[PPT] - Active exterior cloaking Fernando Guevara Vasquez University of PowerPoint Presentation

SLIDE 1

Active exterior cloaking

Fernando Guevara Vasquez

University of Utah

June 18 2012

Conference in honor of Gunther Uhlmann, UC Irvine

Collaborators Graeme W. Milton (University of Utah) Daniel Onofrei (University of Houston)

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SLIDE 2

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: Al`

u, Engheta.

Passive Exterior
Anomalous resonances: McPhedran, Milton, Nicorovici.
Complementary media: Lai, Chen, Zhang, Chan.
Plasmonic cloaking: Al`

u, Engheta, . . .

Active Interior: Miller
Active Exterior:
Onofrei, Ren: integral equation framework
This work: (Laplace and) Helmholtz equations.

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SLIDE 3

Helmholtz equation ∆u + k2u = 0

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SLIDE 4

Active interior cloaking

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Proposed by Miller 2001, but well known in acoustics since the 60s (Malyuzhinets; Jessel and Mangiante;. . .)

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SLIDE 5

Green’s identity

Let D be a domain in Rd (d = 2 or 3) with Lipschitz boundary. ud(x) =

∂D

dSy {−(n(y) · ∇yui(y))G(x, y) + ui(y)n(y) · ∇yG(x, y)} =

−ui(x),

if x ∈ D 0,

therwise,

where the Green’s function for the Helmholtz equation is G(x, y) =     

i 4H(1)

(k|x − y|) in 2D eik|x−y| 4π|x − y| in 3D 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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SLIDE 6

Active interior cloaking

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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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SLIDE 7

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) =

ndev

j=1

∞

m=−∞

bj,mVm(x − xj), where the radiating solutions to the Helmholtz equation are Vm(x) ≡ H(1)

m (k |x|) exp[im arg(x)].

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SLIDE 8

Designing devices that mimic Green’s identities

udev ≈ 0 γ δ α udev ≈ −uinc

We need: (a) udev(x) ≈ −uinc(x) for |x| α (b) udev(x) ≈ 0 for |x| γ Since utot = ui + ud + uscat, (a) ⇒ utot(x) ≈ 0 for |x| α (b) ⇒ utot(x) ≈ uinc(x) for |x| γ 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?)

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SLIDE 9

Finding the coefficients numerically

γ pα

j

α pγ

j

(a’) udev(x) ≈ −uinc(x) for |x| = α (b’) udev(x) ≈ 0 for |x| = γ Construct matrices A, B s.t. Ab = [udev(pα

1 ), . . . , udev(pα Nα)]T,

Bb = [udev(pγ

1 ), . . . , udev(pγ Nγ)]T,

where b ∈ C(2M+1)D ≡ device coefficients.

1. Find b0 = argmin Ab + uinc(|x| = |α|)2

2 (enforce (a’))

2. Find b∗ = argmin

Ab=Ab0

Bb2

2

(enforce (b’))

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SLIDE 10

Cloaking for one single frequency

Inactive devices Active devices

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c = 3 × 108m/s, λ = 12.5cm, ω/(2π) = 2.4GHz α = 2λ, δ = 5λ, γ = 10λ.

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SLIDE 11

Scattering reduction

Percent reduction

1.2 2.4 3.6 10

−6

10

−4

10

−2

ω/(2π) in GHz

c = 3 × 108m/s, λ0 = 12.5cm, ω/(2π) ∈ [1.2, 3.6]GHz

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SLIDE 12

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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SLIDE 13

Green cloak devices idea

∂D1 x2 ∂D2 x1 x4 ∂D4 x3 ∂D3

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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SLIDE 14

Graf’s addition formula

The Green’s function G(x, y) can be written as a superposition of sources located at xj: G(x, y) = i 4H(1)

0 (k

x − xj − (y − xj)
)

= i 4

∞

m=−∞

Vm(x − xj)Um(y − xj), where the entire cylindrical waves are Um(x) ≡ Jm(k |x|) exp[im arg(x)] and the sum converges uniformly in compact subsets of

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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SLIDE 15

Green cloak devices

The device field ud(x) =

ndev

j=1

∞

m=−∞

bj,mVm(x − xj), with bj,m =

∂Dj

dSy{(−n(y) · ∇yui(y)) Um(y − xj) + ui(y)n(y) · ∇yUm(y − xj)} converges (uniformly in compact subsets) outside of the region R =

ndev

l=1

B

xl, sup

y∈∂Dl

|y − xl|

.

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SLIDE 16

A specific configuration

With D = B(0, σ) and devices |xj| = δ:

x2 x3 x1 σ D δ

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 −

√ 3/2)δ ≈ 0.13δ.

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SLIDE 17

Green’s formula SVD Device’s field ud

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Total field ui + ud + us

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SLIDE 18

Cloak performance

ui + ud/ui on |x| = (1 − √ 3/2)δ ud/ui on |x| = 2δ (percent)

5 25 50 10

−5

10

−4

10

−3

10

−2

10

−1

10 10

1

(percent)

5 25 50 10

−15

10

−10

10

−5

10

(a): δ (in λ) (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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SLIDE 19

Size of the “throats”

cut-off |ud(x)| = 100 cut-off |ud(x)| = 5 (device radius / δ)

5 25 50 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

(device radius / δ)

5 25 50 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

(a): δ (in λ) (b): δ (in λ) Estimated device radius relative to δ for different values of δ.

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SLIDE 20

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

Radius of gray balls: r(σ, δ) =
σ − δ

3

2 + 8

9δ2 1

2 . (green)

Largest ball in cloaked region: reff(σ, δ) = δ − r(σ, δ). (red)
Largest cloaked region: r∗

eff =

1 − 2

√ 2 3

δ ≈ 0.057δ.

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SLIDE 21

z = −2σ z = −σ z = 0 z = σ z = 2σ ud utot (active) utot (inactive)

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SLIDE 22

|ud| = 100 |ud| = 5 Contours of |ud| (gray) and |ud + ui| = 10−2 (red).

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SLIDE 23

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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SLIDE 24

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) =

ndev

l=1

∞

n=0

n

m=−n

bl,n,mVm

n (x − xl).

Movable source: (Graf’s Identity)

G(x, y) = linear combination of Vm

n (x − xl).

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SLIDE 25

Main ingredients for Maxwell active cloaking

Stratton-Chu Formula: magnetic and electric dipole density
n ∂D reproduces incident field Ei, Hi in D.
Device Ansatz:

Ed(x) =

ndev

l=1

∞

n=1

n

m=−n

al,n,m∇×((x − xl)Vm

n (x − xl))

+bl,n,m∇×∇×((x − xl)Vm

n (x − xl))

Movable source: (vector addition theorem)

G(x, y)p = linear combination of ∇×((x − xl)Vm

n (x − xl)),

∇×∇×((x − xl)Vm

n (x − xl)), and

∇Vm

n (x − xl).

(REU with Michael Bentley)

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SLIDE 26

Directionality with stationary phase method

u(x) = exp[ikd · x]

y∗ d x

(with Leonid Kunyansky)

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SLIDE 27

Directionality with stationary phase method

u(x) = 0

y∗ d x y∗∗

(with Leonid Kunyansky)

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SLIDE 28

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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SLIDE 29

Thank you!

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