bernd hüttner dlr stuttgart folie 1 a journey through a strange classical optical world bernd...
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Bernd Hüttner DLR Stuttgart
Folie 1
A journey through a strange classical optical world
Bernd Hüttner CPhys FInstPInstitute of Technical Physics
DLR Stuttgart Left-handed media
Metamaterials
Negative refractive index
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
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
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); R Zengerle (1980)
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
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
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
Bernd Hüttner DLR Stuttgart
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
Bernd Hüttner DLR Stuttgart
Folie 11
By the standard procedure we get for the wave equation
2
22 2
2
2
2
2
2
2
cE 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 cS E H E k E k E·E E k·E
4 4 4
c c k k c kk E·E E·E E·E .
4 4 4k k
Bernd Hüttner DLR Stuttgart
Folie 15
Why is n < 0?
1. Simple explanation n · · · i· ·i ·
2. A physical consideration
n , n , n , n
2 2 2E c k E
2nd order Maxwell equation:
1st order Maxwell equation: 0 k
0 k
k E H n e Ec
k H E n e Hc
RHS: > 0, > 0, n > 0 LHS: < 0, < 0, n < 0
, nn n, n ,
Bernd Hüttner DLR Stuttgart
Folie 17
The averaged density of the electromagnetic energy is defined by
2 2d d1
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
Bernd Hüttner DLR Stuttgart
Folie 18
Kramers-Kronig relation
Titchmarsh‘theorem: KKR causality
2 20
2 20
Im n2Re n( ) 1 P d Im n 0
Re n 12Im n( ) P d
Bernd Hüttner DLR Stuttgart
Folie 19
Because the energy is transported with the group velocity we find
1
* *g
d dS c k 1v E·E E·E H·H
U 16 d d4 k
This may be rewritten as
g
c 2 kv .
kd d
d d
Since the denominator is positive the group velocity is parallel to the
Poynting vector and antiparallel to the wave vector.
Bernd Hüttner DLR Stuttgart
Folie 20
The group velocity, however, is also given by
11
g
d ndk k c kv c
d d k knn
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.
Bernd Hüttner DLR Stuttgart
Folie 21
Dispersion of , and n
Lorentz-model 2pe
2 2Re e
1i
2pm
2 2Rm m
1i
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
Bernd Hüttner DLR Stuttgart
Folie 23
Reflection and refraction
but what is with
2 2
2 2
n 1 kR
n 1 k
µ = 1
Optically speakinga slab of space with thickness 2W is removed.Optical way is zero !
Bernd Hüttner DLR Stuttgart
Folie 24
0 0 1 1 0 2 2 2
1 12
0 2 2
2 11
0 2
k sin sin sinc c
sinif '' and '' 1
sin
sin n. 1
sin n
Snellius law for LHS
Due to homogeneity in space
we have k0x = k1x = k2x
Bernd Hüttner DLR Stuttgart
Folie 26
= 2.6left-measuredright-calculated
= -1.4left-measuredright-calculated
Second example: real part of electric field of a wedge
Bernd Hüttner DLR Stuttgart
Folie 27
General expression for the reflection and transmission
The geometry of the problem is plotted in the figure where r1’ = -r1.
d
n =10
n <01
n 12
0 0
r1t t r1 2 2
t t r r1 2 1 2 t t1 2
1
t11
t r1 2
-t r t1 1 2
Bernd Hüttner DLR Stuttgart
Folie 28
22 22 1 1 0 1 2 2 1 1 01
s 20 2 1 1 0 1 2 2 1 1 0
22
2 1 1 02s 2
0 2 1 1 0 1 2 2 1 1 0
cos sinER
E cos sin
2 cosET .
E cos sin
1 = 1=1, 2 = 2 = -1 and 0 = 0 we get R = 0 & T = 1
1. s-polarized
Bernd Hüttner DLR Stuttgart
Folie 29
2. p-polarized
22 22 1 1 0 1 2 2 1 1 01
p 20 2 1 1 0 1 2 2 1 1 0
22
2 1 1 02p 2
0 2 1 1 0 1 2 2 1 1 0
cos sinER
E cos sin
2 cosET .
E cos sin
R = 0 – why and what does this mean?
Impedance of free space0
0
Impedance for = = -1 0 0
0 0
1
1
invisible!
Bernd Hüttner DLR Stuttgart
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 asinn1 1 1 sin
n2 2 2
2 2 asinn1 1 1 sin 2 2
n3 3 3
Bernd Hüttner DLR Stuttgart
Folie 31
0 0.2 0.4 0.6 0.8 1 1.2 1.45.2128258 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
22 22 1 1 0 1 2 2 1 1 01
s 20 2 1 1 0 1 2 2 1 1 0
22
2 1 1 02s 2
0 2 1 1 0 1 2 2 1 1 0
cos sinER
E cos sin
2 cosET .
E cos sin
22 22 1 1 0 1 2 2 1 1 01
p 20 2 1 1 0 1 2 2 1 1 0
22
2 1 1 02p 2
0 2 1 1 0 1 2 2 1 1 0
cos sinER
E cos sin
2 cosET .
E cos sin
Bernd Hüttner DLR Stuttgart
Folie 32
0 0.2 0.4 0.6 0.8 1 1.2 1.40.57
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
Bernd Hüttner DLR Stuttgart
Folie 33
Reflection of a LHS
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60
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 1.60
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 1.60
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( )
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
Bernd Hüttner DLR Stuttgart
Folie 36
An other miracle: Cloaking of a field
For the cylindrical lens, cloaking occurs for distances r0 less than r# if c=m
in3out# rrr
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.
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 kx and ky the better the resolution but kz becomes imaginary if 2
2 2x y2
0
k kc
How does negative slab avoid this limit?
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
Bernd Hüttner DLR Stuttgart
Folie 41
How can we understand this?
Analogy – enhanced transmission through perforated metallic films
Agd=280nm hole diameterd / = 0.35L=750nm hole distant area of holes 11%h =320nm thicknessdopt=11nm optical depthTfilm~10-13 solid film
Bernd Hüttner DLR Stuttgart
Folie 42
Detailed analysis shows it is a resonance phenomenon with the surface plasmon mode.
Surface-plasmon condition: 0kk 2
2
1
1
2
ps
2p
2 21
Bernd Hüttner DLR Stuttgart
Folie 43
Crouse.mov
Interplay of plasma surface modes and cavity modes
The animation shows how the primarily CM mode at 0.302eV (excited by anormal 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.
Bernd Hüttner DLR Stuttgart
Folie 44
Beyond the diffraction limit: Plane with two slits of width /20
=1 =2.2
=-1µ=-1
=-1+i·10-3
µ=-1+i·10-3
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
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!
Bernd Hüttner DLR Stuttgart
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
Bernd Hüttner DLR Stuttgart
Folie 49
RHS
LHS
RHS
Peak is at z=0 at t=0
t < 0the rephasing length lII inside the medium becomes zero at a position z0 = ct / ng.
At z0 the relative phase difference between different Fourier componentsvanishes 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
LHSn < 0
0 L z
II IIII
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 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 gz L ct n L since 0 >t>ngL/c is z0
’’ > L
0>t’>t: z0’’’ > z0
’’ the peak moves forward
Bernd Hüttner DLR Stuttgart
Folie 52
Gold plates (300nm) and stripes (100nm) on glass and MgF2 as spacer layer
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
Bernd Hüttner DLR Stuttgart
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