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SLIDE 1

Electromagnetic cloaking and super-resolved imaging

Habib Ammari Department of Mathematics and Applications Ecole Normale Sup´ erieure, Paris

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Introduction

Electromagnetic cloaking is to make a target invisible with respect to

probing by electromagnetic waves.

Two schemes: anomalous resonance and change of variables.
Mathematical justification of cloaking due to anomalous localized

resonance (CALR). (with G. Ciraolo, H. Kang, H. Lee, and G. Milton)

New cancellation technique to achieve enhanced near-cloaking using the

change of variables scheme. (with H. Kang, H. Lee, and M. Lim)

Generalized polarization tensors (GPTs) vanishing structures.
Numerical illustrations and statistical stability of near-cloaking: (with J.

Garnier and V. Jugnon).

GPTs for shape description: GPTs can capture topology and

high-frequency shape oscillations. GPTs can be used to achieve super-resolution in electromagnetic imaging (with H. Kang, M. Lim, H. Zribi; with J. Garnier, H. Kang, M. Lim, S. Yu).

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Anomalous resonance

Ω: bounded domain in R2; D ⊂⊂ Ω. Ω and D of class C1,µ, 0 < µ < 1.

For a given loss parameter δ > 0, the permittivity distribution in R2 is given by ǫδ =      1 in R2 \ Ω, −1 + iδ in Ω \ D, 1 in D.

Configuration (plasmonic structure): core with permittivity 1 coated by

the shell Ω \ D with permittivity −1 + iδ.

For a given function f compactly supported in R2 satisfying
R2 fdx = 0

(conservation of charge), consider the following dielectric problem: ∇ · ǫδ∇Vδ = αf in R2, with the decay condition Vδ(x) → 0 as |x| → ∞.

Dielectric problems: models the quasi-static (zero-frequency) transverse

magnetic regime.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 4

Anomalous resonance

Fundamental problem: identify f such that when α = 1

Eδ :=

Ω\D

δ|∇Vδ|2dx → ∞ as δ → 0. |Vδ(x)| < C, when |x| > a for some constants C and a independent of δ.

Eδ: proportional to the electromagnetic power dissipated into heat by the

time harmonic electrical field averaged over time.

Infinite amount of energy dissipated per unit time in the limit δ → 0:

unphysical.

Choose α = 1/√Eδ: αf produces the same power independent of δ and

the new associated solution Vδ approaches zero outside the radius a.

Necessary and sufficient condition for CALR (with α = 1) Vδ/√Eδ goes

to zero outside some radius as δ → 0.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 5

Anomalous resonance

We develop a general method based on the potential theory to study

cloaking due to anomalous resonance.

Using layer potential techniques: we reduce the problem to a singularly

perturbed system of integral equations.

The system is non-self-adjoint ⇒ we introduce a symmetrization

technique in order to express the solution in terms of the eigenfunctions

f a self-adjoint compact operator.
Symmetrization technique: based on a generalization of a Calder´
n

identity to the system of integral equations and a general theorem on symmetrization of non-selfadjoint operators obtained by Khavinson

Putinar-Shapiro (2007).
We provide a necessary and sufficient condition on the source term under

which the blowup of the power dissipation takes place. The condition is given in terms of the Newtonian potential of the source 1 2π

R2 ln |x − y|f (y)dy,

x ∈ R2, which is the solution for the potential in the absence of the plasmonic structure.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Anomalous resonance

In the case of an annulus (D is the disk of radius ri and Ω =: Be is the

concentric disk of radius re), it is known (Milton et al.) that there exists a critical radius (the cloaking radius) r∗ =

r 3

e ri −1.

such that any finite collection of dipole sources located at fixed positions within the annulus Br∗ \ Be is cloaked.

We show that if f is an integrable function supported in E ⊂ Br∗ \ Be,

then CALR takes place provided the Newtonian potential of f is not identically zero in R2 \ E ⇒ most integrable sources αf supported in E will be cloaked. (quadrupole source inside the annulus Br∗ \ Be: cloaked).

Conversely we show that if the source function f is supported outside Br∗

then no cloaking occurs.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Anomalous resonance

Notation: Γi := ∂D and Γe := ∂Ω.
For Γ = Γi or Γe and ϕ ∈ L2(Γ), the single and double layer potentials:

SΓ[ϕ](x) := 1 2π

Γ

ln |x − y|ϕ(y) dσ(y), x ∈ R2, DΓ[ϕ](x) := 1 2π

Γ

∂ ∂ν(y) ln |x − y|ϕ(y) dσ(y) , x ∈ R2 \ Γ. ν(y): the outward unit normal to Γ at y.

Neumann-Poincar´

e operators: KΓ[ϕ](x) := 1 2π

Γ

y − x, ν(y) |x − y|2 ϕ(y) dσ(y), K∗

Γ: L2-adjoint of KΓ.

KΓ and K∗

Γ: compact in L2(Γ) if Γ is C1,α for some α > 0. Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 8

Anomalous resonance

Notation: F Newtonian potential of f ; H = L2(Γi) × L2(Γe); zδ =

iδ 2(2−iδ).

Representation formula:

Vδ(x) = F(x) + SΓi [ϕi](x) + SΓe[ϕe](x).

Introduce:

Φ := ϕi ϕe

,

g :=     ∂F ∂νi − ∂F ∂νe     .

Singularly perturbed equation:

(zδI2 + K∗)Φ = g.

K∗ : H → H Neumann-Poincar´

e-type operator (compact non-self-adjoint in general): K∗ :=     −K∗

Γi

− ∂ ∂νi SΓe ∂ ∂νe SΓi K∗

Γe

    .

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Anomalous resonance

The spectrum of K∗ lies in the interval [−1/2, 1/2].
The operator

S = SΓi SΓe SΓi SΓe

is self-adjoint and −S ≥ 0 on H.
Calder´
n’s-type identity: SK∗ = KS.
K∗ is Hilbert-Schmidt (in 2D; Schatten-von Neumann in 3D).
K∗ is symmetrizable: there is a bounded self-adjoint operator A on

Range(S) such that A √ −S = √ −SK∗.

Khavinson et al : let M ∈ Cp(H). If there exists a strictly positive

bounded self-adjoint operator R such that R2M is self adjoint, then there is a bounded self-adjoint operator A ∈ Cp(H) such that AR = RM.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Anomalous resonance

A self-adjoint ⇒ an orthogonal decomposition: H = KerA ⊕ (KerA)⊥,

and (KerA)⊥ = RangeA.

P and Q = I − P: the orthogonal projections from H onto KerA and

(KerA)⊥, respectively. Let λ1, λ2, . . . with |λ1| ≥ |λ2| ≥ . . . be the nonzero eigenvalues of A and Ψn be the corresponding (normalized)

eigenfunctions. A ∈ C2(H) ⇒

∞

n=1

λ2

n < ∞,

and AΦ =

∞

n=1

λnΦ, ΨnΨn, Φ ∈ H.

Theorem: If P

√ −Sg = 0, then CALR takes place. If Ker(K∗) = {0}, then CALR takes place if and only if δ

n

| √ −Sg, Ψn|2 λ2

n + δ2

→ ∞ as δ → 0.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 11

Anomalous resonance

Anomalous resonance in an annulus:

Eigenvalues λ of A= {±ρ|n|}, ρ = ri

re .

Theorem: (blow-up of power dissipation criterion) For a given source f

supported outside Be (with α = 1), ρN

0=|n|≤N

|g n

e |2

|n|ρ2|n| → ∞ as N → ∞ if and only if

Be\Bi

δ|∇Vδ|2 → ∞ as δ → 0. g n

e : Fourier coefficient of − ∂F ∂νe on Γe.

Theorem: E: a measurable subset of Br∗ \ Be such that R2 \ E is
connected. Suppose that f is an integrable function supported in E

satisfying

E f = 0. If Newtonian potential of f is not identically zero in

R2 \ E, then CALR takes place.

f = c1χB2r (y∗) − c2χBr (y∗), for some r > 0 and y∗, where c1 and c2 are

positive constants to satisfy

R2 fdx = 0. F ≡ 0 on R2 \ B2r(y∗) and f is

not cloaked.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Anomalous resonance

Anomalous resonance in an annulus:

Any source supported outside Br∗ cannot make the blow-up of the power

dissipation happen and is not cloaked. Indeed, in the limit δ → 0 the annulus itself becomes invisible to sources that are sufficiently far away.

Theorem: If f is supported in R2 \ Br∗, then
Be\Bi

δ|∇Vδ|2 < C holds for some constant C independent of δ (with α = 1). Moreover, we have sup

|x|≥r∗

|Vδ(x) − F(x)| → 0 as δ → 0.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Near-cloaking

Cloak a region inside the cloaking device.
Conductivity problem (quasi-static regime): the Dirichlet-to-Neumann

map is nearly the same as the one associated to the constant conductivity distribution.

Helmholtz equation: the scattering cross-section is nearly zero.
Change of variable scheme + structures with vanishing generalized

polarization tensors or scattering coefficients.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Transformation of PDE

Let Λ[σ] be the Dirichlet-to-Neumann map, i.e., Λ[σ](φ) = σ ∂u ∂ν where u is the solution to

∇ · σ∇u = 0,

in Ω, u = φ,

n ∂Ω.

Let F be a diffeomorphism of Ω which is identity on ∂Ω. Push-forward of σ by F to obtain the anisotropic conductivity: F∗σ(y) = DF(x)σ(x)DF(x)t det(DF(x)) , x = F −1(y). Then Λ[σ] = Λ[F∗σ].

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Singular transformation

Greenleaf-Lassas-Uhlmann (2003):

F : {x : 0 < |x| < 2} → {x : 1 < |x| < 2} is given by F(x) :=

1 + |x|

2 x |x|. Then, anything inside the hole {|x| < 1} surround by a suitable anisotropic conductivity is invisible by the DtN map (perfect cloaking).

Pendry et al (2006) used exactly the same transformation optics for

electromagnetic cloaking.

Physically: selective bending of light rays, i.e., a ray is diverted in the

direction of the high conductivity, routed tangentially around |x| = 1, and then ejected out the other side to continue on its way.

And then, many works on cloaking have been produced. (A review in

SIREV by Greeleaf-Lassas-Kurylev-Uhlmann (2009))

Drawback: F∗1 is singular on |x| = 1 (0 in the normal direction, ∞ in

tangential direction, 2D)

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Near cloaking (Regularization)

Blowing-up a small ball (Kohn-Shen-Vogelius-Weinstein (2008))

For a small number ρ, let

σ =

σ1

if |x| < ρ, 1 if ρ ≤ |x| ≤ 2. (σ1 can be 0 (the core is insulated) or ∞ (perfect conductor))

Let

F(x) =

2−2ρ

2−ρ + 1 2−ρ|x|

x

|x|

if ρ ≤ |x| ≤ 2,

x ρ

if 0 ≤ |x| ≤ ρ. Then F maps B2 onto B2 and blows up Bρ onto B1.

Then, approximate cloaking

Λ[F∗σ] − Λ[1] ≤ Cρ2.

Conductivity in the inner cloaking region: O(ρ) in the normal direction,

0(1/ρ) in tangential direction, 2D (product = 1).

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Small volume expansions

Λ[F∗σ] = Λ[σ] and

Λ[σ](φ)(x) = Λ[1](φ)(x) + ∇U(0) · M ∂ ∂νx ∇yG(x, 0) + h.o.t, x ∈ ∂Ω, where G is the Dirichlet Green function and U the solution to

∆U = 0

in Ω, U = φ

n ∂Ω,

M is the polarization tensor of Bρ, and G(x, y) is the Green function for Ω. (The expansion holds uniformly for σ1).

PT for Bρ with conductivity σ1 (proportional to the volume):

M = 2(σ1 − 1) σ1 + 1 |Bρ|I.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Polarization tensor of a two-phase structure

Make PT vanish enhances the cloaking.
Not possible to make PT vanish with two phases.
Multi-phase structures.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Hashin’s neutral inclusion

M = 0 (GPTs vanishing structure of order 1; a disc with a single coating)

x y

u

−0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Enhanced near cloaking

σN (GPTs vanishing structure of order N)

Estimate:

Λ[F∗σN] − Λ[1] = Λ[σN] − Λ[1] ≤ Cρ2N+2 for some C independent of ρ and N.

Keep the conductivity in the inner cloaking O(ρ) in the normal direction,

0(1/ρ) in tangential direction, 2D.

Make the h.o.t. vanish in the asymptotic expansion of the

Dirichlet-to-Neumann map.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Multiply layered structure

Let u be the solution to
∇ ·
σχ(D) + χ(R2 \ D)
∇u = 0

in R2, u(x) − H(x) = O(|x|−1) as |x| → ∞.

Theorem: The far-field expansion holds as |x| → ∞:

(u − H)(x) = −

∞

m,n=1

cos mθ 2πmr m (Mcc

mnac n + Mcs mnas n) + sin mθ

2πmr m (Msc

mnac n + Mss mnas n)

where H(x) = H(0) + ∞

n=1 r n(ac n cos nθ + as n sin nθ).

Mcc

mn, Mcs mn, Msc mn, Mss mn are called (contracted) GPTs. Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 22

Multiply layered structure

Disc with multiple coatings:

For a positive integer N, let 1 = rN+1 < rN < . . . < r1 = 2 and define

Aj := {rj+1 < r ≤ rj}, j = 1, 2, . . . , N.

A0 = R2 \ B2, AN+1 = B1.
Set σj to be the conductivity of Aj for j = 1, 2, . . . , N + 1, and σ0 = 1.

Let σ =

N+1

j=0

σjχ(Aj). (σN+1 may (or may not) be fixed: σN+1 is fixed to be 0 if the core is insulated.)

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 23

GPTs vanishing structure

Let Mcc

mn[σ], etc, denote the GPTs associated with σ. Because of the

symmetry of the disc, Mcs

mn[σ] = Msc mn[σ] = 0

for all m, n, Mcc

mn[σ] = Mss mn[σ] = 0

if m = n, and Mcc

nn[σ] = Mss nn[σ]

for all n.

Let Mn = Mcc

nn, n = 1, 2, . . .. Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 24

GPTs vanishing structure

To compute Mk, we look for solutions uk to

∇ · σ∇u = 0 in R2

f the form

uk(x) = a(k)

j

r k cos kθ + b(k)

j

r k cos kθ in Aj, j = 0, 1, . . . , N + 1, with a(k) = 1 and b(k)

N+1 = 0.

Then uk satisfies

(uk − H)(x) = b(k) r k cos kθ as |x| → ∞. with H(x) = r k cos kθ.

Hence, Mk = −2πkb(k)

0 . Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 25

GPTs vanishing structure

The transmission conditions on the interface {r = rj}:
a(k)

j

b(k)

j

=

1 2σj

σj + σj−1

(σj − σj−1)r −2k

j

(σj − σj−1)r 2k

j

σj + σj−1 a(k)

j−1

b(k)

j−1

,

and hence

a(k)

N+1

=

N+1

j=1

1 2σj

σj + σj−1

(σj − σj−1)r −2k

j

(σj − σj−1)r 2k

j

σj + σj−1 1 b(k)

.
Let

P(k) =

p(k)

11

p(k)

12

p(k)

21

p(k)

22

:=

N+1

j=1

1 2σj

σj + σj−1

(σj − σj−1)r −2k

j

(σj − σj−1)r 2k

j

σj + σj−1

.

Then, b(k) = −p(k)

21

p(k)

22

.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 26

GPTs vanishing structure

GPTs vanishing structure of order N: Mk = 0 for k = 1, . . . , N, or

p(k)

21 = 0,

k = 1, . . . , N.

Solve the equations for rN < . . . < r2 and σN, . . . , σ1.
If N = 1, Hashin’s neutral inclusion.
For N = 2, 3, ..., can be solved by hand.
For arbitrary N, the equation is non-linear algebraic equation and

existence of solution is yet to be proved.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 27

GPTs vanishing structure

1 1.5 2 1 2

r σ

1 3 15 10

−15

10

−10

10

−5

10 10

5

k Mk

Figure: The conductivity of the core is not fixed. N = 3

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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GPTs vanishing structure

1 1.5 2 1 2

r σ

1 6 15 10

−15

10

−10

10

−5

10 10

5

k Mk

Figure: The conductivity of the core is not fixed. N = 6

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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GPTs vanishing structure

1 1.5 2 1 2

r σ

1 9 15 10

−15

10

−10

10

−5

10 10

5

k Mk

Figure: The conductivity of the core is not fixed. N = 9

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 30

GPTs vanishing structure

1 1.5 2 1 5 10 15

r σ

1 3 15 10

−15

10

−10

10

−5

10 10

5

k Mk

Figure: The conductivity of the core is fixed to be 0. N = 3

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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GPTs vanishing structure

1 1.5 2 1 5 10 15

r σ

1 6 15 10

−15

10

−10

10

−5

10 10

5

k Mk

Figure: The conductivity of the core is fixed to be 0. N = 6

Electromagnetic cloaking and super-resolved imaging Habib Ammari

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Enhancement of near cloaking

Theorem: Let σ be a multi-layered structure with r1 = 2 and rN+1 = 1. If

f = ∞

k=−∞ fkeikθ, then

Λ[σ(1

ρx)] − Λ[1]

(f ) =

∞

k=−∞

2|k|ρ2|k|M|k|[σ] 2π|k| − ρ2|k|M|k|[σ]fkeikθ.

Corollary: σ: a GPTs vanishing structure of order N, σN(x) = σ( 1

ρx).

Then

Λ[σN] − Λ[1]
(f ) =
|k|>N

2|k|ρ2|k|M|k|[σ] 2π|k| − ρ2|k|M|k|[σ]fkeikθ.

Lemma: |Mk[σ]| ≤ 2πk22k for all k.
Theorem: Using the transformation blowing up a small ball, we can get a

near-cloaking structure such that Λ[σN] − Λ[1] = Λ[F∗σN] − Λ[1] ≤ Cρ2N+2.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 33

Enhancement of near cloaking

Change of variables (sends the annulus [ρ, 2ρ] onto a fixed annulus): Fρ(x) :=            3 − 4ρ 2(1 − ρ) + 1 4(1 − ρ)|x| x |x| for 2ρ ≤ |x| ≤ 2, 1 2 + 1 2ρ|x| x |x| for ρ ≤ |x| ≤ 2ρ, x ρ for |x| ≤ ρ. Anisotropic conductivity distributions:

log10(σ11), alternative blow−up of a 3 layer structure

−2 −1.5 −1 −0.5 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0.5 1 1.5 2 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

log10(σ11), alternative blow−up of a 6 layer structure

−2 −1.5 −1 −0.5 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0.5 1 1.5 2 −1 −0.5 0.5 1

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 34

Enhancement of near cloaking

log10 of the eigenvalues of Λ[σ( 1

ρx)] − Λ[1] for different values of N:

1 2 3 4 5 6 7 8 −20 −15 −10 −5 5

log10(|λk[σ]−λk[1]|) k Perturbation of the eigenvalues of the DtN map

hole of radius 1 hole of radius ρ=0.25 hole of radius ρ=0.25+1 layer hole of radius ρ=0.25+2 layers hole of radius ρ=0.25+3 layers hole of radius ρ=0.25+4 layers hole of radius ρ=0.25+5 layers hole of radius ρ=0.25+6 layers

N-layer vanishing GPTs structure: same first N DtN eigenvalues as Λ[1].

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 35

Enhancement of near cloaking

Measure of the invisibility of a cloak:

β(σ) = sup

k

|λk

B2,B1[σ] − λk B2[1]|

λk

B2,B1[σ], λk B2[1]: eigenvalues of Λ[σ( 1 ρx)], Λ[1].

To obtain the same invisibility as a 6-layer structure with ρ = 0.25, one

has to use a change of variable cloak with size parameter ρeq ≈ 1.5.10−6 which will result in a much more singular conductivity distribution.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 36

Enhancement of near cloaking

Stability of near-cloaking with respect to noise in the conductivity values: Perturb all values of the conductivity profile with a normal error of standard deviation proportional to the value: σper

k

≡ σk(1 + N(0, η2)) ∀k = 1 . . . N, η ∈ [0, 0.5]. Mean and standard deviation of the invisibility measure as a fct of noise level:

5 10 15 20 25 30 35 40 45 50 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

visibility of the change of variable cloak with same ρ=0.25 noise level (%) on the conductivity of all layers mean visibility β(σ)

5 10 15 20 25 30 35 40 45 50 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

noise level (%) on the conductivity of all layers standard deviation of visibility β(σ)

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 37

Enhancement of near cloaking

Perform a statistical sensitivity analysis of the invisibility measure using Sobol

indices. Goal: explain the fluctuations of the invisibility measure of the

multi-layer cloak in terms of the conductivities of the layers.

6 5 4 3 2 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 layer sobol first order index sobol total index

Figure: Sobol indices where the conductivities are chosen uniform random variables. Outermost layer is the most sensitive.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 38

Enhancement of near cloaking

6 5 4 3 2 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 layer sobol first order index sobol total index

Figure: Sobol indices where the conductivities are chosen uniform random variables around the optimal values with relative variance equal to 0.1 the

ptimal values. Strong interation between the conductivity values.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 39

Near cloaking for the Helmholtz equation

U: plane wave. Helmholtz equation in R2:

     ∇·

χ(R2 \ ¯

D) + 1 µχ(D)

∇u + ω2

χ(R2 \ ¯ D) + ǫχ(D)

u = 0,

(u − U) satisfies the outgoing condition.

Scattering coefficients (concept equivalent to GPTs):

Wnm =

∂D

Jn(k0|y|)e−inθy ψm(y)dσ(y), U replaced with Jm(k0|x|)eimθx and ψm: jump of the normal derivative of the solution on ∂D.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 40

Near cloaking for the Helmholtz equation

θ, θ′: incident and scattered directions. Far-field expansion:

(u −U)(x) = −ie− πi

4

eiω|x|

8πω|x|

A∞[ǫ, µ, ω](θ, θ′)+o(|x|− 1

2 )

as |x| → ∞.

Use of Graf’s formula.
Theorem: Wnm: scattering coefficients

A∞[ǫ, µ, ω](θ, θ′) =

n,m∈Z

(−i)nimeinθ′Wnm[ǫ, µ, ω]e−imθ.

Scattering cross section:

S[ǫ, µ, ω](θ′) := 2π

A∞[ǫ, µ, ω](θ, θ′)
2

dθ

|Wnm| decays to zero fast as |n| + |m| → +∞.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 41

Near cloaking for the Helmholtz equation

We construct scattering coefficients vanishing structures.
Theorem: For ρ small enough, Ψ 1

ρ (x) = 1

ρx, (ǫ, µ): S-vanishing structure

f order N:

S

(Fρ)∗(µ ◦ Ψ 1

ρ ), (Fρ)∗(ǫ ◦ Ψ 1 ρ ), ω

(θ′) = o(ρ4N).
Anisotropic permittivity and permeability distributions are essential in
rder to ensure near cloaking.
If a S-vanishing structure achieves cloaking enhancement at a fixed

frequency ω0, then the cloaking enhancement is also achieved for any frequency ω ≤ ω0.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 42

Near cloaking for the Helmholtz equation

No cloak −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −1 −0.5 0.5 1 1.5 Change of variable cloak −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 Change of variable cloak + 1 layer −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 No cloak −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 Change of variable cloak −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −0.01 −0.005 0.005 0.01 0.015 Change of variable cloak + 1 layer −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −4 −3 −2 −1 1 2 3 4 5 x 10 −4 1 2 3 4 5 6 20 40 60 80 100 120

θ No cloak

1 2 3 4 5 6 0.01 0.02 0.03 0.04 0.05 0.06

θ Change of variable cloak

1 2 3 4 5 6 1 2 3 4 5 6 7 x 10 −5

θ Change of variable cloak + 1 layer

Orders of magnitude: 102, 10−1, 10−5(ρ = 0.05, ω = π, N = 1).

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 43

GPTs for super-resolved imaging

GPTs: obtained in (multistatic wave) imaging from data by solving a

linear system.

Number of computed GPTs: depends only on the signal-to-noise ratio

(SNR) in the data.

Recursive procedure for reconstructing a shape from its GPTs:robust.
Higher is the order of the GPT, more resolved is the encoded shape

information.

Initial guess: the equivalent ellipse.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 44

GPTs for super-resolved imaging

Recursive procedure: minimize the discrepancy between the first l GPTs.

Use the result of step l as an initial guess for step l + 1.

High-frequency oscillations of the boundary of a domain are only

contained in its high-order GPTs.

GPTs can capture the topology of the domain (level set version of the

recursive procedure).

GPTs: translation, rotation, scaling, and monotonicity properties.
GPTs: shape descriptor (contrast with moments).

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 45

GPTs for super-resolved imaging

Shape derivative of contracted GPTs Mc

mn (corresponding to

H(x) = r neinθ, x = (r, θ): polar coordinates) : (σ − 1) ∂un ∂ν

−

∂vm ∂ν

− + 1

σ ∂un ∂T

−

∂vm ∂T

−
(x)

ν: normal, T : tangential.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 46

GPTs for super-resolved imaging

                 ∆un = 0 in D ∪ (R2\D), un|+ − un|− = 0

n ∂D,

∂un ∂ν

+ − σ ∂un

∂ν

− = 0
n ∂D,

un(x) − r neinθ = O(r −1) as r → ∞,                  ∆vm = 0 in D ∪ (R2\D), σvm|+ − vm|− = 0

n ∂D,

∂vm ∂ν

+ − ∂vm

∂ν

− = 0
n ∂D,

vm(x) − r meimθ = O(r −1) as r → ∞. The formula is derived using the layer potential techniques.

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 47

GPTs for super-resolved imaging

ǫ-perturbation of D:

∂Dǫ := {˜ x = x + ǫh(x)ν(x) | x ∈ ∂D}.

D is a disk:

Mc

mn(λ, Dǫ) − Mc mn(λ, D) = 2πǫmn

λ2 ˆ hm+n + O(ǫ2), as ǫ → 0; ˆ hp: Fourier coefficients of h; λ = (σ − 1)/(2(σ + 1)).

High-frequency oscillations of the boundary deformation of a disk-shaped

inclusion are only contained in its high-order contracted GPTs.

Only ˆ

hp for p up to 2N can be reconstructed from the set of contracted GPTs Mc

mn for m, n ≤ N. Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 48

GPTs for super-resolved imaging

Reconsructions from GPTs of order up to N = 6:

−1 1 −1 1 −1 1 −1 1

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 49

GPTs for resolved imaging

Reconsructions from GPTs of order up to N = 5 and N = 6 (higher SNR), respectively:

−1 1 −1 1 −1 1 −1 1

Electromagnetic cloaking and super-resolved imaging Habib Ammari

SLIDE 50

Conclusion

Justification of cloaking by anomalous localized resonance.
Improve near cloaking effect using GPTs and S-vanishing structures.
Extension of near cloaking to full Maxwell’s equations (using results
btained with M. Vogelius and D. Volkov, JMPA, 2001).
GPTs can be obtained from (multistatic) wave measurements by solving

a linear system.

GPTs can be used as a shape descriptor.
GPTs yield super-resolved shape imaging.
Design a fast algorithm which identifies a target using a dictionary of

precomputed GPTs data.

Electromagnetic cloaking and super-resolved imaging Habib Ammari