Basics of single negative and double negative...

Basics of single negative and double negative

metamaterials

Ekaterina Shamonina

UNIVERSITY OF ERLANGEN-NÜRNBERGUNIVERSITY OF OSNABRÜCK

Basics of single negative and double negative metamaterials

Contents

• Definition of single negative and double negative media• Properties and resulting applications• Common ε-negative and μ-negative media• How they work

• Example: Near field imaging with silver superlens


Contents



• Near field imaging with magnetic metamaterials(Anna Radkovskaya’s lecture)

What are metamaterials?

μετά

= beyond

(Greek)


μετά

= beyond

(Greek)

Metamaterials are not natural materials

crystal


μετά

= beyond

(Greek)

Metamaterials

are engineered composites

http://physicsweb.org/articles/world/16/5/3/1#pwpia2_05-03

crystal


μετά

= beyond

(Greek)

Metamaterials

are engineered composites that exhibit superior properties not found in nature and not observed in the constituent materials.

crystal

What is a metamaterial?Metamaterial

is an artificial material in which electromagnetic

properties (ε,μ)

can be controlled.

It is made up of periodic arrays of metallic resonant

elements. Both the size of the element and the unit cell are small

relative

to the wavelength.

dλ>>d

What is a metamaterial?Metamaterial

is an artificial material in which electromagnetic

properties (ε,μ)

can be controlled.

It is made up of periodic arrays of metallic resonant

elements. Both the size of the element and the unit cell are small

relative

to the wavelength.

Why is it important?Because it makes possible the manipulation

of fields and

waves at a subwavelength

scale.

dλ>>d

Near

field: evanescent

wavesAn object of width w can be represented in terms of Fourier components up to 2

xkwπ

=

w x kx

⇔2π/ww

NEAR FIELD IMAGING REQUIREMENT: Transfer Function

T(kx )=1 for

all Fourier

components!

Near

field: evanescent

wavesAn object of width w can be represented in terms of Fourier components up to

If wave is evanescen2 t: xw k k πλ

λ< > =

2xk

wπ

=

w x kx

⇔λ 2π/λ

2π/ww


T(kx )=1 for

all Fourier

components!

Near

field: evanescent

waves

x

z

k

An object of width w can be represented in terms of Fourier components up to

If wave is evanescen2 t: xw k k πλ

λ< > =

2xk

wπ

=

w x kx

⇔

[ ]2 2 2 2 2 2

Wave with .

Wave

so that

If then is imaginary and the wave is evan

( , )

exp[ ] exp ( )

escent

2 .

x z

z x

z x z x x

x z

k k k

jk r j k z k x

k k k k k k k

k k kv

ω με

π ωω με

λ

=

− ⋅ = − +

= + = − = −

> = = =

λ 2π/λ

2π/ww


T(kx )=1 for

all Fourier

components!

Definition of single

negative and double negative media

ε

με>0, μ>0 „double positive“(conventional materials)

real k real

n

ω εμ εμ=± =±, c

k n



ε


real k real

n

ε<0, μ>0 „single negative“(plasmas, rod MM)

complex

kcomplex

n


k n



ε


real k real

n


complex

kcomplex

n

ε>0, μ<0 „single negative“(ferrites, ring MM)

complex

kcomplex

n


k n



ε


real k real

n


complex

kcomplex

n

ε>0, μ<0 „single negative“(ferrites, ring MM)

complex

kcomplex

n

ε<0, μ<0 „double negative“(ring-rod MM)

real kreal

n


k n

air

conventionalmaterialsε>0, μ>0real kreal

n>0

ε

μ


k n

air air


n>0

plasmas,rod metamaterialsε<0, μ>0complex

kcomplex

n

ε

μno propagation!


k n

air air

air


n>0


kcomplex

n

ferrites,ring metamaterialsε>0, μ<0complex

kcomplex

n

ε

μ

no propagation!

no propagation!


k n

air air

air air


n>0


kcomplex

n

ring-rod metamaterials ε<0, μ<0real kreal

n<0


kcomplex

n

ε

μ

no propagation!

no propagation!


k n

air

ε

μ


n>0


k n

air air

air

ε

μ


n>0


kcomplex

n


kcomplex

n

no propagation!

no propagation!


k n

air air

air air

ε

μ


n>0


kcomplex

n

ring-rod metamaterials ε<0, μ<0real kreal

n<0


kcomplex

n

no propagation!

no propagation!negative

refraction!


k n

Various

names, the same

physics

• Double Negative Medium DNM• Veselago Medium• Negative Refractive Index Medium NRIM • Negative Index Medium NIM• Left-Handed Medium LHM• Backward Wave Medium BWM

Double negative media

ε

μ

Double negative media: why “left-handed”?

n<0

“right handed”

“left handed”

n>0

ε

μ


E

k

H“right handed”n>0

n<0 “left handed”

ε

μ


E

k

H

E

kH

n<0

“right handed”

“left handed”

n>0

ε

μE

k

H

E

kH

Double negative media: why “backward wave media”?

“backward wave”

“forward wave”

ε

μ


E

kH

E

k

H

“backward wave”

“forward wave”↑↑S k

×= *1Re[ ]2 E HS

ε

μ


E

kH

↑↑S k “forward wave”

↑↑ pg hv v×= *1Re[ ]2 E HS

phase and group velocities in the same direction

μ


E

kH

“backward wave”


↓↑S k

↑↑ pg hv v

kH

×= *1Re[ ]2 E HS

E


×= *1Re[ ]2 E HS

ε

μ


E

kH

“backward wave”


↓↑S k

↑↑ pg hv v

↑↓ pg hv vkH

×= *1Re[ ]2 E HS

E


×= *1Re[ ]2 E HS

ε

opposite phase and group velocities

More

on terminology

for

single/double negative media

ε

με>0, μ>0 „double positive“„DPS“

ε<0, μ>0 „single negative“„ENG“

ε>0, μ<0 „single negative“„MNG“

ε<0, μ<0 „double negative“„DNG“

H H

Negative refraction

Maxwell‘s equations and appropriate boundary conditions!

Boundary

condition for

the Poynting

vector

H H

Boundary

condition for

the wave

vector

k k

⊥ >0S⊥ >0S

+

Negative refraction

Maxwell‘s equations and appropriate boundary conditions!

Positive refraction

vs

negative refraction

wedge ε=2.2, μ=1 wedge ε=-1, μ=-1

↑↑S k “backward wave”“forward wave” ↑↓S k

Historic remark…

ARTHUR SCHUSTER, An Introduction to the Theory of Optics, 19041904

forward wave

backward wave

Historic remark…

Backward waves, negative refraction

1904 Lamb, Schuster

1944 Mandelstam

1968 Veselago: ε<0 and μ

<0 → n<0

Historic remark…

ε<0silver (ω<ωp )

1961

Rotman: rods1996

Pendry: rods

μ<01956 Thompson: ferrites1981

Hardy: split rings1999

Pendry:

split rings


1904 Lamb, Schuster

1944 Mandelstam


<0 → n<0

ε<0silver (ω<ωp )

1961

Rotman: rods1996

Pendry: rods

μ<01956 Thompson: ferrites1981

Hardy: split rings1999

Pendry:

split rings

2000 Smith et al.: ε<0 and μ

<0 → n<0

2001 Shelby et al.: first experiment


1904 Lamb, Schuster

1944 Mandelstam


<0 → n<0

Historic remark…

Artificial

medium

with

ε<0: realisation?

ωω

ε = −2

21 p

ω ωε

<<0

p

Drude response

ωp

ω

ε

plasma frequencyε

ω =

2

0p

Nem

bulk metal

Artificial

medium

with

ε<0: realisation?

Brown 1950‘s, Rotman 1960‘s, Pendry 1996

ωω

ε = −2

21 p

ω ωε

<<0

p

Drude-like response

ωpω

ε„wire medium“

ωω

ε = −2

21 p

ω ωε

<<0

p

Drude response

ωp

ω

ε

plasma frequencyε

ω =

2

0p

Nem

bulk metal

E

Artificial

medium

with

ε<0: realisation?

Brown 1950‘s, Rotman 1960‘s, Pendry 1996

ωω

ε = −2

21 p

ω ωε

<<0

p

Drude-like response

ωpω

ε„wire medium“

ωω

ε = −2

21 p

ω ωε

<<0

p

Drude response

ωp

ω

ε

plasma frequencyε

ω =

2

0p

Nem

bulk metal

E

plasma frequency

tunable, depends on the geometry!

πε

ω =

2

0

2ln( / )p

ca r

Artificial

medium

with

μ<0: realisation?

single split ring: LC circuit

ω

I

ω0

L

CV

Artificial

medium

with

μ<0: realisation?

Pendry 1999

ωμ πω ω ω ω

ωω

= =− −

=2

020

2

2( 1/ )V V j H rZ j L C j L

I


ω

I

resonant frequency

ω =01LC

L

CV

voltage

ωμ π=∂Φ= −∂

20j H r

tV

impedanceωω ω

ω ω⎛ ⎞

− = −⎜ ⎟⎝ ⎠

=202

1( ) 1j L j LC

Z

ω0

Artificial

medium

with

μ<0: realisation?

Pendry 1999

ωμ πω ω ω ω

ωω

= =− −

=2

020

2


I

strongdiamagnetic response


ω

I

resonant frequency

ω =01LC

ω0

L

CV

voltage

ωμ π=∂Φ= −∂

20j H r

tV

impedanceωω ω

ω ω⎛ ⎞

− = −⎜ ⎟⎝ ⎠

=202

1( ) 1j L j LC

Z

Artificial

medium

with

μ<0: realisation?

Pendry et al.1999

„split ring medium“

ω πω ω ω ω

ωω

= =− −

=2

20

2


I


ω

I

resonant frequency ω =01LC

frequencies

depend on the geometry!

ω0

ωω

ωμ

−= − 2

0

2

21 F

ω ω ωμ

< <<

0

0F

ω

μ

Hω ω0, F

ω0 ωF

μ

strongdiamagnetic response

Exploring

magnetic

„atoms“: Split Ring Resonators

Kafesaki et al. 2005

Aydin et al. 2005

Marques et al. 2003

Exploring

magnetic

„atoms“: the family

grows…

O‘Brien & Pendry 2005

Guo et al. 2005 Hsu et al. 2004 Bulu et al. 2005

Radkovskaya et al. 2007 Kafesaki et al. 2005 Kafesaki et al. 2005

Isotropic magnetic „atoms“

in 2D

in 3D

Gay-Balmaz,Martin 2002

Chen et al. 2006

Gay-Balmaz,Martin 2002

Baena 2005 Padilla 2005

Double negative media: realisations

Combination of SRRs (μ<0) & wires (ε<0)

ω ω ωμ

< <<

0

0F

ω

μ

ω0 ωF

μ

ω ωε

<<0

p

ωpω

ε

H

E

EH k

Smith et al. 2000

Double negative media: realisations

Omega-particlesSRR+rod

chiral

particles

Smith et al. 2000

Shelby et al. 2001

Saadoun, Engheta 1992, 1994

Tretyakov 1996

From

SRR to wire

pairs

…

Svirko et al. 2001 Podolskiy et al. 2002, Panina et al. 2002

Dolling 2005

From

microwave

to optical

frequencies

…

source: Wegener, Linden, von Freymann, lecture course 2007, Karlsruhe

Double negative metamaterials: realisation

Physics

Extension of electromagnetism

“inverse”

electromagnetic phenomena

-

inverse Snell’s law (negative refraction of rays)

-

inverse Doppler shift, Cherenkov

radiation, Goos-Haenchen

shift

-

“growing”

evanescent waves

Why

Metamaterials?

Applications

-

“Perfect lens”, subwavelength

imaging in photonics

-

Invisibility and cloaking

-

“Nano-circuits”, miniaturisied

waveguide components

-

Medical imaging

Far field

dd/2 d/2

Far field

Superlens

2000 Pendry:

metamaterial plate with n<0 acts as a perfect lens

Far field

dd/2 d/2

Far field2000 Pendry:


also for a sub-λ-object?

Superlens

2000 Pendry:


also for a sub-λ-object?

Explanation: Near fieldn>0 exponential decayn<0 exponential growth

Mechanism: surface resonant modescouple to the evanescent part of the object spectrum and lead to the reconstruction of the object in the image plane

Near field

dd/2 d/2

Far field

Superlens

Surface

modes

in metamaterials?

Surface

modes

in metamaterials?

ε

μ

TM: „transverse magnetic“TE: „transverse electric“

Domains of existence of surface modes

Silver

slab

as a superlens: surface

plasmon-polaritons

coordinate x, nm spatial frequency kx/k0

1 10 1000.1-40 40 80-80 0

•Coupled modes of surface plasmon-polariton resonances

•Two spatial resonances; flat transfer function in-between

the thinner the slab, the flatter the transfer function

Shamonina et al., Electr.Lett. (2001)

λ=360nm

d

d/2

d/2

(z=2d)

Surface electromagneticeigenmode of the medium

Silver

slab


plasmon-polaritons

ω ωε

<<0

p

ωpω

ε


Silver

slab


plasmon-polaritons

ω ωε

<<0

p

ωpω

ε

ω =/ x ck

xk

ωωp

light line

Surface plasmon-polariton

slow waves of short wavelength!

cannot interact with propagating waves!


do interact with the evanescent,near field components of the Fourier spectrum of an object!

Silver

slab


plasmon-polaritons

ω ωε

<<0

p

ωpω

ε

slow waves of short wavelength!

cannot interact with propagating waves!

ω =/ x ck

xk

ωωp

light line


SLAB: TWO SURFACES

ARBITRARY VALUES OF μ,ε

SLAB, ARBITRARY VALUES OF μ,ε

In order to achieve a cut-off at 10 k0 (for resolution λ/10) a slab of d=0.1 λ tolerates a loss of 0.002,a slab of d=0.67 λ tolerates a loss of not more than 10-19!

:

Near-perfect? Near-sighted!

Podolskiy and Narimanov (2005)Smith et al. (2003), French et al. (2006).

ε μ− −1, 1

Silver lens: Near-perfect?

d=40 nmthick!

d=20 nm

Silver lens: perfect without losses? No!

T. Taubner et al., SCIENCE 313, 2006

Experiment: SiC

Superlens

for IR

• Resolution λ/20 (λ=10.85μm)• Optical Signal Processing

Experiment: Silver Superlens

for UV

N Fang et al., Science 308, 2005

Shamonina et al., Electron. Lett. (2001)

Silver slab: Poynting

vector optics

ε=-1-jα, α=10-4

Shamonina et al., PIERS 2002

NEGATIVE REFRACTION?! (only ε

is negative!)

NEGATIVE REFRACTION?! (only ε

is negative!)

ω =/ x ck

xk

ω

ωp

light line


Poynting vector

Group velocity near zero!

60nm slab too thick for λ=360nm!

Shamonina et al., Electron. Lett. (2001)

Multilayered superlens

Microscopic picture: „Poynting vector

optics”

λ=360nm, ε=-1-jα, α=0.1 ε=-1-jα, α=10-4

Shamonina et al., PIERS 2002

Multilayered superlens


optics”

Numerical simulation (CST Microwave Studio)

E.Tatartschuk (Erlangen)

Magnifying multilayered superlens

flat „near-sighted“ lens

The image is not magnified

!

cylindrical „far-sighted“ lens

The image is magnified and can be captured with an optical microscope

Microscopic picture: „Poynting vector optics“

Theory: Jacob et al. Opt. Expr. (2006), Salandrino and Engheta, Phys. Rev. B (2006)Experiment: Liu et al. Science (2007), Smolyaninov et al., Science (2007)

Magnifying multilayered superlens: simulation


optics”

Numerical simulation (CST Microwave Studio): E.Tatartschuk (Erlangen)

Cylindrical multilayered superlens: experiment

I.I. Smolyaninov et al., Science 315 (2007)

Cylindrical multilayered superlens: experiment

Z. Liu et al., Science 315 (2007)

Physics

Extension of electromagnetism

“inverse”

electromagnetic phenomena

-

inverse Snell’s law (negative refraction of rays)

-

inverse Doppler shift, Cherenkov

radiation, Goos-Haenchen

shift

-

“growing”

evanescent waves

Metamaterials

Applications

-

“Perfect lens”, subwavelength

imaging in photonics

-

Invisibility and cloaking

-

“Nano-circuits”, miniaturisied

waveguide components

-

Medical imaging


Summary




Summary



• Near field imaging with magnetic metamaterials(Anna Radkovskaya’s lecture)

Invisibility

and cloaking

Positive and negative Goos-Haenchen

Shift

Ziolkowski 2003

Basics of single negative and double negative...

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