Bernd Hüttner CPhys FInstP
DLR Stuttgart
Folie 1
Bernd Hüttner DLR Stuttgart

Overview
1. Short historical background
2. What are metamaterials?
3. Electrodynamics of metamaterials
4. Optical properties of metamaterials
5. Invisibility, cloaking, perfect lens
6. Surface plasmon waves and other waves
7. Faster than light
8. Summary
Folie 2
Bernd Hüttner DLR Stuttgart

Overview
1. Short historical background
2. What are metamaterials?
3. Electrodynamics of metamaterials
4. Optical properties of metamaterials
5. Invisibility, cloaking, perfect lens
6. Surface plamon waves and other waves
7. Faster than light
8. Summary
Folie 3
Bernd Hüttner DLR Stuttgart

A short historical background
V G Veselago , " The electrodynamics of substances with simultaneously negative values of eps and mu ", Usp. Fiz. Nauk 92 , 517-526 (1967)
A Schuster in his book An Introduction to the Theory of Optics
(Edward Arnold, London, 1904).
H Lamb (1904), H C Pocklington (1905), G D Malyuzhinets , (1951),
D V Sivukhin , (1957); R Zengerle (1980)
J B Pendry „Negative Refraction Makes a Perfect Lens”
PHYSICAL REVIEW LETTERS 85 (2000) 3966-3969
Folie 4
Bernd Hüttner DLR Stuttgart

Objections raised against the topic
1. Valanju et al. – PRL 88 (2002) 187401-Wave Refraction in Negative-
Index Media: Always Positive and Very Inhomogeneous
2. G W 't Hooft – PRL 87 (2001) 249701 - Comment on “Negative
Refraction Makes a Perfect Lens”
3. C M Williams arXiv:physics 0105034 (2001) - Some Problems with Negative Refraction
Folie 5
Bernd Hüttner DLR Stuttgart

Overview
1. Short historical background
2. What are metamaterials?
3. Electrodynamics of metamaterials
4. Optical properties of metamaterials
5. Invisibility, cloaking, perfect lens
6. Surface plamon waves and other waves
7. Faster than light
8. Summary
Folie 6
Bernd Hüttner DLR Stuttgart

Folie 7
Bernd Hüttner DLR Stuttgart

Photonic crystals
1995 2003
Folie 8
Bernd Hüttner DLR Stuttgart

Overview
1. Short historical background
2. What are metamaterials?
3. Electrodynamics of metamaterials
4. Optical properties of metamaterials
5. Invisibility, cloaking, perfect lens
6. Surface plamon waves and other waves
7. Faster than light
8. Summary
Folie 9
Bernd Hüttner DLR Stuttgart

Definition:
Left-handed metamaterials (LHMs) are composite materials with effective electrical permittivity, ε , and magnetic permeability, µ , both negative over a common frequency band.
What is changed in electrodynamics due to these properties?
Taking plane monochromatic fields Maxwell‘s equations read c·rotE
 i
 
H
  i·c k E
 c·rotH
  i
 
E
 


Note, the changed signs
Folie 10
Bernd Hüttner DLR Stuttgart

By the standard procedure we get for the wave equation
 
 c k

 
 2 
E
 c

  

  c
2
    k
2 



2
 2
  c
2

 2 2 c k E
  2     
2 n n i .
no change between
LHS and RHS
Poynting vector
S
 c
4


E H

  c
2
4


E k E

  c
2
4

  c
2
4

 
2 c k
4
  k k
 

  c

4
 
  k
E·E .
k
 
Folie 11
Bernd Hüttner DLR Stuttgart

RHS
LHS
S
 k v p
 v g
S
 k v g
 v p
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Bernd Hüttner DLR Stuttgart

Two (strange) consequences for LHM
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Bernd Hüttner DLR Stuttgart

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Bernd Hüttner DLR Stuttgart

1. Simple explanation n
Why is n < 0?
· · · i·

·i ·
2. A physical consideration
  
, n
  
, n
   
, n n
  
2 nd order Maxwell equation:
1 st order Maxwell equation: k E k H
   2 2
E c k E

H n e E k c


E n e H k c

RHS:

> 0,

> 0, n > 0 LHS:

< 0,

< 0, n < 0
Folie 15
Bernd Hüttner DLR Stuttgart

whole parameter space
Folie 16
Bernd Hüttner DLR Stuttgart

3. An other physical consideration
The averaged density of the electromagnetic energy is defined by
U

1
8


 d
  d

 d
  d

 



.
Note the derivatives has to be positive since the energy must be positive and therefore LHS possess in any case dispersion and via KKR absorption
Folie 17
Bernd Hüttner DLR Stuttgart

Kramers-Kronig relation
  2

P


0

 
2 d

2


P


0

 

2
1 d

Titchmarsh‘theorem: KKR
 
0 causality
Folie 18
Bernd Hüttner DLR Stuttgart

Because the energy is transported with the group velocity we find v g

S

U
 c

4
  k k

1
16


 d
  d

E·E
*  d
  d

H·H
*

 


1
This may be rewritten as v g
  c





 d 

2 d

 



 
  d 
 d

 





 k
.
k
Since the denominator is positive the group velocity is parallel to the
Poynting vector and antiparallel to the wave vector.
Folie 19
Bernd Hüttner DLR Stuttgart

The group velocity, however, is also given by v g



 dk d





1
 c



 d 
 n d

 







1 k k
 n c
 

 k k
We see n < 0 for vanishing dispersion of n
This should be not confused with the superluminal, subluminal or negative velocity of light in RHS.
These effects result exclusively from the dispersion of n.
Folie 20
Bernd Hüttner DLR Stuttgart

Lorentz-model
Dispersion of

,
 and n
 
Re
 2 pe
2 i e
 
Rm
 2 pm
2 i m
Folie 21
Bernd Hüttner DLR Stuttgart

Overview
1. Short historical background
2. What are metamaterials?
3. Electrodynamics of metamaterials
4. Optical properties of metamaterials
5. Invisibility, cloaking, perfect lens
6. Surface plamon waves and other waves
7. Faster than light
8. Summary
Folie 22
Bernd Hüttner DLR Stuttgart

Reflection and refraction but what is with R

  
2  k
2
  
2  k
2
µ = 1
Optically speaking a slab of space with thickness 2W is removed.
Optical way is zero !
Folie 23
Bernd Hüttner DLR Stuttgart

Snellius law for LHS
Due to homogeneity in space we have k
0x
= k
1x
= k
2x sin sin

2

0
 
0
 c
 
1 1 sin
 
0
 c
 
2 2 sin

2

 
1 1
 
2 2 if

'' and
 
1 sin sin

2

0

 n
1 n
2
.
 
1
1
Folie 24
Bernd Hüttner DLR Stuttgart

First example water: n = 1.3
„negative“ water: n = -1.3
Folie 25
Bernd Hüttner DLR Stuttgart

Second example: real part of electric field of a wedge

= 2.6
left-measured right-calculated

= -1.4
left-measured right-calculated
Folie 26
Bernd Hüttner DLR Stuttgart

General expression for the reflection and transmission
The geometry of the problem is plotted in the figure where r
1
’ = -r
1
.
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Bernd Hüttner DLR Stuttgart

1. s-polarized
R s

E
1
E
0
2

  
2 1 1 cos
       
0 1 2 2 1 1 sin
2 
0
  
2 1 1 cos
       
0 1 2 2 1 1 sin
2

0
2
T s

E
2
E
0
2

2
  
2 1 1 cos

0
  
2 1 1 cos
       
0 1 2 2 1 1 sin
2 
0
2
.
e
1
=

1
=1, e
2
= m
2
= -1 and u
0
= 0 we get R = 0 & T = 1
Folie 28
Bernd Hüttner DLR Stuttgart

2. p-polarized
R p

E
1
E
0
2

  
2 1 1 cos
       
0 1 2 2 1 1 sin
2 
0
  
2 1 1 cos
       
0 1 2 2 1 1 sin
2 
0
2
T p

E
2
E
0
2

2
  
2 1 1 cos

0
  
2 1 1 cos
       
0 1 2 2 1 1 sin
2 
0
2
.
R = 0 – why and what does this mean?
Impedance of free space
Impedance for e
= m
= -1


0
0
0
0


0

0 invisible!
Folie 29
Bernd Hüttner DLR Stuttgart

Reflectivity of s-polarized beam of one film rs1


2
 
2
 


 
 



1


1


1

1


 cos
 
 cos
 
 
 



2


2



2
2



 cos cos



 
2
2

 
2
 

2
 



 rs2


2
 
2
 


 
 



2


2



2
2



 cos cos



 
2
2




2
2





 

 
 



3


3


3

3



 cos cos



 
2
2

 
2
 

2
 

 


2
 
2
 

 asin

 n1


1 n2




1
2



 sin
2



 
2
 
2
 

 asin 
 n1


1
 
1

 n3
 sin

3


 

3

2
 
2
 
 


Rsf


2
 
2
   d

 rs1


2
 
2
 

2

1

2 rs1


2
 
2
 

 rs2



2
 
2
  
 rs2


2
 
2
  


2
 
2
  
 


2
 
2
   d


 


2
 
2
   d
 
 rs2


2
 
2
 

2 rs1


2
 
2
 

2
 rs2


2
 
2
 

2
Folie 30
Bernd Hüttner DLR Stuttgart

Absorption or reflection of a normal system
Absorption of Al, p- and s-polarized
R s
0.4

E
1
E
0
2

  
2 1 1 cos
       
0 1 2 2 1 1 sin
2 
0
  
2 1 1 cos
       
0 1 2 2 1 1 sin
2 
0
0.35
T s 0.3

E
2
E
0
2

2
  
2 1 1 cos

0
  
2 1 1 cos
       
0 1 2 2 1 1 sin
2 
0
2
2
.
0.25
0.2
R p
0.15

E
1
E
0
2

  
2 1 1
  
2 1 1 cos
       
0 1 2 2 1 1 sin
2 
0 cos
       
0 1 2 2 1 1 sin
2 
0
2
T p
0.1

E
2
E
0
2

2
  
2 1 1 cos

0
  
2 1 1 cos
       
0 1 2 2 1 1 sin
2 
0
2
.
0.051
5.2128258

10
4
0 0.2
0.4
0.6
0.8
1 1.2
1.4
Folie 31
Bernd Hüttner DLR Stuttgart

Reflection of a normal system
Reflectivity of Al, p- and s-polarized
0.82
0.77
0.72
0.67
0.97
0.92
0.87
0.62
0.57
0 0.2
0.4
0.6
0.8
1 1.2
1.4
Folie 32
Bernd Hüttner DLR Stuttgart

1
Reflection of a LHS
Rsf  (
(
Rpf Rpf 
(
(

  

 

1
1
  1
1


  

 5
5
  

5
5
5
5
) 

) 5
5
)
)
  



1
1
0.5
1     

5
5
5  

 5
5
5
) )
)
0.4
0.4
0.2
0.2
0 0
0
0
0
0 0.2
0.2
0.4
0.4
0.6
0.6
0.8


1
1
1.2
1.2
1.4
1.4
1.6
1.6
Folie 33
Bernd Hüttner DLR Stuttgart

Overview
1. Short historical background
2. What are metamaterials?
3. Electrodynamics of metamaterials
4. Optical properties of metamaterials
5. Invisibility, cloaking, perfect lens
6. Surface plamon waves and other waves
7. Faster than light
8. Summary
Folie 34
Bernd Hüttner DLR Stuttgart

Al plate, d=17µm
Z eff

Z
0
1
1
  eff
   eff
2
Folie 35
Bernd Hüttner DLR Stuttgart

For the cylindrical lens, cloaking occurs for distances r
0 than r
# if
 c
=
 m 3 r
#
 r out r in less
The animation shows a coated cylinder with
 in
=1,
 s
=-1+i·10 -7 , r out
=4, r in
=2 placed in a uniform electric field. A polarizable molecule moves from the right. The dashed line marks the circle r=r
#
. The polarizable molecule has a strong induced dipole moment and perturbs the field around the coated cylinder strongly. It then enters the cloaking region, and it and the coated cylinder do not perturb the external field.
Folie 36
Bernd Hüttner DLR Stuttgart

There is more behind the curtain: 1. outside the film perfect lens – beating the diffraction limit
How can this happen?
Let the wave propagate in the z-direction the larger k x and k y the better the resolution but k z becomes imaginary if
 2 c
2
0
 k
2 x
 k
2 y
How does negative slab avoid this limit?
Due to amplification of the evanescent waves
Folie 37
Bernd Hüttner DLR Stuttgart

Amplification of evanescent waves
Folie 38
Bernd Hüttner DLR Stuttgart

Folie 39
Bernd Hüttner DLR Stuttgart

Overview
1. Short historical background
2. What are metamaterials?
3. Electrodynamics of metamaterials
4. Optical properties of metamaterials
5. Invisibility, cloaking, perfect lens
6. Surface plamon waves and other waves
7. Faster than light
8. Summary
Folie 40
Bernd Hüttner DLR Stuttgart

How can we understand this?
Analogy – enhanced transmission through perforated metallic films
Ag d=280nm hole diameter d / l
= 0.35
L=750nm hole distant area of holes 11% h =320nm thickness d opt
=11nm optical depth
T film
~10 -13 solid film
Folie 41
Bernd Hüttner DLR Stuttgart

Detailed analysis shows it is a resonance phenomenon with the surface plasmon mode.
Surface-plasmon condition:

1 k
1


2 k
2

0   
2
1
 2 p
 2
 s

 p
2
Folie 42
Bernd Hüttner DLR Stuttgart

Interplay of plasma surface modes and cavity modes
The animation shows how the primarily CM mode at 0.302eV (excited by a normal incident TM polarized plane wave) in the lamellar grating structure with h=1.25
μ m , evolves into a primarily SP mode at 0.354eV when the contact thickness is reduced to h=0.6
μ m along with the resulting affect on the enhanced transmission.
Folie 43
Bernd Hüttner DLR Stuttgart


=1
Beyond the diffraction limit: Plane with two slits of width l
/20

=2.2

=-1
µ=-1

=-1+i·10 -3
µ=-1+i·10 -3
Folie 44
Bernd Hüttner DLR Stuttgart

Folie 45
Bernd Hüttner DLR Stuttgart

Overview
1. Short historical background
2. What are metamaterials?
3. Electrodynamics of metamaterials
4. Optical properties of metamaterials
5. Invisibility, cloaking, perfect lens
6. Surface plamon waves and other waves
7. Faster than light
8. Summary
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There is more behind the curtain: 2. inside the film
The peak starts at the exit before it arrives the entry
Example. Pulse propagation for n = -0.5
Oje, is this mad?!
No, it isn’t!
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An explanation:
Let us define the rephasing length l of the medium where v g is the group velocity
If the rephasing length is zero then the waves are in phase at

=

0
Remember, Fourier components in same phase interfere constructively
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RHS
I
LHS
RHS
n=1
RHS
0
Peak is at z=0 at t=0
II
LHS n < 0
L
III
RHS
n=1 z t < 0 the rephasing length l
II inside the medium becomes zero at a position z
0
= ct / n g
.
At z
0 the relative phase difference between different Fourier components vanishes and a peak of the pulse is reproduced due to constructive interference and localized near the exit point of the medium such that
0 > t > n g
L/c .
The exit pulse is formed long before the peak of the pulse enters the medium
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At a later time t’ such that 0 > t’ > t, the position of the rephasing point inside the medium z
0
’ = ct’ / n g z
0
’
< z
0 decreases i.e., and hence the peak moves with negative velocity
-v g inside the medium.
t=0: peaks meet at z=0 and interfere destructively.
Region 3: z
''
0
   since 0 >t>n g
L/c is z
0
’’
> L
0>t’>t: z
0
’’’
> z
0
’’ the peak moves forward
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Gold plates (300nm) and stripes (100nm) on glass and
MgF
2 as spacer layer
Folie 52
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Overview
1. Short historical background
2. What are metamaterials?
3. Electrodynamics of metamaterials
4. Optical properties of metamaterials
5. Invisibility, cloaking, perfect lens
6. Surface plamon waves and other waves
7. Faster than light
8. Summary
Folie 53
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Summary
Metamaterials have new properties:
1. S and v g are antiparallel to k and v p
2. Angle of refraction is opposite to the angle of incidence
3. A slab acts like a lens. The optical way is zero
4. Make perfect lenses, R = 0, T = 1
5. Make bodies invisible
6. Can be tuned in many ways
Folie 54
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n
W
= 1.35
n
G
= 1.5
n
W
= 1.35
n
G
= -1.5
n
W
= -1.35
n
G
= 1.5
n
W
= -1.35
n
G
= -1.5
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