metamaterial powerpoint negative refractive index

Bernd Hüttner DLR Stuttgart
Folie 1
A journey through a strange classical
optical world
Bernd Hüttner CPhys FInstP
Institute of Technical Physics
DLR Stuttgart
Left-handed media
Metamaterials
Negative refractive index

Folie 2
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 3
Overview
6. Surface plamon waves and other waves
8. Summary

Folie 4
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).
J B Pendry „Negative Refraction Makes a Perfect Lens”
PHYSICAL REVIEW LETTERS 85 (2000) 3966-3969
H Lamb (1904), H C Pocklington (1905), G D Malyuzhinets, (1951),
D V Sivukhin, (1957); R Zengerle (1980)

Folie 5
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 6
Overview
8. Summary

Folie 7

Folie 8
Photonic crystals
1995 2003

Folie 9
Overview
8. Summary

Folie 10
Left-handed metamaterials (LHMs) are composite materials with effective
electrical permittivity, ε, and magnetic permeability, µ, both negative over a
common frequency band.
Definition:
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 i·c k H .
 
     
 
 


     
 
Note, the changed signs

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

Folie 12
RHS
LHS
p g
S k
v v


g p
S k
v v



Folie 13
Two (strange) consequences for LHM

Folie 14

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

Folie 16
whole parameter space

Folie 17
The averaged density of the electromagnetic energy is defined by
 
 
 
 
 
 
2 2
d d
1
U E H .
8 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
3. An other physical consideration

Folie 18
Kramers-Kronig relation
Titchmarsh‘theorem: KKR causality
 
 
 
 
 
2 2
0
2 2
0
Im n
2
Re n( ) 1 P d Im n 0
Re n 1
2
Im n( ) P d


 
 
 
     
 
 
   
 
 
  
   
   



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

Folie 20
The group velocity, however, is also given by
 
   
1
1
g
d n
dk k c k
v c
d d k k
n
n



 
 
 
   
  
 
   
   
      

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 21
Dispersion of ,  and n
Lorentz-model  
2
pe
2 2
Re e
1
i

   
    
 
2
pm
2 2
Rm m
1
i

   
    

Folie 22
Overview
8. Summary

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

Folie 24
0 0 1 1 0 2 2 2
1 1
2
0 2 2
2 1
1
0 2
k sin sin sin
c c
sin
if '' and '' 1
sin
sin n
. 1
sin n
 
        




   
  

  

Snellius law for LHS
Due to homogeneity in space
we have k0x = k1x = k2x

Folie 25
water: n = 1.3 „negative“ water: n = -1.3
First example

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

Folie 27
General expression for the reflection and transmission
The geometry of the problem is plotted in the figure where r1’ = -r1.

Folie 28
2
2 2
2 1 1 0 1 2 2 1 1 0
1
s
2
0 2 1 1 0 1 2 2 1 1 0
2
2
2 1 1 0
2
s
2
0 2 1 1 0 1 2 2 1 1 0
cos sin
E
R
E cos sin
2 cos
E
T .
E cos sin
           
 
           
   
 
           
e1 = 1=1, e2 = m2 = -1 and u0 = 0 we get R = 0 & T = 1
1. s-polarized

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

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

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

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

Folie 33
Reflection of a LHS
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0
0.2
0.4
0.6
0.8
Rsf 1.
 1

 1
 1
 
 5
 5

( )
Rpf 1.
 1

 1.0
 1
 
 5
 5

( )

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0
0.2
0.4
0.6
0.8
Rsf 1.05
 1

 1
 1
 
 5
 5

( )
Rpf 1.05
 1

 1.0
 1
 
 5
 5

( )

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0
0.2
0.4
0.6
0.8
Rsf 1.25
 1.05

 1
 1
 
 5
 5

( )
Rpf 1.25
 1.05

 1
 1
 
 5
 5

( )

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0
0.2
0.4
0.6
0.8
1
Rsf 0.5
 1.5

 1
 1
 
 5
 5

( )
Rpf 0.5
 0.5

 1
 1
 
 5
 5

( )


Folie 34
Overview
8. Summary

Folie 35
Invisibility
eff 0 eff
eff
1
Z Z 2
1
   
 
Al plate, d=17µm

Folie 36
An other miracle: Cloaking of a field
For the cylindrical lens, cloaking occurs for distances r0 less
than r# if c=m
in
3
out
# r
r
r 
The animation shows a coated cylinder with in=1, s=-1+i·10-7, rout=4,
rin=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 37
There is more behind the curtain: 1. outside the film
Due to amplification of the evanescent waves
perfect lens – beating the diffraction limit
How can this happen?
Let the wave propagate in the z-direction
the larger kx and ky the better the resolution but kz becomes imaginary if
2
2 2
x y
2
0
k k
c

 
How does negative slab avoid this limit?

Folie 38
Amplification of evanescent waves

Folie 39

Folie 40
Overview
8. Summary

Folie 41
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
dopt=11nm optical depth
Tfilm~10-13 solid film

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

Folie 43
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=1.25μm, evolves into a primarily SP mode at 0.354eV when the contact
thickness is reduced to h=0.6μm along with the resulting affect on the enhanced
transmission.

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

Folie 45

Folie 46
Overview
8. Summary

Folie 47
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!

Folie 48
An explanation:
Let us define the rephasing length l of the medium
where vg is the group velocity
Remember, Fourier components in same phase interfere constructively
If the rephasing length is zero then the waves are in phase at =0

Folie 49
RHS
LHS
RHS
Peak is at z=0 at t=0
t < 0
the rephasing length lII inside the medium becomes
zero at a position z0 = ct / ng.
At z0 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 > ngL/c.
The exit pulse is formed long before the peak of the pulse enters the medium
RHS
n=1
RHS
n=1
LHS
n < 0
0 L z
II III
I

Folie 50
At a later time t’ such that 0 > t’ > t, the position of the
rephasing point inside the medium z0’ = ct’/ng decreases i.e.,
z0’ < z0 and hence the peak moves with negative velocity
-vg inside the medium.
t=0: peaks meet at z=0 and interfere destructively.
Region 3: ''
0 g
z L ct n L
   since 0 >t>ngL/c is z0
’’ > L
0>t’>t: z0
’’’ > z0
’’ the peak moves forward

Folie 51

Folie 52
Gold plates (300nm) and
stripes (100nm) on glass and
MgF2 as spacer layer

Folie 53
Overview
8. Summary

Folie 54
Summary
Metamaterials have new properties:
1. S and vg are antiparallel to k and vp
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 55
nW = 1.35
nG = 1.5
nW = 1.35
nG = -1.5
nW = -1.35
nG = 1.5
nW = -1.35
nG = -1.5

metamaterial powerpoint negative refractive index

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