Download Metamaterials as Effective Medium

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
When two or more constituents are mixed at sub-wavelength dimensions
Effective properties can be applied

New type of artificial dielectrics

Negative refraction in non-magnetic metamaterials

Super-resolution imaging
dm dd
  xx


  0 0
 0

0
 yy
0
0

0
 zz 
Pendry’s artificial plasma

Motivation: metallic behavior at GHz frequencies

Problem: the dielectric response is negatively (close to) infinite

Solution: “dilute” the metal
ne e 2
 
 0m
2
p
The electrons density is reduced
nneff  ne
 p2,eff 
r 2
a2
neff e 2
 0 meff
* The effective electron mass is increased due to self inductance
Lowering the plasma frequency, Pendry, PRL,76, 4773 (1996)
Simple analysis of 1D and 2D systems
a  

Periodicity or inclusions much smaller than wavelength

2+1D or 1+2D (dimensions of variations)

Effective dielectric response determined by filling fraction f
1D-periodic (stratified)
2D-periodic (nano-wire aray)
3D?
a

Averaging over the (fast) changing dielectric response
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  
1  2
   ll
Note: parallel=ordinary


For isotropic constituents
effective fields
Di   i Ei
Eeff  Eave  fE1  (1  f ) E2
a
 ll
 ll

Deff  Dave  fD1  (1  f ) D2
Stratified metal-dielectric metamaterial: Parallel polarization
 ll
E
k
a
Boundary conditions
E1  E2  E
Eeff  Eave  fE  (1  f ) E  E
Deff  Dave  fD1  (1  f ) D2  f 1 E  (1  f ) 2 E   eff E
 ll  f1  (1  f ) 2
 ll

Stratified metal-dielectric metamaterial: Normal polarization
 ll
E
 ll

a
D1  D2  D
Eeff  Eave  fE1  (1  f ) E2
Deff  Dave  fD  (1  f ) D  D
Eeff  f
D
1
 (1  f )
D
2

D
 eff
1


f
1

(1  f )
2
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  
 ll


1  2
   ll
Note: parallel=extraordinary
Nanowire metamaterial: Parallel polarization
 ll
E


E1  E2  E
Eeff  Eave  fE  (1  f ) E  E
Deff  Dave  fD1  (1  f ) D2  f 1 E  (1  f ) 2 E   eff E
 ll  f1  (1  f ) 2
Nanowire metamaterial: Normal polarization polarization
 ll


E
• More complicated derivation
• Homogenization (not simple averaging)
• Assume small inclusions (<20%)
• Maxwell-Garnett Theory (MGT)
(metal nanowires in dielectric host)
  ( x   y )   d
(1  f ) m  (1  f ) d
(1  f ) m  (1  f ) d
Strongly anisotropic dielectric Metamaterial
  0 0 



  0 0  0 
0 0  
ll 

 ll


 ll  f m  (1  f ) d
  ( x   y )   d
  ll


  0 0
0

(1  f ) m  (1  f ) d
(1  f ) m  (1  f ) d
0
 ll
0
0

0
  
 ll  f m  (1  f ) d
1


f
1

 ll
 ll

(1  f )
2
For most visible and IR wavelengths  m   d
  ll  0,    0
Example: nanowire medium medium
60nm nanowire diameter
Ag wires
110nm center-center wire distance
4
Al2O3
matrix
2
0
 //
-2
-4
-6
Effective permittivity
from MG theory
 // ( z )  p m  (1  p) d
(1  p) m  (1  p) d
  ( x   y )   d
(1  p) m  (1  p) d
-8
-10
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
um
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
um
30
20

10
0
-10
-20
0.3
Broad band
Wave propagation in anisotropic medium
  xx


  0 0
 0

0
 yy
0
0

0
 zz 
Uniaxial  xx   yy
Maxwell equations for time-harmonic waves
 

k  H  D
 

k  E  0 H
 
D   E   0 ( xx E x xˆ   yy E y yˆ   zz E x zˆ )
  
 
k  k  E  0 k  H  k02 ( xx Ex xˆ   yy E y yˆ   zz Ex zˆ)
  xx k02  (k y2  k z2 )
 E x 
kxk y
kxkz

 
2
2
2
kxk y
 yy k0  (k x  k z )
k ykz

 E y   0

2
2
2 

k
k
k
k

k

(
k

k
)
E
x
z
y
z
zz
0
x
y
z




Det(M)=0,  xx   yy
 k x2  k y2 k z2 
k  k 
 0
 
 x 
z


2
2
x 0

Wave propagation in anisotropic medium
 x


  0 0
0

0
x
0
0

0
 z 
2
2


k

k
k z2
y
2
2
2
2  x
2
k x  k y  k z   x k0
  k0  0
 

x
z




Extraordinary waves (TM)
Ordinary waves (TE)

E
H
• 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

H
E
Extraordinary waves in anisotropic medium
 x


  0 0
0

0

0
 z 
0
x
0
kz
isotropic medium
x  z
1
kx
2
z

kz
kx
2
x
k
2
0
k x  k z  k02
2
2
1.5
anisotropic medium
For x<0
kx
2
z

kz
kz
x  z
n  n( )
‘Hyperbolic’ medium
kz
2
x
 k02
kx
kx
Energy flow in anisotropic medium
isotropic medium
kz
k x  k z  k02
2
normal to the k-surface
2
1
kx
1.5
x  z
‘Indefinite’ medium
anisotropic medium
kz
x  z
kx
2
z

kz
kz
2
x
 k02
kx


S and k are not parallel

S Is normal to the curve!
* Complete proof in “Waves and Fields in Optoelectronics” by Hermann Haus
kx
Refraction in anisotropic medium
kz
What is refraction?
 x

  0 0
0


0
x
0
0

0
 z 
kx
2


kz

2



2
c2
1
kx
1.5
Conservation of tangential momentum
 x  0,  z  0
Sr , z
kr , z H 02

0
 x 2 0
Sr , x
kr , x H 02

0
 z 2 0
kz
Hyperbolic
air
Negative refraction!
kx
Refraction in nanowire medium medium
4
Ag wires
2
0
Al2O3
matrix
-2
-4
 //
-6
-8
-10
0.3
Effective permittivity
from MG theory
 // ( z )  p m  (1  p) d
(1  p) m  (1  p) d
  ( x   y )   d
(1  p) m  (1  p) d
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
um
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
um
30
20
10
0

-10
-20
0.3
Broad band
Negative refraction for >630nm
Refraction in layered semiconductor medium
•SiC
•Phonon-polariton resonance at IR
Negative refraction for 9>>12m
 // ( x   y )  p m  (1  p) d
Hyperbolic metamaterial “phase diagram”
kx
2
z
kz

 x  0, z  0
2
x
 k02
 ll  f m  (1  f ) d
1


f
1

 x  0, z  0
(1  f )
 x  0, z  0
2
 x  0, z  0
dielectric
Type I
Type II
Ag/TiO2 multilayer system
Effective medium at different regimes
We choose propogation by
 x  0,   z  0
kz
2
  2 kx2 
0
  x  2 
 z 
c
X=normal
(suitable for Nanowires)
X=parallel
Suitable for stratified medium
propagation
propagation
  m   d
 //  f m  (1  f ) d
(1  f ) m  (1  f ) d
  d
(1  f ) m  (1  f ) d
x
 m   d
• extreme material properties
• Low-loss
• epsilon near-zero
• Broad-band
• Diffraction management
• resolution limited by periodicity
• Resolution limited by loss
x
Conditions Normal-X direction (kx<</D)
X=normal
(suitable for Nanowires)
propagation
f 
x
kz
2
  2 kx2 
0
    2 
 ll 
c
 //   m   d  0
 3  
  d m d  0
2  m  3 d
1
2
kz
  m  3 d
3
  d
22
kz
2
  2 kx  3  2
  d 2
    2 
 2 c
c

ll 

3 
d
2
c

c
• Low loss
• Low diffraction management
• moderate  values
• diffraction management improves with em
• Limited by periodicity
•no near-0 
kx
Conditions for Normal Z-direction
propagation
x
kz
2
  2 kx2 
0
  ll  2 
  
c
 // 
m  d
2
0
1  m d
 
0
2 m  d
k z  0 For large range of kx
kr
m  d
 //  0
  d
kx
• Good diffraction management
• near-zero 
• Limited by ?
Effective medium with loss…
 m   m  i m
propagation
x
m  d
kz
2
  2 kx2 
0
  ll  2 
  
c
 //   m   d  i m
 2  3i m
   d d
2 2 d  i m
Im( kz )  Re( kz )
High loss!
kz
2
  2 kx2 
0
    2 

c

ll 

  m   d
(Long wavelengths)
 //   m   d   m
 3  
  d m d  0
2  m  3 d
3
  d
2
kz
2
  2 kx2 

    2 
 ll 
c
Very low loss at low k
Moderate loss at high k
Limits of indefinite medium for super-resolution

Open curve vs. close curve

No diffraction limit!

No limit at all…
kx
2
z

kz
2
x
 x  0,  z  0
 k02
kr
 2 kx2 
 k
k z   x  k0 


x

z 


Is it physically valid?
• Reason: approximation to homogeneous medium!
• What are the practical limitations?
• Can it be used for super-resolution?
kx
Exact solution – transfer matrix
Z
Unit Cell
m
...
d
Am
Cm
Bm
m
A  A 
M cell  n 1    n 
 Bn 1   Bn 
Am1
Bm1
Dm
X
X=nD
X=nD+d
U  M (1,1)  e
ikm d m
V  M (1, 2)  e
W  M (2,1)  e
2
2


  diel
km2
i  m2 kdiel
cos
k
d

sin
k
d





diel d
diel d 
2  diel  m kdiel km


 ikm d m
ikm d m
X  M (2, 2)  e
X=(n+1)D
2
2
 i  m2 kdiel

  diel
km2
sin  kdiel d d  

 2  diel  m kdiel km

2
2
 i  m2 kdiel

  diel
km2
sin
k
d



diel d 
 2  diel  m kdiel km

 ikm d m
km  k02 m  kz2 , kdiel  k02 diel  kz2
2
2


  diel
km2
i  m2 kdiel
cos
k
d

sin
k
d





diel d
diel d 
2  diel  m kdiel km


2
2


  diel
km2
1
1  m2 kdiel
K x  arccos  cos  kdiel dd  cos  km d m  
sin  kdiel d d  sin  km d m  
D
2  diel  m kdiel km


Exact solution – transfer matrix
Z
Unit Cell
m
...
d
Am
Cm
Bm
Dm
m
Am1
(1) Maxwell’s equation
Bm1
X
X=nD
X=nD+d
X=(n+1)D

Am  eikm ( x mD )  Bm  eikm ( x mD )

H ( x)  Cm  eikd ( x dmetal mD )  Dm  eikd ( x dmetal mD )

ikm ( x  m 1 D )
 Bm1  e km ( x  m1 D )
 Am1  e
mD  x  mD  d metal
mD  d metal  x   m  1 D
 m  1 D  x   m  1 D  d metal
km  k02 m  kz2 , kdiel  k02 diel  kz2

i

A0ikm  eikm x  B0ikm  e ikm x

 metal


i
E ( x)  
C0ikd  eikd ( x  dmetal )  D0ikd  e ikd ( x  dmetal )
  diel

i
A1ikm  eikm ( x  D )  B1ikm  e  km ( x  D )
 
 metal






0  x  d metal

d metal  x  D
D  x  D  d metal
Exact solution – transfer matrix
Z
Unit Cell
m
...
m
d
Am
Am1
Cm
Bm
(2) Boundary conditions
Bm1
Dm
X
X=nD
H (x  d
X=nD+d

metal

metal
)  H (x  d
E( x  d
 eikm dm

ik d
 ikm  e m m
 
metal

 1

 ikd

 diel
)  E(x  d
X=(n+1)D

metal

metal
)
e ikm dm

ikm  e  ikm dm
 metal

)
1
 metal


 1
  A0  
  B    ikd
  0    diel

1
1   eikm dm
 
ikd   ikm  eikmdm

 diel    metal
eikm dm

ikm  eikmdm
 metal
A0  eikm dm  B0  e ikm dm  C0  D0
1
A0ikm  eikm dm  B0ikm  e ikm dm 
 C0ikd   D0ikd 


  C0 
ikd   

D
 diel   0 
1

  A0   C0 
 B    D 
 0   0 

 diel
Exact solution – transfer matrix
Z
Unit Cell
m
...
d
Am
Cm
Bm
m
Am1
(3) Combining with Bloch theorem
Bm1
Dm
X
X=nD
X=nD+d
X=(n+1)D

 Am   Am1 
M
 cell    


 Bm   Bm1 

eiK x D  Am    Am1 
  


B
B
m
m

1





U  eiK x D
det 
 W

 M cell  e

0
X  eiK x D 
V
iK x D
e

 U  eiK x D
 Am 
 0 
 Bm 
 W
iK x D
  Am 
   0
X  eiK x D   Bm 
V
UX
U  X 

 i 1 

2
 2 
2
2
2


  diel
km2
1
1  m2 kdiel
K x  arccos  cos  kdiel dd  cos  km d m  
sin  kdiel d d  sin  km d m  
D
2  diel  m kdiel km


Beyond effective medium: SPP coupling in M-D-M
• “gap plasmon” mode
• deep sub- “waveguide”
• symmetric and anti-symmetric modes
Metal
Symmetric: k<ksingle-wg
Metal
Antisymmetric: k>ksingle-wg
Beyond effective medium: SPP coupling in M-D-M
metal
dielectric
• Abrupt change of the dielectric function
• variations much smaller than the wavelength
• Paraxial approximation not valid!
•Need to start from Maxwell Equations
z
x
• TM nature of SPPs
• Calculate 3 fields 


 1 E

1 H
 H 
,  E  
c t
c t
Eigenmode problem:
• Eigen vectors  EM field
• Eigen values  Propagation constants
~
Ex ( x, z)  Ex ( x)eiz
~
E z ( x , z )  E z ( x ) e  i z
~
H y ( x, z )  H y ( x)eiz
Hamiltonian-like operator:


ˆ
M ( x) ( x)   ( x)

~ ~
  ( H y , E x )T
2


0

k
1
0
ˆ

M ( x)  
ˆ
k 0  H ( x) 0 
 1 
Hˆ  (
)  k02
x  x
Plasmonic Bloch modes
Kx=/D
Kx=0
1
1
1
Magnetic
Tangential
Electric
Magnetic
Tangential
Electric
-1
0.97
-1
Ag=20nm Air=30 nm =1.5m
kz
k0
k x / k0
Metamaterials at low spatial frequencies
The homogeneous medium perspective
D  

k 
D
Averaged dielectric response
Can be <0
 // ( z )  p m  (1  p) d
(1  p) m  (1  p) d
  ( x   y )   d
(1  p) m  (1  p) d
kx kz  2

 2
z x c
2
2
Hyperbolic dispersion!
Metamaterials at low spatial frequencies
The homogeneous medium perspective
D  

k 
D
Averaged dielectric response
Can be <0
 // ( z )  p m  (1  p) d
(1  p) m  (1  p) d
  ( x   y )   d
(1  p) m  (1  p) d
kx kz 

 2
z x c
2
2
3
2.5
2
1.5
1
2
Hyperbolic dispersion!
0.5
0
0.5
1
1.5
2
2.5
3
3.5
Use of anisotropic medium for far-field super resolution
Conventional lens

Superlens can image near- to near-field
Superlens


Need conversion beyond diffraction limit

Multilayers/effective medium?

Can only replicate sub-diffraction image by diffraction suppression
Solution: curve the space
The Hyperlens
X 
Z r
dm dd
• Metal-dielectric sub-wavelength layers
• No diffraction in Cartesian space
• object dimension at input a
• D is constant D  a
r
•Arc at output
kr 2


k 2
r
    ll  0
R
A  RD  a
r
Magnification ratio determines the resolution limit.
 k0 2
Optical hyperlens view by angular momentum
• Span plane waves in angular momentum base (Bessel func.)
e
ikx


m
im
i
J
(
kr
)
e
 m
m  
• 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