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Metamaterials as Effective Medium. Negative refraction and super-resolution. Previously seen in “optical metamaterials”. Sub-wavelength dimensions with SPP Negative index Use of sub-wavelength components to create effective response Super-resolution imaging. d d. d m. - PowerPoint PPT Presentation
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Metamaterials as Effective Medium
Negative refraction and super-resolution
Previously seen in “optical metamaterials”
Sub-wavelength dimensions with SPP
Negative index
Use of sub-wavelength components to create effective response
Super-resolution imaging
Metamaterials as sub-wavelength mixture of different elements
New type of artificial dielectrics
Negative refraction in non-magnetic metamaterials
Super-resolution imaging
zz
yy
xx
00
00
00
0
dm dd
When two or more constituents are mixed at sub-wavelength dimensions
Effective properties can be applied
Pendry’s artificial plasma
Motivation: metallic behavior at GHz frequencies
Problem: the dielectric response is negatively (close to) infinite
Solution: “dilute” the metal
Lowering the plasma frequency, Pendry, PRL,76, 4773 (1996)
2
2
a
rnn eneff
The electrons density is reduced
m
enep
0
22
eff
effeffp m
en
0
22
,
* The effective electron mass is increased due to self inductance
Simple analysis of 1D and 2D systems
Periodicity or inclusions much smaller than wavelength
2+1D or 1+2D (dimensions of variations)
Effective dielectric response determined by filling fraction f
a
a
1D-periodic (stratified) 2D-periodic (nano-wire aray)
Averaging over the (fast) changing dielectric response
3D?
Stratified metal-dielectric metamaterial
Two isotropic constituents with bulk permittivities
Filling fractions f for 1,1-f for 2
2 ordinary and one extra-ordinary axes (uniaxial)
2 effective permittivities
a
a
1
ll
llll
For isotropic constituents
effective fields
iii ED
21 )1( EffEEE aveeff
21 )1( DffDDD aveeff
Note: parallel=ordinary
2
Stratified metal-dielectric metamaterial: Parallel polarization
a
llll
EEE 21
EEffEEE aveeff )1(
EEfEfDffDDD effaveeff 2121 )1()1(
k
E
21 )1( ffll
Boundary conditions
Stratified metal-dielectric metamaterial: Normal polarization
a
llll
DDD 21
21 )1( EffEEE aveeff
DDffDDD aveeff )1(
E
21
)1(1
ff
effeff
DDf
DfE
21
)1(
Nanowire metal-dielectric metamaterial
Two isotropic constituents with bulk permittivities
Filling fractions f for 1,1-f for 2
2 ordinary and one extra-ordinary axes
2 effective permittivities
a
1
ll
Note: parallel=extraordinary
2
ll
Nanowire metamaterial: Parallel polarization
E
ll
EEE 21
EEffEEE aveeff )1(
EEfEfDffDDD effaveeff 2121 )1()1(
21 )1( ffll
Nanowire metamaterial: Normal polarization polarization
E
ll
• More complicated derivation
• Homogenization (not simple averaging)
• Assume small inclusions (<20%)
• Maxwell-Garnett Theory (MGT)
dm
dmdyx ff
ff
)1()1(
)1()1()(
(metal nanowires in dielectric host)
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
dm
dmdyx
dmz
pp
pp
pp
)1()1(
)1()1()(
)1()(//
Effective permittivity from MG theory
Al2O3 matrix
Ag wires
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2-20
-10
0
10
20
30
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2-10
-8
-6
-4
-2
0
2
4
Broad band
um
um
//
Example: nanowire medium medium
60nm nanowire diameter
110nm center-center wire distance
Wave propagation in anisotropic medium
zz
yy
xx
00
00
00
0
Maxwell equations for time-harmonic waves
)ˆˆˆ(0
0
zEyExEED
HEk
DHk
xzzyyyxxx
)ˆˆˆ(200 zEyExEkHkEkk xzzyyyxxx
0
)(
)(
)(
2220
2220
2220
z
y
x
yxzzzyzx
zyzxyyyx
zxyxzyxx
E
E
E
kkkkkkk
kkkkkkk
kkkkkkk
Uniaxial yyxx
Det(M)=0, yyxx 0222
20
2
x
z
z
yxx
kkkkk
Wave propagation in anisotropic medium
z
x
x
00
00
00
0
0 20
22220
222
k
kkkkkkk
x
z
z
yxxzyx
Ordinary waves (TE) Extraordinary waves (TM)
E
H H
E• Electric field along y-direction
• does not depend on angle
• constant response of x
• Electric field in x-z(y-z) plan
• Depend on angle
• combined response of x,z
Extraordinary waves in anisotropic medium
z
x
x
00
00
00
0
20
22
kkk
x
z
z
x
kx
kz
kx
kz
isotropic medium
zx
anisotropic medium
zx
20
22 kkk zx
)(nn
kx
kz
‘Hyperbolic’ medium
For x<0
20
22
kkk
x
z
z
x
Energy flow in anisotropic medium
kx
kz
kx
kz
isotropic medium
zx
anisotropic medium
zx
20
22 kkk zx
kx
kz
normal to the k-surface
20
22
kkk
x
z
z
x
S
and are not parallelk
‘Indefinite’ medium
* Complete proof in “Waves and Fields in Optoelectronics” by Hermann Haus
S
and are not parallelk
S
Is normal to the curve!
Refraction in anisotropic medium
z
x
x
00
00
00
0
2
222
c
kk zx
What is refraction?
kx
kz
kx
kz
Hyperbolicair
0,0 zx
02 0
20,
, Hk
Sx
zrzr
02 0
20,
, Hk
Sz
xrxr
Conservation of tangential momentum
Negative refraction!
dm
dmdyx
dmz
pp
pp
pp
)1()1(
)1()1()(
)1()(//
Effective permittivity from MG theory
Al2O3 matrix
Ag wires
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2-20
-10
0
10
20
30
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2-10
-8
-6
-4
-2
0
2
4
Broad band
um
um
//
Refraction in nanowire medium medium
Negative refraction for >630nm
dmyx pp )1()(//
Refraction in layered semiconductor medium
•SiC
•Phonon-polariton resonance at IR
Negative refraction for 9>>12m
Hyperbolic metamaterial “phase diagram”
20
22
kkk
x
z
z
x
dmll ff )1(
21
)1(1
ff
Ag/TiO2 multilayer system
0 0,x z
0 0,x z
0 0,x z
0 0,x z
dielectric Type I Type II
We choose propogation by
Effective medium at different regimes
dm
dmd
dm
ff
ff
ff
)1()1(
)1()1(
)1(//
x
propagation
• extreme material properties
• epsilon near-zero
• Diffraction management
• Resolution limited by loss
dm
• Low-loss
• Broad-band
• resolution limited by periodicity
x
propagation
dm
X=parallelSuitable for stratified medium
X=normal (suitable for Nanowires)
0,0 zx 02
2
22
z
xxz
k
ck
Conditions Normal-X direction (kx<</D)
x
propagation
dm 3
X=normal (suitable for Nanowires)
02
2
22
ll
xz
k
ck
03
3
2
0//
dm
dmd
dm
2
1f
• Low loss
• moderate values
• Limited by periodicity
kx
kz
2
22
2
22
2
3
c
k
ck d
ll
xz
cd
2
3
c
d2
3
• Low diffraction management
• diffraction management improves with em
•no near-0
Conditions for Normal Z-direction
x
propagation
dm
02
2
22
x
llz
k
ck
02
1
02//
dm
dm
dm
0//
d
kx
kr
0zk For large range of kx • Good diffraction management
• near-zero
• Limited by ?
Effective medium with loss…
x
propagation
dm
02
2
22
x
llz
k
ck
md
mdd
mdm
i
i
i
2
32
2
//
dm
02
2
22
ll
xz
k
ck
03
3
2
//
dm
dmd
mdm
ll
xz
k
ck
2
2
22
d2
3
mmm i
)Re()Im( zz kk
(Long wavelengths)
Very low loss at low kModerate loss at high k
High loss!
End of class
Limits of indefinite 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 – 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
(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
Beyond effective medium: SPP 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
Beyond effective medium: SPP coupling in M-D-M
t
H
cE
t
E
cH
1,
1 • TM nature of SPPs
• Calculate 3 fields
Eigenmode problem:
zixx exEzxE )(
~),(
ziyy exHzxH )(
~),(
zizz exEzxE )(
~),(
20)
1(ˆ k
xxH
Hamiltonian-like operator:
0)(ˆ01
)(ˆ20
0 xH
k
kxM
)()()(ˆ xxxM
T
xy EH )~
,~
(
• Eigen vectors EM field
• Eigen values Propagation
constants
z
x
metal dielectric
• Abrupt change of the dielectric function• variations much smaller than the wavelength• Paraxial approximation not valid! •Need to start from Maxwell Equations
Plasmonic Bloch modes
Kx=/D
MagneticTangentialElectric
-1
1
Kx=
MagneticTangentialElectric
0.97
1
-1
1
0k
kz
0/ kkx
Ag=20nm Air=30 nm =1.5m
Metamaterials at low spatial frequencies
2
222
c
kk
x
z
z
x
The homogeneous medium perspective
Dk
D
dm
dmdyx
dmz
pp
pp
pp
)1()1(
)1()1()(
)1()(//
Averaged dielectric response
Hyperbolic dispersion!
Can be <0
Metamaterials at low spatial frequencies
2
222
c
kk
x
z
z
x
The homogeneous medium perspective
Dk
D
dm
dmdyx
dmz
pp
pp
pp
)1()1(
)1()1()(
)1()(//
Averaged dielectric response
Hyperbolic dispersion!
Can be <0
0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
2.5
3
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 by diffraction suppression
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.
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