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) Fernando Guevara Vasquez, Active exterior cloaking 1/28
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. Fernando Guevara Vasquez, Active exterior cloaking 2/28
Helmholtz equation ∆u + k 2 u = 0 Fernando Guevara Vasquez, Active exterior cloaking 3/28
Active interior cloaking 20 20 20 0 0 0 −20 −20 −20 −20 0 20 −20 0 20 −20 0 20 Proposed by Miller 2001, but well known in acoustics since the 60s (Malyuzhinets; Jessel and Mangiante;. . .) Fernando Guevara Vasquez, Active exterior cloaking 4/28
Green’s identity Let D be a domain in R d ( d = 2 or 3) with Lipschitz boundary. � u d ( x ) = d S y { −( n ( y ) · ∇ y u i ( y )) G ( x , y ) + u i ( y ) n ( y ) · ∇ y G ( x , y ) } ∂D � − u i ( x ) , if x ∈ D = 0, otherwise, where the Green’s function for the Helmholtz equation is 4 H ( 1 ) i ( k | x − y | ) in 2D 0 G ( x , y ) = e ik | x − y | in 3D 4 π | x − y | � we get a single and double layer potential on ∂D so that • u i + u d = 0 in D • u d = 0 in R d \ D . Fernando Guevara Vasquez, Active exterior cloaking 5/28
Active interior cloaking 20 20 20 0 0 0 −20 −20 −20 −20 0 20 −20 0 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. Fernando Guevara Vasquez, Active exterior cloaking 6/28
Active exterior cloaking (in 2D) Devices’ field u dev must satisfy Helmholtz equation with Sommerfeld radiation condition. For point-like devices located at positions x j : n dev ∞ � � u d ( x ) = b j , m V m ( x − x j ) , m =− ∞ j = 1 where the radiating solutions to the Helmholtz equation are V m ( x ) ≡ H ( 1 ) m ( k | x | ) exp [ im arg ( x )] . Fernando Guevara Vasquez, Active exterior cloaking 7/28
Designing devices that mimic Green’s identities We need: (a) u dev ( x ) ≈ − u inc ( x ) for | x | � α γ (b) u dev ( x ) ≈ 0 for | x | � γ u dev ≈ − u inc α Since u tot = u i + u d + u scat , δ (a) ⇒ u tot ( x ) ≈ 0 for | x | � α (b) ⇒ u tot ( x ) ≈ u inc ( x ) for | x | � γ u dev ≈ 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 8/28
Finding the coefficients numerically (a’) u dev ( x ) ≈ − u inc ( x ) for | x | = α (b’) u dev ( x ) ≈ 0 for | x | = γ Construct matrices A , B s.t. γ Ab = [ u dev ( p α 1 ) , . . . , u dev ( p α N α )] T , p γ Bb = [ u dev ( p γ 1 ) , . . . , u dev ( p γ N γ )] T , j α p α j where b ∈ C ( 2 M + 1 ) D ≡ device coefficients. 1. Find b 0 = argmin � Ab + u inc ( | x | = | α | ) � 2 2 (enforce (a’)) � Bb � 2 2. Find b ∗ = argmin (enforce (b’)) 2 Ab = Ab 0 Fernando Guevara Vasquez, Active exterior cloaking 9/28
Cloaking for one single frequency Inactive devices Active devices 20 20 0 0 −20 −20 −20 0 20 −20 0 20 c = 3 × 10 8 m/s, λ = 12.5cm, ω/ ( 2 π ) = 2.4GHz α = 2 λ , δ = 5 λ , γ = 10 λ . Fernando Guevara Vasquez, Active exterior cloaking 10/28
Scattering reduction −2 10 Percent reduction −4 10 −6 10 1.2 2.4 3.6 ω / ( 2 π ) in GHz c = 3 × 10 8 m/s, λ 0 = 12.5cm, ω/ ( 2 π ) ∈ [ 1.2, 3.6 ] GHz Fernando Guevara Vasquez, Active exterior cloaking 11/28
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). Fernando Guevara Vasquez, Active exterior cloaking 12/28
Green cloak devices idea x 2 ∂D 2 x 3 ∂D 3 ∂D 1 x 1 x 4 ∂D 4 Idea The contribution of portion ∂D j to the single and double layer potentials in Green’s formula is replaced by a multipolar source located at x j / ∈ ∂D . Fernando Guevara Vasquez, Active exterior cloaking 13/28
Graf’s addition formula The Green’s function G ( x , y ) can be written as a superposition of sources located at x j : G ( x , y ) = i 4 H ( 1 ) � � 0 ( k � x − x j − ( y − x j ) � ) ∞ = i � V m ( x − x j ) U m ( y − x j ) , 4 m =− ∞ where the entire cylindrical waves are U m ( x ) ≡ J m ( k | x | ) exp [ im arg ( x )] and the sum converges uniformly in compact subsets of � > � � � � � x − x j � y − x j � . Use summation formula to “move” monopoles and dipoles from a portion of the boundary to the corresponding x j . Fernando Guevara Vasquez, Active exterior cloaking 14/28
Green cloak devices The device field n dev ∞ � � u d ( x ) = b j , m V m ( x − x j ) , m =− ∞ j = 1 with � b j , m = dS y { (− n ( y ) · ∇ y u i ( y )) U m ( y − x j ) ∂D j + u i ( y ) n ( y ) · ∇ y U m ( y − x j ) } converges (uniformly in compact subsets) outside of the region n dev � � � x l , sup . R = B | y − x l | y ∈ ∂D l l = 1 Fernando Guevara Vasquez, Active exterior cloaking 15/28
A specific configuration With D = B ( 0, σ ) and devices | x j | = δ : x 2 δ D x 1 σ x 3 • Gray disks have radius: r ( σ , δ ) = (( σ − δ/ 2 ) 2 + 3 δ 2 / 4 ) 1 / 2 . • Largest disk in cloaked region radius: r eff ( σ , δ ) = δ − r ( σ , δ ) . • Largest cloaked region ( σ ∗ = δ/ 2): √ r ∗ eff ( δ ) = ( 1 − 3 / 2 ) δ ≈ 0.13 δ . Fernando Guevara Vasquez, Active exterior cloaking 16/28
Green’s formula SVD 20 20 Device’s field u d 0 0 −20 −20 −20 0 20 −20 0 20 Total field u i + u d + u s 20 20 0 0 −20 −20 −20 0 20 −20 0 20 Fernando Guevara Vasquez, Active exterior cloaking 17/28
Cloak performance √ � u i + u d � / � u i � on | x | = ( 1 − 3 / 2 ) δ � u d � / � u i � on | x | = 2 δ 1 10 0 10 0 10 (percent) (percent) −1 −5 10 10 −2 10 −3 −10 10 10 −4 10 −5 −15 10 10 5 25 50 5 25 50 (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 2 M ( δ ) terms Fernando Guevara Vasquez, Active exterior cloaking 18/28
Size of the “throats” cut-off | u d ( x ) | = 100 cut-off | u d ( x ) | = 5 (device radius / δ ) (device radius / δ ) 0.9 0.9 0.85 0.85 0.8 0.8 0.75 0.75 0.7 0.7 0.65 0.65 0.6 0.6 0.55 0.55 5 25 50 5 25 50 (a): δ (in λ ) (b): δ (in λ ) Estimated device radius relative to δ for different values of δ . Fernando Guevara Vasquez, Active exterior cloaking 19/28
Cloaking for Helmholtz equation in 3D With D = tetrahedron inscribed in B ( 0, σ ) , devices | x j | = δ : (a) suboptimal, σ = δ/ 5 (b) optimal, σ = δ/ 3 9 δ 2 � 1 � 2 + 8 2 . (green) �� σ − δ • Radius of gray balls: r ( σ , δ ) = 3 • Largest ball in cloaked region: r eff ( σ , δ ) = δ − r ( σ , δ ) . (red) √ � � 1 − 2 2 • Largest cloaked region: r ∗ eff = δ ≈ 0.057 δ . 3 Fernando Guevara Vasquez, Active exterior cloaking 20/28
z = − 2 σ z = − σ z = 0 z = σ z = 2 σ u d u tot (active) u tot (inactive) Fernando Guevara Vasquez, Active exterior cloaking 21/28
| u d | = 100 | u d | = 5 Contours of | u d | (gray) and | u d + u i | = 10 − 2 (red). Fernando Guevara Vasquez, Active exterior cloaking 22/28
δ = 6 λ δ = 12 λ δ = 18 λ δ = 24 λ Cross-section of level set | u d | � 10 2 (black) and of the region R (shades of gray) on the sphere | x | = σ for the optimal σ = δ/ 3. Fernando Guevara Vasquez, Active exterior cloaking 23/28
Main ingredients for Helmholtz 3D active cloaking • Green’s identity: mono- and dipole density on ∂D reproduces incident field u i in D . • Device Ansatz: n dev n ∞ � � � b l , n , m V m u d ( x ) = n ( x − x l ) . l = 1 n = 0 m =− n • Movable source: (Graf’s Identity) G ( x , y ) = linear combination of V m n ( x − x l ) . Fernando Guevara Vasquez, Active exterior cloaking 24/28
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