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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
An example: The Luneburg lens
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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 )
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
Jeinc, Jm
inc
Cloaking
0 0sc sc, ,E H ε r μ r
bS
0 0, ,E H
V
aS
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
Virtual space Trasformed (topological) space
Trasformed (topological) space
Trasformed (topological) space
Material in REAL space
Trasformation Optics process for cloaking
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
x
z
yH-plane
E-plane
H-plane
H-plane E-plane
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
' ' : ', ', ' ( ', ', '), ( ', ', '), ( ', ', ')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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
, 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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
'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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
'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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
' ' , ' ' , ' ' , ' 'E r H r D r B r , , ,E r H r D r B r
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
, 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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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.
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
( , , )S tX x k
Modulated Surface Reactance (MSR)
k
xk
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
( , , )S tX x k
Modulated Surface Reactance (MSR)
k
xk
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
'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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
'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)
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
'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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
'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]
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
'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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
%
%
%
%
%
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
'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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
Conformal transformation
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
Sheet impedance b.c. (set the integral equation)
impedance sheet boundary conditions for a Lunenburg Lens
FEKO‐ step‐wiseboundary conditions
Ground plane
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
Implementation of the impedance by circular patches (pixlels)
Use of the local periodicty to implementthe impedance in terms of patches
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
2 1/2( ) [2 ( ) ]eqn R ρ
Printed patch Lunenberg LensFull wave numerical Simulation
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
Solving integral equation with impedance sheet boundary conditions for a Maxwell’s fish eye
FEKO‐ step‐wiseboundary conditionsOur code
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
Maxwell’s fish-eye lens: MoM simulation
2 1/2( ) 2[1 ( ) ]eqn R ρ
Full wave MoM numerical simulation
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
Sorgenti eccitate in controfase, z=5mm
Distance from the sources: 1.3cm Lunghezza d’onda: 4 cmMaxwell’s fish-eye lens: MoM simulation
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
42 14
0
1 1 1 5( ) , ; ;
2 4 41
z
f z dw z F zw
SCHWARZ‐CHRISTOFFEL
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
42 14
0
1 1 1 5( ) , ; ;
2 4 41
z
f z dw z F zw
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
-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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
-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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
-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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
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
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
PLANAR CLOACK
Laboratorio Elettromagnetismo Applicato
Stefano Maci, Antenna Mini Symposium, Tel Aviv University, November 20, 2013
• 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