Analysis And Computation Of Some Mathematical Problems In Electromagnetic Scattering And Inverse Scattering For Chiral Media

Acoustic and electromagnetic scattering and inverse scattering problems is an important field in mathematical physics. The direct scattering problem is to determine the scattering field from a knowledge of the incident field and the differential equation governing the wave motion. The corresponding references are [8] [9] [24] [29] [34] [35] [36] [37] [61] [65] [67] etc.. The inverse scattering problem is to reconstruct the differential equation or the domain of definition from a knowledge of the behavior of the scattering solution. For the references, we refer to [5] [10] [11] [15] [46] [60] [62]etc.Recently, electromagnetic scattering and inverse scattering problems in special media have been payed attention. Chiral media have attracted much attention due to many associated intersting phenomena in optical and elecromagnetic activities and their potential applications in various fields. Ammari, Nédélec, Gerlach, Ammari, G.Bao, Athanasiadis and D.Y.Zhang, F.M.Ma, e.t.[l] [2] [42] [71] [72] have studied some concerned problems.In chiral media, time-harmonic electromagnetic fields are governed by the Maxwell equations and the Drude-Born-Fedorov constitutive equationswhere x∈R³, E,H,D and B are the electric field, the magnetic field, the electric and magnetic displacement vectors in R³, respectively,εis the electric permittivity, is the wave number,μis the magnetic permeability, andβis the Chirality admittance.Together with (1) and (2), we havewhere , we assume kβ< 1.Because of complexity of differential equations (e.g. the coupling of electric field and magnetic field) and the character of computation domain (e.g. unboundedness), there is a lot of difficulties in computational methods. To find effective computational methods is an important subject.In this paper, the theoretical analysis and numerical methods of electromagnetic scattering and inverse scattering problems for chiral media are discussed. What we have done is as follows:(â… ). The problem of electromagnetic scattering and inverse scattering by a perfctly conducting obstacle in chiral media.In chiral media, plane wave Eⁱ and Hⁱ incident on a perfectly conducting obstacle D|～. D| = {x∈R³|(x₁, x₂)∈D, x₃∈R} is an infinitely cylinder parallel to the x₃-direction. Here D is the cross-section of D|～in the x₁ - x₂ plane, D is simply connected with C^{2, -α} boundary (?)D,α∈(0,1). In chiral media, left-handed and right-handed fields can both propagate independently and with different phase speed [55]. To see this, we consider the decomposition of E^s, H^s into suitable Beltrami fields Q_L, Q_R:E = Q_L + Q_R, H = -i(Q_L - Q_R).Assuming additionally that all the fields do not depend on the x₃ variable, denote u^s, v^s the x₃-components of Q_L^s and Q_R^s of the scattering wave E^s and H^s, uⁱ, vⁱ the x₃-components of Q_Lⁱ and Q_Rⁱ of the incident wave Eⁱ and Hⁱ respectively, we obtainthe boundary conditionand uniformly for all directions x/|x|whereη= x/|x|, r = |x|, x∈R².The problem of electromagnetic scattering by a perfctly conducting obstacle in chiral media can be stated as follows:Problem 1. For given incident waves uⁱ, vⁱ with the directions p_l, p_r, find u^s,v^s∈C²(R²\D|-)∩C¹(R²\D) which satisfy the differential equation (4), the boundary condition (5) and the radiation condition (6). Depent on Rellich Lemma, we have the uniqueness resultTheorem 1. Problem 1 has at most one solution.we search the solutions u^s, v^s in the form:with unknown continuous densities ,α∈(0,1), and real parameters c₁≠0, c₂≠0. Here S₀ is the single-layer potential of the fundamental solutionΦ₀(x,y).Let x→(?)D, the jump relations for single- and double-layers and their normal derivatives and the boundary condition (5) lead to the following integral equation:The integral equation (8) has at most one solution .Theorem 2. Problem 1 has a unique solution. For constants c₁≠0, c₂≠0, the solution of problem 1 can be expressed by (7), where is the unique solution of (8). The solution depends continuously on the data (5).Next we give the inverse scattering problem of problem 1.Problem 2. Given the far fields u_∞(·; p_l), v_∞(·; p_r) corresponding to the solutions u^s(·; p_l), v^s(·; p_r) of problem 1 for all incident directions p_l, p_r, with 0 <θ_l,θ_r < 2Ï€find the shape of the scatterer D.Then we obtaion the uniquness of solution for the inverse scattering problem 2.Theorem 3. Assume that the far field patterns coincide, i.e. for all incident plane fields corresponding to distinct directions p_l, p_r with 0 <θ_l,θ_r < 2Ï€and for fixed wave numbers, then D₁ = D₂. (â…¡). The problem of electromagnetic scattering by a bounded chiral obstale.is an infinitely cylinder filled with chiral media parallel to the x₃-direction. Here D is the cross-section of D|～in the x₁-x₂ plane, D is simply connected with C^2,α boundary (?)D,α∈(0,1). Around the cylinder there is the media withεandμ.We assume that the electromagnetic fields are invariant in the x₃ direction. Then all variables in electromagnetic fields are independent of x₃. The x₁- and x₂- components can be expressed in terms of the x₃-components, so the above problem is transformed into a two-dimensional problem.Let E = (e₁,e₂,e₃)^T, H = (h₁, h₂, h₃)^T. eⁱ, hⁱ is the x₃-component of Eⁱ, Hⁱ. Denote e₃ = eⁱ + eⁱ, h₃ = hⁱ + h^s. Through Maxwell equations and the constitutive equations we obtain the differentional equations of e^s and h^s. Let u^s = (e^s,h^s)^T, and define , we have the vector equation as followingwhereand f = (f₁, f₂)^T.And the scattering fields uniformly satisfy the radiation conditon in alldirections whereη= x/|x|, r = |x|.The scattering problem by a bounded chiral obstacle is as following: Problem 3. Given the incident waves uⁱ = (eⁱ, hⁱ)^T incident on a bounded chiral obstacle D, find u^s = (e^s, h^s)^T satisfy (9), and the radiation condition (10).Assume that B_R is the circular domain of radius R which center is the center of domain D and R is sufficient large such that D is involved, (?)B_R is the boundary of B_R. There is vacuum with electric permittivityε₀ and the magnetic permeabilityμ₀ in R²\B_R.By the introduce of the boundary truncate operator Twhere x = re^iθ, , andLemma 1. There exists operator T : , such thatwhere the operator is defined as (11).we have the variational problem of Problem 3 on B_R:Find w = (e^s, h^s)^T∈H¹(B_R)×H¹(B_R) satisfywhereThrough Green's theorem and theradiation condition, we obtain the uniqueness of the variational problem: Theorem 4. The variational problem (12) has at most one solution.By the analysis of the operator corresponding to the sesquilinear forms W and Fredholm alterative, we prove the theorem:Theorem 5. For all k, the variational problem (12) admits a solution u^s∈H¹(B_R)×H¹(B_R).Then we consider the vector PML equation of the chiral obstacle problem.By the PML technique, construct the PMLΩ^PML of Problem 3. OnΩ_p =Ω^PML∪B_R, the PML solution u|^ satisfy:whereNote thatwhere A the matrix function in PML of Helmholtz equation in polar coordinates,andζ,ζ|- are the coresponding coefficients in PML.We have studied the well-posedness of the variation of the PML problem. Next the convergence of PML solution to the solution of the original problem will be given. Theorem 6. For all but possibly a discrete set of wave number k, the variational problem of the PML problem admits a unique solution u|^ in H₀¹(Ω_ρ)×Theorem 7. Under the assumption of Theorem 6, we have the following estimate:where ,α₀ = 1 + iσ₀, the constant C is independent of k, R,ρ,σ₀.Then we present several numerical results of electromagnetic scattering by a obstacle. Through the numerical experiments, the property of the chiral media is clearly shown.(â…¢). The inverse electromagnetic scattering problem by a bounded chiral obstacle.To partly reduce the coupling of the electric and magnetic fields, we use the decomposition similar to Bohren's decomposition and introduce new mathematical model of elecromagnetic scattering by a bounded chiral obstacle. u_c is the electromagnetic in D, u is the total field in R²\D|-,and the boundary conditon, Hereα_a = k_c/(1 - k_cβ),α_b = k_c/(1 + k_cβ),ξ= (ε_cμ)^1/2,ξ₊ =(ξ+ξ^-1)/2 andξ_- = (ξ-ξ^-1)/2.The scattered field u^s satisfies uniformly for all directions x/|x|:,The scattering fields u^s(x) have an asymptotic behaviorDefine u_∞= (a_∞, b_∞)^T is the far field pattern of u^s,where d∈Ω, is the incident direction of incident electromagnetic.From Green's representation and (20), we can obtain the following property of far field pattern of scattering fields by chiral obstacle,Lemma 1. uⁱ is the plane incident field , let u_∞(x|^,d) be the far field pattern corresponding to the scattering of (15)-(19), thenwhere d is the incident direction of uⁱ.The inverse electromagnetic scattering problem by a bounded chiral obstacle is as follows:Problem 4. For given the incident field uⁱ(x, d), and the far field u_∞(x|^, d), d∈Ω, we want to reconstruct the boundary of the chiral obstacle (?)D.The following result on the uniqueness of the inverse scattering problem has been given. Theorem 8. The D₁, D₂ with the interior constantsε_0;j andβ_{0; j}, j=1,2 be the two scatters satisfying the conditions given for the direct problem, and assume thatα_a;j≠kξ_j orα_b;j≠kξ_j. Then, if the far fields to D₁ and D₁ coincide for all incident directions, i.e. a_∞,1(.; d) = a_∞,2(.; d) and b_∞,1(.;d) = b_∞,2(.; d) for all d∈Ω, the scatters coincide D₁ = D₂.By Green's Theorem and the theory of potentials, introduce new integral operators T₁ and T₂.Then we give the integral expreesion of mathematical model of elecromagnetic scattering by a bounded chiral obstacle:We show that the scattering problem (15)-(19) is equivalent to the problem of solving the integral equation (21)-(22).In order to study the theory of construct the boundary of chiral obstacle, we introduce interior transmission problemthe interior transmission problem: where is the fundermenal solution of vector Helmholtz equation, and , r > 0 is sufficient small, y* is on (?)D.Theorem 9. If u_c,n and u_n are the sequences of solutions of problem (23)-(26), thenThrough Theorem 9 we obtain the main theory of reconstruct chiral obstacle:Theorem 10. For , there exists g = g(·, y_n)∈L²(Ω)×L²(Ω), satisfiesand . Moreover, if v_g is the Herglotz vector wavefuction defined bythen also .In numerical experiment, we give the numerical method to reconstruct the chiral obstacle by Theorem 1. And by the operator factorization method developed on the linear sampling method we also give the numerical example. Due to the complexity and the lack of a mathematical theory of problems in optics and electromagnetism, especial in chiral media, there are still many problems to solve.