Metamaterials as Effective Medium

Negative refraction and super-resolution

Previously seen in “optical metamaterials”

Sub-wavelength dimensions with SPP

Negative index

Use of sub-wavelength components to create effective response

Super-resolution imaging

Metamaterials as sub-wavelength mixture of different elements

New type of artificial dielectrics

Negative refraction in non-magnetic metamaterials

Super-resolution imaging

zz

yy

xx

00

00

00

0

dm dd

When two or more constituents are mixed at sub-wavelength dimensions

Effective properties can be applied

Pendry’s artificial plasma

Motivation: metallic behavior at GHz frequencies

Problem: the dielectric response is negatively (close to) infinite

Solution: “dilute” the metal

Lowering the plasma frequency, Pendry, PRL,76, 4773 (1996)

2

2

a

rnn eneff

The electrons density is reduced

m

enep

0

22

eff

effeffp m

en

0

22

,

* The effective electron mass is increased due to self inductance

Simple analysis of 1D and 2D systems

Periodicity or inclusions much smaller than wavelength

2+1D or 1+2D (dimensions of variations)

Effective dielectric response determined by filling fraction f

a

a

1D-periodic (stratified) 2D-periodic (nano-wire aray)

Averaging over the (fast) changing dielectric response

3D?

Stratified metal-dielectric metamaterial

Two isotropic constituents with bulk permittivities

Filling fractions f for 1,1-f for 2

2 ordinary and one extra-ordinary axes (uniaxial)

2 effective permittivities

a

a

1

ll

llll

For isotropic constituents

effective fields

iii ED

21 )1( EffEEE aveeff

21 )1( DffDDD aveeff

Note: parallel=ordinary

2

Stratified metal-dielectric metamaterial: Parallel polarization

a

llll

EEE 21

EEffEEE aveeff )1(

EEfEfDffDDD effaveeff 2121 )1()1(

k

E

21 )1( ffll

Boundary conditions

Stratified metal-dielectric metamaterial: Normal polarization

a

llll

DDD 21

21 )1( EffEEE aveeff

DDffDDD aveeff )1(

E

21

)1(1

ff

effeff

DDf

DfE

21

)1(

Nanowire metal-dielectric metamaterial

Two isotropic constituents with bulk permittivities

Filling fractions f for 1,1-f for 2

2 ordinary and one extra-ordinary axes

2 effective permittivities

a

1

ll

Note: parallel=extraordinary

2

ll

Nanowire metamaterial: Parallel polarization

E

ll

EEE 21

EEffEEE aveeff )1(

EEfEfDffDDD effaveeff 2121 )1()1(

21 )1( ffll

Nanowire metamaterial: Normal polarization polarization

E

ll

• More complicated derivation

• Homogenization (not simple averaging)

• Assume small inclusions (<20%)

• Maxwell-Garnett Theory (MGT)

dm

dmdyx ff

ff

)1()1(

)1()1()(

(metal nanowires in dielectric host)

Strongly anisotropic dielectric Metamaterial

ll

00

00

00

0

dm

dmdyx ff

ff

)1()1(

)1()1()(

dmll ff )1(

ll

llll

00

00

00

0 ll

ll

21

)1(1

ff

dmll ff )1(

For most visible and IR wavelengths dm 0,0 ll

dm

dmdyx

dmz

pp

pp

pp

)1()1(

)1()1()(

)1()(//

Effective permittivity from MG theory

Al2O3 matrix

Ag wires

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2-20

-10

0

10

20

30

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2-10

-8

-6

-4

-2

0

2

4

Broad band

um

um

//

Example: nanowire medium medium

60nm nanowire diameter

110nm center-center wire distance

Wave propagation in anisotropic medium

zz

yy

xx

00

00

00

0

Maxwell equations for time-harmonic waves

)ˆˆˆ(0

0

zEyExEED

HEk

DHk

xzzyyyxxx

)ˆˆˆ(200 zEyExEkHkEkk xzzyyyxxx

0

)(

)(

)(

2220

2220

2220

z

y

x

yxzzzyzx

zyzxyyyx

zxyxzyxx

E

E

E

kkkkkkk

kkkkkkk

kkkkkkk

Uniaxial yyxx

Det(M)=0, yyxx 0222

20

2

x

z

z

yxx

kkkkk

Wave propagation in anisotropic medium

z

x

x

00

00

00

0

0 20

22220

222

k

kkkkkkk

x

z

z

yxxzyx

Ordinary waves (TE) Extraordinary waves (TM)

E

H H

E• Electric field along y-direction

• does not depend on angle

• constant response of x

• Electric field in x-z(y-z) plan

• Depend on angle

• combined response of x,z

Extraordinary waves in anisotropic medium

z

x

x

00

00

00

0

20

22

kkk

x

z

z

x

kx

kz

kx

kz

isotropic medium

zx

anisotropic medium

zx

20

22 kkk zx

)(nn

kx

kz

‘Hyperbolic’ medium

For x<0

20

22

kkk

x

z

z

x

Energy flow in anisotropic medium

kx

kz

kx

kz

isotropic medium

zx

anisotropic medium

zx

20

22 kkk zx

kx

kz

normal to the k-surface

20

22

kkk

x

z

z

x

S

and are not parallelk

‘Indefinite’ medium

* Complete proof in “Waves and Fields in Optoelectronics” by Hermann Haus

S

and are not parallelk

S

Is normal to the curve!

Refraction in anisotropic medium

z

x

x

00

00

00

0

2

222

c

kk zx

What is refraction?

kx

kz

kx

kz

Hyperbolicair

0,0 zx

02 0

20,

, Hk

Sx

zrzr

02 0

20,

, Hk

Sz

xrxr

Conservation of tangential momentum

Negative refraction!

dm

dmdyx

dmz

pp

pp

pp

)1()1(

)1()1()(

)1()(//

Effective permittivity from MG theory

Al2O3 matrix

Ag wires

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2-20

-10

0

10

20

30

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2-10

-8

-6

-4

-2

0

2

4

Broad band

um

um

//

Refraction in nanowire medium medium

Negative refraction for >630nm

dmyx pp )1()(//

Refraction in layered semiconductor medium

•SiC

•Phonon-polariton resonance at IR

Negative refraction for 9>>12m

Hyperbolic metamaterial “phase diagram”

20

22

kkk

x

z

z

x

dmll ff )1(

21

)1(1

ff

Ag/TiO2 multilayer system

0 0,x z

0 0,x z

0 0,x z

0 0,x z

dielectric Type I Type II

We choose propogation by

Effective medium at different regimes

dm

dmd

dm

ff

ff

ff

)1()1(

)1()1(

)1(//

x

propagation

• extreme material properties

• epsilon near-zero

• Diffraction management

• Resolution limited by loss

dm

• Low-loss

• Broad-band

• resolution limited by periodicity

x

propagation

dm

X=parallelSuitable for stratified medium

X=normal (suitable for Nanowires)

0,0 zx 02

2

22

z

xxz

k

ck

Conditions Normal-X direction (kx<</D)

x

propagation

dm 3

X=normal (suitable for Nanowires)

02

2

22

ll

xz

k

ck

03

3

2

0//

dm

dmd

dm

2

1f

• Low loss

• moderate values

• Limited by periodicity

kx

kz

2

22

2

22

2

3

c

k

ck d

ll

xz

cd

2

3

c

d2

3

• Low diffraction management

• diffraction management improves with em

•no near-0

Conditions for Normal Z-direction

x

propagation

dm

02

2

22

x

llz

k

ck

02

1

02//

dm

dm

dm

0//

d

kx

kr

0zk For large range of kx • Good diffraction management

• near-zero

• Limited by ?

Effective medium with loss…

x

propagation

dm

02

2

22

x

llz

k

ck

md

mdd

mdm

i

i

i

2

32

2

//

dm

02

2

22

ll

xz

k

ck

03

3

2

//

dm

dmd

mdm

ll

xz

k

ck

2

2

22

d2

3

mmm i

)Re()Im( zz kk

(Long wavelengths)

Very low loss at low kModerate loss at high k

High loss!

End of class

Limits of indefinite medium for super-resolution

Open curve vs. close curve

No diffraction limit!

No limit at all…

Is it physically valid?

kx

kr

20

22

kkk

x

z

z

x

0,0 zx

xk

z

xxz

kkk

220

• Reason: approximation to homogeneous medium!

• What are the practical limitations?

• Can it be used for super-resolution?

Exact solution – transfer matrix

2 2 2 21 1

arccos cos cos sin sin2

m diel diel mx diel d m m diel d m m

diel m diel m

k kK k d k d k d k d

D k k

Z

X

...

Unit Cell

X=nD X=(n+1)D

dm

X=nD+d

mmA

mB

mC

mD

1mA

1mB

2 2 2 20 0,m m z diel diel zk k k k k k

1

1

n ncell

n n

A AM

B B

2 2 2 2

2 2 2 2

2 2 2 2

(1,1) cos sin2

(1,2) sin2

(2,1) sin2

m m

m m

m m

ik d m diel diel mdiel d diel d

diel m diel m

ik d m diel diel mdiel d

diel m diel m

ik d m diel diel md

diel m diel m

k kiU M e k d k d

k k

k kiV M e k d

k k

k kiW M e k

k k

2 2 2 2

(2,2) cos sin2

m m

iel d

ik d m diel diel mdiel d diel d

diel m diel m

d

k kiX M e k d k d

k k

Exact solution – transfer matrixZ

X

...

Unit Cell

X=nD X=(n+1)D

dm

X=nD+d

mmA

mB

mC

mD

1mA

1mB

( ) ( )

( ) ( )

( 1 ) ( 1 )1 1

( ) 1

1 1

m m

d metal d metal

m m

ik x mD ik x mDm m metalik x d mD ik x d mD

m m metal

ik x m D k x m Dm m metal

A e B e mD x mD d

H x C e D e mD d x m D

A e B e m D x m D d

0 0

( ) ( )0 0

( ) ( )1 1

0

( )

m m

d metal d metal

m m

ik x ik xm m metal

metal

ik x d ik x dd d metal

diel

ik x D k x Dm m metal

metal

iA ik e B ik e x d

iE x C ik e D ik e d x D

iAik e B ik e D x D d

(1) Maxwell’s equation

2 2 2 20 0,m m z diel diel zk k k k k k

Exact solution – transfer matrixZ

X

...

Unit Cell

X=nD X=(n+1)D

dm

X=nD+d

mmA

mB

mC

mD

1mA

1mB

0 0 0 0

0 0 0 0

( ) ( )1 1

( ) ( )

m m m m

m m m m

ik d ik d

metal metalik d ik d

m m d dmetal metalmetal diel

A e B e C DH x d H x d

A ik e B ik e C ik D ikE x d E x d

0 0

0 0

1 1m m m m

m m m m

ik d ik d

ik d ik dd dm m

diel dielmetal metal

e eA C

ik ikik e ik eB D

1

0 0

0 0

1 1 m m m m

m m m m

ik d ik d

ik d ik dd d m m

diel diel metal metal

e eA C

ik ik ik e ik eB D

(2) Boundary conditions

Exact solution – transfer matrixZ

X

...

Unit Cell

X=nD X=(n+1)D

dm

X=nD+d

mmA

mB

mC

mD

1mA

1mB

(3) Combining with Bloch theorem

1

1

1

1

0 0x

x

x

x

m mcell iK D

m m m miK Dcell iK D

m mm miK D

m m

A AM

B B A AU e VM e

B BW X eA Ae

B B

det 0x

x

iK D

iK D

U e V

W X e

2

12 2

xiK D U X U Xe i

2 2 2 21 1

arccos cos cos sin sin2

m diel diel mx diel d m m diel d m m

diel m diel m

k kK k d k d k d k d

D k k

Beyond effective medium: SPP coupling in M-D-M

Metal Metal

Symmetric: k<ksingle-wg Antisymmetric: k>ksingle-wg

• “gap plasmon” mode• deep sub-“waveguide” • symmetric and anti-symmetric modes

Beyond effective medium: SPP coupling in M-D-M

t

H

cE

t

E

cH

1,

1 • TM nature of SPPs

• Calculate 3 fields

Eigenmode problem:

zixx exEzxE )(

~),(

ziyy exHzxH )(

~),(

zizz exEzxE )(

~),(

20)

1(ˆ k

xxH

Hamiltonian-like operator:

0)(ˆ01

)(ˆ20

0 xH

k

kxM

)()()(ˆ xxxM

T

xy EH )~

,~

(

• Eigen vectors EM field

• Eigen values Propagation

constants

z

x

metal dielectric

• Abrupt change of the dielectric function• variations much smaller than the wavelength• Paraxial approximation not valid! •Need to start from Maxwell Equations

Plasmonic Bloch modes

Kx=/D

MagneticTangentialElectric

-1

1

Kx=

MagneticTangentialElectric

0.97

1

-1

1

0k

kz

0/ kkx

Ag=20nm Air=30 nm =1.5m

Metamaterials at low spatial frequencies

2

222

c

kk

x

z

z

x

The homogeneous medium perspective

Dk

D

dm

dmdyx

dmz

pp

pp

pp

)1()1(

)1()1()(

)1()(//

Averaged dielectric response

Hyperbolic dispersion!

Can be <0

Metamaterials at low spatial frequencies

2

222

c

kk

x

z

z

x

The homogeneous medium perspective

Dk

D

dm

dmdyx

dmz

pp

pp

pp

)1()1(

)1()1()(

)1()(//

Averaged dielectric response

Hyperbolic dispersion!

Can be <0

0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

Use of anisotropic medium for far-field super resolution

Superlens can image near- to near-field

Need conversion beyond diffraction limit Multilayers/effective medium?

Can only replicate sub-diffraction image by diffraction suppression

Solution: curve the space

Conventional lens

Superlens

• Metal-dielectric sub-wavelength layers

• No diffraction in Cartesian space

• object dimension at input a

• is constant

•Arc at output

dm dd

The Hyperlens

rZ

X

222

0r

r

kkk

r

a

r

RaRA

0 ll

Magnification ratio determines the resolution limit.

Optical hyperlens view by angular momentum

• Span plane waves in angular momentum base (Bessel func.)

imm

m

mikx ekrJie )(

• resolution detrrmined by mode order

• penetration of high-order modes to the center is diffraction limited

• hyperbolic dispersion lifts the diffraction limit

•Increased overlap with sub-wavelength object