Transcript Metamaterials as Effective Medium

Metamaterials as Effective Medium
Negative refraction and super-resolution
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


0
 ll
0
(1  f ) m  (1  f ) d
(1  f ) m  (1  f ) d
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
Limits of hyperbolic 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 kx 2 
 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 for stratified medium – transfer matrix
Z
Unit Cell
m
...
Am
d
Cm
Bm
Dm
m
A  A 
M cell  n 1    n 
 Bn 1   Bn 
Am1
Bm1
X
X=nD
X=nD+d
X=(n+1)D
2
2


  diel
km2
i  m2 kdiel
U  M (1,1)  e
cos
k
d

sin
k
d





diel d
diel d 
2  diel  m kdiel km


2 2
2
2

 ikm d m  i  m kdiel   diel k m
V  M (1, 2)  e
sin  kdiel d d  

 2  diel  m kdiel km

ikm d m
2
2
 i  m2 kdiel

  diel
km2
W  M (2,1)  e
sin
k
d



diel d 
 2  diel  m kdiel km

2
2

  diel
km2
i  m2 kdiel
 ikm d m 
X  M (2, 2)  e
cos
k
d

sin
k
d





diel d
diel d 
2  diel  m kdiel km


km  k02 m  k z2 , kdiel  k02 diel  k z2
ikm d m
2
2


  diel
km2
1
1  m2 kdiel
K x  arccos  cos  kdiel d d  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  k z2 , kdiel  k02 diel  k z2

i

A0ikm  eikm x  B0ikm  e  ikm x

 metal


i
E ( x)  
C0ikd  eikd ( x  dmetal )  D0ik d  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
X=nD+d

metal

metal
H (x  d
E( x  d
 eikm dm

ik d
 ikm  e m m
 
metal

 1

 ikd

 diel
)  H (x  d
)  E( x  d
X=(n+1)D

metal

metal
eikm dm

ikm  e ikm dm
 metal
)
)
A0  eikmdm  B0  eikmdm  C0  D0

1
 metal

 1
  A0  
  B    ikd
  0    diel

1
1   eikm dm
 
ikd   ikm  eikm dm

 diel    metal
eikm dm

ikm  eikm dm
 metal
 A ik
0
m

 eikmdm  B0ikm  eikmdm 

  C0 
ikd   

D
 diel   0 
1

  A0   C0 
 B    D 
 0   0 

1
 diel
C0ikd  D0ikd 
Exact solution – transfer matrix
Z
Unit Cell
m
Am
...
m
d
Am1
Cm
Bm
(2) Boundary conditions
Bm1
Dm
X
X=nD
X=nD+d

X=(n+1)D

H (x  D )  H (x  D )
E ( x  D )  E ( x  D )
 eikd ddiel

ik d
 ikd  e d diel
 
diel

 1

 ikm

 metal
e  ikd ddiel

ikd  e  ikd ddiel
 diel

1
 diel
1
M cell
C ik
0

 1
  C0  
  D    ikm
  0    metal

1   eikd ddiel
 
ikm   ikd  eikd ddiel

 metal    diel
A  A 
M cell  0    1 
 B0   B1 
C0  eikd ddiel  D0  eikd ddiel  A1  B1
 1
  ikm

 metal
1
 metal
 Aik
1 m  B1ikm 
1 
  A1 
ikm   

B
 metal   1 
e  ikd ddiel


ikd ddiel
 D0ikd  eikd ddiel 
d e
ikd  e  ikd ddiel
 diel

  C0   A1 
 D    B 
 0   1 

1
1   eikd ddiel
 
ikm   ikd  eikd ddiel

 metal    diel
eikd ddiel

ikd  e ikd ddiel
 diel
 1

  ikd
   diel

1
1   eikm dm
 
ikd   ikm  eikm dm

 diel    metal
eikm dm

ikm  e ikm dm
 metal





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 d d  cos  km d m  
sin  kdiel d d  sin  km d m  
D
2  diel  m kdiel km


Effective medium vs. periodic multilayer
kx kz  2


 z  x c2
2
2
2
2


  diel
km2
1
1  m2 kdiel
K x  arccos  cos  kdiel d d  cos  km d m  
sin  kdiel d d  sin  km d m  
D
2  diel  m kdiel km


 m  2.4
 d  3.2
  365nm
  30nm
6
5
kx
k0
4
3
5
 m  9
d  1
  500nm
kz
k0
4
3
2
2
1
0
kz / k0
0
2
4
6
1
kx / k0
2
4
6
8
Effective medium vs. periodic multilayer
kx kz  2


 z  x c2
2
2
2
2


  diel
km2
1
1  m2 kdiel
K x  arccos  cos  kdiel d d  cos  km d m  
sin  kdiel d d  sin  km d m  
D
2  diel  m kdiel km


  365nm
  70nm
 m  2.4
 d  3.2
2.5
3
2
2.5
  500nm
 m  9
d  1
2
1.5
1.5
1
1
0.5
0.5
0
0
0.5
1
1.5
2
2.5
0.5
1
1.5
2
2.5
3
3.5
Surface Plasmons coupling in M-D-M
• symmetric and anti-symmetric modes
• anti-symmetric mode cutoff
• single mode “waveguide”
• deep sub-
Metal
Metal
Metal
Metal
Metal
Metal
H-field
11
Surface Plasmons coupling in M-D-M
• “gap plasmon” mode
• deep sub- “waveguide”
• symmetric and anti-symmetric modes
•No cut-off through the metal
Metal
Symmetric: k<ksingle-wg
Metal
Antisymmetric: k>ksingle-wg
12
Plasmonic waveguide coupling
No counterpart in dielectrics!
Metal
Metal
High contrastDielectric WG – E-field confinement
Diffraction
limit
Michal Lipson, OL (2004)
nlow=1
nhigh=3.5
13
High spatial frequencies with low loss?
• No limitations on the proximity
• deep sub- “waveguide”
• symmetric and anti-symmetric modes
Metal
Metal
Multi-layer plamonic metamaterial
14
Modes in M-D multilayer – beyond EMA
• Sub-WL scale layers
• Strong variation
in the dielectric function
(sign and magnitude)
• Paraxial approximation Is not valid!
Maxwell Equations Time-harmonic solution
Maxwell Equations
  1  E
H 
c t
 
1 H
E  
c t
 

  H  i E
c
 

  E  i H
c


E
 iE
t


H
 iH
t
15
Linear modes in M-D multilayer
TM mode
 iˆ

 x

E
 x
 iˆ

x

0

ˆj
y
0
ˆj
y
Hy

H  Hyˆ

E  Ex xˆ  Ez zˆ

k0 
c
kˆ 
 z    z E x   x E z  ˆj  ik0 H

Ez 

kˆ 
 z     z H y iˆ   x H y x kˆ  ik0 Exiˆ  Ez kˆ

0


i H
k0 x
i H
Ex 
k0 z
Ez  
E x E z

 ik 0 H
z
x
 



H
 ik0E x
z
Ex
i   1 H 


  ik0 H
z
k0 x   x 
16
Modes in M-D multilayer
2


0

k
0
H



 H 

i
       1   2
 

0  Ex 
  k0
z  Ex 
k0  
 x   x 

Looking for SPATIAL eigenmodes (not varying with propagation)
~
H ( x, z)  H ( x)eiz 
 i
z
~
iz
Ex ( x, z)  E ( x)e
~
~
H 
H 
Mˆ  ~     ~ 
E 
E 
 x
 x

0
k02 

1
Mˆ     1   2
0 
  k0
k0  
 x   x 

An “eigenvalue” problem
First-order equation for the vector
~
H 
  ~ 
E 
 x
17
Plasmonic Bloch modes
Kx=p/D
Kx=0
1
1
1
Magnetic
Tangential
Electric
Magnetic
Tangential
Electric
-1
0.97
-1

k0
kx / k0
18
Spatial frequency limited by periodicity large K available even far from the resonance
Plasmonic Bloch modes
Kx=p/D
Kx=0
1
1
1
Magnetic
Tangential
Electric
Magnetic
Tangential
Electric
-1
0.97
-1
Symmetric in dielectric
Symmetric in metal
ETransverse
i H

k0 z
Etan gential
i H

k0 x
Symmetric in dielectric
Antisymmetric in metal
Same symmetry as H
• Symmetry opposite to H
19
-6000
-4000
-2000
0
2000
4000
6000
Anomalous Diffraction and Refraction
Normal diffraction
Negative refraction without
actual negative index
Diffraction
800
600
400
200
0
-200
-400
-600
3400
3500
3600
3700
3800
3900
4000
Anomalous diffraction
d 2y
d 2kz
 2
2 ↔ D
dk x
dx
Direction of energy
  
S  E  H vg
20
2D analog: Metal Nanowires array
Podolskiy, APL 89, 261102 2006
 Show anomalous properties in all directions
 Broad-band response
 Large-scale manufacturing
Averaged dielectric response
d,r<<
Kp/d
kx kz 

 2
z x c
2
2
2
 // ( z )  p m  (1  p) d
(1  p) m  (1  p) d
  ( x   y )   d
(1  p) m  (1  p) d
Hyperbolic dispersion!
21
Metal-dielectric multilayers – dispersion curve
Single
band
22
Periodic metal-dielectric composites
Dispersion relation
 At longer wavelengths metal permittivity grows (negatively)
Less E-field in the metal
Less loss
Less coupling (tunneling)
Less diffraction
Dkz decreases
 Resolution limited by the period
kx/k0 increases
Short 
• strong coupling
• Large wavenumber
• Broad range
Longer 
• weak coupling
• moderate wavenumber
23
• Large Bloch k-vectors
• lower loss
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 near-field to near-field
Solution: curve the space
The Hyperlens
X q
Z r
dm dd
• Metal-dielectric sub-wavelength layers
• No diffraction in Cartesian space
• object dimension at input a
• Dq is constant Dq  a
r
•Arc at output
kr 2
q

kq 2
r
q   ll  0
R
A  R Dq  a
r
Magnification ratio determines the resolution limit.
 k02
Maxwell’s equations in cylindrical coordinates
H , E , E 
TM solution
z
  H  i
 E  i


c
c


E 
H 
Isotropic case



 
0

0
1



 
E

 E
1
      z  
z

1 

 

0 
H z     H z    ik0 E   E 
  

 

Hz

z
1 
1 
0
 E  
E  ik0 H z

 
  
0
1 
1 
 E  
E  ik0 H z

 
  
i 
H
 2  E 
 k0  z
1
 3
E 
i 
H
 k0  z
 1   i 

1   i 

H

H



  ik0 H z
    k0  z      k0  z 
 H
  
 
H z   2 z  k02 2 H z
  
 
Maxwell’s equations in cylindrical coordinates
Separation of variables:
   
H z  R      

  

R      R    2     k02 2 R      

  


   dR        ''
 k02 2


R     
d     
    0eim       H z   m2  k02 2 H z  0




Solution given by Bessel functions
1. Bessel Function of the First kind:

J m k0  
2. Bessel Function of the Second kind:


[cosine functions in Cartesian coordinates]
Ym k0  

[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:

H   k
2. Bessel Function of the Second kind:
3. Hankel Function of the First kind:

J m k0  
4. Hankel Function of the Second kind:

[cosine functions in Cartesian coordinates]

 
Ym k0  
1
m
0

H m 2 k0
5. Modified Bessel Function of the First kind:

[sine functions in Cartesian coordinates]
 1 ikr 
 e  ].
r

1

  [converging cylindrical wave  e ikr  ]
r

[expanding cylindrical wave

I m k0  
6. Modified Bessel Function of the Second kind: w K m
k
0

x
[ equivalent to e ]


[ equivalent to
e x ]
Optical hyperlens view by angular momentum
• Span plane waves in angular momentum base (Bessel func.)
e
ikx


m
imq
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