a fascinating world of invisibility cloakingdjsym/talk/Day2/Harsha-Hutridurga.pdf · such that...

a fascinating world ofinvisibility cloaking

Harsha Hutridurga(IIT Bombay)

Joint work with

H. Ammari (ETH Zurich)

R. Craster (Imperial college London)

A. Dabrowski (RICAM Linz)

A. Diatta (Institut Fresnel Marseille)

S. Guenneau (Imperial college & CNRS)

05 January 2019 - Diamond Jubilee Symposium

Harsha Hutridurga (IIT Bombay) Invisibility cloaking 05/01 IITB 1 / 20

invisibility cloaking and meta-materials

♣ Illusionists use tricks to render objects invisible.

♣ 2006 saw scientific breakthrough in this seemingly impossible task:[Ref.] j.b.pendry, d.schurig, d.r.smith, Science, 312 (2006).

[Ref.] u.leonhardt, Science, 312 (2006).

♣ They showed the possibility of designing a coat around an objectsuch that object becomes invisible to electromagnetic radiation.

electromagnetic waves

♣ Governing equations: Maxwell’s equations.

♣ Unknowns: Electric and Magnetic fields – E(t, x) ∈ R3 and H(t, x) ∈ R3∇× E = −µ∂tH∇×H = ε ∂tE

♣ Here µ, ε ∈ R3×3 are material parameters:magnetic permeability and electric permittivity.

an idea to cloak

♣ The coating material has physical properties (µ, ε) such thatthe electromagnetic waves are bent around the object.

♣ Materials with such exotic physical properties are meta-materials.

an idea to cloak

transformation optics

♣ Geometric setting: B1 – region we wish to hide.B2 \B1 – region we wish to apply the meta-material coat.

♣ Blow-up a point (origin): Take Fp : Ω→ Ω

Fp(x) :=

x for x ∈ Ω \B2(

2|x|)x

|x|for x ∈ B2 \ 0

♣ Fp maps 0 to B1 and maps B2 \ 0 to B2 \B1

♣ Fp is smooth except at 0

Fp(x) :=

x for x ∈ Ω \B2(

2|x|)x

|x|for x ∈ B2 \ 0

Fp(x) :=

x for x ∈ Ω \B2(

2|x|)x

|x|for x ∈ B2 \ 0

Fp(x) :=

x for x ∈ Ω \B2(

2|x|)x

|x|for x ∈ B2 \ 0

Fp(x) :=

x for x ∈ Ω \B2(

2|x|)x

|x|for x ∈ B2 \ 0

transformation optics – recasting in new coordinates

♣ New spatial coordinates: y = Fp(x)

♣ Rewrite Maxwell’s equations in (t, y) coordinates.

♣ For E∗(t, y) := E(t, F−1p (y)) and H∗(t, y) := H(t, F−1

p (y))∇y × E∗ = −µcl(y)∂tH∗

∇y ×H∗ = εcl(y)∂tE∗

♣ The transformed parameters take the following structure

µcl(y), εcl(y) =

µ0, ε0 for y ∈ Ω \B2

function of µ0, ε0, Fp for y ∈ B2 \B1

♣ We will have, irrespective of the values for µ and ε in B1,

E∗(t, ·) = E0(t, ·) and H∗(t, ·) = H0(t, ·) in Ω \B2,

E0 & H0 solutions to Maxwell’s equations with parameters µ0, ε0.

p (y))∇y × E∗ = −µcl(y)∂tH∗

µcl(y), εcl(y) =

p (y))∇y × E∗ = −µcl(y)∂tH∗

µcl(y), εcl(y) =

p (y))∇y × E∗ = −µcl(y)∂tH∗

µcl(y), εcl(y) =

p (y))∇y × E∗ = −µcl(y)∂tH∗

µcl(y), εcl(y) =

p (y))∇y × E∗ = −µcl(y)∂tH∗

µcl(y), εcl(y) =

magic parameters µcl and εcl

♣ Their properties in B2 \B1 are such thatthe EM waves are steered away from the region B1

♣ Singularity in Fp at the origin translates toµcl and εcl being singular at the boundary of B1

♣ Regularisation of Fp to avoid singularities – rather than blowingup a point, blow-up a small ball.

Fδ(x) :=

x for x ∈ Ω \B2(

2− 2δ

2− δ+|x|

2− δ

|x|for x ∈ B2 \Bδ

δfor x ∈ Bδ

♣ This leads to the notion of near-cloaking

Eδ(t, ·) ≈ E0(t, ·) and Hδ(t, ·) ≈ H0(t, ·) in Ω \B2.

♣ Enough to study small volume perturbations in parameters

Fδ(x) :=

x for x ∈ Ω \B2(

2− 2δ

2− δ+|x|

2− δ

δfor x ∈ Bδ

Fδ(x) :=

x for x ∈ Ω \B2(

2− 2δ

2− δ+|x|

2− δ

δfor x ∈ Bδ

Fδ(x) :=

x for x ∈ Ω \B2(

2− 2δ

2− δ+|x|

2− δ

δfor x ∈ Bδ

Fδ(x) :=

x for x ∈ Ω \B2(

2− 2δ

2− δ+|x|

2− δ

δfor x ∈ Bδ

Fδ(x) :=

x for x ∈ Ω \B2(

2− 2δ

2− δ+|x|

2− δ

δfor x ∈ Bδ

what else can be cloaked?

Q1 Can cloaking be translated from electromagnetism to other areasof wave physics? – acoustics.

Q2 Can meta-material ideas be carried over to other physical systemswhere waves are absent altogether?

I static mechanics (elastostatics)I thermodynamics (transient heat propagation)

In the absence of wave behaviour, what it means to cloak?Isotropic homogenous medium

I static mechanics

(elastostatics)

I thermodynamics

(transient heat propagation)

In the absence of wave behaviour, what it means to cloak?

Isotropic homogenous medium

In the absence of wave behaviour, what it means to cloak?Isotropic homogenous medium medium with a defect

In the absence of wave behaviour, what it means to cloak?Isotropic homogenous medium cloak surrounding the defect

material properties and observations

Context Governing equation Material properties Observations

Electromagnetism Maxwell’s equations permittivity Electric field

permeability Magnetic field

Thermodynamics Heat equation density Temperature field

heat capacity

conductivity

Elastostatics Lame (Navier) system Elasticity tensor Displacement field

Thin plate theory Linear Kirchhoff equation Flexural rigidity Flexural displacement

heat capacity

conductivity

heat capacity

conductivity

heat capacity

conductivity

heat capacity

conductivity

experimental validation (thermal cloak)

[Ref.] r.schittny, m.kadic, s.guenneau, m.wegener, Phys. Rev. Lett., (2013).

♣ Wegener’s group designed a thermal cloak in a metal plate.

Hot and cold baths; heat diffuses from left to right.

♣ An infrared (IR) camera films heat diffusion in the plate.

Experimental observation Numerical simulation

Experimental observation

Numerical simulation

our cloaking result for the heat flow

[Ref.] r.craster, s.guenneau, h.hutridurga, g.pavliotis,

SIAM Multiscale Model. Simul., 16(3), (2018).

∂tu−∆u = 0, ρcl(x) scl(x) ∂tuδ −∇ ·(Acl(x)∇uδ

Cloaking Theorem for heat equation

For any η > 0, there exists a T∗ <∞ and we can prescribedensity, heat capacity and conductivity in the cloak (by taking δ = d

√η)

such that the corresponding temperature field uδ(t, x) satisfies

‖u(t, ·)− uδ(t, ·)‖H1 . η ∀t > T∗

irrespective of the physical properties of the region B1.

√η)

‖u(t, ·)− uδ(t, ·)‖H1 . η ∀t > T∗

√η)

‖u(t, ·)− uδ(t, ·)‖H1 . η ∀t > T∗

√η)

‖u(t, ·)− uδ(t, ·)‖H1 . η ∀t > T∗

our cloaking result in elastostatics

[Ref.] r.craster, a.diatta, s.guenneau, h.hutridurga

Proc. R. Soc. A, (in revision); arXiv:1803.01360

∇ ·(C∇u

)= f , ∇ ·

(Ccl(x)∇uδ

)= f .

Cloaking Theorem for Lame (Navier) system

For any η > 0, we can prescribe an elasticity tensor in the cloak (bytaking δ = d

√η) such that the corresponding displacement field uδ(x)

satisfies‖u(·)− uδ(·)‖H1 . η

∇ ·(C∇u

)= f , ∇ ·

(Ccl(x)∇uδ

)= f .

∇ ·(C∇u

)= f , ∇ ·

(Ccl(x)∇uδ

)= f .

∇ ·(C∇u

)= f , ∇ ·

(Ccl(x)∇uδ

)= f .

our cloaking results for thin plates

[Ref.] h.ammari, r.craster, a.diatta, s.guenneau, h.hutridurga

∆2u = f, anisotropic plate equation.

For any η > 0, we can prescribe some material parameters in thegeneralised plate equation such that the corresponding flexuraldisplacement uδ(x) satisfies

‖u(·)− uδ(·)‖H2 . η

some open questions in invisibility cloaking

♣ Design of meta-materials.I Exotic physical properties are not known for ordinary materials.I Design assemblies of small components which may have desired

properties as a whole.I Inverse homogenization: given a property, look for a microstructure.

♣ Invisibility cloaking in nonlinear models.I Prove a parabolic analogue of the static quasilinear cloaking result:

[Ref.] t.ghosh, k.iyer, Inverse Problems & Imaging, (2018)

I Very thin plate theory (von Karman equations)

♣ Rigorous results on invisibility cloaking for wave phenomena.

♣ Is it possible to cloak gravitational waves?I study Einstein’s field equationsI transformation media theory for this very geometric equation?I ligo’s achievement in detecting gravitational waves

♣ Is it possible to cloak gravitational waves?

I study Einstein’s field equationsI transformation media theory for this very geometric equation?I ligo’s achievement in detecting gravitational waves

♣ Is it possible to cloak gravitational waves?I study Einstein’s field equations

I transformation media theory for this very geometric equation?I ligo’s achievement in detecting gravitational waves

♣ Is it possible to cloak gravitational waves?I study Einstein’s field equationsI transformation media theory for this very geometric equation?

I ligo’s achievement in detecting gravitational waves

does cloaking make sense for random phenomena?

♣ Take Xt a diffusion processdXt = b(Xt)dt+ σ(Xt)dWt

X0 = x

with drift b(x) and diffusion matrix Σ(x) = σ(x)σ(x)>

♣ Generator of the above diffusion process is defined as

L := b(x) · ∇+1

2Σ(x) : D2

♣ In particular, the generator of

dXt = b(Xt)dt+√

isb(x) · ∇+ ∆

X0 = x

L := b(x) · ∇+1

2Σ(x) : D2

dXt = b(Xt)dt+√

isb(x) · ∇+ ∆

X0 = x

L := b(x) · ∇+1

2Σ(x) : D2

dXt = b(Xt)dt+√

isb(x) · ∇+ ∆

X0 = x

L := b(x) · ∇+1

2Σ(x) : D2

dXt = b(Xt)dt+√

isb(x) · ∇+ ∆

mean first passage times

♣ First passage time (exit time) is defined as

τxΩ := inft ≥ 0 : Xt 6∈ Ω

♣ Averaged over the probability space gives mean exit time

τ(x) := EτxΩ = E(

inft ≥ 0 : Xt 6∈ Ω

∣∣X0 = x)

♣ Mean exit time solves the following boundary value problemb(x) · ∇τ + ∆τ = −1 in Ω,

τ = 0 on ∂Ω.

inft ≥ 0 : Xt 6∈ Ω

∣∣X0 = x)

τ = 0 on ∂Ω.

inft ≥ 0 : Xt 6∈ Ω

∣∣X0 = x)

τ = 0 on ∂Ω.

inft ≥ 0 : Xt 6∈ Ω

∣∣X0 = x)

τ = 0 on ∂Ω.

stochastic process in a force field

♣ Drift is given by a potential ϕ i.e. b(x) = ∇ϕ(x).

small volume perturbation of passage times

♣ Defects in potential and diffusion

ϕε(x), aε(x) =

ϕ0(x), a0 for x ∈ Ω \ ∪mj=1Dj

ϕj(x), aj for x ∈ Dj := zj + εB

♣ Perturbed mean exit time solves∇ϕε(x) · ∇τ ε(x) + div (aε(x)∇τ ε(x)) = −1 in Ω,

τ ε = 0 on ∂Ω.

Theorem

The perturbed and the homogeneous mean exit times satisfy

‖τ ε − τ‖H1 . εd

ϕε(x), aε(x) =

τ ε = 0 on ∂Ω.

Theorem

‖τ ε − τ‖H1 . εd

ϕε(x), aε(x) =

τ ε = 0 on ∂Ω.

Theorem

‖τ ε − τ‖H1 . εd

ϕε(x), aε(x) =

τ ε = 0 on ∂Ω.

Theorem

‖τ ε − τ‖H1 . εd

ϕε(x), aε(x) =

τ ε = 0 on ∂Ω.

Theorem

‖τ ε − τ‖H1 . εd

ϕε(x), aε(x) =

τ ε = 0 on ∂Ω.

Theorem

‖τ ε − τ‖H1 . εd

some final comments

♣ How about the exit times themselves (i.e. before taking mean)?

♣ A right formulation of this problem and its resolution would openup a world of possibilities in the study of stochastic processes andtheir applications.

♣ Small volume perturbations for eigenvalue problems is another setof problems that we have worked on.

I Dirichlet, Neumann, Steklov – Dirichlet-to-Neumann maps.

thank you

a fascinating world of invisibility cloakingdjsym/talk/Day2/Harsha-Hutridurga.pdf · such that...

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