Negative refraction in photonic crystals

Mike KaliteevskiDurham University

Outine•Photonic Crystals: Introduction

•Negative refraction in left-handed material•Non-diffracting beams •Electromagnetic wiggler

Bragg reflector

r

n1 n2

t

21

21

nn

nnr

21

12

nn

nt

Bragg reflector

n1r n1

d2

n2

tndrndt

2

02

0

2exp

2exp

0

22

02

0

4122

nd

ndnd2

02 4n

d

Bragg reflector

r

n1 n2 n1n2

d2d2

1

01 4n

d

Periodic sequence of the pairs of quarterwave layers is the Bragg reflector. The waves, reflected from different boundaries experience positive interference (enforce each other).

Bragg reflector

BRBRBR ir /)(exp

0,8 1,0 1,20,0

0,2

0,4

0,6

0,8

1,0

-

Arg

(r)

R

Energy, eV

210

21

nnn

nn

Bloch theorem. Dispersion relations

)exp()()( iKzzuzE KK

H

EiKD

H

ET D )exp(ˆ )(

0ˆ)exp(ˆdet )( IiKDT D

KDTT DD cos2ˆˆ )(22

)(11

)sin()sin(2

1)cos()cos()cos( 022011

1

2

2

1022011 kdnkdn

n

n

n

nkdnkdnKD

0

0

0

0 Densityof modes

Densityof modes

k

Im(k)

Im(k)

k /D

/D0

0

1

1 0

0Reflectivity

Reflectivity

BR = c/(n1d1+n2d2)

BRBR

Formation of the photonic band gap in periodic structures

Probability of spontaneous emission

22

22EuedlW

Probability of spontaneous emission

L

LEEnergy 22/

22

22EuedlW

/L )2/(2 LE

Microcavity

Microcavity

L

nRR

n2

n1

Electric field

Magnetic field

0

1

/0

R

Probability of spontaneous emission

L

L

2D Photonic crystal

1D photonic crystal

2D photonic crystal

2D photonic crystal

Dispersion relations in 2D photonic crystal

k

)exp()()( rkirvrH

kk

)()( arvrvkk

)()( arr

Plane waves method

a

)()()(

12

2

rHc

rHr

rGiGr G

exp)()(

1

)exp()()( rkirvrHkk

)(

1

)(

1

arr

)()( arvrvkk

rGkiGkHrHG

k

exp),()(

Bloch theorem

Wave equation

G

Lattice vector

Reciprocal lattice vector

Plane waves method

G

)()()(

12

2

rHc

rHr

Wave equation

Reciprocal lattice vector

),()',(')''(2

2

'

GkHc

GkHGkGkGGG

k

2D photonic crystals

H E

0.0

0.1

0.2

0.3

0.4

0.5Fr

eque

ncy,

c/d

K KM

TE

K M

0.0

0.1

0.2

0.3

k

k

Freq

uenc

y, c

/d

K KM

K M

TM

Disperison relations

H

E

Complete PBG

Transmissiom of light

d=50m

TT

TT

1d=60m

d=70m

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

1

1

0

0

0

d=80m1

0

f, THzExperiment Modelling

PC spectral filter

0.4 0.8 1.2 1.6 2.0 2.4

T

1

0

f, THz

D

G

a

a

Defects in photonic crystals

0,0

0,1

0,2

0,3

0,4

0,5

Fre

quen

cy, c

/d

m = 1

m = 1

m = 2

m = 2

m = 0

m = 3

Photonic crystal waveguide

PC Waveguide

OE_15_12982

3D Photonic crystals

Transmission of light and bandstructure in opals and inverse opals.

Photonic microstructures in nature

Negative refraction in left-handed material

Right - hand materials

2

000 Enk

kHE

00

kS

0grv

0 n

•Usual electromagnetic word

Left - hand materials

V.G.Veselago, Electrodinamics of the materials with negative dielectric and magnetic constant (1967)

2

000 Enk

kHE

00

kS

0grv

0 n

•Inversed Doppler effect•Inversed Vavilov – Cherenkov effect•Negative refraction

Refraction

Kτ

Kτ

Kτ

Kτ

kS

Positive refraction

Kτ

Kτ

kS

Negative refraction

Left - hand materials

kS

0grv

Negative refraction

Flat Lense

L

n1 n2

A

D

ALD

Flat lence

n1

n2 =-n1

Superlence ???

L

n1 n2

A

D

ALD

n1

n2 =-n1

Comment: John Michael Williams, Some Problems with Negative Refraction, Phys. Rev. Lett. 87, 249703 (2001) Comment: G. W. 't Hooft, Comment on “Negative Refraction Makes a Perfect Lens”, Phys. Rev. Lett. 87, 249701 (2001) Reply: M. Nieto-Vesperinas and N. Garcia, Nieto-Vesperinas and Garcia Reply:, Phys. Rev. Lett. 91, 099702 (2003)

J. B. Pendry , Negative Refraction Makes a Perfect Lens, Phys. Rev. Lett. 85, 3966 - 3969 (2000)

Автор ввел понятие "суперлинза", ...утверждая, что для этого устройства отсутсвует дифракционный предел. Наверное, наиболее убедительное доказательство ошибочности подобного рода утверждений можно найти в ... [ В.Г.Веселаго, УФН, 173 (7) 790 (2003) ]

With a conventional lens sharpness of the image is always limited by the wavelength of light. An unconventional alternative to a lens, a slab of negative refractive index material, has the power to focus all Fourier components of a 2D image, even those that do not propagate in a radiative manner. Such “superlenses” .....

Realization of left-hand materials

MetamaterialsPhotonic crystals

Negative refraction in photonic crystals

Band 2

fX

1.0

1.2

1.4

1.6

1.8

Band 1

fJ

XJ

f, T

Hz

vgr<0

vgr>02D hexagonal metallic PC, D =200 microns, d = 60 microns

Negative refraction in 2D hexagonal photonic crystals

Band 2

fX

1.0

1.2

1.4

1.6

1.8

Band 1

fJ

XJ

f, T

Hz

PRF

NRF

IFSOURCE

(a)

PRF

NRF

IFSOURCE

(a)

PRF

NRF

IFSOURCE

(b)

Refraction of wave in photonic crystal prism

vgr<0Band 2

fX

1.0

1.2

1.4

1.6

1.8

Band 1

fJ

XJ

f, T

Hz

vgr>0

Refraction of wave in photonic crystal prism

0.5 1.0 1.5 2.0

T1

0

f, THz

PRF

NRF

IFSOURCE

(a)

PRF

NRF

IFSOURCE

(a)

PRF

NRF

IFSOURCE

(b)

Refraction of wave in photonic crystal prism

0.5 1.0 1.5 2.0

T

1

0

f, THz

PRF

NRF

IF

SOURCE

(c)

PRF

NRF

IF

SOURCE

(c)

Band 2

fX

1.0

1.2

1.4

1.6

1.8

Band 1

fJ

XJ

f, T

Hz

PRF

NRF

IFSOURCE

Refraction of wave in photonic crystal prism

1n

Experimental study of negative refraction

Experimental study of negative refraction of THz

using QCL

Experimental study of negative refraction of THz

using QCL

SIGNAL WITHOUT SAMPLE

Negatively refracted beam

Non-diffracting beams

W

W sin

L

l1l2

A

D

0 AnLD /

An

LALD

22

2

sin

sin1

tan

tan

nsin D

Non-diffracting beams

n1 n2 n1

21 nn

11 n02 n

0 AnLD /

An

LALD

22

2

sin

sin1

tan

tan

nsin D

Non-diffracting beams

21 n

11 n02 n

L

l1l2

A

D

L

l1l2

A

D

n1 n2 n1

1.0

1.2

1.4

1.6

f, T

Hz

-0.5 0 XJneff

L

L

L

D0

D0

A

(c)

(b)

(a)

A

16275 m

4000

m

Non-diffracting beams

L

L

L

D0

D0

A

(c)

(b)

(a)

A

16275 m

4000

m =185m

=180m

-2 -1 0 1 2

I n

t e n

s i

t y, a

. u.

=175m

Position, mm

(a)

(b)

(c)

Non-diffracting beams

1.0

1.2

1.4

1.6

f, T

Hz

-0.5 0 XJneff

Negative refraction in 1D photonic crystals

n1 n2

d1 d2

Problem: Veselago lens based on 1D PC Bragg reflector does not work.

Because system is anisotropic: negative effective mass is required for negative refraction, and for 2nd , 4th , etc bands mz<0, but always mx>0

0 0

0

2 2

( , ) [exp( ) exp( )]exp( )

2

( / )

Ry p p p p

p

p

p p

E x z i x R i x iK z

pK K

D

c K

n1 n2

d1 d 2

x

zK

0

1

( , ) ( ) exp( )Brm m m

m

E x z a u z i x

0 0

0

2 2

( , ) [exp( ) exp( )]exp( )

2

( / )

Ry p p p p

p

p

p p

E x z i x R i x iK z

pK K

D

c K

Field of the wave in the structure

Modes in Bragg reflector

0),,( 2 Kf

1 2 1 2

1( ) cos( )cos( ) sin( )sin( ) cos( ) 0

2f d d d d KD

2 2 21( / )n c

2 2 22 ( / )n c

Amplitude of waves

* * *1,1 0,1 ,1

* * *, 0, ,

1 1,1 1 ,

1 100 0,1 1 0 0,

1 1,1 1 ,

1 0

0 1

1 0 0

0 1

1 0

0 0 1

P P

mP M M P M

PP P P P M M

M M

PP P P P M M

aJ J J

aJ J J

RJ J

RJ J

RJ J

*0,1

*0,

0

1

0

M

J

J

*0[ ]p p pn n

p

R J a

0

1

[ ]q q q m m qmm

R a J

-3000 -2500 -2000 -1500 -1000 -500 0-0.5

0.0

0.5

1.0

1.5

2.0

2.5

log 1

0(|a

m|2 )

f (2 )

21E-3

0.01

0.1

1

-3000 -2500 -2000 -1500 -1000 -500 0 500 1000 1500

-4

-2

0

2

4

log 1

0(|a

m|2 )

f (2

)

21E-6

1E-5

1E-4

1E-3

0.01

0.1

1

10

High contrast:n1=3.7n2=1

Low contrast:n1=1.4n2=1.8

-40 -20 0 20 40

-3

-2

-1

0

1

2

3

KD

(m -1)

6

-40 -30 -20 -10 0 10 20 30 40

-3

-2

-1

0

1

2

3

KD

(m -1)

5

-30 -20 -10 0 10 20 30

-3

-2

-1

0

1

2

3

KD

(m -1)

4

0 1 2 30.0

0.5

1.0

1.5

2.0

2.5

3.0

45

6

3

2

( eV

)

KD

1-20 -15 -10 -5 0 5 10 15 20

-3

-2

-1

0

1

2

3

KD

(m -1)

3

-5 -4 -3 -2 -1 0 1 2 3 4 5-2

-1

0

1

2

KD

(m -1)

1

-8 -6 -4 -2 0 2 4 6 8

-3

-2

-1

0

1

2

3

KD

(m -1)

2

0 1 2 30.0

0.5

1.0

1.5

2.0

2.5

3.0

( eV

)

KD

-5 -4 -3 -2 -1 0 1 2 3 4 5-2

-1

0

1

2

KD

(m-1) aa

0 1 2 30.0

0.5

1.0

1.5

2.0

2.5

3.0

( eV

)

KD

-30 -20 -10 0 10 20 30

-3

-2

-1

0

1

2

3

KD

(m-1)

bb

Negative refraction

cc

0.0 0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

1.5

, <S

z>K

frequency (eV)

photonicband gap

negativerefractionarea

DKS z 0sin~

Normal channelling

Normal channelling

-3000 -2500 -2000 -1500 -1000 -500 0-0.5

0.0

0.5

1.0

1.5

2.0

2.5

log 1

0(|a

m|2 )

f (2 )

21E-3

0.01

0.1

1

Low contrast:n1=1.4n2=1.8

xS z )(cos 21

Electromagnetic wiggler

Electromagnetic wggler

Conclusion:

• One can hardly make Veselago lense based 1D photonic crystal

• But there are some interesting effects like “electromagnetic snake”, normal channeling, etc.