1st PRIMA Congress

Cloaking and Transformation

Optics

Gunther Uhlmann

University of Washington

Sydney, July 6, 2009

H. G. Wells: The Invisible Man (1897)

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Susan Storm Richards: The Invisible Woman (1961)

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Transformation Optics

Two Recent Articles in Science on Invisibility “Control-ling Electromagnetic Fields”, J.B. Pendry, D. Schurig,D.R. Smith, Science 312, pp. 1780-1782, (June 2006).

Related article by Ulf Leonhard “Optical ConformalMapping” in same issue.

Earlier article of A. Greenleaf, M. Lassas and G-U,“Non-uniqueness for Calderón’s problem”, Math. Re-search Letters, 2003.

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Transformation Optics

Science Magazine

No. 5, Breakthrough of 2006: THE ULTIMATE CAMOUFLAGE

“The real breakthrough may lie in the theoretical toolsused to make the cloak. In such ”transformation op-tics,” researchers imagine–à la Einstein–warping emptyspace to bend the path of electromagnetic waves. Amathematical transformation then tells them how tomimic the bending by filling unwarped space with amaterial whose optical properties vary from point topoint. The technique could be used to design anten-nas, shields, and myriad other devices. Any way youlook at it, the ideas behind invisibility are likely to casta long shadow.”

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From Pendry et al’s paper

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All Boundary measurements for the homogeneous con-ductivity γ = 1 and the degenerate conductivity �σ arethe same

Figure: Analytic solutions for the currents

Based on work of Greenleaf-Lassas-U, MRL 20037

White Hole

From Pendry et al’s paper

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Index of Refraction

Fermat’s Principle: Minimize Optical Length

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Mirage

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S. Cummer, B. Pope, D. Schurig, D. Smith and J.Pendry, “Full-wave simulations of electromagnetic cloak-ing structures”, Phys. Rev. E 74 036621 (2006)

“It is open problem whether full-wave cloaking is possible,even in theory”

Answer : It is possible for all frequencies for electro-magnetic waves. This is joint work with A. Greenleaf,M. Lassas and Y. Kurylev. (Comm. Math. Physics,2007).

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Model :Maxwell’s Equations for Time Harmonic Waves

∇× E − ikµ(x)H = 0∇×H + ik�(x)E = 0D = �E, B = µH

divD = 0,divB = 0

J. Maxwell

(E, H) Electromagnetic Field

D=electric displacementB=magnetic displacement

�(x)=electric permittivityµ(x)=magnetic permeability

Index of refraction:√

�µ

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What do we mean by invisibility?

Object to be cloaked in D. (�(x), µ(x)) electromagneticparameters of Ω, arbitrary on D.We want to show that if Maxwell’s equations are solvedin Ω, boundary information of solutions on Ω is same as the case of

�(x) =

1 0 00 1 00 0 1

, µ(x) =

1 0 00 1 00 0 1

(Homogeneous case)

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We will consider a special case of anisotropic materials:

�(x) = µ(x) = g−1√

det g were g = (gij) is a semiposi-tive definite symmetric matrix

Similar (Helmholtz, Acoustic waves, Quantum waves)

∆gE + k2E = 0

∆gH + k2H = 0

∆g = Laplace-Beltrami operator

= 1√det g

�3i,j=1

∂

∂xi(√

det g gij ∂∂xj

)

(gij) = (gij)−1.

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(Helmholtz)

∆gE + k2E = 0

∆gH + k2H = 0

Consider first static case (k = 0)

∆gE = ∆gH = 0

This problem in dimension n ≥ 3 is equivalent to theElectrical Impedance Tomography (Calderón’s Problem)

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CALDERÓN’S PROBLEM

Ω ⊂ Rn(n = 2,3)

Can one determine the electrical conductivity of Ω, γ(x),by making voltage and current measurements at theboundary?(Calderón; Geophysical prospection)

Early breast cancer detectionNormal breast tissue 0.3 mhoCancerous breast tumor 2.0 mho

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ACT3 imaging blood as it leaves the heart( blue) and fills the

lungs (red) during systole.

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CALDERÓN’S PROBLEM (EIT)

Consider a body Ω ⊂ Rn. An electrical potential u(x)causes the current

I(x) = γ(x)∇u

The conductivity γ(x) can be isotropic, that is, scalar,or anisotropic, that is, a matrix valued function. If thecurrent has no sources or sinks, we have

div(γ(x)∇u) = 0 in Ω

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div(γ(x)∇u(x)) = 0u

���∂Ω

= fγ(x) = conductivity,f = voltage potential at ∂Ω

Current flux at ∂Ω = (ν · γ∇u)���∂Ω

were ν is the unitouter normal.

Information is encoded inmap

Λγ(f) = ν · γ∇u���∂Ω

EIT (Calderón’s inverse problem)

Does Λγ determine γ ?

Λγ = Dirichlet-to-Neumann map

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Objects one cannot cloack

a) Isotropic case

0 < C1 ≤ γ(x) ≤ C2 x ∈ ΩΛγ1 = Λγ2 ⇒ γ1 = γ2

• n ≥ 3, Sylvester-U, 1987

Assumption, γ ∈ C2(Ω) improved to γ ∈ C3/2(Ω)Also valid for isotropic Helmoltz at fixed frequency ∆ + k2n(x)

n ∈ L∞ (more generally for the Schrödinger equation ∆+q), q ∈ L∞.

• n = 2, γ ∈ L∞, (Astala-Päivärinta, 2006)

For isotropic Helmholtz equation ∆+k2n(x), n(x) ∈ L∞and Schrödinger equation ∆ + q, q ∈ L∞ have recentlybeen proven by Bukhgeim (2008).

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Objects one cannot cloack

b) Anisotropic case

γ = (γij) positive-definite in Ω, that is there existsC1, C2 > 0 such that

C1|ξ|2 ≤�

γij(x)ξiξj ≤ C2|ξ|2, x ∈ Ω

• n = 3, γij real-analytic (Lassas-U, 2001)

Also true for Helmholtz equation ∆g + k2, g real ana-lytic.

• n = 2, γij ∈ L∞, (Astala-Lassas-Päivärinta, 2005)

True for Helmholtz equation ∆g+k2 (Bukhgeim, 2008)

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DIRICHLET-TO-NEUMANN MAP(M, g) compact Riemannian manifold with boundary.

∆g Laplace-Beltrami operator g = (gij) pos. def. sym-metric matrix

∆gu =1

√det g

n�

i,j=1

∂

∂xi

�

det g gij∂u

∂xj

(gij) = (gij)−1

∆gu = 0 on M

u

���∂M

= f

Conductivity:γ

ij =√

det g gij

Λg(f) =n�

i,j=1ν

jgij ∂u

∂xi

�det g

�����∂M

ν = (ν1, · · · , νn) unit-outer normal26

∆gu = 0

u

���∂M

= f

Λg(f) =∂u

∂νg

=n�

i,j=1ν

jgij ∂u

∂xi

�det g

�����∂M

current flux at ∂M

Inverse-problem (EIT)

Can we recover g from Λg ?

Λg = Dirichlet-to-Neumann map or voltage to currentmap

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ANOTHER MOTIVATION (STRING THEORY)

HOLOGRAPHY

Dirichlet-to-Neumann map is the “boundary-2pt func-tion”

Inverse problem: Can we recover (M, g) (bulk) fromboundary-2pt function ?

M. Parrati and R. Rabadan, Boundary rigidity and holog-raphy, JHEP 0401 (2004) 034

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∆gu = 0

u

���∂M

= fΛg(f) =

∂u

∂νg

�����∂M

Λg ⇒ g ?

Answer: No Λψ∗g = Λg where

ψ : M → M diffeomorphism, ψ���∂M

= Identity and

ψ∗g =

�Dψ ◦ g ◦ (Dψ)T

�◦ ψ

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Theorem (n ≥ 3) (Lassas-U, Lassas-Taylor-U) (M, gi), i =1,2, real-analytic, connected, compact, Riemannian man-ifolds with boundary. Assume

Λg1 = Λg2

Then ∃ψ : M → M diffeomorphism, ψ���∂M

= Identity, sothat

g1 = ψ∗g2

One can also determine topology of M, as well (onlyneed to know Λg, ∂M).

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Non-uniqueness for EIT

Motivation (Greenleaf-Lassas-U, MRL, 2003)

When bridge connecting the two partsof the manifold gets narrower theboundary measurements give less infor-mation about isolated area.

When we realize the manifold in Euclidean space weshould obtain conductivities whose boundary measure-ments give no information about certain parts of thedomain.

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Greenleaf-Lassas-U (2003 MRL)

Let Ω = B(0,2) ⊂ R3,D = B(0,1) where B(0, r) = {x ∈ R

3; |x| < r}

F : Ω \ {0}→ Ω \ D

F (x) = (|x|2

+ 1)x

|x|

F diffeomorphism, F���∂Ω

= Identity

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Letg = identity metric in B(0,2)ĝ = (F−1)∗g on B(0,2) \ B(0,1)σ̂ = conductivity associated to ĝ

In spherical coordinates (r, φ, θ)→ (r sin θ cosφ, r sin θ sinφ, r cos θ)

σ̂ =

2(r − 1)2 sin θ 0 00 2 sin θ 00 0 2(sin θ)−1

Let �g be the metric in B(0,2) (positive definite in B(0,1))s.t. �g = ĝ in B(0,2) \ B(0,1). Then

Theorem (Greenleaf-Lassas-U 2003)

Λ�g = Λg

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Based on work of Greenleaf-Lassas-U, MRL 2003

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Pendry et al’s construction (TRANSFORMATION ME-DIA):

Step1 : Change coordinates and write Maxwell’s equa-tions in new coordinates.

In our notation: (F−1)∗g where g = Identity.

Recall that this preserves boundary measurements.

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From Pendry et al’s paper

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From Pendry et al’s paper

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Pendry et al: Second StepSpherical coordinates, (r, θ, φ). “All fields in the regionr < R2 are compressed into the region R1 < r < R2”

r� = R1 +

r(R2 −R1)R2

φ� = φ

θ� = θ

(F)

Applying the coordinate transformation (F )

��r� = µ

�r� =

R2

R2 −R1(r� −R1)2

r�2

��θ� = µ

�θ� =

R2

R2 −R1��φ� = µ

�φ� =

R2

R2 −R1

were � is electric permittivity and µ is magnetic permeability

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From Pendry et al’s paper

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Helmholtz Equation (Acoustic Cloaking)

1√

det g

n�

i,j=1

∂

∂xi

�

det g gij∂u

∂xj

+ k2u = 0 (gij) = (gij)−1.

Acoustic equation with density ρ =√

det ggij and bulkmodulus λ−1 =

√det g

g−1 =

2(r − 1)2 sin θ 0 00 2 sin θ 00 0 2(sin θ)−1

Theorem (Greenleaf-Kurylev-Lassas-U, CMP 2007)Λg = Λe

Acoustic Cloaking:H. Chen and C. J. Chan, Appl. Phy. Lett., (2007)S. Cummer et al, PRL (2008);

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Tsunami Cloaking

“Broadband cylindrical acoustic cloak for linear surface waves in a

fluid”, M. Farhat et al, PRL (2008).

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Isotropic Transformation Optics (Greenleaf-Kurylev-Lassas-U, PRL 2008, NJP 2008)

Acoustic and conductivity equation:

n�

i,j=1

∂

∂xi

(γij∂u

∂xj

) +k2

√det γ

u = 0 .

F (x) =

x, |x| > 2(1 + |x|2 )

x

|x|, 0 < |x| < 2

�γ =

F∗(δjk), x ∈ B(0,3)−B(0,1)2(δjk), x ∈ B(0,1)

, F∗γ =1

|detF |(DF )◦γ◦(DF )T◦F−1

(�γjkR

)(x) =

(�γjk)(x), |x| > R2(δjk), |x| ≤ R

,1 < R < 2,non-singular anisotropic

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( �γjkR

non-singular anisotropic)

Homogenization (Allaire, Cherkaev, Milton)

Construct isotropic conductivities �γR,ε that give almostcloaking for acoustic and conductivity equation

Λ�γR,ε −−−−−→ε→0,R→1

Λe , e = (δjk)

div(�γR,ε∇u) +k2

�det �γR,ε

u = 0.

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Approximate quantum cloaks (Greenleaf-Kurylev-Lassas-U, PRL 2008, NJP 2008) E is not a Neumann eigen-value in cloaked region

ϕ =�

�γR,εu,

(−∆ + VE,R,ε)ϕ = Eϕ, E = k

VE,R,ε =∆

��γR,ε

��γR,ε

+ E(1−1

det �γR,ε)

W bounded potential on B(0,1). Construct a sequenceVn(E) s.t.

ΛVn(E)+W −→n→∞ Λ0

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Left: Approximate cloak with E not a Neumann eigen-value; matter wave passes unaltered.

Right: E a Neumann eigenvalue: cloak supports almostbound state.

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Cloaking for Maxwell’s equations (Passive Devices)

Model :Maxwell’s Equations for Time Harmonic Waves

∇× E − ikµ(x)H = 0∇×H + ik�(x)E = 0D = �E, B = µH

divD = 0,divB = 0

(E, H) Electromagnetic Field

D=electric displacementB=magnetic displacement

�(x)=electric permittivityµ(x)=magnetic permeability

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We will consider a special case of anisotropic materials:�(x) = µ(x) = g−1

√det g were g = (gij) is a semiposi-

tive definite symmetric matrix

�g =

ĝ on Ω \ D, ĝ = (F−1)∗eˆ̂g on D, ˆ̂g Riemannian metric on D

Theorem (Greenleaf-Lassas-Kurylev-U, CMP, 2007)

Λe = Λ�g

Here

Λg(E ∧ ν) = H ∧ ν

where ν is the inner-unit normal.More generally we define the Cauchy data (E∧ν, H∧ν).

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Cloaking for Maxwell’s equations (Active Devices)Model :Maxwell’s Equations for Time Harmonic Waves

∇× E − ikµ(x)H = 0∇×H + ik�(x)E = JD = �E, B = µH

divD = 0,divB = 0

Where J is a non-zero current supported in cloaked re-gion.

�g =

ĝ on Ω \ D, ĝ = (F−1)∗eˆ̂g on D, ˆ̂g singular on ∂D

Theorem (Greenleaf-Kurylev-Lassas-U, CMP, 2007)

Λ�g = Λe.

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Based on work of A. Greenleaf, Y. Kurylev, M. Lassas–U

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Metamaterials

Negative Refractive Index : −√�µ

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Negative index of refraction for visible light

Nature 2008, Valentine et al, X. Zhang’s group at UC Berkeley

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Metamaterials

Cloaking

At microwave frequencies (D. Smith et al):

”Two-dimensional metamaterial structure exhibiting reduced visi-

bility at 500nm”, I.I. Smolyaninov et al, (Opt. Lett. 2008).

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Electromagnetic Wormholes

Harry Potter’s invisibility sleeve (Scientific American)(A. Greenleaf, Y. Kurylev, M. Lassas–U, PRL, 2007)

How to construct a device that functions like a worm-hole for electromagnetic fields?

Figure: Schematic two-dimensional figure and an artis-tic interpretation of the wormhole device by ScientificAmerican.

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Blueprints of a Wormhole for Electromagnetic Waves

Take a cylinder

make parallel corrugation on its surface and coat it withsuitable metamaterials. Such material has already beenconstructed for microwaves. Studies on optical frequen-cies are going on.

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A ray coming back

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Ray tracing simulations: the end of a wormhole

Length of handle

Nature News (Nov. 15, 2007)

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Possible applications in future:

• Invisible optical cables.

• Components for opticalcomputers.

• 3D video displays: ends ofinvisible tunnels work as light source in 3D voxels.

• Light beam collimation.

• Virtual magnetic monopoles.

• Scopes for Magnetic Resonance Imaging devices.

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Optics: Watch your back

(Kosmas L. Tsakmakidis and Ortwin Hess)Nature 451, 27(January 3, 2008)

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