On Approximate Cloaking by Nonsingular Transformation Media

Transcription

1 On Approximate Cloaking by Nonsingular Transformation Media TING ZHOU MIT Geometric Analysis on Euclidean and Homogeneous Spaces January 9, 2012

2 Invisibility and 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)

3 Outline Cloaking for Electrostatics 1 Cloaking for Electrostatics 2 Regularization scheme Cloaking a passive medium Cloaking an active medium Cloak-busting inclusion and lossy layer 3 Ideal cloaking for the Helmholtz equation Nonsingular regularization and non-local boundary conditions

4 Outline Cloaking for Electrostatics 1 Cloaking for Electrostatics 2 Regularization scheme Cloaking a passive medium Cloaking an active medium Cloak-busting inclusion and lossy layer 3 Ideal cloaking for the Helmholtz equation Nonsingular regularization and non-local boundary conditions

5 Non-uniqueness for Calderón s problem Conductivity equation: γ u = 0 in Ω. Dirichlet to Neumann map: Λ γ : f ν γ u Ω. Anisotropic conductivity: γ = (γ ij ) pos. def. sym. matrix Non-uniqueness Calderón s problem: Λ γ1 = Λ γ2 γ 1 = γ 2? If ψ : Ω Ω is a diffeomorphism, ψ Ω = Identity, then ( (Dψ) T ) γ (Dψ) Λ ψ γ = Λ γ where ψ γ = ψ 1. det (Dψ)

6 Cloaking for EIT Cloaking for Electrostatics F : B 2 \{0} B 2 \B 1 ( F(y) = 1 + y ) y 2 y. 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. Removable singularity argument (only works for Laplacian).

7 Outline Cloaking for Electrostatics 1 Cloaking for Electrostatics 2 Regularization scheme Cloaking a passive medium Cloaking an active medium Cloak-busting inclusion and lossy layer 3 Ideal cloaking for the Helmholtz equation Nonsingular regularization and non-local boundary conditions

8 Maxwell s equations and Impedance map Maxwell s equations of electromagnetic fields E(x) and H(x) (as 1-forms): E = iωµh H = iωεe + J in Ω with permittivity ε(x) and permeability µ(x) (as linear operators that map 1-forms to 2-forms). For regular (Nonsingular) medium: ε and µ are pos. def. sym. and bounded from below, (E, H) H(curl) H(curl). Impedance map: Λ µ,ε : ν E Ω ν H Ω.

9 Transformation invariance Let F : Ω Ω be a diffeomorphism. Denote y = F(x). Pullback fields by F 1 : Ẽ(y) = (F 1 ) E := (DF T ) 1 E F 1 (y) similarly define H(y), J(y) = (F 1 ) J := [det (DF)] 1 DF J F 1 Push-forward of medium by F: 1 µ(y) = F µ := det (DF(x)) DF(x) µ(x) x=f DF(x)T similarly define ε(y). 1 (y) Then Ẽ = iω µ H, H = iω εẽ + J in Ω Moreover, if F Ω = Identity, we have Λ µ, ε = Λ µ,ε

10 Electromagnetic ideal cloaking design in B 2 F : B 2 \{0} B 2 \B 1 ( F(x) = 1 + x ) x 2 x. F B2 = Identity. Greenleaf-Kurylev-Lassas-Uhlmann (2007) (µ, ε) = { (I, I) in B 2 (F I, F ( µ, ε) = I) in B 2 \B 1 (µ 0, ε 0 ) arbitrary in B 1 Λ µ, ε = Λ I,I.

11 Singular cloaking medium Cloaking device medium: 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 +! Notions of solutions for singular systems: Physically meaningful Forms with degenerate singular weighted energy may not be distributions In what sense the equations are satisfied? The cloaked region is considered isolated from the cloaking device or not? Finite energy solutions (Ẽ, H) (Greenleaf-Kurylev-Lassas-Uhlmann).

12 Blow-up-a-small-ball regularization scheme Nonsingular 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 Regularized cloaking for Scalar optics and Acoustics (Helmholtz equations): Kohn-Onofrei-Vogelius-Weinstein Scattering estimates of small inhomogeneity. Analysis as ρ 0 + (Nguyen; Nguyen-Vogelius) Truncation-based regularization scheme for Helmholtz equations in 3D: Greenleaf-Kurylev-Lassas-Uhlmann

13 EM approximate cloaking Construct regular 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). Is Λ µρ, ε ρ Λ I,I? Yes and No.

14 EM waves in physical and virtual space Physical space: Virtual space: Ẽ + ρ = iω µ ρ H + ρ, H + ρ = iω ε ρ Ẽ + ρ + J, in B 2\B 1, Ẽ ρ = iωµ 0 H ρ, H ρ = iωε 0 Ẽ ρ + J, in B 1, ν Ẽ + ρ B + 1 ν H ρ + B + 1 ν Ẽ ρ + B2 = f. = ν Ẽρ B, 1, = ν H ρ B 1 E + ρ = iωh + ρ, H + ρ = iωe + ρ + J, in B 2\B ρ, Eρ = iω((fρ 1 ) µ 0 )Hρ, Hρ = iω((fρ 1 ) ε 0 )Eρ + J, in B ρ, ν E ρ + B + ρ = ν Eρ B ρ, ν H + ρ B + ρ = ν H ρ B ρ, ν E + ρ B2 = f.

15 Cloaking a passive medium: J = 0 Assume µ 0 and ε 0 are positive constants. Spherical expansion: E + ρ = Ẽ ρ = ε 1/2 0 n n=1 m= n n n=1 m= n α m n M m n,kω + β m n M 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. Systems of linear equations for coefficients with coefficients matrix A 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 ).

16 Cloaking a passive medium: J = 0 Theorem 1 (Liu-Z) Suppose ω is not an eigenvalue of the transmission problems. Then we have Λ µ, ε Λ I,I = O(ρ 3 ) as ρ 0 +. boundary condition from the interior ν Ẽ ρ B 1 0, ν H ρ B 1 0.

17 Demonstration (passive) 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.

18 Cloaking an active medium: J 0 supported in B 1 Given an internal point current J = α <K ( α x δ 0 (x))v α at the origin, Spherical expansion: Ẽ ρ = ε 1/2 0 E + ρ = n n=1 m= n n n=1 m= n Convergence order as ρ 0: 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. γ m n = O(1), η m n = O(1); c m n = O(ρ n+1 ), d m n = O(ρ n+1 ); α m n = O(1), β m n = O(1).

19 Cloaking an active medium: J 0 supported in B 1 Theorem 2 (Liu-Z) With an internal point current J = α <K ( α x δ 0 (x))v α at the origin, if ω is not an eigenvalue of the transmission problems, we have Λ µ, ε Λ I,I = O(ρ 2 ) as ρ 0 +.

20 Demonstration (active) 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.

21 Resonance and Cloak-busting inclusion 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. For example, when (ω, µ 0, ε 0 ) satisfies for some n j n (kω) µ 0 J n (kω) = ρ j n(ωρ)h (1) n (2ω) h (1) n (ωρ)j n (2ω) J n (ωρ)h (1) (2ω) H n (ωρ)j n (2ω) where k = (µ 0 ε 0 ) 1/2. ω is the resonant frequency; (µ 0, ε 0 ) is called cloak-busting inclusion. n

22 Demonstration (Resonance and Cloak-busting inclusion) Er(ρ) µ x 10 3 Lossless passive cloaking (ρ=0.01) ω ρ Er(ρ) Cloak busting inclusion at ω=14, n=1, k=1 ε 0 Lossless active cloaking (ρ=0.01) ω ρ Boundary error Er(ρ) for mode n = 1, when ρ = 0.01 and µ 0 = ε 0 = 2, against frequency ω [1, 3] (Left: passive; Right: active). Cloak-busting inclusion medium µ 0 (left) and ε 0 (right) against ρ (0, 0.1) at ω = 14 for mode n = 1. Notice the singular behavior of the coefficients near the cloaking surface in resonant modes!

23 Remedy: Lossy layer F 2ρ blows up B 2ρ to B 1, τ is the damping parameter (conductivity). Spherical expansion of EM fields in three layers. Lossy regularization for the Helmholtz equations (Kohn-Onofrei-Vogelius-Weinstein, Kohn-Nguyen). Remedies by SS lining, SH lining (FSH lining) (Liu).

24 Demonstration (lossy) Re(Ẽ ρ) 1 when ω = 5, ε 0 = µ 0 = 2, ρ = 1/6 and τ = 3 for the lossy cloaking of a passive medium. The boundary convergence order is the same as the lossless cloaking (3 for passive, 2 for active).

25 Demonstration (lossy) Er(ρ) Lossy passive approximate cloaking (ρ=0.01) ω Resonant frequencies disappear. Boundary error Er(ρ) for mode n = 1 when ρ = 0.01 of lossy approximate cloaking (passive), against frequency ω [1, 10]. Observation: At some frequencies Er(ρ) is relatively large. Are such frequencies the poles or transmission eigenvalues in the complex plane?

26 Outline Cloaking for Electrostatics Ideal cloaking for the Helmholtz equation Nonsingular regularization and non-local boundary conditions 1 Cloaking for Electrostatics 2 Regularization scheme Cloaking a passive medium Cloaking an active medium Cloak-busting inclusion and lossy layer 3 Ideal cloaking for the Helmholtz equation Nonsingular regularization and non-local boundary conditions

27 Ideal cloaking for the Helmholtz equations Ideal cloaking for the Helmholtz equation Nonsingular regularization and non-local boundary conditions 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 in the whole space R 2 : ( σ, λ) (I, 1) x > 2 = (F I, F 1) 1 < x 2 (σ a, λ a ) x 1 where F λ(x) := [det(df)λ] F 1 (x).

28 Singular cloaking medium Ideal cloaking for the Helmholtz equation Nonsingular regularization and non-local boundary conditions Cloaking device medium: 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 +!

29 Truncation based regularization scheme Ideal cloaking for the Helmholtz equation Nonsingular regularization and non-local boundary conditions Regular approximate inverse of mass tensor and bulk modulus with regularization parameter 1 < R < 2 ( σ R, λ R ) = { ( σ, λ) x > R (σ a, λ a ) x R Case without internal source p = 0 in B R : Greenleaf-Kurylev-Lassas-Uhlmann

30 Ideal cloaking for the Helmholtz equation Nonsingular regularization and non-local boundary conditions 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.

31 Ideal cloaking for the Helmholtz equation Nonsingular regularization and non-local boundary conditions Cloaking a homogeneous medium with an internal source Given p C (R 2 ) with supp(p) B R0 (0 < R 0 < 1) 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.

32 Resonance observations due to the internal source Ideal cloaking for the Helmholtz equation Nonsingular regularization and non-local boundary conditions Resonant frequency limit ω 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 + ( 2 θ) 1/2 V] r=1 + = 0. A non-local boundary condition

33 Non-resonant result: non-local boundary conditions Ideal cloaking for the Helmholtz equation Nonsingular regularization and non-local boundary conditions 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. 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.

34 Ideal cloaking for the Helmholtz equation Nonsingular regularization and non-local boundary conditions Conclusions Electromagnetic non-singular approximate cloaking Convergence orders are 3 for passive cloaking and 2 for active cloaking for both lossless and lossy cloaking; 2D acoustic cloaking: A non-local boundary condition is obtained as R 1 +.

35 Ideal cloaking for the Helmholtz equation Nonsingular regularization and non-local boundary conditions Thank you!