The Imaging of Anisotropic Media in Inverse Electromagnetic Scattering

Transcription

1 The Imaging of Anisotropic Media in Inverse Electromagnetic Scattering Fioralba Cakoni Department of Mathematical Sciences University of Delaware Newark, DE 19716, USA Research supported in part by grants from AFOSR and NSF

2 Introduction Many materials in real life exhibit anisotropic properties to electromagnetic interrogation. Mathematically, the presence of anisotropy raises interesting and challenging questions. Can qualitative methods in inverse scattering provide information about the physical properties of the scatterer in addition to the support? We will discuss here two problems: Inverse electromagnetic scattering for a mixed Dirichlet - impedance boundary value problem with matrix impedance function. Inverse electromagnetic scattering problem for an anisotropic inhomogeneous medium

3 Partially Coated Obstacles i E D! I! D " s E We first consider the scattering by a perfect conductor partially coated with a thin layer of anisotropic conducting material. The first order approximate model for the forward scattering problem leads to an exterior mixed Dirichlet - impedance boundary value problem with matrix surface impedance.

4 Direct Scattering Problem The direct scattering problem reads: E k 2 E = 0 in R 3 \ D ν E = 0 on Γ D ν ( E) ikλ (ν E) ν = 0 on Γ I E(x) = E s (x) + E i (x) lim ( E s x ik x E s ) = 0 x where E i (x) = ik(d p) d e ikx d, k 2 = ɛ 0 µ 0 ω 2 and Λ is the real valued tensor impedance which maps a vector tangential to Γ I at a point x to a vector tangential to Γ I at the same point x.

5 Tensor Impedance Assume that D is Lipschitz curvilinear polyhedron. For x Γ j I (a smooth face), let ν(x) be the outward unit normal, and ˆt 1 (x) and ˆt 2 (x) two unit perpendicular tangential vectors. If ξ(x) = αˆt 1 (x) + βˆt 2 (x) then Λ(x) ξ(x) = (αλ 11 (x) + βλ 21 (x))ˆt 1 (x) + (αλ 12 (x) + βλ 22 (x))ˆt 2 (x). Λ L (Γ I ) C(Γ j I ), real valued, symmetric i.e. λ 12 = λ 21 and uniformly positive definite i.e. ξ (x)λ(x) ξ(x) = α 2 λ 11 (x) + 2αβλ 12 (x) + β 2 λ 22 (x) γ ξ(x) 2 where γ > 0 is a constant independent of x.

6 The Inverse Problem The scattered field E s, which is in H loc (curl, R 3 \ D) with ν E s in L 2 t (Γ I), has the asymptotic behavior E s (x) = eik x x { ( )} 1 E (ˆx, d, p) + O x where E (, d, p) is the far field pattern defined on the unit sphere Ω. The inverse scattering problem is Determine both D and Λ from a knowledge of E (ˆx, d, p) for d, ˆx Ω 0 Ω and two linearly independent polarizations p tangential to Ω

7 Uniqueness Uniqueness of D For fixed k, E (ˆx, d, p) for d, ˆx Ω 0 Ω and two linearly independent polarizations p perpendicular to d uniquely determine D Uniqueness of Λ For fixed k, E (ˆx, d, p) for ˆx, d Ω 0 Ω and two linearly independent polarizations p perpendicular to d, uniquely determine Γ I and the tensor Λ.

8 Uniqueness Assume that D is known. The far field pattern corresponding to one incident wave does not uniquely determine the matrix Λ (uniquely determines Γ I ) Theorem Given the far field pattern corresponding to one incident wave E i, there exist a matrix Λ depending on the geometry of D and the corresponding total field E i + E s, and a unknown scalar function a defined on Γ I such that Λ = a Λ on Γ I. Conjecture: The far field pattern corresponding to two (appropriately chosen) incident waves uniquely determine Λ The incident waves must be such that the corresponding total electric fields have linearly independent tangential components almost everywhere on Γ I.

9 Far Field Equation Reconstruction is based on the far field equation (F g)(ˆx) = E e, (ˆx, z, q) where E e, (ˆx, z, q) denotes the far field pattern of the electric dipole E e (x, z, q) := i k x x q 1 e ik x z 4π x z, and the far field operator F : L 2 t (Ω) L2 t (Ω) is defined by (Fg)(ˆx) := E (ˆx, d, g(d)) ds(d) Ω which is the far field pattern produced by the Herglotz wave function E g ɛ z (x) := ik e ikx d gz(d)ds(d). ɛ Ω

10 Solution of Far-Field Equation Theorem For z D and an arbitrary ɛ > 0 there exists an approximate solution g ɛ z of the far field equation Fg ɛ z E e, (, z, q) L 2 t (Ω) < ɛ is such that lim ɛ 0 E g ɛ z L 2 (D) exists For z / D every approximate solution g ɛ z Fg ɛ z E e, (, z, q) L 2 t (Ω) < ɛ is such that lim ɛ 0 E g ɛ z L 2 (D) and lim ɛ 0 g ɛ z L 2 (Ω).

11 Solution of Far-Field Equation Theorem (cont.) Furthermore, for z D, E g ɛ z converges to the unique solution E z of E z k 2 E z = 0 in D ν [E z + E e (, z, q)] = 0 on Γ D ν [E z + E e (, z, q)] ikλ ν [E z + E e (, z, q)] ν = 0 on Γ I

12 The Linear Sampling Method The support D can be reconstructed using this behavior of the approximate (regularized) solution of the far field equation. In particular, Construct a grid G and for z i G, solve (αi + F F )g zi,q = F E e, For z i G and three linearly independent q 1, q 2, q 3 R 3 evaluate G(z i ) = 1 3 ( ) g zi,q g l 2 zi,q g l 2 zi,q 3 1. l 2 Choose an appropriate C > 0 and visualize the boundary by plotting G(z) = C max z i G G(z i).

13 Determination of Λ Recall that the Herglotz wave function E gz with kernel g z the (regularized) solution of the far field equation converges to E z. Two important properties of W z := E z + E e (, z, q) E gz + E e (, z, q). Let z 1, z 2 D. Then 2 (ν W z1 ) Λ (ν W z2 ) ds = q 2 A(z 1, z 2, k, q) Γ I +k(q E z1 (z 2 ) + q E z2 (z 1 )) where A(z 1, z 2, k, q) is a computable. The set of ν W z ΓI for z in a ball B r D is complete in L 2 t (Γ I). Note that Γ D := {z D : ν W z = 0}.

14 Examples of Reconstruction λ = 1, wavelength= 0.7

15 Examples of Reconstruction / g z z z z

16 Examples of Reconstruction Face ( Exact Λ ) ( Matrix Reconstruction ) x =.5 (.05.1 ) ( ) y =.5 ( ) ( ) z =.5 ( 0 1 ) ( ) z =.5 ( 0.3 ) ( ) y =.5 ( 0.4 ) ( ) x =

17 Literature These results are presented in F. Cakoni and P. Monk, The determination of anisotropic surface impedance in electromagnetic scattering, J. Methods and Applications of Analysis, 17, no 4, (2010). which generalizes the approach in F. Cakoni and D. Colton, The determination of the surface impedance of a partially coated obstacle from far field data, SIAM J. Appl. Math. 64, (2004).

18 The Scattering Problem Consider now the scattering by an inhomogeneous anisotropic dielectric media. The goal is to determine the support of inhomogeneity and information about the anisotropy. E i E s! D The relative electric permittivity N(x) = ɛ(x), x D, is a positive definite symmetric matrix-valued function with bounded entries. The magnetic permeability is constant µ 0 and the wave number k > 0 is such that k 2 = ɛ 0 µ 0 ω 2 where ω > 0 is the frequency. ɛ 0

19 Maxwell s Equations After eliminating the magnetic field, the scattering problem can be written in terms of electric fields as: E s k 2 E s = 0 E k 2 N(x)E = 0 ν (E s + E i ) = ν E ν (E s + E i ) = ν E in R 3 \ D in D on D on D where the scattered electric field satisfies the Silver-Muller radiation condition lim x ( E s x ik x E s ) = 0. and E i (x, d, p, k) = i k p eikx d.

20 Inverse Problem The inverse scattering problem is Determine both D and (some information about) N from a knowledge of E (ˆx, d, p) for d, ˆx Ω 0 Ω and two linearly independent polarizations p perpendicular to d, and a range of wave numbers k [k 0 k 1 ] Theorem (Uniqueness) If N 2 > 1 or N 1 2 > 1. Then D is uniquely determined from the above data. There is no uniqueness proof for the determination of the matrix N for the above data.

21 Far Field Operator Again, the scattered field E s has the asymptotic behavior ( ) E s (x, d, p, k) = eikr 1 E (ˆx, d, p, k) + O r r 2 as r, ˆx = x/ x, r = x. Then far field operator F : L 2 (Ω) L 2 (Ω) (Fg)(ˆx) := E (ˆx, d, g(d), k) ds(d) Ω is the far field pattern of the scattered field corresponding to the electric Helglotz wave function E g (x) := e ikx d g(d) ds(d) as incident field. Ω

22 Transmission Eigenvalues Theorem The far field operator F : L 2 (Ω) L 2 (Ω) is injective and has dense range if and only if there does not exists a nontrivial solution to the homogeneous interior transmission problem E k 2 N(x)E = 0 in D E 0 k 2 E 0 = 0 in D ν E = ν E 0 on D ν ( E) = ν ( E 0 ) on D such that E 0 := E g is an electric Herglotz wave function with kernel g.

23 Transmission Eigenvalues Values of k for which the homogenoeus interior transmission problem has a non trivial solution are called transmission eigenvalues. Transmission eigenvalues are related to non scattering waves if E 0 can be extended as a solution to the Maxwell s equation outside D then this extension is an incident wave that does not scatterer. Since Herglotz wave functions E g (x) := ik e ikx d g(d)ds(d), for g L 2 t (Ω) Ω are dense in the space { W L 2 (Ω) : W k 2 W = 0 } then at a transmission eigenvalue there is an incident field that produces arbitrarily small scattered field.

24 Far Field Equation We again consider the far field equation (F g)(ˆx) = E e, (ˆx, z, q) where E e, (ˆx, z, q) denotes the far field pattern of the electric dipole E e (x, z, q) := i k x x q 1 e ik x z 4π x z, and the far field operator F : L 2 t (Ω) L2 t (Ω) is defined by (Fg)(ˆx) := E (ˆx, d, g(d)) ds(d). Ω

25 Solution of Far Field Equation Theorem 1 Let z D. If k is not a transmission eigenvalue then for every ɛ > 0 there exists g z,ɛ,k L 2 (Ω) satisfying Fg z,ɛ,k E e, L2 (Ω) < ɛ and lim ɛ 0 E gz,ɛ,k L2 (D) = E 0 L2 (D). If k is a transmission eigenvalue, every g z,ɛ,k L 2 (Ω) satisfying Fg z,ɛ,k E e, L 2 (Ω) < ɛ is such that lim ɛ 0 E gz,δ,k L 2 (D) =. 2 Let z / D. For all k > 0, every g z,ɛ,k L 2 (Ω) satisfying Fg z,ɛ,k E e, L 2 (Ω) < ɛ is such that lim ɛ 0 E gz,ɛ,k L 2 (D) =.

26 Solution of Far Field Equation In practice one has available the noisy far field operator F δ, (F δ g)(ˆx) := E (ˆx, δ d, g(d), k) ds(d) Ω i.e. the far field operator with kernel the noisy measurements with δ being the noise level. For various of z R 3 and k > 0, g z,k is computed using Tikhonov regularization technique, i.e by solving α(δ) 0 as δ 0. (α(δ)i + F δ F δ )g δ z,k = F δ E e, (, z, q) The behavior of g in the above theorem is inherited by g δ z,k as δ 0. This behavior for different z and k can be used to solve the inverse problem.

27 Reconstruction of the Support D For fixed frequency k > 0 not a transmission eigenvalue we have g z,k ( ) L2 (Ω)is bounded for z D and very large for z / D. Using band limit multifrequency data for k [k 0, k 1 ]. Note: If [k 0, k 1 ] contains at least a transmission eigenvalue than g z,k ( ) L2 (Ω) as function of k is not integrable in [k 0, k 1 ]. In this case we can use G(z) := k1 as an indicator function for D. 1 k 0 g z,k ( ) L 2 (Ω)

28 Examples of Reconstruction N = 5I k is not a TE k is a TE

29 Examples of Reconstruction N = 5I k is not a TE Reconstruction for k [2, 3.5]

30 Examples of Reconstruction N = 16I, k is not TE

31 Examples of Reconstruction N = 16I, k is a TE

32 Computation of Real Transmission Eigenvalues Results for an isotropic sphere of unit radius. Solving the far-field equation for several source points z inside the sphere (using 42 incoming directions and 42 measurements) gives obvious peaks at the first transmission eigenvalue. 0.7 averaged over 81 sources 5 averaged over 81 sources L 2 norm of g L 2 norm of g Wave number k N = 16I Wave number k N = 5I Red dots indicate exact transmission eigenvalues. Using the far field pattern, peaks in g indicate transmission eigenvalues.

33 Non-destructive Testing of Anisotropic Materials What do transmission eigenvalues say about N(x) = ɛ(x)/ɛ 0? To this end let λ 1 (x) λ 2 (x) λ 3 (x) be the eigenvalues of N and define λ min = inf λ 1(x) and λ max = sup λ 3 (x). x D Let k 1,D,N(x) be the first transmission eigenvalue for our scattering problem and let k 1,D,λmin and k 1,D,λmax be the first transmission eigenvalues for the case of N = λ min I and N = λ max I, respectively. x D

34 Non-destructive Testing of Anisotropic Materials Theorem Let α and β be positive constants. 1 If N(x) 2 α > 1 then k 1,D,λmax k 1,D,N(x) k 1,D,λmin 2 If N(x) 2 β < 1 then k 1,D,λmin k 1,D,N(x) k 1,D,λmax Given k 1,D,N(x) we now compute a constant n such that k 1,D,N(x) = k 1,D,n. Then the above theorem implies that λ min n λ max and that k 1,D,n depends monotonically on n. It can in fact be shown that k 1,D,n depends strictly monotonically on n.

35 Numerical Examples 1.18 N=diag([16 a,16,16+a]) 2.7 N=diag([5 a,5,5+a]) Lowest real transmission eigenvalue N=diag([16,16,16+a]) Lowest real transmission eigenvalue N=diag([5,5,5+a]) Perturbation parameter a Perturbation of N = 16I Perturbation parameter a Perturbation of N = 5I

36 Numerical Examples (cont) Having the first transmission eigenvalue for anisotropic N then compute the isotropic n discussed previously Lowest real transmission eigenvalue λ min n λ max Isotropic Permittivity N Lowest transmission eigenvalue against N (isotropic) N k 1,D,N(x) n diag([15.5, 16, 16.5]) diag([15, 16, 17]) diag([16, 16, 16.5]) diag([16, 16, 17])

37 Numerical Examples (cont) 2.8 The same procedure can be carried out at lower N Lowest real transmission eigenvalue Isotropic Permittivity N Lowest transmission eigenvalue against N (isotropic) N λ 1,D,N(x) n diag([4.5, 5, 5.5]) diag([4, 5, 6]) diag([5, 5, 5.5]) diag([5, 5, 6])

38 Absorbing Media There is no material that is a perfect dielectric. So what happens to the above theory if there is a small amount of absorption in D? Let N 1 (x) = ɛ(x) + i σ(x) describe the absorbing media. Using ɛ 0 ɛ 0 ω Kato s perturbation theory for linear operators, it is possible to prove the following theorem. Theorem Let k > 0 be a real transmission eigenvalue corresponding to N(x) = ɛ(x). Then if σ(x) < ɛ for ɛ sufficiently small there exists at ɛ 0 least one (complex) transmission eigenvalue near k corresponding to N 1 (x).

39 Numerical Example: Absorbing Media Norm of g Wave number k D is a sphere of diameter 1 and N = (16 + i)i

40 Anisotropic Media with Cavities Can the assumption N > I or 0 < N < I in D be relaxed? D Do Do N = I in D 0 N I δ > 0 in D \ D 0 The case when there are regions D 0 in D where N = I (i.e. cavities) is more delicate but the same type of analysis as above can be carried through.

41 Anisotropic Media with Cavities Let k 1,D,D0,N(x) be the first transmission eigenvalue for D, D 0, N(x). Monotonicity with respect to the index of refraction k 1,D,D0,N(x) k 1,D,D0,Ñ(x), Ñ(x) N(x). Monotonicity with respect to voids k 1,D,D0,N(x) k 1,D, D0,N(x), D 0 D 0. Other type of inclusions such as a perfect conductor or crack could also be considered inside the inhomogeneity.

42 6.2.5 Numerical examples Numerical Example: Media with Cavities We first compute the transmission eigenvalues for a spherical geometry and we compare the values with the values computed from analytical solutions to the ITP (see Appendix F). It is represented in figure We then add a cavity inside the sphere with the form of a box. One can remark that the first transmission eigenvalue as been shifted to the right. Due to A. Cossonniere, Ph.D. Thesis Figure 6.20: Sphere of radius 1 and index of refraction N =4. The sphere with radius 1 and N = 4I k 1 = 3.16

43 Numerical Example: Media with Cavities Figure 6.20: Sphere of radius 1 and index of refraction N =4. Due to A. Cossonniere, Ph.D. Thesis LSM Integral equations Figure 6.21: Sphere of radius 1 and index of refraction N =4containing a cavity of the form of a box. We can remark that the first transmission eigenvalues is shifted comparing to the previous example. The sphere with radius 1, N = 4I containing a cubic cavity k 1 = 3.33, i.e. k 1 is shifted to the right

44 Case with Contrast in ɛ and µ If µ µ 0 in D then the corresponding transmission eigenvalue problem written in terms of magnetic fields is A H k 2 nh = 0 in D H 0 k 2 H 0 = 0 in D ν H = ν H 0 on D ν (A H 0 ) = ν ( H 0 ) on D. where A 1 = ɛ(x)/ɛ 0 and n = µ/µ 0 Here the electric permittivity ɛ(x) in the medium is a positive definite matrix with bounded entries wheres the magnetic permeability µ µ 0 in the medium is a constant.

45 Case with Contrast in ɛ and µ The support D can be reconstructed by using the Linear Sampling Method. The following results can be obtained on transmission eigenvalues: If the contrasts A I and n 1 have the same fixed sign, then there exists an infinite discrete set of real transmission eigenvalues accumulating at +. If the contrasts A I and n 1 have the opposite fixed sign, then there exits at least one real transmission eigenvalue providing that n is small enough. The first transmission eigenvalue in this case provides estimates for ɛ(x)/µ.

46 Literature F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42, no 1, (2010). F. Cakoni, D. Colton, P. Monk and J. Sun, The inverse electromagnetic scattering problem for anisotropic media, Inverse Problems, 26, (2010). F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem, Int. Jour. Comp. Sci. Math., 1-2, , (2010). F. Cakoni, D. Colton and H. Haddar, The interior transmission problem for regions with cavities, SIAM J. Math. Anal. 42, no 1, (2010). F. Cakoni, D. Colton and H. Haddar, The interior transmission eigenvalue problem for absorbing media, Inverse Problems, (to appear).

47 Literature Most of the results on the Qualitative Methods for Maxwell s equations can be found in the following monograph: F. Cakoni, P. Monk and P. Monk, The linear Sampling Method in Inverse Electromagnetic Scattering, CBMS Series, SIAM Publications 80, (2011). A comprehensive study on transmission eigenvalues for Maxwell s equation can be found in the following thesis: A. Cossoniere, Transmission Eigenvalues for Maxwell s equation and their Application in Non-destructive Testing, Ph.D Thesis (2011).