MetasurfaceTransformation Optics · 2014. 1. 29. · Laboratorio Elettromagnetismo Applicato...

Laboratorio Elettromagnetismo Applicato

Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013

S. MaciDept. Information Engineering and Math Science, University of Siena,Via Roma 56, 53100 Siena, Italymacis.dii.unisi.it

Metasurface Transformation Optics



Acknowledgements

A Toccafondi (Associate Professor)M. Albani (Assistant Professor)

Post DocE. MartiniD. GonzalezM. Casaletti (now in UR1, Rennes)F. CaminitaG. CarluccioC. Della Giovampaola (now UPenn)A Mazzinghi (with UNIFI)S. Skokic (host from UNIZAG)M. Bosiljevac (host from UNIZAG)M. Violetti

PhD studentsG. Minatti (now in ESTEC)G. SardiF. PugelliM. Balasubramanian (now in Fraunhofer) M. FaenziV. Sozio

M. Mencagli

TechnicianD. Rossi



1) Backward‐forward leaky wave antennas

2) Backward TXL and miniaturized resonators

3) True time delay lines and modulators

4) Dispersive delay lines

5) Imaging TXL devices

1) Perfect lenses

2) Focusing systems

3) Negative refraction devices

4) Enhanced transmission through holes

5) Plasmonic devices

6) ENZ devices (tunnelling of guided waves)7) ENZ LW lenses

1) Volumetric Cloaking

2) Under the carpet cloaking

3) TO‐designed lens antennas

4) Field concentrators

5) Illusion devices

ENG ‐ENZ‐DNG MTM

Perfect focusing

Anomalous refraction

Plasmonic phenomena

TRANSFORMATION OPTICS MTM

Addressing waves in MTM

Transmission‐Line MTM

New RF devices

MTM Technology and Phenomenology

METASURFACES

EBG phenomena

Magnetic surfaces,

Hard and Soft surfaces

1) Low side‐lobe vertical monopoles

2) Low‐profile dipole antennas

3) Reduction of surface wave coupling

4) Improvement of planar‐antenna efficiency

5) Quasi‐TEM propagation waveguides

6) Miniaturized cavities

7) Reduction of scan blindness effects

8) Partilally reflective antennas

9) Enhancing bandwidth of small antennas

10) Flat Absorbers



1) Backward‐forward leaky wave antennas

2) Backward TXL and miniaturized resonators

3) True time delay lines and modulators

4) Dispersive delay lines

5) Imaging TXL devices

1) Perfect lenses

2) Focusing systems

3) Negative refraction devices

4) Enhanced transmission through holes

5) Plasmonic devices

6) ENZ devices (tunnelling of guided waves)7) ENZ LW lenses

1) Volumetric Cloaking

2) Under the carpet cloaking

3) TO‐designed lens antennas

4) Field concentrators

5) Illusion devices

ENG ‐ENZ‐DNG MTM

Perfect focusing

Anomalous refraction

Plasmonic phenomena

TRANSFORMATION OPTICS MTM

Addressing waves in MTM

Transmission‐Line MTM

New RF devices

MTM Technology and Phenomenology

METASURFACES

EBG phenomena

Magnetic surfaces,

Hard and Soft surfaces

1) Low side‐lobe vertical monopoles

2) Low‐profile dipole antennas

3) Reduction of surface wave coupling

4) Improvement of planar‐antenna efficiency

5) Quasi‐TEM propagation waveguides

6) Miniaturized cavities

7) Reduction of scan blindness effects

8) Partilally reflective antennas

9) Enhancing bandwidth of small antennas

10) Flat Absorbers



Snell law and Fermat principle

If the change of refraction index is smooth, the ray runs along a curved path.

Optical path = ( )B

A

n l dl

BA

The ray run along the trajectory that minimize the optical path (Fermat principle)

1l2l

3l

B

A



An example: The Luneburg lens



The metric coefficients of the transformation are interpreted in terms of the constitutive parameters of the medium.

Transformation Optics

Virtual space (symplest choice: homogeneous, euclidean, cartisia coordinates)

Transformed space(real space )



Jeinc, Jm

inc

Cloaking

0 0sc sc, ,E H ε r μ r

bS

0 0, ,E H

V

aS



Virtual space Trasformed (topological) space

Trasformed (topological) space

Trasformed (topological) space

Material in REAL space

Trasformation Optics process for cloaking



x

z

yH-plane

E-plane

H-plane

H-plane E-plane



' ' : ', ', ' ( ', ', '), ( ', ', '), ( ', ', ')x y z x x y z y x y z z x y z r r

' : , , '( , , ), '( , , ), '( , , )x y z x x y z y x y z z x y z r Γ r

', ', 'x y z

1'Γ Γ

, ,x y z

1' Γ Γ

Transformation of space variable



1'Γ Γ

1' Γ Γ

1 2 3

1 2 3

, , , ,

ˆ ˆ ˆ ˆ ˆ ˆ, , , , ,

x y z x x x

x y z x x x

r x1, x2,x3

' ' , ' 'E r H r ,E r H r

Transformation of space variable

x ', y ', z' x1 ',x2 ',x3 ' x ', y', z ' x1 ', x2 ', x3 '

1 2 3' ', ', 'x x xr

j E Hμ

j H Eε



Virtual space (cartesian Coordinates)

11 2 3 1 1 2 3 2 1 2 3 3 1 2 3' : ', ', ' ( ', ', '), ( ', ', '), ( ', ', ')x x x x x x x x x x x x x x x r Γ r

Co‐variant and contravariant basis set

Co-variant basis

Contra-variant basis

'iix

rg

'iix g

3

12

2

1 2

3

3 1

/

/ ;

/

g

g

g

g g

g g

g gg

g

g

1 2

2 3

3

3

1

21

g

g

g

g

g

g g

g g g

g g

'iix

rg

3 30' 'x x

'iix g

1g2g

3g

2g

1g

3g

2 20' 'x x1 10' 'x x

1' Γ Γ

1'x

2'x

3'x

2ˆ 'x

3ˆ 'x

1ˆ 'x



Co-variant basis

Contra-variant basis

'i

ix

r

γ

'iix γ

1 2

2 3

3 1

3

1

2

;

g

g

g

γ

γ

γ

γ

γ γ γ

γ

γ

1 2

2 3

3

1

1

3

2

/

/

/

g

g

g

γ γ

γ γ γ

γγ

γ

γ

Co‐variant and contravariant basis set

Virtual space (cartesian Coordinates)

1γ2γ

3γ

2γ

1γ

3γ

2 20x x1 10x x

1'Γ Γ

3 30x x1x

2x

3x

2x

3x

1x

1 2 3 1 1 2 3 2 1 2 3 3 1 2 3' : , , '( , , ), '( , , ), '( , , )x x x x x x x x x x x x x x x r Γ r



, 1,3 1,3 1,3

1

, 1,3 1,3 1,3

''ˆ ˆ ˆ ˆ' '

ˆ ˆ ˆ ˆ' '' '

iii j i j j

ji j i j

j jj i i i j

ii j i j

x

x

x

x

rM x x x g γ x

r

rM x x g x x γ

r

ˆ' '

ˆ' '

i i j i jj

ii i j jj

x x x

x x x

g r x

g x

ˆ' ' '

ˆ' ' '

j j i j ii

jj j i ii

x x x

x x x

γ r x

γ x

Jacobian of the transformation

1det

det

1 1[ ]

[ ]g MM

1

1

''

''

rM M

r

rM M

r



10

10

( ) ( ) '( ')

( ) ( ) '( ')

det( ) ' '

det( ) ' '

T

T

E r M r E r

H r M r H r

D r M M E r

B r M M H r

E jB

H jD



11' [ ]det( ) T

M M M

1 1 2 2 3 3ˆ ˆ ˆ' ' ' ' ' ' 'x x x x x x

1 1 2 2 3 3ˆ ˆ ˆx x x x x x

' ' , ' 'E r H r ,E r H r

Curl operator when changing the coordinates

'( '( )) E r A r E r r

'( '( )) H r A r H r r

j E Hμ

j H Eε



j E Hμ

j H Eε

0

0

1

1

'( ') ' ( ')

'( ') ' ( '

'

' )

A E A E

A H A H

r r μ r r

r r ε r r

'( ')A r E r

'( ')A r H r '( ')A r E r

'( ')A r H r

0

0

'

'

' ' ' '

' ' ' '

j

j

H E

E H

r r

r r

0

0

'

'

' ' /

' ' /

j

j

EA r

A r H

r

r

11' [ ]det( ) T

M M M

α

1 1 1

1 1 1

1

1

[ ]

[ ]

det

det

( ) ' ' ' '

( ) ' ' ' '

T

T

A

A

α M M M A E E

α M M M A H H

Invariant form of Mxw’s equation‐1



1 1 11[ ] ( ') ( ')det( ) ' ' ' 'T

A α M M M A E r E r

;

Is satisfy if and only if

Invariant form of Mxw’s equation‐2

1 1 1det( )

A α M M I

0 0

1 111

det( ) T

α μ ε M M M

TA M

1[ ]T M A I



'iix

rg

0' 'z z

iix g

1g2g

3g

2g

1g

3g

0' 'y y0' 'x x

Field and Inductions: tensoral form

1' Γ Γ

' ' , ' ' , ' ' , ' 'E r H r D r B r , , ,E r H r D r B r

10

10

( ) ( ) '( ')

( ) ( ) '( ')

det( ) ' '

det( ) ' '

T

T

E r M r E r

H r M r H r

D r M M E r

B r M M H r

0 0

11det

1 1 ;

[ ] T

μ ε α

α M M M

' '

' '

' '

' '

E r

H r

D r

B r

1'x

2'x

3'x

2ˆ 'x

3ˆ 'x

1ˆ 'x

1det [ '] ' 'T M M M



'iix

rg

0' 'z z

iix g

1g2g

3g

2g

1g

3g

0' 'y y0' 'x x

Field and Inductions: tensoral form

1' Γ Γ


10

10

( ) ( ) '( ')

( ) ( ) '( ')

det( ) ' '

det( ) ' '

T

T

E r M r E r

H r M r H r

D r M M E r

B r M M H r

0 0

11det

1 1 ;

[ ] T

μ ε α

α M M M

' '

' '

' '

' '

E r

H r

D r

B r

1'x

2'x

3'x

2ˆ 'x

3ˆ 'x

1ˆ 'x

1det [ '] ' 'T M M M



1'x

2'x

3'x

2ˆ 'x

3ˆ 'x

1ˆ 'x

'iix

rg

0' 'z z

iix g

1g2g

3g

2g

1g

3g

0' 'y y0' 'x x

1' Γ Γ


1,3

1,3

1,3

1,3

ˆ( ) ' '( ')

ˆ( ) ' '( ')

1ˆ ' ' '

1ˆ ' ' '

ii

i

ii

i

i ii

i ii

g

g

E r g x E r

H r g x H r

D r g x D r

B r g x D r

Field and Inductions: covariant and contravariant form

' '

' '

' '

' '

E r

H r

D r

B r

0 0

, 1,3

1,3 1,3

det

1 1 ;

1ˆ ˆ

1[ ]

i ji j

i j

i ji j

i j

g

g

μ ε α

α x x γ γ

g g γ γ

M



E

H

k

3

3

'( )1

'( )2

33

( ) '

( ) '

( ) '

jkx

jkx

E e

H e

kx k

r

r

E r g

H r g

k r g

3

3

'1

'2

23

ˆ( ) ' '

ˆ( ) ' '

1ˆ' '

2

jkx

jkx

E e

H e

E

E r x

H r x

S r x

Plane‐wave in the transformed space

3 '( )1 0

32 0

2 2

31 2

1'

'( )1'

' '

2 2( )

jkxE eg

jkxH e

g

E E

g

rD r g

rB r g

S r g g g

E’

H’

k’

D’

S’

B’2ˆ 'x

3ˆ 'x

1ˆ 'xD

B

S



, μ ε , μ ε

METAMATERIALArtificial material realized by small periodically displaced particles, thatexhibits anomalous EM properties(properties that do not exhist in nature)

Homogenization of Metamaterials



Microwave metamaterials

Metamaterials at microwave frequencies

MetasurfaceMetamaterials

Characterized by surface boundary Conditions (impedance tensor)

Characterized by Constitutive relationship (constitutive tensors)

(*) E. Martini, S. Maci, “Metasurface Transformation Theory,” in Transformation Electromagnetics and Metamaterials: Fundamental Principles and Applications, eds. Douglas. H. Werner e Do‐Hoon Kwon, Springer, 2013



addressing Surface or Guided waves

( )sjX ρ

metasurfacereactance(lossless)

“Metasurfing”

1ρ

2ρ

tk

Possible transition to Leaky‐Waves

1ρ

2ρ

tk

Variable Metasurface

Variable Metasurfaces



How to realize a variable metasurface?

1‐Small sizes of the elements in terms of the wavelength (pixel‐like approach) 2‐Gradual variation of a geometry (local periodicity)

Rectangular patches with variablesizes

Circular patches with variable sizes

isotropic anisotropic

ˆ( , ) ( , , ) ( , )t t S t t tZ E k k z H k ˆ( , ) ( , ) ( , )st t t t tZ E k k z H k

Zs (kt,) Zs

T*

(kt,)( , , ) ( , , )S t S tZ jX k k



Inhomogeneous Metasurfaces‐ local periodicity

Synthesis of the metasurface

Local periodicity.

The texture and the relevant reactance are locally indentified with those of a periodic structure that locally matches the geometry (smooth, gradual variation)

Use of periodic Green’s Function.

The local periodic problem can be analyzed very easy by a periodic MoM. The number of unknown are those of a single cell.



z

x

Inductive impedance support propagation of TM SW with zero cut off frequency

TM Surface waves (SW)

UNIFORM Isotropic Reactance Propagation of SW and Guided modes

kx k (phase velocity less than speed of light)

EzEx

Hy

2

1St

Xk k

Lz

Xk

ˆt s tjX E z H

sjX kt

xk

yk

21 /sk X



40 60 80 100 120 140 160 1802

2.5

3

3.5

4

4.5

5

5.5x 10

9

kd (°)

f (G

Hz)

=0°

=45°

=90°

Patch on a grounded slab (effect of variation of the capacitance)

Patches (capacitive below resonance)

Ground plane

Evanescent SW mode capacitance

k

Almost independent of the wavenumber direction



( , , )S tX x k

Modulated Surface Reactance (MSR)

k

xk



( , , )S tX x k

Modulated Surface Reactance (MSR)

k

xk



Anysotropic elements‐ local periodicity

Design of anisotropic b.c.

40 60 80 100 120 140 160 1802

2.5

3

3.5

4

4.5

5

5.5x 10

9

kd (°)

f (G

Hz)

=0°

=45°

=90°

40 60 80 100 120 140 160 1802

2.5

3

3.5

4

4.5

5

5.5x 10

9

kd (°)

f (G

Hz)

=0°

=45°

=90°k

t

Ximp

Xxx

Xxy

Xyx

Xyy

Xxx

0

0 X yy



'tk

kx

ky

k 1 Xs

/ 2

Xs

tk

yk1e

2e

k 1 X2

/ 2

k 1 X1

/ 2

xk

XS X1e1e1 X2e2e2

tk

yk1e

2e

k 1 X2

/ 2

2

11 /k X

xk

XS X1e1e1 X2e2e2

a

Isofrequency dispersion ellipse in the spectral plane

k 1 X1

/ 2k 1 X

1/ 2



'tk

kx

ky v

g

k t k

t ReS

Yimp

yxx

yxy

yyx

yyy

R()

jCxx

0

0 jCyy

R()

Yimp

0TMY 0

TEY

Y1TEY

1TM

Isofrequency dispersion curves and group velocity

Poynting vector and wavevectorsare alligned only in the principalPlanes (simmetry plane)



S

g1g2g

1

g2

M t xi '

xj

xi '

xj

xi 'i , j1,2

x j xi 'i1,2

gi g j x jj1,2

Mt

1

xj

xi '

xj

xi 'x j

i , j1,2

xi ' gi xi 'i1,2

x jj1,2

g j

'S

1ˆ 'x2ˆ 'x

' ( 1)

Transformation optics for metasurfaces (*)

sjX eq eq

s sjZ X

ˆ' ' '

ˆ' ' '

j j i j ii

jj j i ii

x x x

x x x

γ ρ x

γ x

'1

(*) E. Martini, S. Maci, “Metasurface Transformation Theory,” in Transformation Electromagnetics and Metamaterials: Fundamental Principles and Applications, eds. Douglas. H. Werner e Do‐Hoon Kwon, Springer, 2013

Co‐variant

Contra‐variant



0 0

'x

z 'y

x

z ysjX sjXS

S'S

't eq eq

s sjZ X

'ρ ρ

0 0

Note: we want free‐space here

Metasurface Transformation Optics

Objective: finding the surface tensor able to support the SW field associated with this potential

OrdinaryverticalDebyepotential

k t 'k t ' k 1 XS 2for TM-SW

Dispersion equationimposed by b.c.

2' '' 'ˆ ˆ' ', ' t tt

z kjz TMz A I e e k kk ρA ρ z z�=

Arising by thetransformation

' '( ) ( ) ˆ, t zj zTMz I e e k ρ ρ ρA ρ z



'tT

tt kM k

0 0 0 0

'x

z 'y

x

z ysjX

sjX

S

S

'S

eq eq

s sjZ X

'ρ ρ

'tktk

Condition for fulfilment of the wave equation

2 2

0lim t z t z t t zz

A k A j A

k

t t t t k k k

't

2 '

x 'y ' k 1 XS 2 1



'S 'S

S'tk

tk

'tT

tt kM k

22' ' 1t t Sk X k k 22 1t t Stg k X k α k

Local dispersion equation

11

, 1,2

1 1ˆ ˆ( ) T i j

i jt tti jg g

α ρ M M x x γ γ

ˆ' ' '

ˆ' ' '

j j i j ii

jj j i ii

x x x

x x x

γ ρ x

γ x 1 11 1 2 2 2ˆ ˆ ˆ ˆt

α e e e e

1tΓ

1

g det[M

t]



S

kt

eq eq

s sjZ X

Metasurface Transformation Optics Theory:

Sr

Co‐variant

Contra‐variant

Co‐variant

g1 v

gS

r

Contra‐variant

g1 kt



'S

'S

S

'tk

'xk

'yk

2

2 1 /sk k X

ˆ 'y

ˆ 'x

'tktk vg Sr

tk

xk

yk1e

2e

y

x

2

2

2

11

Sk k X

g

2

1

1

11

Sk k X

g

'tT

tt kM k

22' ' 1t t Sk X k k 22 1t t Stg k X k α k

Local dispersion equation

11

, 1,2

1 1ˆ ˆ( ) T i j

i jt tti jg g

α ρ M M x x γ γ

1 11 1 2 2 2ˆ ˆ ˆ ˆt

α e e e e



S

tk

Matching the local dispersion equation with the one of an anysotropic impedance

tk

xk

yk1e

2e

2

21 /k X 2

11 /k X

y

x

1 1 1 2 2 2ˆ ˆ ˆ ˆeq

SX X X e e e e

tk

tk

xk

yk1e

2e

y

x

2

2

2

11

Sk k X

g

2

1

1

11

Sk k X

g

21

1 1

n S

n

X X

g

1 11 1 2 2 2ˆ ˆ ˆ ˆt

α e e e e



%

%

%

%

%

1 2 1 2

2 11 2

1 2

1 2

22

, 14

X X X XX XX X

X XX X

f

Maximum percent error for ellipse‐approximation

1 1 1 2 2 2ˆ ˆ ˆ ˆeq

SX X X e e e e

tk

tk

1e

2e

y

x



'S

S

Condition of the transformation

22 2

max

max2

11

11

t t S

S

k X kg

g

X

k k

Non radiating condition

2 2

0lim 0t z t zz

A k A

t t t t k k k

2 '

x ' y ' sin 'k 1 XS 2 1 Condition for fulfilment of

wave equation



Conformal transformation



Conformal and quasi‐conformal transformation

1 2 ' ' 0t tx y γ γ

2 1't t S

t

kX

x

k k

Cauchy Rieman Condition

21( )

'

Seq

t

Xn

x

ρisotropic medium of equivalent refractive index

It is locally supported by the isotropic surface reactance

22

1 /1

'

eq S

t

X X

x

211

' ' sineq S

t t

X X

x y

't x

't y

Quasi orthogonaltransformation

2

1

1 2, eqs n ds ρ

ρ

ρ ρ ρ

Fermat Principle

1ρ

2ρ

tk



Intel Xeon X5667 @ 3.07GHz, X64, 96 GB RAM

Parallel code (8 threads)

Multiscale analysis

• Antenna diameter: 54 cm• Period of impedance modulation: 2cm• Patches average diameters: 4 mm• Slots: 0,4 mm

Isoflux antenna for space(ESA Project) 8.5 GHz



3) Microscale(Slot widths and rwg basis function): /100

Computational Models:

• Use of sheet impedance MoM for macroscale• Use of local periodic MoM for implementing the impedance• For microscale: analytical functions or aggregation of RWG in MoM analysis (in

addition to all acceleration techniques like adaptive Integral methods)

Multiscale problem

1) Macroscale: Antenna dimension (15

2) HomogeneizedImpedance

Variation:

3) Patch scale: /10



2

2nR

radial coordinate

lens radius

Luneburg law for the refractive index

Design of a Luneburg lens

Sheet impedance b.c.Ground plane

Z0TM

jXs()

Z1TM

jXs()

MoM with impedance surface

0ˆ ˆ ˆ ˆinceq s eqn n E Z n n J Z J



Sheet impedance b.c. (set the integral equation)

impedance sheet boundary conditions for a Lunenburg Lens

FEKO‐ step‐wiseboundary conditions

Ground plane



Implementation of the impedance by circular patches (pixlels)

Use of the local periodicty to implementthe impedance in terms of patches



2 1/2( ) [2 ( ) ]eqn R ρ

Printed patch Lunenberg LensFull wave numerical Simulation



Solving integral equation with impedance sheet boundary conditions for a Maxwell’s fish eye

FEKO‐ step‐wiseboundary conditionsOur code



Maxwell’s fish-eye lens: MoM simulation

2 1/2( ) 2[1 ( ) ]eqn R ρ

Full wave MoM numerical simulation



Sorgenti eccitate in controfase, z=5mm

Distance from the sources: 1.3cm Lunghezza d’onda: 4 cmMaxwell’s fish-eye lens: MoM simulation



42 14

0

1 1 1 5( ) , ; ;

2 4 41

z

f z dw z F zw

SCHWARZ‐CHRISTOFFEL





42 14

0

1 1 1 5( ) , ; ;

2 4 41

z

f z dw z F zw



-2 0 2-5

-4

-3

-2

-1

0

1

2

3

4

5

-2 0 2-5

-4

-3

-2

-1

0

1

2

3

4

5

X’

X

Y’

Y

Real SpaceVirtual Space

x' x

y' 0.4x y

Beam shifter



-20 0 20-20

-10

0

10

20

x (centimeters)

y (c

enti

met

ers)

Real(Ez)

-0.5

0

0.5

εr=14hd=1.55mm

w

aLp

-80 -60 -40 -20 0 20 40 60 80

-60

-40

-20

0

20

40

60

Kx*a (degrees)

Ky*

a (d

egre

es)

Beam shifter



-b -a a b

-b

-a

a

b

x

y

-b -a a bx

0

20

40

60

80

100

Planar Cloak

Percent error in reconstractingthe ellipses



4.4 4.5 4.6 4.7 4.8

1.5

2

2.5

3

3.5

4

Dp (mm)

G (

mm

)

kmin

200

300

400

500

4.4 4.5 4.6 4.7 4.8

1.5

2

2.5

3

3.5

4

Dp (mm)

G (

mm

)

kmax

200

300

400

500

Dispersion maps

4.4 4.5 4.6 4.7 4.8

1.5

2

2.5

3

3.5

4

Dp (mm)

G (

mm

)

1

1.2

1.4

1.6

1.8

2

2.2

kmax/kmin

w=Dp/4

a=5mm

f=7.5GHz

a Dp

w

G

h=1.575mm

r=9.8



PLANAR CLOACK



• We have presented a theory for transforming metasurfaces which extends TO to control the wavefront (rays) of a SW propagating along modulated metasurfaces

• The transformation is chosen in order to generate a desired curved wavefront.

• The impedance tensor becomes a scalar (isotropic) impedance in the case of nearly‐orthogonal transformation

• In conventional TO, the output are the metamaterial constituent parameters, herethe outcome are the components of an anysotropic modulated impedance tensor.

• The process suggested here is not exact; however we have individuated the constraints on the transformation for having a good approximation and for avoiding radiation losses.

Conclusions

• Planar cloak implementation is affected by the insufficient dynamic range of impedance, which renders difficult the practical implementation

MetasurfaceTransformation Optics · 2014. 1. 29. · Laboratorio Elettromagnetismo Applicato...

Documents

Transcript of MetasurfaceTransformation Optics · 2014. 1. 29. · Laboratorio Elettromagnetismo Applicato...