Photonic crystals (I) Bloch's theorem,

photonic band structure, and energy flow

Pi-Gang Luan & Wave Engineering Lab

( 欒丕綱 & 波動工程實驗室 )Institute of Optical Sciences

National Central University

( 中央大學光電科學研究所 )

Photonic/Sonic CrystalsPhotonic/Sonic Crystals

1D Crystal

2D Crystal

3D Crystal

3D Phononic Crystal?

Photonic/Sonic Band Structure

90

Applications

Photonic crystals as optical components

P. Halevi et.al.Appl. Phys. Lett.75, 2725 (1999)

See alsoSee alsoPhys. Rev. Lett. Phys. Rev. Lett. 8282, 7, 719 (1999)19 (1999)

Periodic function, Fourier Series and Reciprocal Lattice

: ( ) ( ) ( ), , : period f x a f x f x R R na a 1D periodic function

Try ( ) ( )exp( )G

f x C G iGxWe have exp( ) 1 exp( 2 )iGa i n

2 2, , ( ) expn

n

nG nb b f x C inbx

a a

0 0

1exp exp ( )exp

a a

mn nimbx inbx dx a C f x inbx dxa

0

1Or ( ) ( )exp

aC G f x iGx dx

a

1 2

1 1 2 1 1 2

: ( ) ( ) ( ) ( )

, and : fundamental translation vectors

f f f f

n n

2D periodic function r a r a r r R

R a a a a

1 2(Primitive) Cell Area: | |cA a a

1 2Try ( ) ( )exp( ) exp( ) exp( ) 1f C i i i G

r G G r G a G a

1 1 2 2Define 2 ; , 1,2i j ijn n i j G b b a b

1 1 2 2 1 2In a cell: ; 0 1, 0 1 r a a

1 2

1 2

1 2 1 1 2 2Thus ( ) ( , ) exp 2n nn n

f f C i n n

r

21 1 2 2

1 21 2

| ( ) ( ) |Using the fact:

| |c

d dd rd d

A

a a

a a

1 2

2

1 1

1 2 1 1 2 2 1 20 0

1We have ( ) ( )exp( )

Or ( , )exp 2

c cell

n n

C f i d rA

C f i n n d d

G r G r

31Fourier Component: ( ) ( )exp( )

c cell

C f i d rV

G r G r

1 2 3Cell Volumn: | ( ) |cV a a a

1 1 2 2 3 3Reciprocal Lattice: , 2i j ijn n n G b b b a b

2 3 3 1 1 21 2 3RL Bases: 2 , 2 , 2

c c cV V V

a a a a a a

b b b

Fourier Expansion: ( ) ( )exp( )f C i G

r G G r

1 1 2 2 3 3: ( ) ( ), f f n n n 3D periodic function r r R R a a a

1 1 1

1 2 3 1 1 2 2 3 3 1 2 3

0 0 0

( ) ( , , )exp 2C f i n n n d d d G

1 1 2 2 3 3 1 2 3Point in a Unit Cell: , 0 , , 1 r a a a

Lattice Bases vs. Reciprocal Lattice Bases (2D)

Square Lattice

1

2

ˆ ˆ32

ˆ ˆ32

a

a

a x y

a x y

1

2

2 1ˆ ˆ

3

2 1ˆ ˆ

3

a

a

b x y

b x y

1

2

ˆ

ˆ

a

a

a x

a y1

2

2ˆ

2ˆ

a

a

b x

b y

Triangular Lattice

Binary System and Structure Factor (2D)

if region ( )

if region a

b

a

b

rr

r( ) ( )exp( )i

G

r G G r

2

2 2

2 2

1( ) ( )exp( )

1 1exp( ) exp( )

( )exp( ) exp( )

c cell

a bc ca b

a b b

c ca a b cell

i d rA

i d r i d rA A

i d r i d rA A

G r G r

G r G r

G r G r

( ) if ( )

(1 ) if a b

S

f f

G G 0G

G 0

2

region

1Structure factor: ( ) exp( )

Filling Fraction:

c a

a

c

S i d rA

Af

A

G G r

Example 1: Square Lattice, Circular Rods/Holes

2

0 0

0 020 0

21

12

1

1: ( ) exp( 'cos ) ' '

2 2' ( ') ' ( )

2 2( )

2

r

c

r Gr

c c

c c

S iGr d r drA

r J Gr dr xJ x dxA G A

J GrrGrJ Gr

G A A Gr

J Grf

Gr

Structure Factor G

1 0: exp( cos ) ( ) , ( ) ( )n inn

n

dix i J x e xJ x xJ x

dx

Identities

2

2

rf

a

2 2 2 2

1 2 1 2

24

rGr n n f n n

a

Example 2: Triangular Lattice, Circular Rods/Holes

1: ( ) 2J Gr

S fGr

Structure Factor G

1 2

1 3ˆ ˆ ˆ, ,

2 2a a

a x a x y

1 2

2 1 2 2ˆ ˆ ˆ,

3 3a a

b x y b y

2

1 2

2

2

2

2

| |

sin( / 3)

2

3

rf

r

a

r

a

a a

1 2

2

2 2 11

2 21 1 2 2

2 21 1 2 2

2 1 2 2ˆ ˆ ˆ

3 3

22

3

24

3

8

3

Gr n n ra a

n nrn

a

rn n n n

a

fn n n n

x y y

Example 3: Simple Cubic Lattice, Spheres

1 2

0 1

2

0

3

:

2( ) exp( 'cos ) cos ' '

4 sin( ')' '

'

sin( ) cos( )3

( )

r

c

r

c

S G iGr d r drV

Grr dr

V Gr

Gr Gr Grf

Gr

Structure Factor

2 1/ 3 2 2 2(6 ) x y zGr f n n n

3

3

4

3

sphere

c

Vf

V

r

a

1 2 3ˆ ˆ ˆ, ,a a a a x a y a z 1 2 3

2 2 2ˆ ˆ ˆ, ,

a a a

b x b y b z

Bloch’s Theorem and Brillouin zone

Bloch’s Theorem (Electron Systems):2

2: ( ) ( ) ( ) ( ), ( ) ( ) 2

V E V Vm

Eigenvalue problem r r r r r r R

: ( ) ( )exp( ), ( ) ( )

( )exp

u i u u

u i

k k k k

kG

Solution r r k r r r R

k First Brillouin Zone G G r

First Brillouin Zone

3Try ( ) ( )exp( )d K C i r K K r

Here ( ) ( )exp( )V V i G

r G G r

2 23Thus we have ( ) ( ) ( ) exp( ) 0

2

KE C V C i d K

m

G

K G K G K r

22Since ( ) ( ) ( ) ( )

2V E

m r r r r

3

3

( ) ( ) exp ( )

( ) ( ) exp

V C i d K

V C i d K

G

G

G K K G r

G K G K r

2 2

Or ( ) ( ) ( ) 02

KE C V C

m

G

K G K G

3

. .

3

. .

3

. .

3

. .

3

. .

( ) ( ) exp ( )

( ) exp ( )

exp( ) ( ) exp( )

exp( ) ( )

( )

B Z

B Z

B Z

B Z

B Z

d K C i

d k C i

d k i C i

d k i u

d k

G G

G

G

k

k

r k G k G r

k G k G r

k r k G G r

k r r

r

Define

( ) ( )exp( )

( ) exp( ) ( )

u C i

i u

kG

k k

K k G

r k G G r

r k r r

Proving Bloch’s Theorem

Band Structure (Electron System)Band Structure (Electron System)

2 2

'

( )We have ( ) ( ') ( ') ( )

2C V C E C

m

k

G

k Gk G G G k G k G

2 2

' '

( )Difine ( )= ( '), ( ) ( )

2M V C C

m

GG GG G

k Gk G G k k G

' ''

So we have the eigenvalue equation

( ) ( )= ( ) ( )M C E C GG G GG

k k k k

'''

More explicitly:

( ) ( )= ( ) ( ),

: Band Index;

: Bloch Wave Vector

( ) : Energy (Hyper)Surfaces

n nn

n

M C E C

n

E

G GGGG

k k k k

k

k

2-1 -1

2

( )1( ) ( ), ( ) ( )exp( )

( )n ic

k kG

kH r H r r G G r

r

( ) ( )exp ( )i k kG

H r h G k G r

Bloch’s Theorem and Photonic Band Structure

2

1' 2

'

( )( ') ' , = + , '= + 'n

c

G GG

kG G P P h h P k G P k G

22 2

1' ' ' 2

' ' 1 1

( )ˆ ˆ ˆ( ') ' , nh h h

c

G G G GG

kG G P e P e h e

Energy Flow in Photonic Crystal

*1Time averaged Poynting vector: ( ) Re ( ) ( )

2 S r E r H r

2 21Time averaged energy density: ( ) ( ) | ( ) | ( ) | ( ) |

4U r r E r r H r

Harmonic fields: ( , ) ( ) , ( , ) ( )i t i tt e t e E r E r H r H r

Bloch Eigenmodes: ( ) ( ) , ( ) ( )i ie e k r k rk k k kE r e r H r h r

ˆPhase velocity: | |

Group velocity:

( )Energy velocity:

( )

p

g

e gU

k

v kk

v

S rv v

r

See Sakoda

Frequency Contours

Square Lattice

Two-Dimensional Inhomogeneous Wave Systems

2

2 2

1

c t

( ), ( )c c r r

2

2 2

2

2 2

2

2 2

2

2 2

1, Acoustic wave in fluid

1ˆ, -polarized shear wave: =

1ˆ, E-polarized EM wave: .

1, H-polarized EM wa

t

p p

c t

u uz u

c t

E EE

c t

H H

c t

u z

E z

ˆve: .HH z

0cc

2 2

1 1( ) ,

( )e

e t tc c

rr

Classical Waves

Unified Treatment

Binary System, Harmonic Waves

2

2

( ) ( )

( ) ( ) ( )c

r r

r r r

1 2

1 21 2

| |

1 1

Bd Bd

Bd Bdn n

1 1 1, ,c k

, ,c k

Scattered waveIncident wave

2 2 2 21 10, 0.k k

(1)

(1)1 1 1

( , ) ( ) ( )

( , ) ( ) ( )

inn n n n

n

inn n n n

n

r A H kr B J kr e

r C H k r D J k r e

0nC

(1)1

(1) 11

1

( ) ( ) ( )

'( ) '( ) '( )

n n n n n n

n n n n n n

A H ka B J ka D J k a

kkA H ka B J ka D J k a

nnB i

(1) (1)

'( ) ( / ) ( ) '( / )

( ) '( / ) '( ) ( / )n n n n n

nn n n n

ghJ ka J ka h J ka J ka hA i

H ka J ka h ghH ka J ka h

(1) (1)

2 /( )

( ) '( / ) '( ) ( / )n

nn n n n

gh i kaD i

H ka J ka h ghH ka J ka h

Band Structure Calculation (2D, Scalar Wave)

2

2 2

( , ) 1 ( , )Wave Equation: 0

( ) ( ) ( )

t t

c t

r r

r r r

Harmonic wave: ( , ) ( )exp( )t i t r r

2

2

( ) ( )Thus

( ) ( ) ( )c

r r

r r r

1

12

Media Parameters:

( ) ( )exp( ),

( ) ( ) ( )

( )exp( )

i

c

i

G

G

r G G r

r r r

G G r

1 2or ( ) ( ) ( ) ( ) (*) k k kr r r r

(1 ) if 0( )

( ) ( ) if 0a b

a b

f f

S

GG

G G

(1 ) if 0( )

( ) ( ) if 0a b

a b

f f

S

GG

G G

Substitute ( ) ( )exp into Eq. (*)i k kG

r G k G r

2

' '

We get ( ')( ) ( ') ( ') ( ') ( ') k k kG G

G G k G k G G G G G

2Generalized Eigenvalue Problem: k k k k kM v N v

'

'

where

( ) ( ')( ) ( ')

( ) ( ')

( ) ( )

k GG

k GG

k G

M G G k G k G

N G G

v G

2 1

2 1

Transform

(2 1)( ) ( 1)

(2 1)( ) ( 1)

i N n N n N

j N n N n N

1 1 2 2

1 1 2 2' ' '

n n

n n

G b b

G b b

1 2

1 2

Choose

,

,

N n N N n N

N n N N n N

' '( ) ( ) , ( ) ( ) , ( ) ( )ij ij i k GG k k GG k k G kM M N N v v

1 2

2 2

2

We can also solve the Eigenvalue Problem:

The dimension of , : (2 1) (2 1)

There are (2 1) plane waves or eigenvalues

M N N N

N

k k k k kN M v v

Reduced Brillouin Zone and Dimensionless Frequency

/ 2 / /a c a c a

Photonic Band Gaps

Left: Photonic Band , Right: Transmission (20layers)