bernd hüttner dlr stuttgart folie 1 a journey through a strange classical optical world bernd...

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


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


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Overview






6. Surface plamon waves and other waves


8. Summary


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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)


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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


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Overview








8. Summary


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Photonic crystals

1995 2003


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Overview








8. Summary


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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


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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


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RHS

LHS

p g

S k

v v

g p

S k

v v


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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 , 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 ,


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whole parameter space


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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


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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


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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.


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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.


Folie 21

Dispersion of , and n

Lorentz-model 2pe

2 2Re e

1i

2pm

2 2Rm m

1i


Folie 22

Overview








8. Summary


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 !


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


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water: n = 1.3 „negative“ water: n = -1.3

First example


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= 2.6left-measuredright-calculated

= -1.4left-measuredright-calculated

Second example: real part of electric field of a wedge


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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


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


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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!


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


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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


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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


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( )


Folie 34

Overview








8. Summary


Folie 35

Invisibility

eff 0 effeff

1Z 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

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.


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?


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Amplification of evanescent waves


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Folie 40

Overview








8. Summary


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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


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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


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.


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


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Overview








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 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 < 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


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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: ''0 gz L ct n 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


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Overview








8. Summary


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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 = 1.35nG = 1.5

nW = 1.35nG = -1.5

nW = -1.35nG = 1.5

nW = -1.35nG = -1.5

bernd hüttner dlr stuttgart folie 1 a journey through a strange classical optical world bernd...

Documents

summary slide

negative refraction

lhm slide

rhs lhs slide

lefthanded metamaterials

surface plamon waves

rhs poynting vector

surface plasmon waves