in special media have been payed attention to. When the electromagnetic wave propagation is between the dif-ferent medium, we may divide problems into the periodic problems and non-periodic problems. We discuss the periodic diffractive structures, also known as diffraction gratings. The diffraction grating problems are different from the electromagnetic scat-tering problems in radiation condition and properties of solution. The research is more difficult because of unbounded property of the calculation region.For non-periodic diffraction models, we discuss the electromagnetic scattering problems by a homogeneous chiral environment. A three-dimensional chiral object cannot be brought into congruence with its mirror image by rotation or translation, so collection of chiral objects will form a material which is characterized by right-handedness or left-handedness. In general, the electromagnetic wave propagation in a chiral medium is governed by Maxwell's equations and a set of constitutive equations known as the Drude-Born-Fedorov constitutive equations, in which the electric and magnetic fields are coupled. The coupling is responsible for the chiral of the medium.Above problems have not only theoretical research value, but also very impor-tant practical significance. Research on method of calculating the problems is one of the current hot topic. To consider direct and inverse acoustic scattering problems of grating, as well as scattering problems in chiral medium are as follows:(1) Numerical calculation of the scattering problem for grating by integral equation methodConsidering a time-harmonic electromagnetic plane wave incident on a slab of some optical material in R3, which is periodic in x1 direction and invariant in x3 direction. We study the diffraction problem in the TM (traverse magnetic) polarization, then the scattering problem is governed by Helmholtz equation in R2.Assume x=(x1,x2) is point in R2, S is simple curve imbedded in the strip function f∈C2 is periodic by L, i.e. f(x1+L) = f(x1), (?)x1∈R. LetΩis region in R2Given a plane wave ui as incident wave. i.e. ui(x)= eiαx1-iβx2, where incident angle|θ|<(?) andα=ksinθ,β= k cosθ. Let Imk≥0, the scattering problem of grating can be stated as follows:Problem 1. Given incident wave ui and the a simple curve S, find scattering field u(x)∈C2(Ω)∩C(Ω), satisfying boundary condition outgoing condition and quasi-periodic condition where An∈R are coefficients in (0-3).We assume that k2≠(αn-α)2 for all n∈Z,αn=2nπ/L. This condition excludes "resonance" cases and ensures that a fundamental solution for (0-1) exists insideΩ.A fundamental solution for (0-1) is where By [19] we can give by following several lemmas:Lemma 0.1 LetΨsatisfy is Dirac measure in (nL,0),(?)'is dual space of Frechet.Lemma 0.2 LetΨ(x)-e-iax1Φ0(x) is smooth function inxWe consider scattering problem of grating in one periodic. LetBy [19] we can see quasi-periodic solution of (0-1)-(0-4) that can be written u(x)= eiαx1uα(x), where uα(x) is periodic function of L in x1 direction and satisfy where density functionΦ∈C(S0), such that normal vector of y is ny and the direction of the external region ofΩ0.Considering the integral kernel, we can get about the boundary integral equation where g=2e-iβf(x1),Operator A defined as followsThe solvability of (0-8) are given by following theorems:Theorem 0.1 Except for a possibly discrete set composed of countable number of k, the integral equation (0-8) has a unique solution.Theorem 0.2 If equation (0-8) has a solutionΦ∈C(S0), such that equation (0-7) has a solution u∈C2(Ω)∩C(Ω),In the numerical method, We first solve the equation (0-8) can be similar to the approximate density functionΦ. then use (0-9) get the approximate scattered field uα.Considering the convergence and Error estimates of the solution of Imk> 0. ThroughΦsatisfy the integral equation we can getΦ∈Hs[0,2π], s≥1. In numerical computation, respectively, truncation the fundamental solution and integral. The sup-position after the interruption density function isΦmn, To know that we are given the numerical method is at least linear convergence. We have the following result:When the wave number Imk=0, although in theory has not been proven, however, we can see that the solution of scattering problems is also convergent in numerical experiments. (2)Inverse scattering problem of gratingAssume x=(x1,x2)is point in R.,S is simple curve imbedded in the strip function f∈C2 is periodic by L,i.e.f(x1+L)=f(x1),(?)x1∈R.LetΩis region in R2Given a plane wave ui as incident wave,i.e.ui(x)=eiαx1-iβx2,where incident angle |θ|<(?) andα=k sinθ,β=k cosθ.Let Imt≥0,the inverse scattering problem of grating can be stated as follows:Problem 2. Given incident wave ui and scattering data onГ={x2=b,b> max f},find S,satisfying boundary condition where(0-11)is Dirichlet boundary condition.outgoing wave condition and quasi-periodic condition where An∈R are coefficients in(0-12).We assume that k2≠(αn-α)2 for all n∈z,αn=2nπ/L. This condition excludes"resonance" cases and ensures that a fundamental solution for(0-10)exists insideΩ.A fundamental solution for(0-10)is where In general, properties of uniqueness and stability are very hard, if not impossible, to establish. But because of the important impact of these properties on applications, characterizations of uniqueness and stability are required.Here, we present a uniqueness result for inverse problem. Let us assume that for the given incident field ui. u1 and u2 solve the direct scattering problem of grating with respect to S1={x2=f1(x1)} and S2={x2=f2(x1)}, respec-tively. The functions f1,f2∈C2 are assumed to be sufficiently smooth and L-periodic. Let b> max{f1(x1),f2(x1)} be a fixed constant, min{f1(x1),f2(x1)}, we have following theorem [40]:Theorem 0.3 Assume that u1(x1,b)= u2(x2.b). Assume further that one of the following condition is satisfied:(i) k has a nonzero imaginary part;(ii) k is real and T satisfies k2< 2(T-2+L-2),Then f1(x1)＝f2(x1).Next, We consider the scattered fields for incident plane waves with respect to an interval of wave numbers and one fixed direction are sufficient to determine f [45].Theorem 0.4 Letθ= (sinθ, -cosθ) be a fixed direction with|θ|<(?). Let f,g∈C2(R) be 2π-periodic functions andα> max{f(t),g(t):t∈R}. The scattered fields are assumed to coincide, i.e. for all incident plane waves corresponding to wave numbers for some 0 max{f(t),g(t):t∈R} and N distinct wave numbers ki∈(0, kmax],l= 1,...,N.Then f and g coincide.In application, it is impossible to make exact measurements. Stability is crucial in the practical reconstruction of profiles since it contains necessary information to determine to what extent the data can be trusted.Before stating the stability result, let us first introduce some notations [40]. For any two domain D1,D2,∈R2,define d(D1,D2), the Hausdorff distance between them by whereDenote and a sequence of domain Dh= normal to S={x2= f(x1)}. Assume the boundary is periodic of the same period L. The functionσh satisfies|σh(x1)|≤C. So, for ho sufficiently small, the sequence of domains is assumed to satisfy where C1, C2 are positive constants.For the fixed incident plane wave ui, assume that us, uhs solve the scattering prob-lem with respect to periodic structures S, Sh.Then we have the following stability result ([40]).Theorem 0.6 Under the above assumptions, where the constant C may depend on the family{σh}.Without loss of generality, Let the boundary S is periodic of the period 2πГ={x2=-b} is the auxiliary curve of period 2πbelow S. Assume we have the Rayleigh expansion aboveГ, then scattered field can be approximately expressed aboveГas whereAccording to classical scattering theory [49] and quasi-periodic, we have following result Because of uapprox(x1, b)= us(x1,b),uapproxs is equivalent to solving whereBy Tikhonv regularization method, (0-16) is equivalent to whereγ>0 is regularization factor. Mn indicated that Mn complex conjugate. We have following result from (0-17).Assume A is the auxiliary curve of period 2πaboveГ. In the admissible class U, we are minimizing the following formula where ||·|| is the Euclidean norm, S and A in U.(3) An integral equation method for two-dimensional scattering by obstacles in a homogeneous chiral environment Let us consider the time-harmonic (time dependence e-iωt) Maxwell's equations where E, H, D and B denote the electric field, the magnetic field, the electric and magnetic displacement vectors in R3, respectively. For chiral media, E,H,D and B satisfy with the Drude-Born-Fedorov constitutive equations: whereε> 0 is the electric permittivity,μ> 0 is the magnetic permeability, andβ≥0 is the chirality admittance. Further, the Maxwell's equations may be written as Sinceβis generally small, we make the additional assumption that kβ< 1.Assume that an infinitely long cylinder parallel to the x3-direction. The space outside D is filled with homogeneous chiral medium, and the chiral or a chiral medium inside D is also homogeneous. Here D is the cross-section of D in the x1-X2-plane. We assume that D is simply connected with C2,α-boundary (?)D,α∈(0,1).Consider plane waves with pL·qL=0, pL×qL=-iqL, pR·qR=0,pR×qR=iqR incident on the cylinder, whereThe scattering problem is to find the solutions E, H of equations (0-19), which and the radiation conditions uniformly for all directions x/|x|. In the chiral environment, we consider the de-composition of ES,HS into suitable Beltrami fields Assume that the electromagnetic fields E and H depend only on x1 and x2 variable. We can easily obtain the differential equations and the radiation conditions where u and v are the x3-components of QLs and QRs,respectively.r=|x|.Here△= the two-dimensional Laplace operator.The x1-and x2-components of QLs and QRs can be expressed in terms of the x3-components.In the obstacle,we can similarly deduce the differential equations for the x3-components of QL and QR.respectively, where QL and QR are the Beltrami fields in D with E=QL+QR,H=-1η1-1(QL-QR),andThe continuity of the tangential components of the fields E and H across the boundary leads to the following transmission conditions for u,v,a and b On (?)D: whereλ=η0/η1,n is the unit outward normal to (?)D. Here uI=(eI+1η0hI)/2,vI= (eI-1η0hI)/2,eI and hI are the x3-components of EI and HI,respectively.Now,the direct scattering problem can be reformulated as follows.Problem 3. For given incident plane waves eI and hI,find u,v which satisfy the differential equations(0-20), (0-21) and (0-24), (0-25), the boundary conditions (0-26) and the radiation condition (0-22), (0-23).For problem 3, we have the following uniqueness result.Theorem 0.7 Problem 3 has at most one solution.For the existence of a solution, we use a combination of single-and double-layer potentials. And we will show the corresponding second kind integral equations admits a unique solution.We denote by the fundamental solution of the Helmholtz equation with wave number k, where H01(K|x-y|) is the Hankel function of the first kind of order zero.Define a single-layer potential by and a double-layer potential whereΦandψare density functions. We also need the following boundary integral operators Sk, Kk,Kk*:C((?)D)→C((?)D), and T*k:C1,α((?)D)→C((?)D),We search for the solution to Problem 1 in the form with unknown densitiesΦ1,Φ2∈C0,α((?)D),ψ1,ψ2∈C1,α((?)D),α∈(0,1) and real constants c1≠0, c2≠0.The jump relations for single-and double-layers, and the boundary conditions (0-26) lead to the following integral equations:By the continuous dependence of the densities on the boundary data and the dependence of the potentials on the densities(see [49, Theorem 3.3,3.4]), it follows that where constant C is independent of the data. We summarize these results in the following theorem.Theorem 0.8 Problem 3 has a unique solution. For real constants c1≠0, C2≠0, the solution of problem 3 can be expressed by (0-27), where (Φ1,Φ2,ψ1,ψ2)T is the unique solution of (0-28). The solution depends continuously on the data as expressed in (0-29).