The Mathematics of Invisibility: Cloaking Devices, Electromagnetic Wormholes, and Inverse Problems. Lectures 1-2

Transcription

1 CSU Graduate Workshop on Inverse Problems July 30, 2007 The Mathematics of Invisibility: Cloaking Devices, Electromagnetic Wormholes, and Inverse Problems Lectures 1-2 Allan Greenleaf Department of Mathematics, University of Rochester joint with Yaroslav Kurylev, Matti Lassas and Gunther Uhlmann

2 A movie... mjlassas/movie1.avi And a picture... from Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith(2006

3 And another picture... from G., Kurylev, Lassas and Uhlmann (March, 2007)

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5 Q. What is invisiblity, or cloaking?

6 Q. What is invisiblity, or cloaking? A. Hiding something from observation, with the fact of hiding being imperceptible as well.

7 Invisibility ( or Cloaking ) Devices, circa May, 2006 U. Leonhardt: dimension 2 via conformal mapping J. Pendry, D. Schurig and D. Smith: via singular transformations

8 Invisibility ( or Cloaking ) Devices, circa May, 2006 U. Leonhardt: dimension 2 via conformal mapping J. Pendry, D. Schurig and D. Smith: via singular transformations July Cummer, Popa, Schurig, Smith and Pendry: numerics

9 Invisibility ( or Cloaking ) Devices, circa May, 2006 U. Leonhardt: dimension 2 via conformal mapping J. Pendry, D. Schurig and D. Smith: via singular transformations July Cummer, Popa, Schurig, Smith and Pendry: numerics October Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith: Physical experiment - Cloaking a copper disc from microwaves.

10 Invisibility ( or Cloaking ) Devices, circa May, 2006 U. Leonhardt: dimension 2 via conformal mapping J. Pendry, D. Schurig and D. Smith: via singular transformations July Cummer, Popa, Schurig, Smith and Pendry: numerics October Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith: Physical experiment - Cloaking a copper disc from microwaves. November G., Kurylev, Lassas and Uhlmann: Rigorous treatment of spherical and cylindrical cloaks, necessity of linings.

11 April, June 2007 Ruan, Yan, Neff, Qiu: Approximate cloaking; Experimental cloak ( reduced parameters ) is visible.

12 April, June 2007 Ruan, Yan, Neff, Qiu: Approximate cloaking; Experimental cloak is visible April Weder: Time domain analysis, self-adjoint extensions

13 April, June 2007 Ruan, Yan, Neff, Qiu: Approximate cloaking; Experimental cloak is visible April Weder: Time domain analysis, self-adjoint extensions March, April G.-K.-L.-U.: Electromagnetic wormholes

14 April, June 2007 Ruan, Yan, Neff, Qiu: Approximate cloaking; Experimental cloak is visible April Weder: Time domain analysis, self-adjoint extensions March, April G.-K.-L.-U.: Electromagnetic wormholes July G.-K.-L.-U.: Approximate cloaking, improvement with SHS lining

15 Cloaking made feasible by metamaterials:

16 Cloaking made feasible by metamaterials: Manmade materials having electromagnetic properties not found in nature, or even in conventional composite materials.

17 Cloaking made feasible by metamaterials: Manmade materials having electromagnetic properties not found in nature, or even in conventional composite materials. Metamaterials allow variable electromagnetic (EM) material parameters to be custom designed.

18 Cloaking made feasible by metamaterials: Manmade materials having electromagnetic properties not found in nature, or even in conventional composite materials. Metamaterials allow variable electromagnetic (EM) material parameters to be custom designed. Fabricated and used for microwaves since late 90s, e.g., NIMs (negative index materials).

19 Cloaking made feasible by metamaterials: Manmade materials having electromagnetic properties not found in nature, or even in conventional composite materials. Metamaterials allow variable electromagnetic (EM) material parameters to be custom designed. Fabricated and used for microwaves since late 90s, e.g., NIMs (negative index materials). Now becoming available at optical wavelengths.

20 Note: Metamaterials are monochromatic/ narrow band: Need to be fabricated and assembled for particular frequency. = Cloaking devices and wormholes should be considered as being very narrow band. = Applications will probably be in controlled, engineered settings.

21 See picture of experimental device from Schurig, et al.

22 Q: (1) Can invisibility from observation by EM waves hold (theoretically) for waves at some (or all) frequencies k?

23 Q: (1) Can invisibility from observation by EM waves hold (theoretically) for waves at some (or all) frequencies k? (2) Does mathematics have anything to contribute to the discussion? What does invisibility have to do with inverse problems?

24 Q: (1) Can invisibility from observation by EM waves hold (theoretically) for waves at some (or all) frequencies k? (2) Does mathematics have anything to contribute to the discussion? What does invisibility have to do with inverse problems? (3) What other artificial wave phenomena can one produce?

25 Q: (1) Can invisibility from observation by EM waves hold (theoretically) for waves at some (or all) frequencies k? (2) Does mathematics have anything to contribute to the discussion? What does invisibility have to do with inverse problems? (3) What other artificial wave phenomena can one produce? A: (2) Yes - see these lectures.

26 Q: (1) Does invisibility from observation by EM waves hold (theoretically) for waves at some (or all) frequencies k? (2) Does mathematics have anything to contribute to the discussion? What does invisibility have to do with inverse problems? (3) What other artificial wave phenomena can one produce? A: (2) Yes - see these lectures. (1) Yes. Hidden boundary conditions = Linings are necessary if want to hide EM active objects; Desirable even for passive objects Prevent blow-up of EM fields near cloaking surface Reduce far-field pattern in approximate cloaking

27 Q: (1) Does invisibility from observation by EM waves hold (theoretically) for waves at some (or all) frequencies k? (2) Does mathematics have anything to contribute to the discussion? What does invisibility have to do with inverse problems? (3) What other artificial wave phenomena can one produce? A: (2) Yes - see these lectures. (1) Yes. hidden boundary conditions = modifications are needed if want to hide EM active objects, desirable even for passive objects Prevent blow-up of EM fields near cloaking surface Reduce far-field pattern in approximate cloaking (3) Electromagnetic wormholes: invisible optical tunnels Change the topology of space vis-a-vis EM wave propagation

28 Q. What does invisibility have to do with inverse problems?

29 Q. What does invisibility have to do with inverse problems? Visibility = uniqueness in certain inverse problems

30 Q. What does invisibility have to do with inverse problems? Visibility = uniqueness in certain inverse problems Invisibility = counterexamples to uniqueness in certain inverse problems

31 Q. What does invisibility have to do with inverse problems? Visibility = uniqueness in certain inverse problems Invisibility = counterexamples to uniqueness in certain inverse problems Invisibility at frequency k = 0 electrostatics counterexamples to Calderón s inverse conductivity problem

32 Q. What does invisibility have to do with inverse problems? Visibility, detectability = uniqueness in inverse problems Invisibility = counterexamples to uniqueness in certain inverse problems Invisibility at frequency k = 0 electrostatics counterexamples to Calderón s inverse conductivity problem Invisibility at frequency k>0 optics, E&M counterexamples to inverse problems for Helmholtz, Maxwell

33 Conductivity equation: Electrostatics Interior of a region Ω R n,n=2, 3, filled with matter with electrical conductivity σ(x) (Ω = human body, airplane wing,...) Place voltage distribution on the boundary Ω, inducing electrical potential u(x) throughout Ω u(x) satisfies the conductivity equation, (σ )u(x) =0onΩ, u Ω = f = prescribed voltage Note: σ(x) can be a scalar- or symmetric matrix-valued function. (isotropic or anisotropic conductivity, resp.)

34 Physics: in the time or frequency domain? We live in the time domain: Waves u(x, t). But: Metamaterials are designed with particular frequencies in mind. Natural to consider invisibility in the frequency domain: Time-harmonic waves of frequency k 0: u(x)e ikt

35 Helmholtz: Scalar optics at frequency k>0 Material with index of refraction n(x) Speed of light = c/n(x) ;n(x) = 1 in vacuum or air. Optical wave function satisfies ( +k 2 n(x) ) u(x) =ρ(x) Anisotropic media Riemannian metric g ( g + k 2) u(x) =ρ(x)

36 Maxwell: Electricity & magnetism E = ikµ(x)h, H = ikɛ(x)e + J E(x) = electric, H(x) = magnetic fields; J(x) = internal current. Permittivity ɛ(x), permeability µ(x) scalar- or 2-tensor fields.

37 Direct problem for conductivity equation Standard assumptions = BVP has a unique solution. Voltage-to-current or Dirichlet-to-Neumann map for σ: f σ u ν := Λ σ(f) Inverse problem (Calderón, 1980): Does Λ σ determine σ? I.e., is σ(x) visible to electrostatic measurements at the boundary?

38 Yes in many cases. Isotropic: Sylvester-Uhlmann, n 3; Nachman, n = 2;... Ola-Paivarinta-Somersalo: Maxwell... Anisotropic: NOT SO FAST... Observation of Luc Tartar 1982: σ(x) transforms as a contravariant two-tensor of weight one = infinite dimensional loss of uniqueness

39 If F :Ω ΩisaC smooth transformation, then σ(x) pushes forward to give a new conductivity, σ = F σ, (F σ) jk (y) = 1 det[ F j x k (x)] n p,q=1 F j x p (x) Fk x q (x)σpq (x) with the RHS evaluated at x = F 1 (y), or σ = 1 DF (DF)t σ(df)

40 If F :Ω ΩisaC smooth transformation, then σ(x) pushes forward to give a new conductivity, σ = F σ, (F σ) jk (y) = 1 det[ F j x k (x)] n p,q=1 F j x p (x) Fk x q (x)σpq (x) with the RHS evaluated at x = F 1 (y). If F is a diffeomorphism, then ( σ )ũ =0 (σ )u =0, where u(x) =ũ(f (x)). If F fixes Ω, then Λ σ =Λ σ

41 Material parameters of optics and E&M: n(x), ɛ(x), µ(x) all transform in the same way.

42 Material parameters of optics and E&M: n(x), ɛ(x), µ(x) all transform in the same way. This is the basis for transformation optics : specifying parameter fields as pushforwards of simple fields

43 Material parameters of optics and E&M: n(x), ɛ(x), µ(x) all transform in the same way. This is the basis for transformation optics : specifying parameter fields as pushforwards of simple fields Allows theoretical design of devices with wide range of behaviors

44 Material parameters of optics and E&M: n(x), ɛ(x), µ(x) all transform in the same way. This is the basis for transformation optics : specifying parameter fields as pushforwards of simple fields Allows theoretical design of devices with wide range of behaviors Cloaking = extreme transformation optics :

45 Material parameters of optics and E&M: n(x), ɛ(x), µ(x) all transform in the same way. This is the basis for transformation optics : specifying parameter fields as pushforwards of simple fields Allows theoretical design of devices with wide range of behaviors Cloaking = extreme transformation optics : F is singular at some point(s) for cloaking, along curves for wormholes

46 Calderón problem for anisotropic conductivity σ: Can only hope to recover σ up to infinite-dimensional group of diffeomorphisms of Ω fixing the boundary. This already is a form of invisibility: Certain inhomogeneous/anisotropic media are indistinguishable (But nothing is being hidden yet...)

47 Best uniqueness result can hope for is: Theorem If Λ σ =Λ σ, then there is an F :Ω Ω such that σ = F σ. n = 2: True: Sylvester,..., Astala-Lassas-Paivarinta: σ L n 3: True for σ C ω ;evenσ C major open problem. BUT: All are under the assumption that σ is bounded above and below: 0 <c 1 σ(x) c 2 <. SO: Cloaking has to violate this.

48 [ Intermission I ]

49 First examples of invisibility Counterexamples to uniqueness for the Calderón problem Lassas, Taylor, Uhlmann (2003) ; G., Lassas, Uhlmann (2003)

50 First examples of invisibility Counterexamples to uniqueness for the Calderón problem Lassas, Taylor, Uhlmann (2003) ; G., Lassas, Uhlmann (2003) Note: Conductivity equation no longer uniformly elliptic. Need to replace the D2N map with the set of Cauchy data: {( u Ω, u ) ν Ω : u ranges over a space of solutions } Boundary measurements rather than scattering data.

51 Anisotropic conductivities that cannot be detected by EIT Domain Ω = B(0, 2) R 3, boundary Ω =S(0, 2) D = B(0, 1) region to be made invisible ( cloaked region )

52 Anisotropic conductivities that cannot be detected by EIT Domain Ω = B(0, 2) R 3, boundary Ω =S(0, 2) D = B(0, 1) region to be made invisible ( cloaked region ) Start with homogeneous, isotropic conductivity σ on Ω: σ(x) I 3 3 Blow up a point, say 0 Ω singular, anisotropic σ 1 on Ω \ D. One eigenvalue is (r 1) 2, the other two [c 1,c 2 ].

53 Anisotropic conductivities that cannot be detected by EIT Domain Ω = B(0, 2) R 3, boundary Ω =S(0, 2) D = B(0, 1) region to be made invisible ( cloaked region ) Start with homogeneous, isotropic conductivity σ on Ω: σ(x) I 3 3 Blow up a point, say 0 Ω singular, anisotropic σ 1 on Ω \ D. One eigenvalue is (r 1) 2, the other two [c 1,c 2 ]. Fill in the hole with any nonsingular conductivity σ 2 on D σ =( σ 1, σ 2 ) form a conductivity on Ω, singular on D + Spherical cloak with the single coating.

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55 Singular transformation F :Ω\{0} Ω \ D, F (x) = ( 1+ x ) x 2 x or F (rω) = ( 1+ r ) ω, 0 <r 2, ω S 2 2

56 Singular transformation F :Ω\{0} Ω \ D, F (x) = ( 1+ x ) x 2 x Showed that the Dirichlet problem for ( σ u) on Ω has a unique bounded L (weak) solution. Removable singularity theory = one-to-one correspondence { Solutions of ( σ ũ) =0} {Solutions of (σ u) =0} = σ and σ have the same Cauchy data The material in D is invisible to EIT. (Recall: σ 2 can be any smooth nonsingular conductivity.)

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59 Pendry, Schurig and Smith (2006) proposed the same construction, using the same F, for cloaking with respect to electromagnetic waves, i.e., solutions of Maxwell s equations.

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61 Pictures: Simulations and measurements from Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith(2006)

62 Leonhardt used conformal mapping in two-dimensions to define n(x) giving rise to cloaking. [PSS] and [Le] justified using both ray-tracing and transformation law (chain rule) for fields on the complement, Ω \ D. They did not (1) Specify what is allowable inside the cloaked region D: Material parameters (ɛ, µ)? Sources or sinks / internal currents? (In Duke experiment, ɛ(x) ɛ 0,µ(x) µ 0 in D and J = 0.) (2) Consider waves inside D or at D Q. Can cloaking be made rigorous? Yes, but...

63 Rigorous treatment = Hidden boundary conditions Need to understand fields on all of Ω, not just Ω \ D Singular equations = need to use weak solutions Appropriate notion: finite energy, distributional solutions Prove: FE solutions on Ω automatically satisfy certain boundary conditions at the cloaking surface D. Bonus: Understanding this has real world implications for improving the effectiveness of cloaking.

64 Cylindrical cloaking Experimental setup = thin slice of an infinite cylinder Note: Actual experiment used reduced parameters. See Yan, Ruan and Qiu (arxiv.org: ) for analysis.

65 Cylindrical cloaking Experimental setup = thin slice of an infinite cylinder Cylindrical coordinates (x 1,x 2,x 3 ) (r, θ, z) E =(E θ,e r,e z ), etc.

66 Cylindrical cloaking Experimental setup = thin slice of an infinite cylinder Cylindrical coordinates (x 1,x 2,x 3 ) (r, θ, z) E =(E θ,e r,e z ), etc. Hidden BC: E θ = H θ =0on cloaking surface {r =1}

67 Can this be fixed? Yes: (1) Insert physical surface (lining) at D to kill angular components of E,H: SHS (soft-and-hard surface)

68 Can this be fixed? Yes: (1) Insert physical surface (lining) at D to kill angular components of E,H: or (2) The double coating SHS (soft-and-hard surface) Make ɛ, µ be singular from both sides of D covering both inside and outside of D with appropriately matched metamaterials.

69 Can this be fixed? Yes: (1) Insert physical surface (lining) at D to kill angular components of E,H: or (2) The double coating SHS (soft-and-hard surface) Make ɛ, µ be singular from both sides of D covering both inside and outside of D with appropriately matched metamaterials. Use both in the wormhole design!

70 Approximate cloaking Impossible to build device with ideal material parameters ɛ, µ. Q. What happens to invisibility for a physically realistic cloaking device? Extreme values of ideal ɛ, µ can t be attained Can only discretely sample smoothly varying ideal parameter See also Ruan, Yan, Neff and Qiu (2007)

71 Approximate cloaking Impossible to build device with ideal material parameters ɛ, µ. Q. What happens to invisibility for a physically realistic cloaking device? Extreme values of ideal ɛ, µ can t be attained Can only discretely sample smoothly varying ideal parameter Soft-and-Hard Surface lining Q. How does including SHS improve approximate cloaking?

72 Total field for truncated cloaking, R =1.05 Red = with SHS lining, blue = without SHS

73 Scattered field for truncated cloaking, R =1.05 Red = with SHS lining, blue = without SHS

74 Electromagnetic wormholes Invisible tunnels (optical cables, waveguides) From and

75 Electromagnetic wormholes Invisible tunnels (optical cables, waveguides) Change the topology of space vis-a-vis EM wave propagation

76 Electromagnetic wormholes Invisible tunnels (optical cables, waveguides) Change the topology of space vis-a-vis EM wave propagation Based on blowing up a curve rather than blowing up a point Inside of wormhole can be varied to get different effects

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84 [ Intermission II ]