Active exterior cloaking
Fernando Guevara Vasquez
University of Utah
June 14 2012
Waves and Imaging, Heraklion
Collaborators
Graeme W. Milton (University of Utah)
Daniel Onofrei (University of Houston)
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Active exterior cloaking
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Cloaking types
Interior/Exterior: Is the object hidden inside or outside a device?
Passive/Active: Are sources needed to cloak?
• Passive Interior
• Transformation based cloaking: Leonhardt; Cummer, Pendry,
Schurig, Smith; Greenleaf, Kurylev, Lassas, Uhlmann; Farhat,
Enoch, Guenneau; Kohn, Onofrei, Shen, Vogelius, Weinstein;
Cai, Chettiar, Kildishev, Shalaev; . . .
• Plasmonic cloaking: Alù, Engheta.
• Passive Exterior
• Anomalous resonances: McPhedran, Milton, Nicorovici.
• Complementary media: Lai, Chen, Zhang, Chan.
• Plasmonic cloaking: Alù, Engheta, . . .
• Active Interior: Miller
• Active Exterior:
• Onofrei, Ren: integral equation framework
• This work (Laplace and Helmholtz equations)
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Active exterior cloaking
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Laplace equation
∆u = 0
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Cloaking for the 2D Laplace equation
Im
• Solutions to the Laplace equation
(∆u = 0) in 2D are the real part of
analytic functions in C.
Re
• We want to cloak an object (say a
conductor) from an incident field
ui = analytic.
Specifications for the device field ud :
• ud analytic outside gray disk.
• ud (z) ≈ 0 far from device (green).
• ud (z) ≈ −ui (z) in the cloaked region (red).
Objects in the cloaked region (red) are invisible because the
total field ud + ui is very small in the cloaked region.
Fernando Guevara Vasquez,
Active exterior cloaking
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Idea for solution: Kelvin (inversion) transform
Im
Im
w = 1/z
Re
Re
z = 1/w
z−plane
ud (z) analytic outside grey disk
and
• ud (z) ≈ 0 far from device
(green).
w−plane
e d (w) analytic in white disk and
u
• u
e d (w) ≈ 0 in green disk.
• u
e d (w) ≈ −e
ui (w) in red
disk.
• ud (z) ≈ −ui (z) in the
cloaked region (red).
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Active exterior cloaking
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A non-constructive solution
A classic result in harmonic approximation theory:
Lemma (Walsh)
Let K be a compact set in R2 such that R2 \ K is connected. Then
for any function v harmonic on an open set containing K, and for
each > 0, there is a harmonic polynomial q such that |v − q| < on K.
• Applying lemma to
K = red disk ∪ green disk
there is an (analytic) polynomial q approximating
0
in green disk
v(w) =
−e
ui (w) in red disk
• Use as cloaking device: ud (z) = q(1/z).
• Unfortunately this does not give an explicit expression.
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Active exterior cloaking
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An explicit solution
Im
Re
w−plane
e d analytic in white disk s.t.
Find u
e d ≈ 0 in green disk
•u
e d ≈ −e
•u
ui in red disk
Find polynomial p(w) s.t.
• p(w) ≈ 1 in green disk
• p(w) ≈ 0 in red disk
and define cloaking device by
e d = (1 − p)(−e
u
ui ).
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Active exterior cloaking
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Designing cloaking device: first attempt
Here pφ,ψ is the unique polynomial of degree 2n − 1 satisfying for
j = 0, . . . , n − 1
pφ,ψ (eiφ wj ) = 1,
pφ,ψ (β + eiψ wj ) = 0,
with wj = exp[2iπj/n].
Im
× ×
×
0
×
× ×
×
×
×
× ×
×
×
β
|
×
×
×
× ×
Re
(interpolation polynomial)
Fernando Guevara Vasquez,
Active exterior cloaking
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Designing cloaking device: first attempt
Here pφ,ψ is the unique polynomial of degree 2n − 1 satisfying for
j = 0, . . . , n − 1
pφ,ψ (eiφ wj ) = 1,
pφ,ψ (β + eiψ wj ) = 0,
with wj = exp[2iπj/n].
2
2
1
0
1
0
1
1
1
1
1
1
−1
−2
−2
1
1
1
0
0
0
0
0
0
1
0
0
0
0
−1
−2
−1
0
1
2
3
4
5
6
(interpolation polynomial)
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Ensemble averaging
If we use instead the ensemble average
Z
Z 2π
1 2π
p(w) = 2
dφ
dψpφ,ψ (w),
4π 0
0
it is possible to show that p(w) is the unique polynomial of degree
2n − 1 such that
p(0) = 1
p
(k)
(0) = 0, for k = 1, . . . , n − 1,
p(k) (β) = 0, for k = 0, . . . , n − 1.
(Hermite interpolation polynomial). Explicitly:
p(z) =
n−1 z n X z j n+j−1
1−
β
β
j
j=0
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Hermite interpolation polynomial
0.6
0
−0.6
−0.5
0
1
1.5
• Where do we expect p(0) ≈ 1 and p(1) ≈ 0?
• Can we cloak larger regions if we increase degree?
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Active exterior cloaking
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Hermite interpolation polynomial
0.6
0
−0.6
−0.5
0
1
1.5
• Where do we expect p(0) ≈ 1 and p(1) ≈ 0?
• Can we cloak larger regions if we increase degree?
Answer: Cloaked region size is limited. As n → ∞:
• p(z) → 1 in the left lobe of peanut
• p(z) → 0 in the right lobe of peanut
• p(z) diverges outside peanut
2
where peanut = {z ∈ C | |z||z − β| < β4 }
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Static cloaking device in action
Inactive device
Active device
10
10
0
0
−10
−10
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0
10
−10
−10
Active exterior cloaking
0
10
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Helmholtz equation
∆u + k2u = 0
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Active interior cloaking
20
20
20
0
0
0
−20
−20
0
20
−20
−20
0
20
−20
−20
0
20
Proposed by Miller 2001, but well known in acoustics since the 60s
(Malyuzhinets; Jessel and Mangiante;. . .)
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Green’s identity
Let D be a domain in Rd (d = 2 or 3) with Lipschitz boundary.
Z
ud (x) =
dSy {−(n(y) · ∇y ui (y))G(x, y) + ui (y)n(y) · ∇y G(x, y)}
∂D
−ui (x), if x ∈ D
=
0,
otherwise,
where the Green’s function for the Helmholtz equation is

(1)

 4i H0 (k|x − y|) in 2D
G(x, y) =
eik|x−y|

in 3D

4π|x − y|
we get a single and double layer potential on ∂D so that
• ui + ud = 0 in D
• ud = 0 in Rd \D.
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Active exterior cloaking
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Active interior cloaking
20
20
20
0
0
0
−20
−20
0
20
−20
−20
0
20
−20
−20
0
20
• With Green’s identities: The object is completely surrounded
by the cloak.
• To get exterior cloaking: replace the single and double layer
potential in Green’s identities by a few devices.
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Active exterior cloaking (in 2D)
Devices’ field udev must satisfy Helmholtz equation with
Sommerfeld radiation condition. For point-like devices located at
positions xj :
ud (x) =
n
dev
X
∞
X
bj,m Vm (x − xj ),
j=1 m=−∞
where the radiating solutions to the Helmholtz equation are
(1)
Vm (x) ≡ Hm (k |x|) exp[im arg(x)].
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Active exterior cloaking
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Designing devices that mimic Green’s identities
We need:
(a) udev (x) ≈ −uinc (x) for |x| 6 α
γ
udev ≈ −uinc
α
δ
(b) udev (x) ≈ 0 for |x| > γ
Since utot = ui + ud + uscat ,
(a) ⇒ utot (x) ≈ 0 for |x| 6 α
(b) ⇒ utot (x) ≈ uinc (x) for |x| > γ
udev ≈ 0
Caveats
• We need to know the incident field in advance, from e.g.
sensors.
• Information from sensors needs to travel faster than incident
field (OK for acoustics. For electromagnetics: periodicity?)
• Need very accurate reproduction of incident field
(OK in controlled environments like MRI?)
Fernando Guevara Vasquez,
Active exterior cloaking
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Finding the coefficients numerically
(a’) udev (x) ≈ −uinc (x) for |x| = α
(b’) udev (x) ≈ 0
γ
pγj
α
pα
j
for |x| = γ
Construct matrices A, B s.t.
α
T
Ab = [udev (pα
1 ), . . . , udev (pNα )] ,
γ
T
Bb = [udev (pγ
1 ), . . . , udev (pNγ )] ,
where b ∈ C(2M+1)D
≡ device coefficients.
1. Find b0 = argmin kAb + uinc (|x| = |α|)k22 (enforce (a’))
2. Find b∗ = argmin kBbk22
(enforce (b’))
Ab=Ab0
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Active exterior cloaking
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Cloaking for one single frequency
Inactive devices
Active devices
20
20
0
0
−20
−20
0
c=3×
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20
−20
−20
0
20
108 m/s,
λ = 12.5cm, ω/(2π) = 2.4GHz
α = 2λ, δ = 5λ, γ = 10λ.
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Scattering reduction
−2
Percent reduction
10
−4
10
−6
10
1.2
2.4
3.6
ω /(2π ) in GHz
c = 3 × 108 m/s, λ0 = 12.5cm, ω/(2π) ∈ [1.2, 3.6]GHz
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Active exterior cloaking
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Devices for many frequencies (pulse)
By superposition principle: sum device fields for many ω to get
cloaking in a bandwidth (i.e. in the time domain).
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Active exterior cloaking
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Green cloak devices idea
x2
∂D2
x3
∂D3
∂D1
x4
x1
∂D4
Idea The contribution of portion ∂Dj to the single and double
layer potentials in Green’s formula is replaced by a multipolar
source located at xj ∈
/ ∂D.
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Graf’s addition formula
The Green’s function G(x, y) can be written as a superposition of
sources located at xj :
i (1) G(x, y) = H0 (k x − xj − (y − xj ))
4
∞
i X
=
Vm (x − xj )Um (y − xj ),
4 m=−∞
where the entire cylindrical waves are
Um (x) ≡ Jm (k |x|) exp[im arg(x)]
and
the sum
uniformly in compact subsets of
converges
x − xj > y − xj .
Use summation formula to “move” monopoles and dipoles from a
portion of the boundary to the corresponding xj .
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Active exterior cloaking
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Green cloak devices
The device field
ud (x) =
n
dev
X
∞
X
bj,m Vm (x − xj ),
j=1 m=−∞
with
Z
dSy {(−n(y) · ∇y ui (y)) Um (y − xj )
bj,m =
∂Dj
+ ui (y)n(y) · ∇y Um (y − xj )}
converges (uniformly in compact subsets) outside of the region
!
n[
dev
R=
B xl , sup |y − xl | .
l=1
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y∈∂Dl
Active exterior cloaking
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A specific configuration
With D = B(0, σ) and devices |xj | = δ:
x2
δ
D
x1
σ
x3
• Gray disks have radius: r(σ, δ) = ((σ − δ/2)2 + 3δ2 /4)1/2 .
• Largest disk in cloaked region radius: reff (σ, δ) = δ − r(σ, δ).
• Largest cloaked√region (σ∗ = δ/2):
r∗eff (δ) = (1 −
Fernando Guevara Vasquez,
3/2)δ ≈ 0.13δ.
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Device’s field ud
Green’s formula
20
0
0
−20
−20
Total field ui + ud + us
SVD
20
0
20
−20
−20
20
20
0
0
−20
−20
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0
20
−20
−20
0
20
0
20
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Cloak performance
kui + ud k/kui k on |x| = (1 −
√
3/2)δ
kud k/kui k on |x| = 2δ
1
0
10
10
0
(percent)
(percent)
10
−1
10
−2
10
−3
10
−5
10
−10
10
−4
10
−5
10
5
−15
25
10
50
5
(a): δ (in λ)
25
50
(b): δ (in λ)
• blue: SVD method with M(δ) terms
• red: Green’s identity method with M(δ) terms
• green: Green’s identity method with 2M(δ) terms
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Active exterior cloaking
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Size of the “throats”
cut-off |ud (x)| = 5
(device radius / δ)
(device radius / δ)
cut-off |ud (x)| = 100
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
5
25
(a): δ (in λ)
50
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
5
25
50
(b): δ (in λ)
Estimated device radius relative to δ for different values of δ.
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Active exterior cloaking
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Cloaking for Helmholtz equation in 3D
With D =tetrahedron inscribed in B(0, σ), devices |xj | = δ:
(a) suboptimal, σ = δ/5
(b) optimal, σ = δ/3
21
2
• Radius of gray balls: r(σ, δ) = σ − δ3 + 89 δ2 . (green)
• Largest ball in cloaked region: reff (σ, δ) = δ − r(σ, δ). (red)
√
• Largest cloaked region: r∗eff = 1 − 2 3 2 δ ≈ 0.057δ.
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Active exterior cloaking
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z = −σ
z=0
z=σ
z = 2σ
utot (inactive) utot (active)
ud
z = −2σ
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Active exterior cloaking
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|ud | = 100
|ud | = 5
Contours of |ud | (gray) and |ud + ui | = 10−2 (red).
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δ = 6λ
δ = 12λ
δ = 18λ
δ = 24λ
Cross-section of level set |ud | > 102 (black) and of the region R
(shades of gray) on the sphere |x| = σ for the optimal σ = δ/3.
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Active exterior cloaking
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Main ingredients for Helmholtz 3D active cloaking
• Green’s identity: mono- and dipole density on ∂D reproduces
incident field ui in D.
• Device Ansatz:
ud (x) =
n
∞
dev X
X
n
X
m
bl,n,m Vn
(x − xl ).
l=1 n=0 m=−n
• Movable source: (Graf’s Identity)
m
G(x, y) = linear combination of Vn
(x − xl ).
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Main ingredients for Maxwell active cloaking
• Stratton-Chu Formula: magnetic and electric dipole density
on ∂D reproduces incident field Ei , Hi in D.
• Device Ansatz:
Ed (x) =
n
∞
dev X
X
n
X
m
al,n,m ∇×((x − xl )Vn
(x − xl ))
l=1 n=1 m=−n
m
+bl,n,m ∇×∇×((x − xl )Vn
(x − xl ))
• Movable source: (vector addition theorem)
G(x, y)p = linear combination of
m
∇×((x − xl )Vn
(x − xl )),
m
∇×∇×((x − xl )Vn
(x − xl )), and
m
∇Vn
(x − xl ).
(REU with Michael Bentley)
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Directionality with stationary phase method
Case 1: u(x) = exp[ikd · x]
x
y∗
d
(with Leonid Kunyansky)
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Active exterior cloaking
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Directionality with stationary phase method
Case 2: u(x) = 0
x
y∗∗
y∗
d
(with Leonid Kunyansky)
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Active exterior cloaking
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Directionality with stationary phase method
Case 3: u(x) = 0
x
y∗
d
(with Leonid Kunyansky)
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Active exterior cloaking
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Future work
• Time domain problems (active control of waves)
• Approximate Green’s identities with a few devices while
enforcing a constraint (e.g. penalize size of devices)
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Active exterior cloaking
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Thank you!
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Active exterior cloaking
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