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A journey through a strange classical
optical world
Bernd Hüttner CPhys FInstP
Institute of Technical
Physics
Left-handed
media
DLR Stuttgart
Metamaterials
Negative refractive index
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
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Bernd Hüttner DLR Stuttgart
Objections raised against the topic
1. Valanju et al. – PRL 88 (2002) 187401-Wave Refraction in NegativeIndex 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
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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  i·c  k  H  .
Note, the changed signs
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By the standard procedure we get for the wave equation


c
E  c  k  
kE 





   
2 E  c 2 k· E·k  k·k E  c 2 k 2 E

k  k ' i·k ''
2

2
2
 2 
c
  n 2   n  i   .
2
no change between
LHS and RHS
Poynting vector




   
c
c2
c2 
S
EH  
EkE  
k E·E  E k·E 

4
4
4 
c2
c2 k
k
c 
k

k E·E  
E·E

E·E
.
4
4
4 
k
k
 
 
 
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Bernd Hüttner DLR Stuttgart
RHS
S  k
v p  vg
LHS
S  k
vg  v p
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Two (strange) consequences for LHM
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Why is n < 0?
1. Simple explanation
n  ·  ·   ·   i·  ·i    ·
2. A physical consideration
n    , nn   
,, nn  

, , nn  
 
 
2nd order Maxwell equation:
1st
2E  c2 k 2 E

ek  E
c

k  H   0E  n ek  H
c
order Maxwell equation:
k  E   0H  n
RHS:  > 0,  > 0, n > 0
LHS:  < 0,  < 0, n < 0
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whole parameter space
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3. An other physical consideration
The averaged density of the electromagnetic energy is defined by
d      2 
1  d      2
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
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Kramers-Kronig relation
2  Im  n    
Re  n()   1  P 
d
2
2
 0  

Im  n      0

2 Re  n      1
Im  n()   
P
d
2
2
 0  
Titchmarsh‘theorem: KKR
causality
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Because the energy is transported with the group velocity we find

d     
S
c 
k  1  d     
*
* 
vg   
E·E
E·E 
H·H  


U
d
d
4 
k 16 
 
 
1
This may be rewritten as
vg  
c 
2
k
.
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.
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The group velocity, however, is also given by
1
 d  n    k
 dk 
c
vg     c 
 

d  k
n  
 d 

n    
1

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.
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Bernd Hüttner DLR Stuttgart
Dispersion of ,  and n
Lorentz-model
    1 
2pe
2Re  2  i e
    1 
2pm
2Rm  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
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Reflection and refraction
Optically speaking
a slab of space with
thickness 2W is
removed.
Optical way is zero !
 n  1  k 2
R
2
 n  1  k 2
2
but what is with
µ=1
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Bernd Hüttner DLR Stuttgart
Snellius law for LHS
Due to homogeneity in space
we have k0x = k1x = k2x


11 sin 0 
 2  2 sin 2
c
c
11
sin 2

if  '' and  '' 1
sin 0
2 2
k 0 sin 0 
sin 2
n1

. 1  1
sin 0  n 2
Folie 24
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First example
water: n = 1.3
„negative“ water: n = -1.3
Folie 25
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Second example: real part of electric field of a wedge
= 2.6
left-measured
right-calculated
= -1.4
left-measured
right-calculated
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General expression for the reflection and transmission
The geometry of the problem is plotted in the figure where r1’ = -r1.
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1. s-polarized
E
Rs  1
E0
E2
Ts 
E0
2
 2 11 cos 0  1  2  2  11 sin 0
2

2

2
 2 11 cos 0  1  2  2  11 sin 2 0
2
2 2 11 cos 0
 2 11 cos 0  1  2  2  11 sin 0
2
.
e1 = 1=1, e2 = m2 = -1 and u0 = 0 we get R = 0 & T = 1
Folie 28
Bernd Hüttner DLR Stuttgart
2. p-polarized
Rp 
Tp 
E1
E0
E2
E0
2
 2 11 cos 0  1  2  2  11 sin 0
2

2

2
 2 11 cos 0  1  2  2  11 sin 2 0
2
2 2 11 cos 0
 2 11 cos 0  1  2  2  11 sin 0
2
.
R = 0 – why and what does this mean?
Impedance of free space
Impedance for e = m = -1
0
0
1 0

1  0
0
0
invisible!
Folie 29
Bernd Hüttner DLR Stuttgart
Reflectivity of s-polarized beam of one film

 
  

  n    cos    n     cos      
  2 2 
2 1 1 1
1 2 2 2

  
  2n33 3cos 2 2 
  n    cos        n    cos     
3 2 2 2
2 2
2 3 3 3
2 2
rs1 2   2   
rs2 2 2  
 2 n 1 1   1  cos      1 n 2 2   2  cos  2   2  
3 n2 2 2  cos  2 2 
 n11 1 sin   
 2   2     asin 
 n22 2 


 n11 1 sin 2 2   

n 33   3

 2   2    asin 


2
2
 2 rs12  2    rs22   2    cos 2 2  2    d   rs22   2  

Rsf 2   2    d 
2
2
1  2 rs12   2    rs22  2    cos 2 2   2    d   rs12   2     rs22   2  
rs1 2  2  
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Bernd Hüttner DLR Stuttgart
Absorption or reflection of a normal system
Absorption of Al, p- and s-polarized
0.4
Rs 
E1
E0
2
 2 11 cos 0  1  2  2  11 sin 0
2

2
 2 11 cos 0  1  2  2  11 sin 2 0
0.35
Ts 0.3
E2
E0
2

2
2 2 11 cos 0
.
 2 11 cos 0  1  2  2  11 sin 0
2
0.25
0.2
Rp 
0.15
Tp 
0.1
E1
E0
E2
E0
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

2
2
2 2 11 cos 0
 2 11 cos 0  1  2  2  11 sin 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
0.97
0.92
0.87
0.82
0.77
0.72
0.67
0.62
0.57
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Folie 32
Bernd Hüttner DLR Stuttgart
Reflection of a LHS
1
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
Rsf (( 
0.5
 11.5
 1  1    5  5)
1. 
1.05
Rsf
RR
( sf
( 1.25
1.05
1 11 111 1 55 555))5)
sf
Rpf (( 
0.5
 10.5
 1  1   55 55) )
1. 
1.05
Rpf
RR
( 1.25
1.05
1 1.0
1.0
 1111
5555))
pf(pf
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
000 0
000
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8




1
11
1.2
1.2
1.2
1.4
1.4
1.4
1.6
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
Invisibility
Al plate, d=17µm Zeff  Z0
1
1  eff
eff  2
Folie 35
Bernd Hüttner DLR Stuttgart
An other miracle: Cloaking of a field
For the cylindrical lens, cloaking occurs for distances r0 less
than r# if c=m
3
r#  rout rin
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 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
2
2
2
the larger kx and ky the better the resolution but kz becomes imaginary if 2  k x  k y
c0
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
dopt=11nm optical depth
Tfilm~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  2

0
k1 k 2
2  1 
s 
2p
2
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=1.25μm, 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 43
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Beyond the diffraction limit: Plane with two slits of width l/20
=1
=-1
µ=-1
=2.2
=-1+i·10-3
µ=-1+i·10-3
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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 46
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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 vg 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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II
I
RHS
III
LHS
n<0
LHSRHS
n=1
RHS
n=1
RHS
0
L
z
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
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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:
z''0  L  ct  n g L
since 0 >t>ngL/c is z0’’ > L
0>t’>t: z0’’’ > z0’’ the peak moves forward
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Gold plates (300nm) and
stripes (100nm) on glass and
MgF2 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 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
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nW = 1.35
nG = 1.5
nW = 1.35
nG = -1.5
nW = -1.35
nG = 1.5
nW = -1.35
nG = -1.5
Folie 55
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