Electromagnetic cloaking and super-resolved imaging Habib Ammari - PowerPoint PPT Presentation

Electromagnetic cloaking and super-resolved imaging Habib Ammari Department of Mathematics and Applications Ecole Normale Sup´ erieure, Paris Electromagnetic cloaking and super-resolved imaging Habib Ammari

Introduction • Electromagnetic cloaking is to make a target invisible with respect to probing by electromagnetic waves. • Two schemes: anomalous resonance and change of variables. • Mathematical justification of cloaking due to anomalous localized resonance (CALR). (with G. Ciraolo, H. Kang, H. Lee, and G. Milton) • New cancellation technique to achieve enhanced near-cloaking using the change of variables scheme. (with H. Kang, H. Lee, and M. Lim) • Generalized polarization tensors (GPTs) vanishing structures. • Numerical illustrations and statistical stability of near-cloaking: (with J. Garnier and V. Jugnon). • GPTs for shape description: GPTs can capture topology and high-frequency shape oscillations. GPTs can be used to achieve super-resolution in electromagnetic imaging (with H. Kang, M. Lim, H. Zribi; with J. Garnier, H. Kang, M. Lim, S. Yu). Electromagnetic cloaking and super-resolved imaging Habib Ammari

Anomalous resonance • Ω: bounded domain in R 2 ; D ⊂⊂ Ω. Ω and D of class C 1 ,µ , 0 < µ < 1. For a given loss parameter δ > 0, the permittivity distribution in R 2 is given by in R 2 \ Ω ,  1   ǫ δ = − 1 + i δ in Ω \ D ,  1 in D .  • Configuration (plasmonic structure): core with permittivity 1 coated by the shell Ω \ D with permittivity − 1 + i δ . • For a given function f compactly supported in R 2 satisfying � R 2 fdx = 0 (conservation of charge), consider the following dielectric problem: in R 2 , ∇ · ǫ δ ∇ V δ = α f with the decay condition V δ ( x ) → 0 as | x | → ∞ . • Dielectric problems: models the quasi-static (zero-frequency) transverse magnetic regime. Electromagnetic cloaking and super-resolved imaging Habib Ammari

Anomalous resonance • Fundamental problem: identify f such that when α = 1 � δ |∇ V δ | 2 dx → ∞ E δ := as δ → 0 . Ω \ D | V δ ( x ) | < C , when | x | > a for some constants C and a independent of δ . • E δ : proportional to the electromagnetic power dissipated into heat by the time harmonic electrical field averaged over time. • Infinite amount of energy dissipated per unit time in the limit δ → 0: unphysical. • Choose α = 1 / √ E δ : α f produces the same power independent of δ and the new associated solution V δ approaches zero outside the radius a . • Necessary and sufficient condition for CALR (with α = 1) V δ / √ E δ goes to zero outside some radius as δ → 0. Electromagnetic cloaking and super-resolved imaging Habib Ammari

Anomalous resonance • We develop a general method based on the potential theory to study cloaking due to anomalous resonance. • Using layer potential techniques: we reduce the problem to a singularly perturbed system of integral equations. • The system is non-self-adjoint ⇒ we introduce a symmetrization technique in order to express the solution in terms of the eigenfunctions of a self-adjoint compact operator. • Symmetrization technique: based on a generalization of a Calder´ on identity to the system of integral equations and a general theorem on symmetrization of non-selfadjoint operators obtained by Khavinson -Putinar-Shapiro (2007). • We provide a necessary and sufficient condition on the source term under which the blowup of the power dissipation takes place. The condition is given in terms of the Newtonian potential of the source 1 � x ∈ R 2 , R 2 ln | x − y | f ( y ) dy , 2 π which is the solution for the potential in the absence of the plasmonic structure. Electromagnetic cloaking and super-resolved imaging Habib Ammari

Anomalous resonance • In the case of an annulus ( D is the disk of radius r i and Ω =: B e is the concentric disk of radius r e ), it is known (Milton et al.) that there exists a critical radius (the cloaking radius) � r 3 e r i − 1 . r ∗ = such that any finite collection of dipole sources located at fixed positions within the annulus B r ∗ \ B e is cloaked. • We show that if f is an integrable function supported in E ⊂ B r ∗ \ B e , then CALR takes place provided the Newtonian potential of f is not identically zero in R 2 \ E ⇒ most integrable sources α f supported in E will be cloaked. (quadrupole source inside the annulus B r ∗ \ B e : cloaked). • Conversely we show that if the source function f is supported outside B r ∗ then no cloaking occurs. Electromagnetic cloaking and super-resolved imaging Habib Ammari

Anomalous resonance • Notation: Γ i := ∂ D and Γ e := ∂ Ω. • For Γ = Γ i or Γ e and ϕ ∈ L 2 (Γ), the single and double layer potentials: S Γ [ ϕ ]( x ) := 1 � x ∈ R 2 , ln | x − y | ϕ ( y ) d σ ( y ) , 2 π Γ D Γ [ ϕ ]( x ) := 1 � ∂ x ∈ R 2 \ Γ . ∂ν ( y ) ln | x − y | ϕ ( y ) d σ ( y ) , 2 π Γ ν ( y ): the outward unit normal to Γ at y . • Neumann-Poincar´ e operators: K Γ [ ϕ ]( x ) := 1 � � y − x , ν ( y ) � ϕ ( y ) d σ ( y ) , 2 π | x − y | 2 Γ K ∗ Γ : L 2 -adjoint of K Γ . Γ : compact in L 2 (Γ) if Γ is C 1 ,α for some α > 0. • K Γ and K ∗ Electromagnetic cloaking and super-resolved imaging Habib Ammari

Anomalous resonance • Notation: F Newtonian potential of f ; H = L 2 (Γ i ) × L 2 (Γ e ); z δ = i δ 2(2 − i δ ) . • Representation formula: V δ ( x ) = F ( x ) + S Γ i [ ϕ i ]( x ) + S Γ e [ ϕ e ]( x ) . • Introduce: ∂ F   � ϕ i � ∂ν i   Φ := , g :=  .   ϕ e − ∂ F  ∂ν e • Singularly perturbed equation: ( z δ I 2 + K ∗ )Φ = g . • K ∗ : H → H Neumann-Poincar´ e-type operator (compact non-self-adjoint in general): − ∂   −K ∗ ∂ν i S Γ e Γ i K ∗ :=    .   ∂  K ∗ ∂ν e S Γ i Γ e Electromagnetic cloaking and super-resolved imaging Habib Ammari

Anomalous resonance • The spectrum of K ∗ lies in the interval [ − 1 / 2 , 1 / 2]. • The operator � S Γ i � S Γ e S = S Γ i S Γ e is self-adjoint and − S ≥ 0 on H . on’s-type identity: SK ∗ = KS . • Calder´ • K ∗ is Hilbert-Schmidt (in 2D; Schatten-von Neumann in 3D). • K ∗ is symmetrizable: there is a bounded self-adjoint operator A on √ √ − SK ∗ . Range ( S ) such that A − S = • Khavinson et al : let M ∈ C p ( H ). If there exists a strictly positive bounded self-adjoint 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 . Electromagnetic cloaking and super-resolved imaging Habib Ammari

Anomalous resonance • A self-adjoint ⇒ an orthogonal decomposition: H = Ker A ⊕ ( Ker A ) ⊥ , and ( Ker A ) ⊥ = Range A . • P and Q = I − P : the orthogonal projections from H onto Ker A and ( Ker A ) ⊥ , 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. A ∈ C 2 ( H ) ⇒ ∞ � λ 2 n < ∞ , n =1 and ∞ � A Φ = λ n � Φ , Ψ n � Ψ n , Φ ∈ H . n =1 √ − S g � = 0, then CALR takes place. If Ker( K ∗ ) = { 0 } , • Theorem : If P then CALR takes place if and only if √ − S g , Ψ n �| 2 |� � → ∞ as δ → 0 . δ λ 2 n + δ 2 n Electromagnetic cloaking and super-resolved imaging Habib Ammari

Anomalous resonance Anomalous resonance in an annulus: • Eigenvalues λ of A = {± ρ | n | } , ρ = r i r e . • Theorem : (blow-up of power dissipation criterion) For a given source f supported outside B e (with α = 1), e | 2 | g n ρ N � | n | ρ 2 | n | → ∞ as N → ∞ 0 � = | n |≤ N if and only if � δ |∇ V δ | 2 → ∞ as δ → 0 . B e \ B i g n e : Fourier coefficient of − ∂ F ∂ν e on Γ e . • Theorem : E : a measurable subset of B r ∗ \ B e such that R 2 \ E is connected. Suppose that f is an integrable function supported in E � satisfying E f = 0. If Newtonian potential of f is not identically zero in R 2 \ E , then CALR takes place. • f = c 1 χ B 2 r ( y ∗ ) − c 2 χ B r ( y ∗ ) , for some r > 0 and y ∗ , where c 1 and c 2 are R 2 fdx = 0. F ≡ 0 on R 2 \ B 2 r ( y ∗ ) and f is � positive constants to satisfy not cloaked. Electromagnetic cloaking and super-resolved imaging Habib Ammari

Anomalous resonance Anomalous resonance in an annulus: • Any source supported outside B r ∗ cannot make the blow-up of the power dissipation happen and is not cloaked. Indeed, in the limit δ → 0 the annulus itself becomes invisible to sources that are sufficiently far away. • Theorem : If f is supported in R 2 \ B r ∗ , then � δ |∇ V δ | 2 < C B e \ B i holds for some constant C independent of δ (with α = 1). Moreover, we have sup | V δ ( x ) − F ( x ) | → 0 as δ → 0 . | x |≥ r ∗ Electromagnetic cloaking and super-resolved imaging Habib Ammari

Near-cloaking • Cloak a region inside the cloaking device. • Conductivity problem (quasi-static regime): the Dirichlet-to-Neumann map is nearly the same as the one associated to the constant conductivity distribution. • Helmholtz equation: the scattering cross-section is nearly zero. • Change of variable scheme + structures with vanishing generalized polarization tensors or scattering coefficients. Electromagnetic cloaking and super-resolved imaging Habib Ammari