metamaterials as effective medium
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Metamaterials as Effective Medium
Negative refraction and super-resolution
Strongly anisotropic dielectric Metamaterial
ll
00
00
00
0
�
dm
dmdyx ff
ff
)1()1(
)1()1()(
dmll ff )1(
ll
llll
00
00
00
0 ll
ll�
21
)1(1
ff
dmll ff )1(
For most visible and IR wavelengths dm 0,0 ll
Limits of hyperbolic medium for super-resolution
Open curve vs. close curve
No diffraction limit!
No limit at all…
Is it physically valid?
kx
kr
20
22
kkk
x
z
z
x
0,0 zx
xk
z
xxz
kkk
220
• Reason: approximation to homogeneous medium!
• What are the practical limitations?
• Can it be used for super-resolution?
Exact solution for stratified medium – transfer matrix
2 2 2 21 1
arccos cos cos sin sin2
m diel diel mx diel d m m diel d m m
diel m diel m
k kK k d k d k d k d
D k k
Z
X
...
Unit Cell
X=nD X=(n+1)D
dm
X=nD+d
mmA
mB
mC
mD
1mA
1mB
2 2 2 20 0,m m z diel diel zk k k k k k
1
1
n ncell
n n
A AM
B B
2 2 2 2
2 2 2 2
2 2 2 2
(1,1) cos sin2
(1,2) sin2
(2,1) sin2
m m
m m
m m
ik d m diel diel mdiel d diel d
diel m diel m
ik d m diel diel mdiel d
diel m diel m
ik d m diel diel md
diel m diel m
k kiU M e k d k d
k k
k kiV M e k d
k k
k kiW M e k
k k
2 2 2 2
(2,2) cos sin2
m m
iel d
ik d m diel diel mdiel d diel d
diel m diel m
d
k kiX M e k d k d
k k
Exact solution – transfer matrixZ
X
...
Unit Cell
X=nD X=(n+1)D
dm
X=nD+d
mmA
mB
mC
mD
1mA
1mB
( ) ( )
( ) ( )
( 1 ) ( 1 )1 1
( ) 1
1 1
m m
d metal d metal
m m
ik x mD ik x mDm m metalik x d mD ik x d mD
m m metal
ik x m D k x m Dm m metal
A e B e mD x mD d
H x C e D e mD d x m D
A e B e m D x m D d
0 0
( ) ( )0 0
( ) ( )1 1
0
( )
m m
d metal d metal
m m
ik x ik xm m metal
metal
ik x d ik x dd d metal
diel
ik x D k x Dm m metal
metal
iA ik e B ik e x d
iE x C ik e D ik e d x D
iAik e B ik e D x D d
(1) Maxwell’s equation
2 2 2 20 0,m m z diel diel zk k k k k k
Exact solution – transfer matrixZ
X
...
Unit Cell
X=nD X=(n+1)D
dm
X=nD+d
mmA
mB
mC
mD
1mA
1mB
0 0 0 0
0 0 0 0
( ) ( )1 1
( ) ( )
m m m m
m m m m
ik d ik d
metal metalik d ik d
m m d dmetal metalmetal diel
A e B e C DH x d H x d
A ik e B ik e C ik D ikE x d E x d
0 0
0 0
1 1m m m m
m m m m
ik d ik d
ik d ik dd dm m
diel dielmetal metal
e eA C
ik ikik e ik eB D
1
0 0
0 0
1 1 m m m m
m m m m
ik d ik d
ik d ik dd d m m
diel diel metal metal
e eA C
ik ik ik e ik eB D
(2) Boundary conditions
Exact solution – transfer matrixZ
X
...
Unit Cell
X=nD X=(n+1)D
dm
X=nD+d
mmA
mB
mC
mD
1mA
1mB
(2) Boundary conditions
0 0 1 1
0 0 1 1
( ) ( )1 1
( ) ( )
d diel d diel
d diel d diel
ik d ik d
ik d ik dd d m m
diel metal
C e D e A BH x D H x D
C ik e D ik e Aik B ikE x D E x D
0 1
0 1
1
1 1
1 1
d diel d diel
d diel d diel
d diel d diel
d diel d
ik d ik d
ik d ik dm md d
metal metaldiel diel
ik d ik d
ik d ik dm m d d
metal metal diel
e eC A
ik ikik e ik eD B
e e
ik ik ik e ik e
0 1
0 1
diel
diel
C A
D B
0 1
0 1cell
A AM
B B
1 11 1 1 1d diel d diel m m m m
d diel d diel m m m m
ik d ik d ik d ik d
ik d ik d ik d ik dcell m m d dd d m m
metal metal diel dieldiel diel metal metal
e e e e
M ik ik ik ikik e ik e ik e ik e
Exact solution – transfer matrixZ
X
...
Unit Cell
X=nD X=(n+1)D
dm
X=nD+d
mmA
mB
mC
mD
1mA
1mB
(3) Combining with Bloch theorem
1
1
1
1
0 0x
x
x
x
m mcell iK D
m m m miK Dcell iK D
m mm miK D
m m
A AM
B B A AU e VM e
B BW X eA Ae
B B
det 0x
x
iK D
iK D
U e V
W X e
2
12 2
xiK D U X U Xe i
2 2 2 21 1
arccos cos cos sin sin2
m diel diel mx diel d m m diel d m m
diel m diel m
k kK k d k d k d k d
D k k
Effective medium vs. periodic multilayer
2
222
c
kk
x
z
z
x
2 2 2 21 1
arccos cos cos sin sin2
m diel diel mx diel d m m diel d m m
diel m diel m
k kK k d k d k d k d
D k k
500nm
2 4 6 8
1
2
3
4
5
6
0 2 4 60
1
2
3
4
5
9
1m
d
365nm
2 4
3 2
.
.m
d
0/ kkx
0k
kz
0/zk k
0
xk
k
30nm
Effective medium vs. periodic multilayer
2
222
c
kk
x
z
z
x
2 2 2 21 1
arccos cos cos sin sin2
m diel diel mx diel d m m diel d m m
diel m diel m
k kK k d k d k d k d
D k k
0.5 1 1.5 2 2.5 3 3.5
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
500nm 9
1m
d
365nm
2 4
3 2
.
.m
d
70nm
11
Surface Plasmons coupling in M-D-M
Metal Metal
• symmetric and anti-symmetric modes• anti-symmetric mode cutoff• single mode “waveguide” • deep sub-
MetalMetal MetalMetal
H-field
12
Surface Plasmons coupling in M-D-M
Metal Metal
Symmetric: k<ksingle-wg Antisymmetric: k>ksingle-wg
• “gap plasmon” mode• deep sub-“waveguide” • symmetric and anti-symmetric modes•No cut-off through the metal
13
Plasmonic waveguide coupling
Metal Metal
No counterpart in dielectrics!
High contrastDielectric WG – E-field confinement
Michal Lipson, OL (2004)
nlow=1nhigh=3.5
Diffractionlimit
14
High spatial frequencies with low loss?
Metal Metal
• No limitations on the proximity• deep sub-“waveguide” • symmetric and anti-symmetric modes
Multi-layer plamonic metamaterial
15
Modes in M-D multilayer – beyond EMA
t
H
cE
t
E
cH
1
1
Maxwell Equations
Eit
E
Hit
H
Time-harmonic solution
• Sub-WL scale layers
• Strong variation in the dielectric function (sign and magnitude)
• Paraxial approximation Is not valid!
Hc
iE
Ec
iH
Maxwell Equations
16
Linear modes in M-D multilayer
TM mode yHH ˆ
zExEE zx ˆˆ
z
H
k
iEx
0
x
H
k
iEz
0
Hikx
E
z
E zx0
ck
0
Hikx
H
xk
i
z
Ex0
0
1
xEikz
H 0
HikjEE
EE
kji
zxxz
zx
zyx 0ˆ
0
ˆˆˆ
kEiEikkHiH
H
kji
zxxyxyz
y
zyxˆˆˆˆ
00
ˆˆˆ
0
17
Modes in M-D multilayer
Looking for SPATIAL eigenmodes (not varying with propagation)
ziexHzxH )(~
),(
zix exEzxE )(
~),(
xx E
H
E
HM ~
~
~
~ˆ
01
01ˆ
20
20
0k
xx
k
kM
An “eigenvalue” problem
First-order equation for the vector
xx E
H
kxx
k
k
iE
H
z 01
0
20
20
0
iz
xE
H~
~
18
Plasmonic Bloch modesKx=/D
MagneticTangentialElectric
-1
1
Kx=
MagneticTangentialElectric
0.97
1
-1
1
Spatial frequency limited by periodicity large K available even far from the resonance
0k
0/ kkx
19
Plasmonic Bloch modesKx=/D
MagneticTangentialElectric
-1
1
Kx=
MagneticTangentialElectric
0.97
1
-1
1
• Symmetry opposite to H
z
H
k
iETransverse
0
x
H
k
iE gential
0tan
Same symmetry as H
Symmetric in metalAntisymmetric in metal
Symmetric in dielectric Symmetric in dielectric
20
Anomalous Diffraction and Refraction
3400
3500
3600
3700
3800
3900
4000
-600
-400
-200 0
200
400
600
800
-6000
-4000
-2000
02000
4000
6000
Anomalous diffraction Normal diffraction
2
2
x
z
k
kD
d
d2
2
xd
d ↔Diffraction Direction of energy
gvHES
Negative refraction without actual negative index
21
dm
dmdyx
dmz
pp
pp
pp
)1()1(
)1()1()(
)1()(//
2D analog: Metal Nanowires array
Podolskiy, APL 89, 261102 2006
Averaged dielectric response
2
222
c
kk
x
z
z
x
Hyperbolic dispersion!
Show anomalous properties in all directions
Broad-band response
Large-scale manufacturing
d,r<<
d
22
Metal-dielectric multilayers – dispersion curve
Singleband
23
At longer wavelengths metal permittivity grows (negatively)Less E-field in the metal
Less lossLess coupling (tunneling)
Less diffractionkz decreases
Resolution limited by the periodkx/k0 increases
Periodic metal-dielectric composites
Dispersion relation
Short • strong coupling• Large wavenumber• Broad range
Longer • weak coupling• moderate wavenumber• Large Bloch k-vectors• lower loss
Use of anisotropic medium for far-field super resolution
Superlens can image near- to near-field
Need conversion beyond diffraction limit Multilayers/effective medium?
Can only replicate sub-diffraction image near-field to near-field
Solution: curve the space
Conventional lens
Superlens
• Metal-dielectric sub-wavelength layers
• No diffraction in Cartesian space
• object dimension at input a
• is constant
•Arc at output
dm dd
The Hyperlens
rZ
X
222
0r
r
kkk
r
a
r
RaRA
0 ll
Magnification ratio determines the resolution limit.
Maxwell’s equations in cylindrical coordinates
, ,zH E E z
0
0
1 10
0 0
1 1 10
0
z z
z
z
z
H i E H H ik E Ec
H
z
E i H E E ik Hc
E E
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
0
0
0
1 11
2
3
z
z
z
E E ik H
iE H
k
iE H
k
00 0
2 202
1 1z z z
zz z
i iH H ik H
k k
HH k H
TM solution Isotropic case
Maxwell’s equations in cylindrical coordinates
zH R
2 202
2 20
''
R R k R
dRk
R d
0ime
Separation of variables:
2 2 20 0z zH m k H
Solution given by Bessel functions
1. Bessel Function of the First kind: 0mJ k [cosine functions in Cartesian coordinates]
2. Bessel Function of the Second kind: 0mY k [sine functions in Cartesian coordinates]
• penetration of high-order modes to the center is diffraction limited
Maxwell’s equations in cylindrical coordinates
1. Bessel Function of the First kind: 0mJ k [cosine functions in Cartesian coordinates]
2. Bessel Function of the Second kind: 0mY k [sine functions in Cartesian coordinates]
3. Hankel Function of the First kind: 1
0mH k [expanding cylindrical wave 1 ikrer
].
4. Hankel Function of the Second kind: 2
0mH k [converging cylindrical wave 1 ikrer
]
5. Modified Bessel Function of the First kind: 0mI k [ equivalent toxe ]
6. Modified Bessel Function of the Second kind: w 0mK k [ equivalent to xe ]
Optical hyperlens view by angular momentum
• Span plane waves in angular momentum base (Bessel func.)
imm
m
mikx ekrJie )(
• resolution detrrmined by mode order
• penetration of high-order modes to the center is diffraction limited
• hyperbolic dispersion lifts the diffraction limit
•Increased overlap with sub-wavelength object
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