Analysis of the anomalous localized resonance

Transcription

1 Hyundae Lee(Inha University, Korea) Joint work with Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Graeme Milton. UCI, June 22, 2012 A conference on inverse problems in honor of Gunther Uhlmann

2 Outline Introduction Integral operators and its symmetry Spectral analysis of ALR ALR in annulus region

3 Surface plasmon Let ǫ = { 1 in {(x,y) : y 0}, 1 in {(x,y) : y < 0}. Consider ǫ u = 0 in R 2. Then one solution is u = { e y+ix in {(x,y) : y 0}, e y+ix in {(x,y) : y < 0}.

4 Let Ω be a smooth domain in R 2 and let D Ω. The permittivity distribution in R 2 is given by 1 in R 2 \Ω, ǫ δ = 1+iδ in Ω\D, 1 in D. 1 1+iδ 1 f

5 Problem For a given function f compactly supported in R 2 satisfying R 2 fdx = 0, we consider the following equation: ǫ δ V δ = f in R 2, with decaying condition V δ (x) 0 as x. Since the equation degenerates as δ 0, we can expect some singular behavior of the solution, depending on the source term f.

6 Milton-Nicorovici(2006) Figure: Anomalous resonance, Milton et al (2006). Energy concentration near interfaces, depending on the location of source. Associated with the cloaking effect of polarizable dipole. Generalized to a small inclusion with a specific boundary condition by Bouchitté and B. Schweizer(2010).

7 Numerical simulation by Bruno-Linter(2007). There is some cloaking effect even in the presence of a small dielectric inclusion, not perfect. Blow-up may not depend on the location of the source in a layer of general shape.

8 A fundamental problem is to find a region Ω containing Ω such that if f is supported in Ω \Ω, then δ V δ 2 dx as δ 0. Ω\D Such a region Ω \Ω is called the anomalous resonance region or cloaking region. The quantity Ω\D δ V δ 2 dx is a part of the absorbed energy. The blow-up of the energy may or may not occur depending on f. So the problem is not only finding the anomalous resonance region Ω \Ω but also characterizing those source terms f which actually make the anomalous resonance happen.

9 Relation to cloaking Suppose f is a polarizable dipole at x 0, i.e., V δ (x) = U δ (x)+a δ G(x x 0), A δ = k U δ (x 0), for some given coefficient k. If ALR happens, then we should have A δ 0 as δ 0. Otherwise Ω\D δ V δ 2 dx blows up, which is not physical.

10 Let F be the Newtonian potential of f, i.e., F(x) = G(x y)f(y)dy, x R 2. R 2 Then F satisfies F = f in R 2, and the solution V δ may be represented as V δ (x) = F(x)+S Γi [ϕ i](x)+s Γe [ϕ e](x) for some functions ϕ i L 2 0(Γ i) and ϕ e L 2 0(Γ e) (L 2 0 is the collection of all square integrable functions with the integral zero). The transmission conditions along the interfaces Γ e and Γ i satisfied by V δ read ( 1+iδ) V δ = V δ on Γ i ν + ν V δ = ( 1+iδ) V δ on Γ e. ν + ν

11 Using the jump formula for the normal derivative of the single layer potentials, the pair of potentials (ϕ i,ϕ e) is the solution to z δ I KΓ i S Γi ν e on L 2 0(Γ i) L 2 0(Γ e), where we set S Γe ν i z δ I +KΓ e z δ = [ ϕi ϕ e iδ 2(2 iδ). ] = F ν i F ν e Note that the operator can be viewed as a compact perturbation of the operator [ ] zδ I KΓ i 0 0 z δ I +KΓ. e.

12 We now recall Kellogg s result on the spectrums of K Γ i and K Γ e. The eigenvalues of K Γ i and K Γ e lie in the interval ] 1 2, 1 2 ]. Observe that z δ 0 as δ 0 and that there are sequences of eigenvalues of K Γ i and K Γ e approaching to 0 since K Γ i and K Γ e are compact. So 0 is the essential singularity of the operator valued meromorphic function λ C (λi +K Γ e ) 1. This causes a serious difficulty in dealing with (11). We emphasize that K Γ e is not self-adjoint in general. In fact, K Γ e is self-adjoint only when Γ e is a circle or a sphere.

13 Properties of K Let H = L 2 (Γ i) L 2 (Γ e). Let the Neumann-Poincaré-type operator K : H H be defined by K := K Γ i S Γi ν e Then the integral equation can be written as S Γe ν i. K Γ e (z δ I+K )Φ δ = g and the L 2 -adjoint of K, K, is given by [ ] KΓi D Γe K =. D Γi We may check that the spectrum of K lies in the interval [ 1/2,1/2]. K Γe

14 Let S be given by [ ] SΓi S Γe S =. S Γi S Γe The operator S is self-adjoint and S 0 on H. The Calderón s identity is generalized. SK = KS, i.e., SK is self-adjoint. K C 2(H), Schatten-von Neumann class of compact operators.

15 We recall the result of Khavinson et al(2007) Let M C p(h). If there exists a strictly positive bounded operator R such that R 2 M is self adjoint, then there is a bounded self-adjoint operator A C p(h) such that AR = RM. Theorem There exists a bounded self-adjoint operator A C 2(H) such that A S = SK.

16 Limiting properties of the solution ALR occurs if and only if δ V δ 2 dx δ Ω\D Ω\D (S Γi [ϕ δ i ]+S Γe [ϕ δ e]) 2 dx as δ. One can use A S = SK to obtain (S Γi [ϕ δ i ]+S Γe [ϕ δ e]) 2 dx = 1 2 Φ δ,sφ δ + K Φ δ,sφ δ Ω\D = 1 2 SΦ δ, SΦ δ A SΦ δ, SΦ δ.

17 Since A is self-adjoint, we have an orthogonal decomposition H = KerA (KerA), and (KerA) = RangeA. Let P and Q = I P be the orthogonal projections from H onto KerA and (KerA), respectively. Let λ 1,λ 2,... with λ 1 λ 2... be the nonzero eigenvalues of A and Ψ n be the corresponding (normalized) eigenfunctions. Since A C 2(H), we have and λ 2 n <, n=1 AΦ = λ n Φ,Ψ n Ψ n, Φ H n=1

18 We apply S to (z δ I+K )Φ δ = g to obtain (z δ S+ SK )Φ δ = Sg. Then (z δ I+A) SΦ δ = Sg. Projecting onto KerA and (KerA), we have P SΦ δ = 1 z δ P Sg, Q SΦ δ = n Q Sg,Ψ n λ n +z δ Ψ n. We also get A SΦ δ = n λ n Q Sg,Ψ n λ n +z δ Ψ n.

19 We have (S Γi [ϕ δ i ]+S Γe [ϕ δ e]) 2 dx = 1 2 SΦ δ, SΦ δ A SΦ δ, SΦ δ Ω\D 1 δ 2 P Sg 2 + n Let Φ n be the (normalized) eigenfunctions of K. Theorem Q Sg,Ψ n 2 λ n 2 +δ 2. If P Sg 0, then LR takes place. If Ker(K ) = {0}, then ALR takes place if and only if δ Sg,Φ n 2 as δ 0. λ 2 n +δ 2 n

20 Anomalous resonance in annulus The above theorem gives a necessary and sufficient condition on the source term f for the blow up of the electromagnetic energy in Ω\D. This condition is in terms of the Newton potential of f. We explicitly compute eigenvalues and eigenfunctions of A for the case of an annulus configuration. We consider the anomalous resonance when domains Ω and D are concentric disks. We calculate the explicit form of the limiting solution. Throughout this section, we set Ω = B e = { x < r e} and D = B i = { x < r i}, where r e > r i. Lemma Let ρ := r i r e. Then and the eigenvalues of A are {±ρ n }. KerK = {0}

21 Let F ν e = n 0 gn ee inθ. There exists δ 0 such that E δ := δ V δ 2 δ ge n 2 B e\b i n (δ 2 +ρ 2 n ) n 0 uniformly in δ δ 0. g limsup e n 2 = implies only limsupe n n ρ n δ = δ 0 (pointed out by J. Lu and J. Jorgensen).

22 GP : There exists a sequence {n k } with n 1 < n 2 < such that e 2 lim k ρ n k+1 n k g nk n k ρ =. n k Lemma If {g n e} satisfies the condition GP, then lim E δ =. δ 0 If lim n g n e 2 n ρ n =, then lim δ 0 E δ =.

23 Suppose that the source function is supported inside the radius r = rer 3 1 i. Then its Newtonian potential cannot be extended harmonically in x < r in general. So, if F is given by F = c n 0 a nr n e inθ, r < r e, then the radius of convergence is less than r. Thus we have lim sup n a n 2 r 2 n =, n g and limsup e n 2 = holds. The GP condition is equivalent to that there n n ρ n exists {n k } with n 1 < n 2 < such that lim k ρ n k+1 n k n k a nk 2 r 2 n k =.

24 The following is the main theorem. Theorem Let f be a source function supported in R 2 \B e and F be the Newtonian potential of f. (i) If F does not extend as a harmonic function in B r, then weak ALR occurs, i.e., lim supe δ =. δ 0 (ii) If the Fourier coefficients of F satisfy GP, then ALR occurs, i.e., lim E δ =. δ 0 (iii) If F extends as a harmonic function in a neighborhood of B r, then ALR does not occur, i.e., E δ < C for some C independent of δ.

25 Examples If f is a dipole source in B r \B e, i.e., f(x) = a δ y(x) for a vector a and y B r \B e where δ y is the Dirac delta function at y. Then F(x) = a G(x y) and the ALR takes place. This was found by Milton et al. If f is a quadrapole, i.e., f(x) = A : δ y(x) = 2 i,j=1 aij 2 x i x j δ y(x) for a 2 2 matrix A = (a ij) and y B r \B e. Then F(x) = 2. Thus the ALR takes place. i,j=1 aij 2 G(x y) x i x j

26 If f is supported in R 2 \B r, then F is harmonic in a neighborhood of B r, and hence the ALR does not occur. In fact, we can say more about the behavior of the solution V δ as δ 0. Theorem If f is supported in R 2 \B r, then Moreover, B e\b i δ V δ 2 < C. sup x r V δ (x) F(x) 0 as δ 0.

27 Problems How can we describe the cloaking effect when some inclusion is immersed? How can we analyze ALR explicitely in terms of the source term when the given geometry is general?

28 References G. Milton and N.-A. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. A 462 (2006), G. Milton, N.-A. Nicorovici, R.C. McPhedran, and V.A. Podolskiy, A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance, Proc. R. Soc. A 461 (2005), N.-A. Nicorovici, G. Milton, R.C. McPhedran, and L.C. Botten, Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance, Optics Express 15 (2007), O.P. Bruno and S. Lintner, Superlens-cloaking of small dielectric bodies in the quasi-static regime, J. Appl. Phys. 102 (2007), G. Bouchitté and B. Schweizer, Cloaking of small objects by anomalous localized resonance, Quart. J. Mech. Appl. Math. 63 (2010),

29 Thank you!