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A journey through a strange classical optical world

Bernd Hüttner CPhys FInstP 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

, 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

(2000) 3966-3969 Folie 4 Bernd Hüttner DLR Stuttgart

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 5 Bernd Hüttner DLR Stuttgart

Overview

1. Short historical background 2. What are metamaterials?

Folie 7 Bernd Hüttner DLR Stuttgart

Photonic crystals

1995 2003 Folie 8 Bernd Hüttner DLR Stuttgart

Overview

1. Short historical background 2. What are metamaterials?

Definition: Left-handed metamaterials (LHMs) are composite materials with effective electrical permittivity,

ε ,

and magnetic permeability,

, 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  c·rotH   i

 

H     E      Note, the changed signs Folie 10 Bernd Hüttner DLR Stuttgart

By the standard procedure we get for the wave equation  2  E  c k     c 2  c 

 

 

   

k 2 

 

2   2  c 2   2 2 c k E   n 2   n  i   2 .

no change between LHS and RHS Poynting vector S  c 4     c 2 4     c 2 4     2 c k 4      c 2 4  k k   c  4    k .

k Folie 11 Bernd Hüttner DLR Stuttgart

RHS LHS S



k v p



v g S



k v g



v p Folie 12 Bernd Hüttner DLR Stuttgart

Two (strange) consequences for LHM Folie 13 Bernd Hüttner DLR Stuttgart

Folie 14 Bernd Hüttner DLR Stuttgart

1. Simple explanation n Why is n < 0?

· · i·  ·i 2. A physical consideration    , n 2 nd order Maxwell equation: 1 st order Maxwell equation: RHS:  > 0,  > 0, n > 0

  

, n n

  

k E k H E 2 2 c k E H E n n  c  e k e k c   E H LHS:  < 0,  < 0, n < 0 Folie 15 Bernd Hüttner DLR Stuttgart

whole parameter space Folie 16 Bernd Hüttner DLR Stuttgart

3. An other physical consideration The averaged density of the electromagnetic energy is defined by U



1 8

   

d  d







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 Folie 17 Bernd Hüttner DLR Stuttgart

Kramers-Kronig relation

    1 2  P  0   2 d  2   P  0  

 

 2 1 d  Titchmarsh‘theorem: KKR

 

0 causality Folie 18 Bernd Hüttner DLR Stuttgart

Because the energy is transported with the group velocity we find v g  S U   c  4   k k  1 16     d  d   E·E *  d  d   H·H *      1 This may be rewritten as v g   c     d d   2     d d    k .

k Since the denominator is positive the group velocity is parallel to the Poynting vector and antiparallel to the wave vector.

Folie 19 Bernd Hüttner DLR Stuttgart

The group velocity, however, is also given by v g     dk d      1  c     d  

 

  d       1 k k  n c

 

  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.

Folie 20 Bernd Hüttner DLR Stuttgart

Lorentz-model

Dispersion of



,



and n

  Re  2 pe 2 i e   Rm  2 pm 2 i m Folie 21 Bernd Hüttner DLR Stuttgart

Overview

1. Short historical background 2. What are metamaterials?

Reflection and refraction but what is with R     2  2  k 2  k 2 µ = 1 Optically speaking a slab of space with thickness 2W is removed.

Optical way is zero !

Folie 23 Bernd Hüttner DLR Stuttgart

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

  1 1 Folie 24 Bernd Hüttner DLR Stuttgart

First example water: n = 1.3

„negative“ water: n = -1.3

Folie 25 Bernd Hüttner DLR Stuttgart

Second example: real part of electric field of a wedge  = 2.6

left-measured right-calculated  = -1.4

left-measured right-calculated Folie 26 Bernd Hüttner DLR Stuttgart

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

1. s-polarized R s  E 1 E 0 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 T s  E 2 E 0 2  2    2 1 1 cos  0    2 1 1 cos         0 1 2 2 1 1 sin 2  0 2 .

e 1 =  1 =1, e 2 = m 2 = -1 and u 0 = 0 we get R = 0 & T = 1 Folie 28 Bernd Hüttner DLR Stuttgart

2. p-polarized R p  E 1 E 0 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 T p  E 2 E 0 2  2    2 1 1 cos  0    2 1 1 cos         0 1 2 2 1 1 sin 2  0 2 .

R = 0 – why and what does this mean?

Impedance of free space Impedance for e = m = -1   0 0 0 0   0  0 invisible!

Folie 29 Bernd Hüttner DLR Stuttgart

Reflectivity of s-polarized beam of one film

rs1   2   2      2 n1  2 n1    1   1   1   1   cos  cos    1 n2  1 n2    2   2   2   2    cos cos    2  2    2   2        rs2   2   2      3 n2  3 n2    2   2   2    2   cos  cos    2   2    2   2           2 n3  2 n3    3   3   3   3    cos cos    2  2    2   2         2   2     asin   n1   1 n2    1    2  sin  2     2   2     asin   n1   1   1   sin n3   3    2   3   2       Rsf   2   2    d   rs1   2  1   2    2    2   2     rs2   2   2       2   2     rs2   2   2        2   2     d     2   2    d    rs2   2   2    2  rs1   2   2    2  rs2   2   2    2 Folie 30 Bernd Hüttner DLR Stuttgart

Absorption or reflection of a normal system Absorption of Al, p- and s-polarized R s 0.4

 E 1 E 0 2   2  2 0.35

T s 0.3

 E 2 E 0 2   2   1 1 cos         0 1 2 2 1 1 sin 2  0   1 1 cos         0 1 2 2 1 1 sin 2  0 2 2  2   1 1 cos  0   1 1 cos         0 1 2 2 1 1 sin 2  0 2 .

0.25

0.2

R p 0.15

 E 1 E 0 2   2  2 T p 0.1

0.051

 E 2 E 0 2   2   1 1   1 1 cos         0 1 2 2 1 1 sin 2  0 cos         0 1 2 2 1 1 sin 2  0 2 2  2   1 1 cos  0   1 1 cos         0 1 2 2 1 1 sin 2  0 2 .

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

1 Reflection of a LHS Rsf  ( ( Rpf Rpf  ( (        1 1  1.5

1   1 1 1  1 1   5 5     5 5 5 5 )   ) 5 5 ) )          1 1 1 1 1       5 5 5     5 5 5 ) ) ) 0.4

0.4

0.2

0 0 0 0 0 0 0.2

0.2

0.4

0.6

0.8

  1 1 1.2

1.2

1.4

1.6

Folie 33 Bernd Hüttner DLR Stuttgart

Overview

1. Short historical background 2. What are metamaterials?

Invisibility

Al plate, d=17µm Z eff  Z 0 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 r 0 than r # if  c =  m r #



r 3 out r in less 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.

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 the larger k x and k y the better the resolution but k z becomes imaginary if  2 c 2 0  k 2 x  k 2 y 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?

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 Folie 41 Bernd Hüttner DLR Stuttgart

Detailed analysis shows it is a resonance phenomenon with the surface plasmon mode.

Surface-plasmon condition:  1 k 1   2 k 2  0    2 1  2 p  2  s   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=1.25

, evolves into a primarily SP mode at 0.354eV when the contact thickness is reduced to

h=0.6

along with the resulting affect on the enhanced transmission.

Folie 43 Bernd Hüttner DLR Stuttgart

 =1 Beyond the diffraction limit: Plane with two slits of width l /20  =2.2

 =-1 µ=-1  =-1+i·10 -3 µ=-1+i·10 -3 Folie 44 Bernd Hüttner DLR Stuttgart

Folie 45 Bernd Hüttner DLR Stuttgart

Overview

1. Short historical background 2. What are metamaterials?

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 47 Bernd Hüttner DLR Stuttgart

An explanation: Let us define the

rephasing length l

of the medium where

v g

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 Folie 48 Bernd Hüttner DLR Stuttgart

RHS

LHS RHS n=1 RHS 0 Peak is at z=0 at t=0

LHS n < 0 L

III

RHS n=1 z

t <

0 the

rephasing length l II

inside the medium becomes zero at a position

0 = ct / n

0 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 > n g L/c

The exit pulse is formed long before the peak of the pulse enters the medium Folie 49 Bernd Hüttner DLR Stuttgart

At a later time t’ such that 0

t’

t, the position of the rephasing point inside the medium

0 ’ = ct’

n g

0 ’

< z

0 decreases

i.e.,

and hence the peak moves with negative velocity

-v g

inside the medium.

t=0: peaks meet at z=0 and interfere destructively.

Region 3: z '' 0 since 0 >t>n g L/c is z 0 ’’ > L 0>t’>t: z 0 ’’’ > z 0 ’’ the peak moves forward Folie 50 Bernd Hüttner DLR Stuttgart

Folie 51 Bernd Hüttner DLR Stuttgart

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

Overview

1. Short historical background 2. What are metamaterials?

Summary

Metamaterials have new properties: 1. S and v g are antiparallel to k and v p 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 54 Bernd Hüttner DLR Stuttgart

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

Folie 55 Bernd Hüttner DLR Stuttgart

PowerPoint-Präsentation - eLib

Transcript PowerPoint-Präsentation - eLib

A journey through a strange classical optical world

Bernd Hüttner CPhys FInstP DLR Stuttgart

Metamaterials Negative refractive index

Overview

Overview

Overview

 

 

   

 









  

  



   







 

 

 

 

 

Dispersion of

,

and n

Invisibility

An other miracle: Cloaking of a field



Summary

Directory