Metamaterials as Effective Medium

Negative refraction and super-resolution

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

Limits of hyperbolic 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 for stratified medium – 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

(2) Boundary conditions

0 0 1 1

0 0 1 1

( ) ( )1 1

( ) ( )

d diel d diel

d diel d diel

ik d ik d

ik d ik dd d m m

diel metal

C e D e A BH x D H x D

C ik e D ik e Aik B ikE x D E x D

0 1

0 1

1

1 1

1 1

d diel d diel

d diel d diel

d diel d diel

d diel d

ik d ik d

ik d ik dm md d

metal metaldiel diel

ik d ik d

ik d ik dm m d d

metal metal diel

e eC A

ik ikik e ik eD B

e e

ik ik ik e ik e

0 1

0 1

diel

diel

C A

D B

0 1

0 1cell

A AM

B B

1 11 1 1 1d diel d diel m m m m

d diel d diel m m m m

ik d ik d ik d ik d

ik d ik d ik d ik dcell m m d dd d m m

metal metal diel dieldiel diel metal metal

e e e e

M ik ik ik ikik e ik e ik e ik e

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

Effective medium vs. periodic multilayer

2

222

c

kk

x

z

z

x

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

500nm

2 4 6 8

1

2

3

4

5

6

0 2 4 60

1

2

3

4

5

9

1m

d

365nm

2 4

3 2

.

.m

d

0/ kkx

0k

kz

0/zk k

0

xk

k

30nm

Effective medium vs. periodic multilayer

2

222

c

kk

x

z

z

x

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

0.5 1 1.5 2 2.5 3 3.5

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

500nm 9

1m

d

365nm

2 4

3 2

.

.m

d

70nm

11

Surface Plasmons coupling in M-D-M

Metal Metal

• symmetric and anti-symmetric modes• anti-symmetric mode cutoff• single mode “waveguide” • deep sub-

MetalMetal MetalMetal

H-field

12

Surface Plasmons 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•No cut-off through the metal

13

Plasmonic waveguide coupling

Metal Metal

No counterpart in dielectrics!

High contrastDielectric WG – E-field confinement

Michal Lipson, OL (2004)

nlow=1nhigh=3.5

Diffractionlimit

14

High spatial frequencies with low loss?

Metal Metal

• No limitations on the proximity• deep sub-“waveguide” • symmetric and anti-symmetric modes

Multi-layer plamonic metamaterial

15

Modes in M-D multilayer – beyond EMA

t

H

cE

t

E

cH

1

1

Maxwell Equations

Eit

E

Hit

H

Time-harmonic solution

• Sub-WL scale layers

• Strong variation in the dielectric function (sign and magnitude)

• Paraxial approximation Is not valid!

Hc

iE

Ec

iH

Maxwell Equations

16

Linear modes in M-D multilayer

TM mode yHH ˆ

zExEE zx ˆˆ

z

H

k

iEx

0

x

H

k

iEz

0

Hikx

E

z

E zx0

ck

0

Hikx

H

xk

i

z

Ex0

0

1

xEikz

H 0

HikjEE

EE

kji

zxxz

zx

zyx 0ˆ

0

ˆˆˆ

kEiEikkHiH

H

kji

zxxyxyz

y

zyxˆˆˆˆ

00

ˆˆˆ

0

17

Modes in M-D multilayer

Looking for SPATIAL eigenmodes (not varying with propagation)

ziexHzxH )(~

),(

zix exEzxE )(

~),(

xx E

H

E

HM ~

~

~

~ˆ

01

01ˆ

20

20

0k

xx

k

kM

An “eigenvalue” problem

First-order equation for the vector

xx E

H

kxx

k

k

iE

H

z 01

0

20

20

0

iz

xE

H~

~

18

Plasmonic Bloch modesKx=/D

MagneticTangentialElectric

-1

1

Kx=

MagneticTangentialElectric

0.97

1

-1

1

Spatial frequency limited by periodicity large K available even far from the resonance

0k

0/ kkx

19

Plasmonic Bloch modesKx=/D

MagneticTangentialElectric

-1

1

Kx=

MagneticTangentialElectric

0.97

1

-1

1

• Symmetry opposite to H

z

H

k

iETransverse

0

x

H

k

iE gential

0tan

Same symmetry as H

Symmetric in metalAntisymmetric in metal

Symmetric in dielectric Symmetric in dielectric

20

Anomalous Diffraction and Refraction

3400

3500

3600

3700

3800

3900

4000

-600

-400

-200 0

200

400

600

800

-6000

-4000

-2000

02000

4000

6000

Anomalous diffraction Normal diffraction

2

2

x

z

k

kD

d

d2

2

xd

d ↔Diffraction Direction of energy

gvHES

Negative refraction without actual negative index

21

dm

dmdyx

dmz

pp

pp

pp

)1()1(

)1()1()(

)1()(//

2D analog: Metal Nanowires array

Podolskiy, APL 89, 261102 2006

Averaged dielectric response

2

222

c

kk

x

z

z

x

Hyperbolic dispersion!

Show anomalous properties in all directions

Broad-band response

Large-scale manufacturing

d,r<<

d

22

Metal-dielectric multilayers – dispersion curve

Singleband

23

At longer wavelengths metal permittivity grows (negatively)Less E-field in the metal

Less lossLess coupling (tunneling)

Less diffractionkz decreases

Resolution limited by the periodkx/k0 increases

Periodic metal-dielectric composites

Dispersion relation

Short • strong coupling• Large wavenumber• Broad range

Longer • weak coupling• moderate wavenumber• Large Bloch k-vectors• lower loss

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 near-field to near-field

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.

Maxwell’s equations in cylindrical coordinates

, ,zH E E z

0

0

1 10

0 0

1 1 10

0

z z

z

z

z

H i E H H ik E Ec

H

z

E i H E E ik Hc

E E

>>>>>>>>>>>>>>

>>>>>>>>>>>>>>

0

0

0

1 11

2

3

z

z

z

E E ik H

iE H

k

iE H

k

00 0

2 202

1 1z z z

zz z

i iH H ik H

k k

HH k H

TM solution Isotropic case

Maxwell’s equations in cylindrical coordinates

zH R

2 202

2 20

''

R R k R

dRk

R d

0ime

Separation of variables:

2 2 20 0z zH m k H

Solution given by Bessel functions

1. Bessel Function of the First kind: 0mJ k [cosine functions in Cartesian coordinates]

2. Bessel Function of the Second kind: 0mY k [sine functions in Cartesian coordinates]

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

Maxwell’s equations in cylindrical coordinates

1. Bessel Function of the First kind: 0mJ k [cosine functions in Cartesian coordinates]

2. Bessel Function of the Second kind: 0mY k [sine functions in Cartesian coordinates]

3. Hankel Function of the First kind: 1

0mH k [expanding cylindrical wave 1 ikrer

].

4. Hankel Function of the Second kind: 2

0mH k [converging cylindrical wave 1 ikrer

]

5. Modified Bessel Function of the First kind: 0mI k [ equivalent toxe ]

6. Modified Bessel Function of the Second kind: w 0mK k [ equivalent to xe ]

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