The Mathematics Cloaking Devices, Electromagnetic

The Mathematicsof Invisibility:

Cloaking Devices,Electromagnetic Wormholes,

andInverse Problems

Lectures 1 - 2

Allan GreenleafDepartment of Mathematics, University of Rochester

joint with

Yaroslav Kurylev, Matti Lassas and Gunther Uhlmann

CSU Graduate Workshop on Inverse ProblemsJuly 30, 2007

A movie ...

http://math.tkk.fi/ mjlassas/movie1.avi

And a picture ...

from Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith(2006

And another picture ...

from G., Kurylev, Lassas and Uhlmann (March, 2007)

http://arxiv.org/abs/math-ph/0703059

Q. What is invisiblity, or “cloaking”?

A. Hiding something from observation,with the fact of hiding being imperceptible as well.

Invisibility ( or “Cloaking” ) Devices, circa 2006-07

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 ofspherical and cylindrical cloaks, necessity of linings.

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

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

• Cloaking made feasible by metamaterials:

Manmade materials havingelectromagnetic properties not found in nature,

or even in conventional composite materials.

• Metamaterials allow variable electromagnetic (EM) material parametersto be custom designed.

• Fabricated and used for microwaves since late ’90s,e.g., NIMs (negative index materials).

• Now becoming available at optical wavelengths.

Note: Metamaterials are monochromatic/ narrow band:Need to be fabricated and assembled for particular frequency.

Cloaking devices and wormholes should be consideredas being very narrow band.

Applications will probably be in controlled, engineered settings.

See picture of experimental device from Schurig, et al.

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.

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

(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

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

(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

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

Visibility = uniqueness in certain inverse problems

Invisibility = counterexamples to uniquenessin certain inverse problems

Invisibility at frequency k = 0 ⇐⇒ electrostatics ⇐⇒counterexamples to Calderon’s inverse conductivity problem

Visibility, detectability = uniqueness in inverse problems

Invisibility at frequency k = 0 ⇐⇒ electrostatics ⇐⇒counterexamples to Calderon’s inverse conductivity problem

Invisibility at frequency k > 0 ⇐⇒ optics, E&M ⇐⇒counterexamples to inverse problems for Helmholtz, Maxwell

Conductivity equation: Electrostatics

• Interior of a region Ω ⊂ Rn, n = 2, 3, filled with matter withelectrical 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) = 0 on Ω,

u|∂Ω = f = prescribed voltage

Note: σ(x) can be a scalar- or symmetric matrix-valued function.(isotropic or anisotropic conductivity, resp.)

• Physics: in the “time” or “frequency” domain?

• We live in the time domain: Waves u(x, t).

• 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)eikt

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(∆ + k2n(x)

)u(x) = ρ(x)

• Anisotropic media ⇐⇒ Riemannian metric g(∆g + k2

)u(x) = ρ(x)

Maxwell: Electricity & magnetism

∇× E = ikµ(x)H, ∇× H = −ikε(x)E + J

• E(x) = electric, H(x) = magnetic fields; J(x) = internal current.

• Permittivity ε(x), permeability µ(x) scalar- or 2-tensor fields.

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:= Λσ(f)

Inverse problem (Calderon, 1980):

Does Λσ determine σ?

I.e., is σ(x) “visible” to electrostatic measurements at the boundary?

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

• If F : Ω −→ Ω is a C∞ smooth transformation, thenσ(x) pushes forward to give a new conductivity, σ = F∗σ,

(F∗σ)jk(y) =1

det[∂F j

∂xk (x)]

n∑p,q=1

∂F j

∂xp(x)

∂F k

∂xq(x)σpq(x)

with the RHS evaluated at x = F−1(y), or

|DF | (DF )tσ(DF )

• If F : Ω −→ Ω is a C∞ smooth transformation, thenσ(x) pushes forward to give a new conductivity, σ = F∗σ,

(F∗σ)jk(y) =1

det[∂F j

∂xk (x)]

n∑p,q=1

∂F j

∂xp(x)

∂F k

∂xq(x)σpq(x)

with the RHS evaluated at x = F−1(y).

• If F is a diffeomorphism, then

∇(σ · ∇)u = 0 ⇐⇒ ∇(σ · ∇)u = 0,

where u(x) = u(F (x)). If F fixes ∂Ω, then Λσ = Λσ

• Material parameters of optics and E&M:n(x), ε(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

Calderon problem for anisotropic conductivity σ:

Can only hope to recover σ up to infinite-dimensional group ofdiffeomorphisms of Ω fixing the boundary.

This already is a form of invisibility:

Certain inhomogeneous/anisotropic media are indistinguishable

(But nothing is being hidden yet...)

Best uniqueness result can hope for is:

“Theorem” If Λσ = Λσ, then there is an F : Ω −→ Ωsuch that σ = F∗σ.

n = 2: True: Sylvester,..., Astala-Lassas-Paivarinta: σ ∈ L∞

n ≥ 3: True for σ ∈ Cω; even σ ∈ C∞ major open problem.

BUT: All are under the assumption that σ isbounded above and below: 0 < c1 ≤ σ(x) ≤ c2 < ∞.

SO: Cloaking has to violate this.

[ Intermission I ]

First examples of invisibility

Counterexamples to uniqueness for the Calderon problemLassas, Taylor, Uhlmann (2003) ; G., Lassas, Uhlmann (2003)

First examples of invisibility

Counterexamples to uniqueness for the Calderon problemLassas, 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,

∂ν|∂Ω

): u ranges over a space of solutions

↔ “Boundary measurements” rather than scattering data.

“Anisotropic conductivities that cannot be detected by EIT”

• Domain Ω = B(0, 2) ⊂ R3, 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) ≡ I3×3

• Blow up a point, say 0 ∈ Ω → singular, anisotropic σ1 on Ω \ D.

One eigenvalue is ∼ (r − 1)2, the other two ∈ [c1, c2].

• Start with homogeneous, isotropic conductivity σ on Ω: σ(x) ≡ I3×3

• Blow up a point, say 0 ∈ Ω → singular, anisotropic σ1 on Ω \ D.

One eigenvalue is ∼ (r − 1)2, the other two ∈ [c1, c2].

• Fill in the “hole” with any nonsingular conductivity σ2 on D

• σ = (σ1, σ2) form a conductivity on Ω, singular on ∂D+

→ “Spherical cloak with the single coating”.

• Singular transformation F : Ω \ 0 −→ Ω \ D,

F (x) =(

1 +|x|2

F (rω) =(1 +

)ω, 0 < r ≤ 2, ω ∈ S2

• Singular transformation F : Ω \ 0 −→ Ω \ D,

F (x) =(

1 +|x|2

• Showed that the Dirichlet problem for ∇ · (σ∇u) on Ω has aunique bounded L∞ (weak) solution.

• Removable singularity theory =⇒ ∃ one-to-one correspondence

Solutions of ∇ · (σ∇u) = 0 ↔ Solutions of ∇ · (σ∇u) = 0

• =⇒ σ and σ have the same Cauchy data

• The material in D is invisible to EIT.(Recall: σ2 can be any smooth nonsingular conductivity.)

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.

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

• Leonhardt used conformal mapping in two-dimensionsto 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 (ε, µ) ?Sources or sinks / internal currents?(In Duke experiment, ε(x) ≡ ε0, µ(x) ≡ µ0 in D and J = 0.)

(2) Consider waves inside D or at ∂D

Q. Can cloaking be made rigorous? Yes, but . . .

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 boundaryconditions at the cloaking surface ∂D.

• Bonus: Understanding this has real world implications forimproving the effectiveness of cloaking.

Cylindrical cloaking

• Experimental setup = thin slice of an infinite cylinder

Note: Actual experiment used “reduced” parameters.See Yan, Ruan and Qiu (arxiv.org:0706.0655) for analysis.

• Cylindrical coordinates (x1, x2, x3) ↔ (r, θ, z)E = (Eθ, Er, Ez), etc.

• Hidden BC: Eθ = Hθ = 0 on cloaking surface r = 1

Can this be fixed? Yes:

(1) Insert physical surface (lining) at ∂D to killangular components of E, H:

SHS (soft-and-hard surface)

(2) The “double coating”

Make ε, µ be singular from both sides of ∂D⇐⇒ covering both inside and outside of ∂Dwith appropriately matched metamaterials.

(2) The “double coating”

Make ε, µ be singular from both sides of ∂D⇐⇒ covering both inside and outside of ∂Dwith appropriately matched metamaterials.

Use both in the wormhole design!

Approximate cloaking

Impossible to build device with ideal material parameters ε, µ.

Q. What happens to invisibility for a physically realisticcloaking device?

• Extreme values of ideal ε, µ can’t be attained

• Can only discretely sample smoothly varying ideal parameter

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