metamaterial powerpoint negative refractive index

metamaterial powerpoint negative refractive index

1. 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

2. Bernd Hüttner DLR Stuttgart
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

3. Bernd Hüttner DLR Stuttgart
Folie 3
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

4. Bernd Hüttner DLR Stuttgart
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); RZengerle(1980)

5. Bernd Hüttner DLR Stuttgart
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

6. Bernd Hüttner DLR Stuttgart
Folie 6
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

7. Bernd Hüttner DLR Stuttgart
Folie 7

8. Bernd Hüttner DLR Stuttgart
Folie 8
Photonic crystals
1995 2003

9. Bernd Hüttner DLR Stuttgart
Folie 9
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

10. Bernd Hüttner DLR Stuttgart
Folie 10
Left-handed metamaterials (LHMs) are composite materials with effective
electrical permittivity, ε, and magnetic permeability, µ, both negativeover 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 changedsigns

11. Bernd Hüttner DLR Stuttgart
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    
  
22
22
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 .
44 4kk
      
  

  
 

 

12. Bernd Hüttner DLR Stuttgart
Folie 12
RHS
LHS pg
Sk
vv

 gp
Sk
vv



13. Bernd Hüttner DLR Stuttgart
Folie 13
Two (strange) consequences forLHM

14. Bernd Hüttner DLR Stuttgart
Folie 14

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

16. Bernd Hüttner DLR Stuttgart
Folie 16
whole parameter space

17. Bernd Hüttner DLR Stuttgart
Folie 17
The averaged density of the electromagnetic energy is defined by 

 

22
dd1
U E H .
8 d d
     

  
Note the derivatives has to be positivesince the energy must be positive
and therefore LHS possess in any case dispersion and via KKR absorption
3. An other physical consideration

18. Bernd Hüttner DLR Stuttgart
Folie 18
Kramers-Kronig relation
Titchmarsh‘theorem: KKR causality




22
0
22
0
Im n2
Re n( ) 1 P d Im n 0
Re n 12
Im n( ) P d




      

   
 
   
   



19. Bernd Hüttner DLR Stuttgart
Folie 19
Because the energy is transported with the group velocity we find 
   
1
**
g
ddS c k 1
v E·E E·E H·H
U 16 d d4 k

    
   
   


This may be rewritten as


g
c 2 k
v.
kdd
dd


       
   

  



Since the denominator is positive the group velocity is parallel to the
Poynting vector and antiparallel to the wave vector.

20. Bernd Hüttner DLR Stuttgart
Folie 20
The group velocity, however, is also given by


1
1
g
dndk k c k
vc
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.

21. Bernd Hüttner DLR Stuttgart
Folie 21
Dispersion of , and n
Lorentz-model
2
pe
22
Re e
1
i

   
     
2
pm
22
Rm m
1
i

   
    

22. Bernd Hüttner DLR Stuttgart
Folie 22
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

23. Bernd Hüttner DLR Stuttgart
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!

24. Bernd Hüttner DLR Stuttgart
Folie 24 0 0 1 1 0 2 2 2
112
0 22
21
1
02
k sin sin sin
cc
sin
if '' and '' 1
sin
sin n
.1
sin n

        



   
 

  

Snellius law for LHS
Due to homogeneity in space
we have k
0x= k
1x= k
2x

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

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

27. Bernd Hüttner DLR Stuttgart
Folie 27
General expression for the reflection and transmission
The geometry of the problem is plotted in the figurewherer
1’ = -r
1.

28. Bernd Hüttner DLR Stuttgart
Folie 28 2
2
2
2 1 1 0 1 2 2 1 1 01
s
2
0
2 1 1 0 1 2 2 1 1 0
2
2
2 1 1 02
s
2
0
2 1 1 0 1 2 2 1 1 0
cos sinE
R
E cos sin
2 cosE
T.
E cos sin
       

       
  

       
e
1= 
1=1, e
2= m
2= -1 and u
0= 0we get R = 0 & T = 1
1. s-polarized

29. Bernd Hüttner DLR Stuttgart
Folie 29
2. p-polarized2
2
2
2 1 1 0 1 2 2 1 1 01
p
2
0
2 1 1 0 1 2 2 1 1 0
2
2
2 1 1 02
p
2
0
2 1 1 0 1 2 2 1 1 0
cos sinE
R
E cos sin
2 cosE
T.
E cos sin
         

         
   

         
R = 0 –why and what does this mean?
Impedance of free space0
0


Impedance for e= m= -100
00
1
1
  

   invisible!

30. Bernd Hüttner DLR Stuttgart
Folie 30
Reflectivity of s-polarized beam of one filmr
s1

2

2
 

2
n
1

1

1
  cos 
1
n
2

2

2
  cos
2

2
  

2
n
1

1

1
  cos 
1
n
2

2

2
  cos
2

2
  
 r
s2

2

2
 

3
n
2

2

2
  cos
2

2
   
2
n
3

3

3
  cos
2

2
  

3
n
2

2

2
  cos
2

2
   
2
n
3

3

3
  cos
2

2
  
 R
sf

2

2
d 
r
s1

2

2
 
2
2r
s1

2

2
  r
s2

2

2
  cos2
2

2
d   r
s2

2

2
 
2

12r
s1

2

2
  r
s2

2

2
  cos2
2

2
d   r
s1

2

2
 
2
r
s2

2

2
 
2

 
2

2
 asin
n
1

1

1
 sin
n
2

2

2
 








 
2

2
 asin
n
1

1

1
 sin
2

2
  
n
3

3

3
 










31. Bernd Hüttner DLR Stuttgart
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 system2
2
2
2 1 1 0 1 2 2 1 1 01
s
2
0
2 1 1 0 1 2 2 1 1 0
2
2
2 1 1 02
s
2
0
2 1 1 0 1 2 2 1 1 0
cos sinE
R
E cos sin
2 cosE
T.
E cos sin
       

       
  

        2
2
2
2 1 1 0 1 2 2 1 1 01
p
2
0
2 1 1 0 1 2 2 1 1 0
2
2
2 1 1 02
p
2
0
2 1 1 0 1 2 2 1 1 0
cos sinE
R
E cos sin
2 cosE
T.
E cos sin
         

         
   

         

32. Bernd Hüttner DLR Stuttgart
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

33. Bernd Hüttner DLR Stuttgart
Folie 33
Reflection of a LHS0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0
0.2
0.4
0.6
0.8
R
sf
1. 11155( )
R
pf
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
R
sf
1.05 11155( )
R
pf
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
R
sf
1.25 1.05 1155( )
R
pf
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
R
sf
0.5 1.5 1155( )
R
pf
0.5 0.5 1155( )


34. Bernd Hüttner DLR Stuttgart
Folie 34
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

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

36. Bernd Hüttner DLR Stuttgart
Folie 36
An other miracle: Cloaking of a field
For the cylindrical lens, cloaking occurs for distances r
0less
than r
#if 
c=
m in
3
out#
rrr
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.

37. Bernd Hüttner DLR Stuttgart
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 k
xand k
ythe better the resolution butk
zbecomes imaginary if2
22
xy2
0
kk
c


How does negative slab avoid this limit?

38. Bernd Hüttner DLR Stuttgart
Folie 38
Amplification of evanescent waves

39. Bernd Hüttner DLR Stuttgart
Folie 39

40. Bernd Hüttner DLR Stuttgart
Folie 40
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

41. Bernd Hüttner DLR Stuttgart
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
d
opt=11nm optical depth
T
film~10
-13
solid film

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

43. Bernd Hüttner DLR Stuttgart
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.

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

45. Bernd Hüttner DLR Stuttgart
Folie 45

46. Bernd Hüttner DLR Stuttgart
Folie 46
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

47. Bernd Hüttner DLR Stuttgart
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!

48. Bernd Hüttner DLR Stuttgart
Folie 48
An explanation:
Let us define the rephasing length lof the medium
where v
gis 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

49. Bernd Hüttner DLR Stuttgart
Folie 49
RHS
LHS
RHS
Peak is at z=0 at t=0
t < 0
the rephasing length l
II inside the medium becomes
zero at a positionz
0= ct / n
g.
At z
0the 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
gL/c.
The exit pulse is formed long before the peak of the pulse enters the mediumRHS
n=1
RHS
n=1
LHS
n < 0
0 L z

II IIII

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

51. Bernd Hüttner DLR Stuttgart
Folie 51

52. Bernd Hüttner DLR Stuttgart
Folie 52
Gold plates (300nm) and
stripes (100nm) on glass and
MgF
2as spacer layer

53. Bernd Hüttner DLR Stuttgart
Folie 53
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

54. Bernd Hüttner DLR Stuttgart
Folie 54
Summary
Metamaterials have new properties:
1. S and v
gare antiparallelto k and v
p
2. Angle of refraction is oppositeto the angle of incidence
3. A slab acts like a lens. The optical way is zero
4. Make perfectlenses, R = 0, T = 1
5. Make bodies invisible
6. Can be tunedin many ways

55. Bernd Hüttner DLR Stuttgart
Folie 55
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