On Transformation Optics based Cloaking

Transcription

1 On Transformation Optics based Cloaking TING ZHOU Northeastern University CIRM

2 Outline Inverse Problem and Invisibility 1 Inverse Problem and Invisibility 2 Singular cloaking medium 3 Regularization and discussion Limiting Behavior at the Interface 4 for 2D Helmholtz equations Regularization and the limiting behavior at the interface

3 Deflecting light Inverse Problem and Invisibility

4 Black holes and optical white holes

5 Mirage Inverse Problem and Invisibility Fermat s principle: minimize optical length in a medium with variable refractive index.

6 Cloaking for acoustics Inverse Problem and Invisibility

7 Transformation Optics based Cloaking From Pendry et al s paper J. B. Pendry, D. Schurig and D. R. Smith (2006) U. Leonhard (2006) Transformation Optics and Metamaterials A. Greenleaf, M. Lassas and G. Uhlmann (2003)

8 Inverse Problem and Invisibility

9 Visibility: Inverse problems of EIT (Calderón problem) Conductivity equation: γ u = 0 in Ω. Electrical Impedance Tomography (EIT) (0 < C 1 γ C 2 ) Boundary measurements: Dirichlet-to-Neumann (DN) map Λ γ : u Ω ν γ u Ω Calderón problem: Λ γ1 = Λ γ2 γ 1 = γ 2? Isotropic γ scalar: uniqueness results - Visibility [Sylvester-Uhlmann, Nachman,...] Anisotropic γ = (γ ij ) (pos. def. sym. tensor): non-uniqueness.

10 Transformation law and nonuniqueness Ω ( u f γ u f dy x=ψ(y),ψ Ω=I (Dψ) T ) γ (Dψ) = v f ψ 1 v f dx Ω det(dψ) }{{} ψ γ Λ γ (f )f ds y = Λ ψ γ(f )f ds x Ω DN-map: γ u f = 0 u f Ω = f Λ γ (f ) = ν γ u f Ω v f =u f ψ 1 = Ω ψ γ v f = 0 v f Ω = f Λ ψ γ(f ) = ν ψ γ v f Ω Conclusion: ψ: a diffeomorphism on Ω and ψ Ω = Id. Λ γ = Λ ψ γ

11 Cloaking for EIT Inverse Problem and Invisibility F : B 2 \{0} B 2 \B 1 ( F(y) = 1 + y ) y 2 y. 1 F B2 = Identity. Greenleaf-Lassas-Uhlmann (2003) γ = { I : Identity matrix in B 2, F γ in B γ = 2 \B 1 arbitrary γ a in B 1 Λ γ = Λ γ. F I is anisotropic. Removable singularity argument.

12 Currents (vacuum space vs. cloaking) All Boundary measurements for the homogeneous conductivity γ = I and the conductivity γ = (F I, γ a ) are the same Analytic solutions for the currents Based on work of Greenleaf-Lassas-Uhlmann, 2003

13 Singular Ideal Electromagnetic Cloaking

14 Wave theory of light: Electromagnetic waves and Maxwell s equations E iωµh = 0 H + iωεe = J (E, H): electromagnetic field µ(x): magnetic permeability ε(x): electric permittivity J(x): electric current source Refractive index: µε. What is invisibility? W D Arbitrary object to be cloaked in D surrounded by the cloak Ω\D with electromagnetic parameters ( µ(x), ε(x)). We want to show that if Maxwell s equations are solved in Ω, the boundary information of solutions is the same as that of the case with µ = ε = Id.

15 Invisibility and Cloaking Inverse Problem and Invisibility From Pendry et al s paper J. B. Pendry, D. Schurig and D. R. Smith (2006) U. Leonhard (2006) Transformation Optics and Metamaterials A. Greenleaf, M. Lassas and G. Uhlmann (2003)

16 Metamaterials for electromagnetic cloaking Invisibility cloak for 4 cm EM waves Schurig et al, Science 2006.

17 Metamaterials for acoustic cloaking Zhang et al, PRL 2011

18 Tsunami cloaking Inverse Problem and Invisibility Broadband cylindrical cloak for linear surface waves in a fluid, M. Farhat et al, PRL (2008).

19 Electromagnetic waves in regular media Singular cloaking medium Time harmonic Maxwell s equations E = iωµh H = iωεe + J in Ω. with permittivity ε(x) and permeability µ(x). Regular (Nonsingular) medium: ε = (ε ij ) and µ = (µ ij ) are pos. def. sym. matrices, that is, there exists C > 0 such that µ ij (x)ξ i ξ j C ξ 2, ε ij (x)ξ i ξ j C ξ 2 i,j for ξ R n and x Ω. Then (E, H) H(curl) H(curl). i,j

20 Singular cloaking medium Imaging and inverse problems with electromagnetic waves Boundary observation: Impedance map Λ µ,ε : ν E Ω ν H Ω. Inverse problem: Is (µ, ε) Λ µ,ε injective? [Ola-Päivärinta-Somersalo], [Ola-Somersalo]: C 2 isotropic.

21 Transformation law for Maxwell s equations Let ψ : Ω Ω be a diffeomorphism. Pullback of fields by ψ 1 : Singular cloaking medium Ẽ = (ψ 1 ) E := (Dψ T ) 1 E ψ 1 H = (ψ 1 ) H := (Dψ T ) 1 H ψ 1 J = (ψ 1 ) J := [det (Dψ)] 1 Dψ J ψ 1 Push-forward of medium by ψ: ( (Dψ) T µ (Dψ) µ = ψ µ := det(dψ) ( (Dψ) T ε (Dψ) ε = ψ ε := det(dψ) Then ) ψ 1, ) ψ 1. Ẽ = iω µ H, H = iω εẽ + J in Ω Moreover, if ψ Ω = Identity, we have Λ µ, ε = Λ µ,ε

22 Electromagnetic cloaking medium Singular cloaking medium F : B 2 \{0} B 2 \B 1 ( F(y) = 1 + y ) y 2 y. 1 F B2 = Identity. Cloaking medium { (F I, F ( µ, ε) = I) in B 2 \B 1 (µ a, ε a ) arbitrary in B 1 Heterogeneous, anisotropic and singular in the cloaking layer.

23 Transformation Optics for Rays Singular cloaking medium

24 Singular cloaking medium Singular cloaking medium 3D cloaking device medium in B 2 \B 1 : µ = ε = F I = 2 ( x 1)2 x 2 Π(x) + 2(I Π(x)) where Π(x) = ˆxˆx T = xx T / x 2 is the projection along the radial direction. Degenerate singularity at x = 1 +! v.s. Non-singular (Regular) medium: for some C > 0, γ ij (x)ξ i ξ j C ξ 2, ξ R n, x Ω i,j

25 Singular cloaking medium Finite energy solutions [Greenleaf-Kurylev-Lassas-Uhlmann] Finite energy solution (FES) to Maxwell s equations for (B 2, µ, ε): Ẽ, H, D = εẽ and B = µ H are forms in B 2 with L 1 (B 2, dx)-coefficients such that ε ij Ẽ i Ẽ j dx <, B 2 µ ij H i H j dx <, B 2 Maxwell s equations hold in a neighborhood of B 2, and B 2 ( h) Ẽ h iω µ H dx =0 B 2 ( ẽ) H + ẽ (iω εẽ J) dx =0 for all ẽ, h C 0 (B 2). Hidden boundary condition: ν Ẽ B 1 Then cloaking a source ( J B1 = ν H B 1 0) is problematic! = 0.

26 Regularization and discussion Limiting Behavior at the Interface Regularized Electromagnetic Approximate Cloaking

27 Blow-up-a-small-ball regularization Regularization and discussion Limiting Behavior at the Interface Regularized transformation that blows up B ρ (0 < ρ < 1) to B 1 and fixes the boundary B 2. F ρ (y) := { ( 2(1 ρ) 2 ρ ) y y, ρ < y < 2,, y < ρ. + y 2 ρ y ρ Construct non-singular EM anisotropic material ( µ ρ, ε ρ ) := { ((Fρ ) I, (F ρ ) I), 1 < x < 2, (µ 0, ε 0 ), x < 1. Blow-up-a-small-ball regularization scheme for Helmholtz equations [Kohn-Onofrei-Vogelius-Weinstein]

28 Regularized cloaking medium Regularization and discussion Limiting Behavior at the Interface Construct non-singular EM anisotropic material ( µ ρ, ε ρ ) := { ((Fρ ) I, (F ρ ) I), 1 < x < 2, (µ 0, ε 0 ), x < 1. ( (2 ρ) x 2 + 2ρ ) 2 (F ρ ) I = (2 ρ) x 2 Π(x) + (2 ρ)(i Π(x)) Well-posedness: well-defined H(curl) solutions satisfying transmission problems in both physical space (cloaking device + cloaked region) and virtual space (pullback of physical space).

29 Virtual space vs. Physical space Regularization and discussion Limiting Behavior at the Interface Virtual space Same Λ µ, ε Physical space Is Λ µ, ε Λ I,I? Yes and No! What is the limiting behavior (as ρ 0) of the EM waves in the physical space at the interface x = 1?

30 Regularization and discussion Limiting Behavior at the Interface Transmission problems in physical and virtual spaces for y B 2 \B ρ : Virtual space: E ρ + = iωh ρ + H ρ + = iωe ρ + + J for y B ρ : Eρ = iω((fρ 1 ) µ 0 )Hρ Hρ = iω((fρ 1 ) ε 0 )Eρ + J ν E ρ + B + ρ = ν Eρ B ρ ν H ρ + B + ρ = ν Hρ B ρ ν E ρ + B2 = f Physical space: for x B 2 \B 1 : Ẽ + ρ H + ρ for x B 1 : Ẽ ρ H ρ ν Ẽ + ρ B + 1 = iω µ ρ H + ρ = iω ε ρ Ẽ + ρ + J, = iωµ 0 H ρ ν H ρ + B + 1 ν Ẽ ρ + B2 = f. = iωε 0 Ẽ ρ + J = ν Ẽρ B, 1, = ν H ρ B 1

31 Cloaking a passive medium: J = 0 Regularization and discussion Limiting Behavior at the Interface Assume µ 0 and ε 0 are positive constants, k = µ 0 ε 0. Spherical expansion of E s: E + ρ = Ẽ ρ = ε 1/2 0 n n=1 m= n Expansion of H s: H ρ = 1 ikω µ 1/2 0 H + ρ = 1 iω n n=1 m= n α m n M m n,kω + β m n M m n,kω in B 1, c m n N m n,ω+d m n N m n,ω+γ m n M m n,ω+η m n M m n,ω in B 2\B ρ. n n=1 m= n n n=1 m= n k 2 ω 2 β m n M m n,kω + α m n M m n,kω in B 1, ω 2 d m n N m n,ω +c m n N m n,ω +ω 2 η m n M m n,ω +γ m n M m n,ω,

32 Cloaking a passive medium: J = 0 Regularization and discussion Limiting Behavior at the Interface Plug in to the boundary conditions and transmission conditions: { c m n h (1) n (2ω) + γn m j n (2ω) = f nm (1), dn m H n (2ω) + ηn m J n (2ω) = 2f nm (2). { { ρc m n h (1) n (ωρ) + ργ m n j n (ωρ) = ε 1/2 0 α m n j n (kω), d m n H n (ωρ) + η m n J n (ωρ) = ε 1/2 0 β m n J n (kω). kc m n H n (ωρ) + kγn m J n (ωρ) = µ 1/2 0 αn m J n (kω), ρdn m h (1) n (ωρ) + ρηn m j n (ωρ) = µ 1/2 0 kβn m j n (kω). Systems of linear equations for X = (α m n, β m n, c m n, d m n, γ m n, η m n ) A n,ρ,ω,µ0,ε 0 X = b f

33 Cloaking a passive medium: J = 0 Regularization and discussion Limiting Behavior at the Interface Convergence order as ρ 0: γ m n = O(1), η m n = O(1); c m n = O(ρ 2n+1 ), d m n = O(ρ 2n+1 ); α m n = O(ρ n+1 ), β m n = O(ρ n+1 ). Λ µρ, ε ρ Λ I,I Inside B 1, (Ẽ ρ, H ρ ) 0

34 Regularization and discussion Limiting Behavior at the Interface Cloaking a medium with a source: J 0 supported in B 1 Given an internal current source J supported in B r1 where r 1 < 1, Spherical expansion: Ẽ ρ = ε 1/2 0 for r 1 < x < 1. E + ρ = n n=1 m= n n n=1 m= n for ρ < y < 2. α m n M m n,kω +β m n M m n,kω +p m n N m n,kω +q m n N m n,kω c m n N m n,ω + d m n N m n,ω + γ m n M m n,ω + η m n M m n,ω

35 Regularization and discussion Limiting Behavior at the Interface Cloaking a medium with a source: J 0 supported in B 1 Convergence order as ρ 0: γn m = O(1), ηn m = O(1); c m n = O(ρ n+1 ), dn m = O(ρ n+1 ); αn m = O(1), βn m = O(1). Λ µρ, ε ρ Λ I,I

36 Demonstration (passive) Inverse Problem and Invisibility Regularization and discussion Limiting Behavior at the Interface Re(Ẽ ρ) 1 (sliced at x = 0, 1, 2), ω = 5, ε 0 = µ 0 = 2, ρ = 1/6. ρ Er(ρ) e e e e 08 r(ρ) Boundary errors and convergence order when ω = 5, ε 0 = µ 0 = 2.

37 Demonstration (active) Inverse Problem and Invisibility Regularization and discussion Limiting Behavior at the Interface Re(Ẽ ρ) 1 (sliced at x = 0, 1, 2), ω = 5, ε 0 = µ 0 = 2, ρ = 1/12, with a point source. ρ Er(ρ) e e 04 r(ρ) Boundary errors and convergence order ω = 5, ε 0 = µ 0 = 2, with a point source.

38 Regularization and discussion Limiting Behavior at the Interface Resonance

39 Resonance and Cloak-busting inclusions Regularization and discussion Limiting Behavior at the Interface For a fixed cloaking scheme, i.e., fixed ρ > 0, there exists some frequency ω and cloaked medium (µ 0, ε 0 ) such that the transmission problems are NOT well-posed. Therefore, the boundary measurement Λ µ, ε is significantly different from Λ I,I. µ 1/2 0 ρh (1) n (ωρ)j n (kω) ε 1/2 0 kh n (ωρ)j n (kω) = 0 ω is the resonant frequency; (µ 0, ε 0 ) is called cloak-busting inclusion;

40 Demonstration (Resonance) Regularization and discussion Limiting Behavior at the Interface Er(ρ) x 10 3 Lossless passive cloaking (ρ=0.01) ω Er(ρ) Lossless active cloaking (ρ=0.01) ω Boundary error Er(ω) = ν H + ρ B2 ν H B2 for mode n = 1, when ρ = 0.01 and µ 0 = ε 0 = 2, against frequency ω [1, 3] (Left: passive; Right: active).

41 Regularization and discussion Limiting Behavior at the Interface Remedy to resonance: Cloaking with a lossy layer 1. 1 [Kohn-Onofrei-Vogelius-Weinstein] for Helmholtz equations

42 Remedy: Lossy layer Inverse Problem and Invisibility Regularization and discussion Limiting Behavior at the Interface F 2ρ blows up B 2ρ to B 1, F 2ρ (y) := { ( 2(1 2ρ) 2 2ρ + y y 2ρ τ is the damping parameter (conductivity). ) y 2 2ρ y, 2ρ < y < 2,, y < 2ρ.

43 Demonstration (lossy) Inverse Problem and Invisibility Regularization and discussion Limiting Behavior at the Interface 0.1 Lossy passive approximate cloaking (ρ=0.01) Er(ρ) ω Boundary error Er for mode n = 1 when ρ = 0.01 of lossy approximate cloaking (passive), against frequency ω [1, 10]. Resonant frequencies disappear. Complex poles? Damping effect: τ depending on ρ.

44 Damping parameter τ Inverse Problem and Invisibility Regularization and discussion Limiting Behavior at the Interface Lossy regularization for Scalar optics and Acoustics (Helmholtz equations) (Kohn-Onofrei-Vogelius-Weinstein, Kohn-Nguyen). γ u + k 2 qu = 0 lossy cloaking medium ((F 2ρ ) I, (F 2ρ ) 1) 1 < x < 2 ( γ ρ, q ρ ) = ((F 2ρ ) I, (F 2ρ ) (1 + ic 0 ρ 2 )) 1/2 < x < 1 (γ 0, q 0 ) x < 1/2 where Then F q := q det(df) F 1 { ln ρ Λ γρ, q ρ Λ I,1 1, in R 2 ρ, in R 3.

45 Extreme case: Enhanced cloaking by lining Regularization and discussion Limiting Behavior at the Interface τ = sound-soft lining [Liu]. Then { ln ρ Λ γρ, q ρ Λ I,1 1, in R 2 ρ, in R 3. (High loss makes detection easier in infrared regime!) Finite-sound-hard layer [Liu]: ((F 2ρ ) I, (F 2ρ ) 1) 1 < x < 2 ( γ ρ, q ρ ) = (ρ 2 δ (F 2ρ ) I, (F 2ρ ) (α + iβ)) 1/2 < x < 1 (γ 0, q 0 ) x < 1/2 Then Λ γρ, q ρ Λ I,1 ρ n in R n. In FSH, let δ, we have sound-hard lining.

46 Regularization and discussion Limiting Behavior at the Interface Normal limits at the interface due to an internal source 2 2 This is a joint work with Prof. Matti Lassas

47 Regularization and discussion Limiting Behavior at the Interface Radiation at the interface due to the internal source Given a current source J supported on B r1 for r 1 < 1. No resonance. As ρ 0, degenerate singularity arises at B + 1. Consider the limit of ˆx Ẽ + ρ as ρ 0 +.

48 Hint Inverse Problem and Invisibility Regularization and discussion Limiting Behavior at the Interface Formally ˆx Ẽ ρ + p dx = (2 ρ) p ŷ E ρ + p det(df ρ ) dy B 2/(2 ρ) \B 1 B 2ρ\B ρ spherical expansion of E ρ + { = O(ρ 1 ) p = 2, O(1) p = 1. suggesting a superposition of Delta functions at the interface!

49 Distributional limits Inverse Problem and Invisibility Regularization and discussion Limiting Behavior at the Interface Theorem [Lassas-Z] where Ẽ ρ ρ 0 Ẽ + α[ J]δ B1, (Ẽ, H) = H ρ ρ 0 H + β[ J]δ B1 { (F E, F H) 1 < x < 2, (E 0, H 0 ) x < 1 with (E, H) denotes the background waves in the vacuum space and { E0 = iωµ 0 H 0, H 0 = iωε 0 E 0 on B 1 ν E 0 B1 = ν H 0 B1 = 0 extraordinary surface voltage effect [Zhang etc.].

50 for 2D Helmholtz equations Regularization and the limiting behavior at the interface Two dimensional approximate cloaking and non-local (pseudo-differential) boundary conditions

51 for 2D Helmholtz equations Regularization and the limiting behavior at the interface Cloaking for scalar optics and acoustics: Helmholtz equations The Helmholtz equation for acoustics or scalar optics, with a source term p, inverse of the anisotropic mass density σ = (σ jk ) and the bulk modulus λ λ σ u + ω 2 u = p in Ω. Dirichlet to Neumann map: Λ σ,λ : u Ω ν σ u Ω. Cloaking medium ( σ, λ) = where F λ(x) := [det(df)λ] F 1 (x). ũ = u F 1 in B 2 \B 1. { (F I, F 1) 1 < x 2 (σ a, λ a ) arbitrary x 1

52 medium in R 2 for 2D Helmholtz equations Regularization and the limiting behavior at the interface Cloaking layer: in B 2 \B 1 σ = F I = x 1 x Π(x) + x x 1 (I Π(x)) λ = F 1 = x 4( x 1) Both degenerate and blow-up singularities at x = 1 +!

53 Truncation based regularization scheme for 2D Helmholtz equations Regularization and the limiting behavior at the interface Regularized medium with regularization parameter 1 < R < 2 ( σ R, λ R ) = { ( σ, λ) x > R (σ a, λ a ) x R We are interested in the limiting behavior of the solution near the interface when an internal source is present.

54 for 2D Helmholtz equations Regularization and the limiting behavior at the interface Cloaking a homogeneous medium with an internal source Suppose (σ a, λ a ) is constant. Set κ 2 = (σ a λ a ) 1 and ρ = F 1 (R) Physical space: ( λ σ + ω 2 )u + R = p, in B 2\B R ( + κ 2 ω 2 )u R = κ2 p in B R Virtual space: v + R = u+ R F, ( + ω 2 )v + R = p F in B 2\B ρ Transmission conditions and boundary conditions: v + R B + = u ρ R B, ρ r v + R R B + = κr ru ρ R B, R v + R B 2 = f.

55 for 2D Helmholtz equations Regularization and the limiting behavior at the interface Cloaking a homogeneous medium with an internal source Given p C (R 2 ) with supp(p) B R0 (0 < R 0 < 1) and suppose f = 0 on B 2 Spherical expansions: u R ( r, θ) = v + R (r, θ) = n= (a n J n (κω r) + p n H (1) n (κω r))einθ, r (R 0, R) n= (b n J n (ωr) + c n H (1) n (ωr))einθ, r (ρ, 2) Linear system about a n, b n and c n by the transmission conditions and boundary condition.

56 for 2D Helmholtz equations Regularization and the limiting behavior at the interface a n = R(H(1) n ) (κωr)l 1 ρh (1) D n n (κωr)l 2 p n := A n D n p n b n = R{ (H (1) n ) (κωr)j n (κωr) J n (κωr)h(1) n (κωr)} H (1) n (3ω) c n = R{ (H (1) n ) (κωr)j n (κωr) J n (κωr)h(1) n (κωr)} J n (3ω) where D n D n D n = i2n ω n 1 (n 1)! J n (3ω) [ κ 2 ωrj π n(κωr) + nj n (κωr) ] ρ n +O(ρ n+1 ), p n p n

57 Observations for resonant case for 2D Helmholtz equations Regularization and the limiting behavior at the interface Resonant frequency ω for mode n cloak-busting inclusion limit κ = (σ 0 λ 0 ) 1/2 a n, b n, c n as R 1 + (ρ 0 + ) (n 1) [ωκ 2 R(J n ) (κωr) + n J n (κωr)] R=1 = 0 V ±n ( r, θ) := J n (κω r)e ±inθ are eigenfunctions of ( + κ 2 ω 2 )V = 0 in B 1, [κ r r V + ( θ) 2 1/2 V] r=1 + = 0.

58 Non-local boundary conditions for 2D Helmholtz equations Regularization and the limiting behavior at the interface [κ r r V + ( 2 θ) 1/2 V] r=1 + = 0. Operator A := ( 2 θ )1/2 is a pseudo-differential operator: Square root of positive laplacian over S 1. Symbol of P: Pu = Sym(P)û Sym( ) = iξ, Sym( ) = ξ 2, ξ R n A non-local boundary condition: Sym(A) = ξ Au = F 1 ( ξ û)

59 Non-resonant result: non-local boundary conditions for 2D Helmholtz equations Regularization and the limiting behavior at the interface Suppose ω and (σ a, λ a ) satisfy { [ωκ 2 R(J n ) (κωr) + n J n (κωr)] R=1 0, J n (2ω) 0, for n Z. Theorem [Lassas-Z] As R 1 +, u R (the solution in the physical space) converges uniformly in compact subsets of B 2 \ B 1 to the limit u 1 satisfying ( + κ 2 ω 2 )u 1 = κ 2 p in B 1, [κ r u 1 + ( 2 θ) 1/2 u 1 ] B1 = 0. possiblely due to the fact that the phase velocity of the waves in the invisibility cloak approaches infinity near the interface, even though the group velocity stays finite.

60 for 2D Helmholtz equations Regularization and the limiting behavior at the interface Thank you for your attention!