Invisibility cloaks and transformation optics

Invisibility cloaks and transformation optics

A. Nicolet

Institut Fresnel (UMR CNRS 6133), Marseille, FranceCNRS, Aix-Marseille Université, École Centrale de Marseille

Outline

1 Introduction : open problem in electromagnetism with thefinite elements (old stuff in a new bottle)

2 Geometrical transformations3 Open wave propagation problems : Perfectly Matched

Layers (PML)

4 Twisted Microstructured Optical Fibre

5 Invisibility cloak

6 Masking or “polyjuice” effect

7 NRI and superscattering

8 Metamaterials in electromagnetism

9 Wire MediaDescriptionNon local homogenization

10 Conclusion

In collaboration with...

Prof. Frédéric Zolla, Université de Provence,Dr. Sébastien Guenneau, former Ph. D. student and CR1CNRS,Dr. Yacoub Ould Agha, former Ph. D. student,Dr. Alexandru I. Cabuz, former Postdoc.

Open geometries in static and quasistatic problems

Various attempts to set up infinite elements...

D. A. Lowther, E. M. Freeman, B. Forghani, A SparseMatrix Open Boundary Method for the Finite ElementAnalysis, IEEE Transactions on Magnetics, Vol. 25, no. 4,July 1989, pp 2810-2812.Analytical map of the interior of a circle (disk) on theexterior of a circle (infinite domain), connecting theboundary to the boundary of the region of interest.Limited to 2D Laplace operator (harmonic problems).


Various attempts to set up infinite elements...D. A. Lowther, E. M. Freeman, B. Forghani, A SparseMatrix Open Boundary Method for the Finite ElementAnalysis, IEEE Transactions on Magnetics, Vol. 25, no. 4,July 1989, pp 2810-2812.

Analytical map of the interior of a circle (disk) on theexterior of a circle (infinite domain), connecting theboundary to the boundary of the region of interest.Limited to 2D Laplace operator (harmonic problems).


Various attempts to set up infinite elements...D. A. Lowther, E. M. Freeman, B. Forghani, A SparseMatrix Open Boundary Method for the Finite ElementAnalysis, IEEE Transactions on Magnetics, Vol. 25, no. 4,July 1989, pp 2810-2812.Analytical map of the interior of a circle (disk) on theexterior of a circle (infinite domain), connecting theboundary to the boundary of the region of interest.

Limited to 2D Laplace operator (harmonic problems).


Various attempts to set up infinite elements...D. A. Lowther, E. M. Freeman, B. Forghani, A SparseMatrix Open Boundary Method for the Finite ElementAnalysis, IEEE Transactions on Magnetics, Vol. 25, no. 4,July 1989, pp 2810-2812.Analytical map of the interior of a circle (disk) on theexterior of a circle (infinite domain), connecting theboundary to the boundary of the region of interest.Limited to 2D Laplace operator (harmonic problems).


Solve a 2D harmonic problem on the disk.


Generalization : surround the Region of Interest by a shell.

J.-F. Imhoff, G. Meunier, X. Brunotte, J.C. Sabonnadière,On Original Solution for Unbounded Electromagnetic 2D-and 3D- problems Throughout the Finite Element Method,IEEE Transactions on Magnetics, Vol. 26, no. 5, September1990.Adapted to 3D problems and to various shapes, naturaldescription of the geometry...Requires a modification of the operator.


Generalization : surround the Region of Interest by a shell.J.-F. Imhoff, G. Meunier, X. Brunotte, J.C. Sabonnadière,On Original Solution for Unbounded Electromagnetic 2D-and 3D- problems Throughout the Finite Element Method,IEEE Transactions on Magnetics, Vol. 26, no. 5, September1990.

Adapted to 3D problems and to various shapes, naturaldescription of the geometry...Requires a modification of the operator.


Generalization : surround the Region of Interest by a shell.J.-F. Imhoff, G. Meunier, X. Brunotte, J.C. Sabonnadière,On Original Solution for Unbounded Electromagnetic 2D-and 3D- problems Throughout the Finite Element Method,IEEE Transactions on Magnetics, Vol. 26, no. 5, September1990.Adapted to 3D problems and to various shapes, naturaldescription of the geometry...

Requires a modification of the operator.


Generalization : surround the Region of Interest by a shell.J.-F. Imhoff, G. Meunier, X. Brunotte, J.C. Sabonnadière,On Original Solution for Unbounded Electromagnetic 2D-and 3D- problems Throughout the Finite Element Method,IEEE Transactions on Magnetics, Vol. 26, no. 5, September1990.Adapted to 3D problems and to various shapes, naturaldescription of the geometry...Requires a modification of the operator.


How to modify simply the code to introduce the newcoefficients ?

Differential geometry of p-forms

Differential geometry of p-formsp-forms are rank p totally covariant tensors.Exterior derivative d = differential or gradient of functions,curl and divergence of vector fields...dd = 0 for curl grad = 0, div curl = 0 ...Exterior product ∧ = multilinear antisymmetric map.Geometric integration

∫Σ α = line integrals of 1-forms,

surface (flux) integrals of 2-forms, volume integrals of3-forms...Stokes theorem

∫Σ dα =

∫∂Σ α

All are metric free operations !

Electrodynamics

Maxwell’s equationscurlH = J + ∂tDcurlE = −∂tBdiv D = ρdiv B = 0

Poynting identity

div(E × H) = J · E + E · ∂tD + H · ∂tB

Electrodynamics

Maxwell’s equationsdH = J + ∂tDdE = −∂tBdD = ρdB = 0

Poynting identity

d(E ∧ H) = J ∧ E + E ∧ ∂tD + H ∧ ∂tB

These relations are metric free !

Example of the Faraday equation

In a general coordinate system u, v ,w :The electric field is the 1-form E = Eudu + Evdv + Ewdw .The magnetic flux density is the 2-formB = Budv ∧ dw + Bvdw ∧ du + Bwdu ∧ dv .And dE = −∂tB means(∂uEv − ∂vEu + ∂tBw )du ∧ dv+

(∂vEw − ∂wEv + ∂tBu)dv ∧ dw+(∂wEu − ∂uEw + ∂tBv )dw ∧ du = 0

Metric

Distance, angle...Hodge star operator ∗ maps p-forms on (3− p)-forms.Example of Euclidean metric in Cartesian coordinates :

∗dx = dy ∧ dz∗dy = dz ∧ dx∗dz = dx ∧ dy

This simplicity hides metric aspects in Cartesiancoordinates but the relations are more complicated with ageneral coordinate system...

Metric

... and electromagnetic constitutive laws !For example in free space :

D = ε0∗E

B = µ0∗HThe Hodge star operator is necessary to transform fields(1-forms) into flux densities (2-forms) !

Coordinate transformation

Considering a map from the coordinate system u, v ,w tothe coordinate system x , y , z given by the functionsx(u, v ,w), y(u, v ,w), and z(u, v ,w),All the information is in the transformation of thedifferentials and is therefore given by the chain rule :

dx = ∂x∂udu + ∂x

∂v dv + ∂x∂w dw

dy = ∂y∂udu + ∂y

∂v dv + ∂y∂w dw

dz = ∂z∂udu + ∂z

∂v dv + ∂z∂w dw


All the information is in the Jacobian matrix J : dxdydz

= J

dudvdw

with J defined as

J(u, v ,w) =∂(x , y , z)

∂(u, v ,w)=

∂x∂u

∂x∂v

∂x∂w

∂y∂u

∂y∂v

∂y∂w

∂z∂u

∂z∂v

∂z∂w


For a 1-form E, the transformation from x , y , z to u, v ,wcoordinates is performed as follows :

E = Exdx + Eydy + Ezdz = (Ex Ey Ez)

dxdydz

= (Ex Ey Ez)J

dudvdw


We have also :

E = Eudu + Evdv + Ewdw = (Eu Ev Ew )

dudvdw

and the following relation is obtained :

(Ex Ey Ez) J = (Eu Ev Ew )

Pullback

Direct map :New coordinates −→ Cartesian coordinatesu, v ,w x(u, v ,w), y(u, v ,w), z(u, v ,w)=Modelling space =“Physical space”

Pullback :Forms in ←− Forms innew coordinates Cartesian coordinatesEudu + Evdv + Ewdw Exdx + Eydy + Ezdzetc.

Scalar product of 1-forms

E ∧ ∗E′ = (Ex Ey Ez)(E ′x E ′y E ′z)Tdx ∧ dy ∧ dz(thanks to the simplicity of Cartesian coordinates)= (Eu Ev Ew )J−1[(E ′u E ′v E ′w )J−1]Tdx ∧ dy ∧ dz(using the previous relations between the coordinate systems)= (Eu Ev Ew )J−1J−T (E ′u E ′v E ′w )T det(J)du ∧ dv ∧ dw .(transforming the volume form)

A transformation matrix (related to metric tensor !) T isdefined by :

T−1 = J−1J−T det(J)


For two 1-forms E and E′, the scalar product is defined asfollows : ∫

R3

E ∧ ∗E′

Practically, it may be computed as :∫R3

E · T−1E′dV

where · denotes the “dot product in Cartesian coordinates” anddV the Lebesgue measure...

Everything behaves as if T were an anisotropicinhomogeneous tensor material property (inversepermittivity) !


For 2-forms e.g. :dx ∧ dy =[∂x∂udu + ∂x

∂v dv + ∂x∂w dw ] ∧ [∂y

∂udu + ∂y∂v dv + ∂y

∂w dw ] =

(∂x∂u

∂y∂v − ∂x

∂v∂y∂u )du ∧ dv + (∂x

∂v∂y∂w − ∂x

∂w∂y∂v )dv ∧ dw + ( ∂x

∂w∂y∂u −

∂x∂u

∂y∂w )dw ∧ du.

The cofactors of J are now involved in the transformation.These are the elements of J−T det(J). dx ∧ dy

dy ∧ dzdz ∧ dx

= J−T det(J)

du ∧ dvdv ∧ dwdw ∧ dv

Note : J−T means inverse of the transpose of J.


Given a 2-form :

D = Dxdy ∧ dz + Dydz ∧ dx + Dzdx ∧ dy= Dudv ∧ dw + Dvdw ∧ du + Dwdu ∧ dv

the following relation is obtained :

(Dx Dy Dz) J−T det(J) = (Du Dv Dw )

and the matrix involved in the scalar product is here T (stillequivalent to an inverse permittivity !) :∫

R3

D · TD′dV

Coordinate transformations...

...can be encapsulated in material properties.Weak formulations (considering here possibly anisotropic materials) involveterms (3-forms) like

E · εE′, H · µH′ (scalar product of 1-forms type)

curl E · µ−1 curl E′, curlH · ε−1 curlH′

(scalar product of 2-forms type).

Introducing :1-form transformation : (Ex Ey Ez) J = (Eu Ev Ew )2-form transformation : (Dx Dy Dz) J−T det(J) = (Du Dv Dw )and a Jacobian det(J) factor for the measure transformation...All the terms in the weak formulations can be equivalentlycomputed by introducing

equivalent material properties εeq, µ

eq:

εeq

= J−1εJ−T det(J)

µeq

= J−1µJ−T det(J)

Consequences

Map the transformed domain Ω′ (modelling space) on theoriginal domain Ω (physical space usually in Cartesiancoordinates) and pullback the covariant objects (physicalequations) to the transformed domain (new modelequations).

Maxwell’s equations, that express topological anddifferential metric free relations, may be written in acompletely covariant coordinate free form.The material properties, involving all the metric aspects,are written with the Hodge operator and contain all thecoordinate dependent information.

Any change of coordinates can be translated into equivalentmaterials given by :

ε′ = εT−1 , and µ′ = µT−1 .

with the matrix T=JTJ/det(J).

Consequences

Map the transformed domain Ω′ (modelling space) on theoriginal domain Ω (physical space usually in Cartesiancoordinates) and pullback the covariant objects (physicalequations) to the transformed domain (new modelequations).Maxwell’s equations, that express topological anddifferential metric free relations, may be written in acompletely covariant coordinate free form.

The material properties, involving all the metric aspects,are written with the Hodge operator and contain all thecoordinate dependent information.


ε′ = εT−1 , and µ′ = µT−1 .


Consequences

Map the transformed domain Ω′ (modelling space) on theoriginal domain Ω (physical space usually in Cartesiancoordinates) and pullback the covariant objects (physicalequations) to the transformed domain (new modelequations).Maxwell’s equations, that express topological anddifferential metric free relations, may be written in acompletely covariant coordinate free form.The material properties, involving all the metric aspects,are written with the Hodge operator and contain all thecoordinate dependent information.


ε′ = εT−1 , and µ′ = µT−1 .


Consequences

Map the transformed domain Ω′ (modelling space) on theoriginal domain Ω (physical space usually in Cartesiancoordinates) and pullback the covariant objects (physicalequations) to the transformed domain (new modelequations).Maxwell’s equations, that express topological anddifferential metric free relations, may be written in acompletely covariant coordinate free form.The material properties, involving all the metric aspects,are written with the Hodge operator and contain all thecoordinate dependent information.


ε′ = εT−1 , and µ′ = µT−1 .


Coordinate transformation and equivalent materials

Equivalent materials mean thatyou can work in the new coordinate system justas if you were still in Cartesian coordinates butfor the ε and µ that have been turned to newequivalent material properties

ε′ = εT−1 and µ′ = µT−1

with the matrix T=JTJ/det(J) !


In the more general case where the initial ε and µ aretensors corresponding to anisotropic properties, theequivalent properties become

ε′ = J−1εJ−Tdet(J) , and µ′ = J−1µJ−Tdet(J) .

where J−T denotes the transpose of the inverse of J.

The two successive changes of coordinates are given bythe Jacobian matrices JxX and JXu so that

Jxu = JxXJXu .

This rule naturally applies for an arbitrary number ofcoordinate systems.


In the more general case where the initial ε and µ aretensors corresponding to anisotropic properties, theequivalent properties become

ε′ = J−1εJ−Tdet(J) , and µ′ = J−1µJ−Tdet(J) .

where J−T denotes the transpose of the inverse of J.The two successive changes of coordinates are given bythe Jacobian matrices JxX and JXu so that

Jxu = JxXJXu .

This rule naturally applies for an arbitrary number ofcoordinate systems.

All fields change...

1-form transformation : (Ex Ey Ez) J = (Eu Ev Ew ) for theelectric field E and also for the magnetic field H, the vectorpotential A...2-form transformation : (Dx Dy Dz) J−T det(J) = (Du Dv Dw )for the electric displacement D and also for the magneticinduction B, the current density J, the Poynting vector S...What does it REALLY means "equivalent" then ?

Global integral quantities are conserved !

Line integrals of 1-forms are conserved (because curvesand 1-forms experience dual transformations) such aselectromotive force, magnetomotive force, magnetic flux(evaluated with A)...Surface flux integrals of 2-forms are conserved (becausesurfaces and 2-forms experience dual transformations)such as electric flux, magnetic flux (evaluated with B), totalcurrent, power flow...Volume integrals of 3-forms are conserved (becausevolumes and 3-forms experience dual transformations)such as total charge, powers...Measurements are preserved !

T−1 for an open domain (exterior of a disk) mapped ona circular annulus.

Consider the radial transformationr = f (r ′) = (R1 − R2)r ′/(r ′ − R2)so that r ′ = R1 ⇒ r = R1 and r ′ = R2 ⇒ r →∞ .Define c11(r ′) = df (r ′)

dr ′ and

T−1 = R(θ′)diag(f (r ′)

c11(r ′)r ′,c11(r ′)r ′

f (r ′),c11(r ′)f (r ′)

r ′)R(θ′)T

where r ′ and θ′ are the well known functionsr ′(x ′, y ′) =

√x ′2 + y ′2 and θ′(x ′, y ′) = 2 arctan( y ′

x ′+√

x ′2+y ′2) and

R(θ′) is a rotation matrix.

x ′, y ′ are the Cartesian coordinates in the annulusconfiguration.

Equivalent material for unbounded electrostaticproblem

Electrostatic potential : circular cylinder V = 1, V (r →∞) = 0⇒ circular equipotential lines !

Open propagation problems : Perfectly MatchedLayers (PML)

Previous technique in not workable for wave propagationproblems because the wavelength is going to zero.

J.-P. Bérenger, A Perfectly Matched Layer for theAbsorption of Electromagnetic Waves, Journal ofComputational Physics, 1994, 114, pp. 185-200.Nowadays, may be introduced as a complex mapping


Previous technique in not workable for wave propagationproblems because the wavelength is going to zero.J.-P. Bérenger, A Perfectly Matched Layer for theAbsorption of Electromagnetic Waves, Journal ofComputational Physics, 1994, 114, pp. 185-200.

Nowadays, may be introduced as a complex mapping


Previous technique in not workable for wave propagationproblems because the wavelength is going to zero.J.-P. Bérenger, A Perfectly Matched Layer for theAbsorption of Electromagnetic Waves, Journal ofComputational Physics, 1994, 114, pp. 185-200.Nowadays, may be introduced as a complex mapping

A Complex Change of Coordinates

Consider polar coordinates with ρ the radial coordinate.

Perform the complex change of coordinates ρ =∫ ρ

0sρ(ρ′)dρ′ with

sρ a complex valued function (complex radial stretch).

Roughly turns e−iγρ into e−iγρ exponentially decreasing becauseρ is complex (with the good sign of the imaginary part of ρ) and itdamps outgoing waves and other leaky modes...

The global coordinate transformation is taken to be identity(sρ = 1) inside a region of interest (ρ < R∗) and the complexchange of coordinates (sρ complex valued) outside theregion of interest (ρ > R∗) so that the transformed problemprovides directly the required fields in the region of interest (Thiswill be more clear on an example !)






















PML TPML matrix

The inverse matrix T−1PML corresponding to the complex stretch

is given by :ρsρ sin(θ)2

ρ + ρ cos(θ)2

ρsρ

sin(θ) cos(θ)( ρρsρ− ρsρ

ρ ) 0

sin(θ) cos(θ)( ρρsρ− ρsρ

ρ )ρsρ cos(θ)2

ρ + ρ sin(θ)2

ρsρ0

0 0ρsρ

ρ

.

All the quantities involved in the previous expression can begiven as explicit functions of x and y “pseudo Cartesian

modelling coordinates” : θ = 2 arctan

(y

x+√

x2+y2

),

ρ =√

x2 + y2, sρ(ρ) = sρ(√

x2 + y2), and ρ =∫√x2+y2

0 sρ(ρ′)dρ′.

A strong test for PML : leaky modes inMicrostructured Optical Fibres

Cross section of a six air hole in silica MOF structureThe hole structure is with Λ = 6.75µm, rs = 2.5µm, the surroundingannulus used to set up the PML has R∗ = 30µm,R trunc = 40µm

Rtrunc

R∗

x

y

Λ

x

y

n = 1nSi

2rs O

Comparison with the Multipole Method

λ0 = 1.55µm is considered for which the index of silica is about√εSi = nSi = 1.444024.

The corresponding complex effective index neff = β/k0 is

multipole methods :

1.4387741 + 4.3257457 10−8i

finite element method :

1.43877448 + 4.325885 10−8i

COMSOL Multiphysicsr, about 16, 800 second order triangular elements, 150seconds on a Pentium M 1.86 GHz, 1Go RAM laptop computer.




multipole methods :

1.4387741 + 4.3257457 10−8i


1.43877448 + 4.325885 10−8i





multipole methods :

1.4387741 + 4.3257457 10−8i


1.43877448 + 4.325885 10−8i





multipole methods :

1.4387741 + 4.3257457 10−8i


1.43877448 + 4.325885 10−8i


What happens if a Microstructured Optical Fibre istwisted ?

From translational invariance...

... to twisted structures :

We are going to look for a simple and efficient model i.e. still 2D andrigorous !









Helicoidal coordinates

Helicoidal coordinates :x1 = ξ1 cos(αξ3) + ξ2 sin(αξ3) ,

x2 = −ξ1 sin(αξ3) + ξ2 cos(αξ3) ,

x3 = ξ3 ,

(1)

where α is a parameter which characterizes the torsion ofthe structure.

A twisted problemis a problem whose cross section is independent from ξ3.

Helicoidal coordinates

Helicoidal coordinates :x1 = ξ1 cos(αξ3) + ξ2 sin(αξ3) ,

x2 = −ξ1 sin(αξ3) + ξ2 cos(αξ3) ,

x3 = ξ3 ,

(1)

where α is a parameter which characterizes the torsion ofthe structure.

A twisted problemis a problem whose cross section is independent from ξ3.

Jacobian matrix of the Cartesian/helicoidal coordinatetransformation

This coordinate system is characterized by the Jacobianmatrix of the transformation :

J(ξ1, ξ2, ξ3) =∂(x1, x2, x3)

∂(ξ1, ξ2, ξ3)

=

cos(αξ3) sin(αξ3) αξ2 cos(αξ3)− αξ1 sin(αξ3)− sin(αξ3) cos(αξ3) −αξ1 cos(αξ3)− αξ2 sin(αξ3)

0 0 1

,

which does depend on the three variables ξ1, ξ2 and ξ3.

Jacobian matrix of the Cartesian/helicoidal coordinatetransformation

This coordinate system is characterized by the Jacobianmatrix of the transformation :

J(ξ1, ξ2, ξ3) =∂(x1, x2, x3)

∂(ξ1, ξ2, ξ3)

=

cos(αξ3) sin(αξ3) αξ2 cos(αξ3)− αξ1 sin(αξ3)− sin(αξ3) cos(αξ3) −αξ1 cos(αξ3)− αξ2 sin(αξ3)

0 0 1

,

which does depend on the three variables ξ1, ξ2 and ξ3.

Th matrix for equivalent material in helicoidalcoordinates

The transformation matrix Th for helicoidal coordinates is :

Th(ξ1, ξ2) =JTJ

det(J)=

1 0 αξ2

0 1 −αξ1

αξ2 −αξ1 1 + α2(ξ21 + ξ2

2)

,

which only depends on the first two variables ξ1 and ξ2...... allowing a two-dimensional formulation of the twistedwaveguide problem !



Th(ξ1, ξ2) =JTJ

det(J)=

1 0 αξ2

0 1 −αξ1

αξ2 −αξ1 1 + α2(ξ21 + ξ2

2)

,

which only depends on the first two variables ξ1 and ξ2...

... allowing a two-dimensional formulation of the twistedwaveguide problem !



Th(ξ1, ξ2) =JTJ

det(J)=

1 0 αξ2

0 1 −αξ1

αξ2 −αξ1 1 + α2(ξ21 + ξ2

2)

,

which only depends on the first two variables ξ1 and ξ2...... allowing a two-dimensional formulation of the twistedwaveguide problem !

Twisted PML ThPML matrix

Compose the transformation into helicoidal coordinateswith a complex stretch to obtain the twisted PML inversematrix T−1

hPML given by :ρ cos2(ϕ)ρsρ

+ρ(1+α2ρ2)sρ sin2(ϕ)

ρ

sin(2ϕ)(ρ2−ρ2(1+α2ρ2)s2ρ)

2ρρsρ−αρsρ sin(ϕ)

sin(2ϕ)(ρ2−ρ2(1+α2ρ2)s2ρ)

2ρρsρ

ρ sin2(ϕ)ρsρ

+ρ(1+α2ρ2)sρ cos2(ϕ)

ρ α cos(ϕ)ρsρ

−αρsρ sin(ϕ) α cos(ϕ)ρsρρsρ

ρ

All the quantities involved in the previous expression can begiven as explicit functions of the two “helicoidal pseudoCartesian modelling coordinates ” ξ1, ξ2 :

ϕ = 2 arctan

(ξ2

ξ1+√ξ2

1+ξ22

), ρ =

√ξ2

1 + ξ22 , sρ(ρ) = sρ(

√ξ2

1 + ξ22),

and ρ =∫√ξ2

1+ξ22

0 sρ(ρ′)dρ′.

Evolution of the Real Part of Effective Index

xz

y

Λ

Silicaair

d

ParametersΛ = 6.75µm

d = 5µm

nSi . = 1.444024

λ = 1.55µm

1.438

1.4382

1.4384

1.4386

1.4388

1.439

1.4392

1.4394

1.4396

0 500 1000 1500 2000 2500 3000

Re(n

eff)

Torsion parameter α, in m -1

Evolution of the Imaginary Part of Effective Index

xz

y

Λ

Silicaair

d

ParametersΛ = 6.75µm

d = 5µm

nSi . = 1.444024

λ = 1.55µm

4.2e − 008

4.4e − 008

4.6e − 008

4.8e − 008

5e − 008

5.2e − 008

5.4e − 008

0 500 1000 1500 2000 2500 3000

Im(n

eff)

Torsion parameter α, in m -1

Harry Potter’s physics !

From invisibility cloak...

... to polyjuice potion !

Pendry’s map for cylindrical invisibility cloaks

Map an annulus on a circle and let the EQUIVALENTMATERIAL to become a NEW PHYSICAL MATERIAL !Consider a geometric transformation which maps the fieldwithin the disk r ≤ R2 onto the annulus R1 ≤ r ≤ R2 :

r = f (r ′) = (r ′ − R1)R2/(R2 − R1) for R1 ≤ r ′ ≤ R2,

θ = θ′, z = z ′.

where r ′, θ′ and z ′ are “radially contracted cylindricalcoordinates”. Moreover, this transformation maps the field forr ≥ R2 onto itself by the identity transformation.

Pendry’s map for cylindrical invisibility cloaks

Cloaking

The material properties of the invisibility cloak are given by :

T−1 = R(θ′)diag(r ′ − R1

r ′,

r ′

r ′ − R1, c2

11

r ′ − R1

r ′)R(θ′)T

where c11 = R2/(R2 − R1), r ′ and θ′ are the well knownfunctions r ′(x ′, y ′) =

√x ′2 + y ′2 and

θ′(x ′, y ′) = 2 arctan( y ′

x ′+√

x ′2+y ′2), and R(θ′) is a rotation matrix.

x ′, y ′ are the Cartesian coordinates in the annulusconfiguration.

Cloaking

... and we have got the recipe for the cylindrical circularcloak !

Cloaking

In Photonics conferences, courtesy requires that you leaveyour cloak in an appropriate place !

Mirage effect

The light source can also be inside the cloak !In this case, the object in the cavity is still invisible and the light seemsto be emitted from a shifted position (mirage effect).

Cloak of arbitrary shape

Make the radii depend on θ : R1(θ), R2(θ).The geometric transformation which maps the field within thefull domain r ≤ R2(θ) onto the hollow domainR1(θ) ≤ r ≤ R2(θ) :

r ′(r , θ) = R1(θ) + r(R2(θ)− R1(θ))/R2(θ) , 0 ≤ r ≤ R2(θ)

θ′ = θ , 0 < θ ≤ 2π

z ′ = z , z ∈ R ,

and the transformation maps the field for r ≥ R2(θ) onto itselfby the identity transformation. This leads to

Jrr ′ =∂(r(r ′, θ′), θ, z)

∂(r ′, θ′, z ′)=

c11(r ′, θ′) c12(r ′, θ′) 00 1 00 0 1


Jrr ′(r′, θ′) =

∂(r(r ′, θ′), θ, z)

∂(r ′, θ′, z ′)=

c11(r ′, θ′) c12(r ′, θ′) 00 1 00 0 1

where c11(r ′, θ′) = R2(θ′)/(R2(θ′)− R1(θ′)) for 0 ≤ r ′ ≤ R2(θ′)and c11 = 1 for r ′ > R2(θ′)

and c12(r ′, θ′) =(r ′−R2(θ′))R2(θ′)

dR1(θ′)dθ′ −(r ′−R1(θ′))R1(θ′)

dR2(θ′)dθ′

(R2(θ′)−R1(θ′))2

for 0 ≤ r ′ ≤ R2(θ′)and c12 = 0 for r ′ > R2(θ′) .


Finally, the properties of the cloak are given by :

T−1 = R(θ′)

c2

12+f 2r

c12fr r ′− c12

fr0

− c12fr

c11r ′

fr0

0 0 c11frr ′

R(θ′)T

with

fr =(r ′ − R1)R2

(R2 − R1).

The central matrix depends on θ′ !

Elliptical cloak as a particular case

Parametric representation of the ellipseR(θ) = ab√

a2 cos(θ)2+b2 sin(θ)2


Use Fourier series R(θ) = a0 +∑n

i=1(ai cos(iθ) + bi sin(iθ)) to obtain generalshapes :R1 is with a0 = 1, b1 = 0.1, a2 = −0.15, b3 = 0.2, a4 = 0.1,R2 is with a0 = 2, a2 = −0.1, a3 = −0.15, b3 = 0.3, a4 = 0.2,all the other coefficients = 0.


Zoom on the cloak and the source...

Geometric transformation of non homogeneous media

The geometric transformation is performedpointwise on the disk :

ε′(x′) = J−1(x′)ε(x(x′))J−T (x′) det(J(x′)),

µ′(x′) = J−1(x′)µ(x(x′))J−T (x′) det(J(x′)).

These formulae are valid for any initialcontent of the central disk !

Masking effect

Scattering of cylindrical waves (real part of Ez ) by a conductingobject of triangular cross section :

Masking effect

Transforming a non homogeneous region :

The cross sections of the boundaries of homogeneous regions (jumpsof material properties) are curves (here, the triangle) and are thereforecontravariant.They are pushed forward along the r ′ = f −1(r) inverse map (that isfortunately simple) :

x′(t) = f −1(x(t)) = (R2 − R1

R2+

R1

‖x(t)‖ )x(t),

The anamorphosis of the triangle is described by three splinesinterpolating each 40 points.

Masking effect

Geometry description and meshing are performed withGetDP and Gmsh.The inner boundary of the cloak must be very finelymeshed because of the singularity of the materialproperties.

Masking effect

Outside the cloak, the scattered field is left unchanged withrespect to the initial situation !

Masking effect

Compare the field outside the disks :

A multivalued map

The negative slope corresponds to Negative Refraction Indexmaterials and should provide superlenses.

J B Pendry, Phys. Rev. Lett. 85, 3966 (2000)Ulf Leonhardt and Thomas G Philbin, New J. Phys. 8 247 (2006)Min Yan, Wei Yan, and Min Qiu, Phys. Rev. B 78, 125113 (2008)

Superscatterer

Without losses : anomalous resonances* are dazzling !* N. A. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, OpticsExpress, Vol. 15, Issue 10, pp. 6314-6323 (2007)

Superscatterer

With 1/1000 losses in the superlens : the two images of thesources appear !

Superscatterer

With 1/100 losses in the superlens : the disturbance is stillreasonable and the three copies of the source appear clearly !

Superscatterer

Remote control : a small scatterer close to the interior image ofthe source acts as a large one (magnification factor = 4) on thesource (1/100 losses in the superlens) !

Superscatterer

Remote control : a small scatterer close to the interior image ofthe source acts as a large one (magnification factor = 8) on thesource (1/100 losses in the superlens) !

Superscatterer

Remote control : a small scatterer close to the interior image ofthe source acts as a large one (magnification factor = 8) on thesource (1/100 losses in the superlens) ! (zoomed)

Superscatterer

Can you force the light to go to the left with a device located onthe left of the source ?

Superscatterer

No ! The scatterer is a PERTURBATION of the folded geometryand its presence prevents the correct formation of the imagesource !

Metamaterials in electromagnetism

Periodic structures

1D : gratings, Bragg mirrors...2D : biperiodic gratings, microstructured optical fibres...3D : Photonic Crystals (photonic band gap)...

Plasmonic - metallic nanostructures for optics (Re(ε) < 0,Im(ε) very small, size < λ),Split Ring Resonators for microwavesWire media.


Periodic structures1D : gratings, Bragg mirrors...

2D : biperiodic gratings, microstructured optical fibres...3D : Photonic Crystals (photonic band gap)...



Periodic structures1D : gratings, Bragg mirrors...2D : biperiodic gratings, microstructured optical fibres...

3D : Photonic Crystals (photonic band gap)...



Periodic structures1D : gratings, Bragg mirrors...2D : biperiodic gratings, microstructured optical fibres...3D : Photonic Crystals (photonic band gap)...




Plasmonic - metallic nanostructures for optics (Re(ε) < 0,Im(ε) very small, size < λ),Split Ring Resonators for microwaves

Wire media.





Microstructured optical fibre made ofchalcogenide glass manufactured atthe Université de Rennes (ext. diam.= 139 µm).

Split Ring Resonator device withprinted copper "double C" formicrowave experiments.


Purpose : obtain new artificial properties :Negative Refraction Index for Perfect Lenses

Photonic Band Gap and Epsilon Near Zero for guiding lightin defectsInvisibility Cloaking

Homogenization :

Find equivalent material properties ( ε, µ ... ) in order torepresent the metamaterial as an homogeneous medium !


Purpose : obtain new artificial properties :Negative Refraction Index for Perfect LensesPhotonic Band Gap and Epsilon Near Zero for guiding lightin defects

Invisibility Cloaking

Homogenization :



Purpose : obtain new artificial properties :Negative Refraction Index for Perfect LensesPhotonic Band Gap and Epsilon Near Zero for guiding lightin defectsInvisibility Cloaking

Homogenization :



Purpose : obtain new artificial properties :Negative Refraction Index for Perfect LensesPhotonic Band Gap and Epsilon Near Zero for guiding lightin defectsInvisibility Cloaking

Homogenization :


Wire Media

Set of parallel metallic or dielectric wires of FINITE lengthperiodically disposed.This media is completely transparent for waves without acomponent of the electric field transverse to the wires !

L

d

r

Incident field

Wire Media : Bouchitté - D. Felbacq approach

In

[BF] Homogenization of a Wire Photonic Crystal : the Case osSmall Volume Fraction, G. Bouchitté and D. Felbacq, SIAM J.APPL. MATH., Vol. 66, No. 6, pp. 2061-2084, 2006.

the medium is not represented by equivalent ε and µ but by nonlocal homogeneous properties : equivalent sources given by aPDE coupled to the field and involving homogeneouscoefficients !

Wire Media : asymptotic model

Define : k0 := 2π/λ (wave number), η := d/λ (relative size ofthe cell), f := πr2/d2 (filling factor), and Z0 :=

√µ0ε0

(free spaceimpedance)

λ, L are finite

and

d , r , σ−1 → 0 with

κ =σfZo

k

γ = −[1

2log(

f

π)η2]−1

remaining FINITE (critical behaviour).

Wire Media : equivalent system [BF]

E electric field, H magnetic field, y unit vector , j equivalentsources (y component = direction of the parallel wires)

curlE = iωµ0H

curlH = −iωε0 (E + ijy)

∂2j

∂y2+ K 2j = 2iπγE · y

with K 2 := k20 + 2iπγ

κ .+ Boundary Conditions (cf. [BF])

Note : γ is geometrical and therefore real while κ depends onthe medium and is possibly complex and even purely imaginaryin the case of high permittivity dielectrics (σ + ωε′′ − iωε′).

Wire Media : direct validation (A. Cabuz)

0 20 40 60 800

0.2

0.4

0.6

0.8

1

Angle of incidence θ [deg]

Energy efficiencies

FIG.: Transmission (solid), reflection (dot-dashed) and absorption (dashed) efficiencycurves comparing the finite element solution (dot markers) and the effective medium(no markers) as a function of angle of incidence. The wire conductivity is that of TorayT300 R© carbon fibers σ = 5.89 · 104(Ωm)−1. The structure has period d0 = 0.01m, anddimensionless parameters L/d0 = 80, λ/d0 = 20, r/d0 = 3.5 · 10−4, and δ/r = 15.Energy conservation of the finite element model (× markers) is respected to withinbetter than one percent for most angles of incidence. The departure around 80 isexplained by the poor performance of the PML absorbing layers when close to grazingincidence.

Wire Media : direct validation (A. Cabuz)

FIG.: Squared magnitude of the current density for the effective medium model(dashed) and the finite element solution (solid) as a function of position within the slab(which is positioned in z ∈ (0, L)). The structure is the same as in previous Fig.,illuminated at an angle of incidence θ = 40 from the top. Note that the surface areasunder the two curves are the same because they are proportional to the Jouledissipation rates, which are seen to be equal from previous Fig. at the given angle ofincidence.

Ch. Bourel Ph. D. Thesis, Dec. 2010 supervised by G.Bouchitté, USTV

Theoretical tool : two-scale convergence.Theorem : Given a frequency, a real number h, and anarbitrary real symmetric tensor M, by the homogenizationof a periodic structure made of parallelepipeds containinginclusions that are an homogenized wire media (iteratedhomogenization), it is possible to build a media such thatthe effective permittivity has a real part equal to M and animaginary part bounded by h (and the effectivepermeability is equal to 1).It is possible to obtain strong artificial magnetism with aperiodic structure of dielectric inclusions.

Conclusion

Coordinate transformations with equivalent materialsprovide a useful tool to set up several problems :

Modelling of open problems including propagation problemsand leaky modes, helicoidal geometries...Design of new device/EM exotic properties : invisibilitycloaks, masking devices, perfect lenses with negativerefractive index materials...

They also provide another argument for the use ofdifferential geometry to manipulate Maxwell’s equations !

Thank you for your attention [email protected] ://www.fresnel.fr/perso/nicolet/This work has been performed in the framework of theOPTRANS project - ANR-10-BLAN-0124-02.

Conclusion


Modelling of open problems including propagation problemsand leaky modes, helicoidal geometries...

Design of new device/EM exotic properties : invisibilitycloaks, masking devices, perfect lenses with negativerefractive index materials...



Conclusion





Conclusion





Conclusion





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Invisibility cloaks and transformation optics