1

Alex Figotin & Ilya Vitebskiy

University of California at Irvine

Supported by AFOSR

Slow light and resonance phenomena in photonic crystals

September 2005

2What are photonic crystals?Simplest examples of periodic dielectric structures

1D periodicity 2D periodicity

n1 n2

Each constitutive component is perfectly transparent, while their periodic array may not transmit E.M. waves of certain frequencies (frequency gaps).

3

(k)

k

* *

Electromagnetic eigenmodes in periodic media are Bloch waves

Propagating Bloch modes: . Evanescent Bloch modes:

exp .

- d

.

r L r ik L

k k k k

k

ispersion relation.

- group velocity of propagating Bloch mode./u k

(k)

k

11

22

Typical k diagram of a photonic crystal for a given direction of Bloch wave vector k

Typical k diagram of a uniform anisotropic medium for a given direction of k. 1 and 2 denote two polarizations.

4

2

/ 0

Strong spatial dispersion results in existence of slow modes:

at and :

and (extreme points of dispersion curves

.

,

):

Examples of stationary points:

s s

a a

a g

k k u k d dk

k k

1/ 2

2 1/ 2

3 2 2 / 3

0 0 0 0

(stationary inflection point)

.

, .

,

0 :

.

a a

g g g g

u k k

k k u k k

k k u k k

Slow light in photonic crystals:stationary points of dispersion relations

Fragment of dispersion relation with stationary points a, g and 0.

ω

k

g0

a

Every stationary point of the dispersion relation (k) is associated with slow light.But there are some important differences between these cases.

5What happens if the incident light frequency coincides with that of a slow mode?

Will the incident light with the slow mode frequency s be converted into the slow mode inside the photonic crystal, or will it be reflected back to space?The answer depends on what kind of stationary point is associated with the slow mode.

Reflected wave

Incident wave of frequency sPassed slow mode

Semi-infinite photonic crystal

6

• In case g of a band edge, all incident light with = g is reflected back to space. The fraction of the incident wave energy converted to the slow mode vanishes as → g .

• In case a of an extreme point, the incident light with = a is partially reflected and partially transmitted inside in the form of the fast propagating mode. The fraction of the incident wave energy converted to the slow mode vanishes as → a.

• In case 0 of stationary inflection point a significant fraction of incident light can be converted to slow mode, constituting the so-called frozen mode regime.

Fragment of dispersion relation with stationary points a, g and 0.

ω

k

g0

a

7Slow mode amplitude at steady-state regime

/ , / 1

Energy fluxes in steady-state regime: .

Transmittance / reflectance:

Energy flux of transmitted propagating Bloch mode:

.

If ,

.

0

T I R

T I R I

T

S S S

S S S S

u

S W u

then , unless ( .0 )TS W the frozen mode regime

Incident waveSI

Reflected wave

SR

Lossless semi-infinite photonic slab

Transmitted slow mode

ST

ω

k

g0

8

2.

The slow mode mode group velocity

Phot

is

.

The slow mode energy density at

The s

onic band edge (generic cas

, 0

e

1.

)

lo

2

2

bg g g

g g g g

g

k k

u

u k kk

W

2 0

w mode energy flux at

as

implying total reflection of the incident wave.

,

g

T g g gS u W

k

ωg

Band edge

9

300 0

2 2 / 300 0

0

The slow mode group velocity vanishe

Stationary inflection point (the froze

s

while its energy density diverge

n mode

.6

,2

re

s a

gim )

e

t

k k

u k kk

W

2 / 3

0

The slow mode energy flux remains finite

implying conversion of the incident light to the frozen mode

with huge amplitude and

.

nearly zero group velocity.

1,S u W

k

ω

0

Stationary inflection point

10

(4)4

(4)3 3/ 4

The slow mode group velocity i

Degenerate band edge (the intermidi

s

while its energy densit

, 0.24

,6

ate c

y diverg

ase)

e s

s a

dd d d

dd d

d

d

k k

u

u k kk

W

1/ 2

1/ 4

The slow mode energy flux vanishes

This case is intermediate between the frozen mode regime at

stationary inflection point and the case of tota

.

l reflection at regula

r

pho

.T dS u W

tonic band edge.

k

ω

d

Degenerate band edge

11

( )

( )( )

( )

( )

x

y

x

y

E z

E zz

H z

H z

Incident waveΨI

Reflected wave

ΨR

Lossless semi-infinite photonic slab

Transmitted slow mode

ΨT

0

1/ 3

0

Boundary conditions:

Bloch composition of the transmitted wave at frequency close to :

At the photonic crystal boundary:

0 0 0 .

0 0 .

T I R

T pr ev

pr ev

z z z

Space structure of the frozen mode

12Distribution of EM field and its propagating and evanescent components inside semi-infinite slab at frequency close (but not equal) to 0 . The amplitude of the incident light is unity !!!

a) resulting field |T (z)|2 = |pr (z) + ev (z) |2, b) extended Bloch component |pr (z) |2 , c) evanescent Bloch component |ev (z) |2 .

As approaches 0 , |pr |2 diverges as (0 )2/3 and the resulting field distribution |T (z) |2 is described by quadratic parabola.

13

Summary of the case of a plane EM wave incident on semi-infinite photonic crystal:

- If slow mode corresponds to a regular photonic band edge, the incident light of the respective frequency is totally reflected back to space without producing the slow mode in the periodic structure.

- The incident light can be linearly converted into a slow mode only in the vicinity of stationary inflection point (the frozen mode regime).

- If slow mode corresponds to degenerate photonic band edge, incident light of the respective frequency is totally reflected back to space. But in a steady-state regime it creates a diverging frozen mode inside the photonic crystal.

14

The question:

Can the electromagnetic dispersion relation of a periodic layered structure (1D photonic crystal) display a stationary inflection point or a degenerate band edge? In other words, can a 1D photonic crystal display the frozen mode regime?

The answer is:

Stationary inflection point and degenerate band edge, along with associated with them the frozen mode regime can only occur in stacks incorporating anisotropic layers.

15Simplest periodic layered arrays supporting stationary inflection point of the dispersion relation

z

L

A B A B A B A B A B A B

z

yx

L

A1 A2 F

Non-magnetic periodic stack with oblique anisotropy in the A layers

Magnetic periodic stack with misaligned in-plane anisotropy in the A layers

16Simplest periodic layered array capable of supporting degenerate photonic band edge

There are three layers in a unite cell L. A pair of anisotropic layers A1 and A2 have misaligned in-plane anisotropy. The misalignment angle must be different from 0 and π/2. B – layers can be made of isotropic material, for example, they can be empty gaps. The k diagram of the periodic stack is shown in the next slide.

L

z

A1 A2 B A1 A2 B A1 A2 B A1 A2 B

17

The first band of the k diagram of the 3-layered periodic stack for four different values of the B - layer thickness. In the case (b) the upper dispersioncurve develops degenerate band edge d. In the case (d) of B - layers absent, the two intersecting dispersion curves correspond to the Bloch waves with different symmetries; the respective eigenmodes are decoupled.

18

Up to this point we considered the frozen mode regime in semi-infinite photonic crystals. How important is the thickness of the photonic slab?

19

EM field distribution inside plane-parallel photonic crystal of thickness D at the frequency ωd of degenerate band edge. The incident wave amplitude is

unity. The leftmost portion of the curves is independent of the thickness D.

N = 256 N = 64

Frozen mode regime in finite periodic stacks

20Transmission band edge resonance(Fabry-Perot cavity resonance in a finite periodic stack

near the edge of a transmission band)

k

ωg

D

A B A B A B A B A B A B

s

Resonant wave lengths:

Resonant wave numbers:

/ 2, 1, 2,3,...

, 1, 2,...s g

Ds s

k k s sNL

21

s

2

Resonant wave lengths:

Resonant wave numbers:

Dispersion relation near regular photonic band edge:

Resonant fr

/ 2, 1,2,3,...

, 1,2,.

u

.2

eq

..s g

gg g g

Ds s

k k s sNL

k k

2

2

encies:

Resonant fie

, 1,2,.

ld

..

amplitude:

2

max

gs g

I

N s sNL

NW W

s

Fabry-Perot cavity resonance in finite periodic stacks:regular band edge

k

ωg

D

A B A B A B A B A B A B

22Transmission band edge resonance: regular band edge

Field intensity distribution at frequency of first transmission resonance

Finite stack transmission vs. frequency

23

20

2

40

4

Regular band edge: :

Degenerate band edge:

''

2

max

''''

24

:

max

g g

I

d d

I

k k

NW W

s

k k

NW W

s

Fabry-Perot cavity resonance in finite periodic stacks:regular band edge vs. degenerate band edge.

k

ωg

k

ω

d

24

Finite stack transmission vs. frequency

Field intensity distribution at frequency of first transmission resonance

Transmission band edge resonance: degenerate band edge

25

Publications

[1] A. Figotin and I. Vitebsky. Nonreciprocal magnetic photonic crystals. Phys. Rev. E 63, 066609, (2001)[2] A. Figotin and I. Vitebskiy. Electromagnetic unidirectionality in magnetic photonic crystals. Phys. Rev. B 67, 165210 (2003).[3] A. Figotin and I. Vitebskiy. Oblique frozen modes in layered media. Phys. Rev. E 68, 036609 (2003).[4] J. Ballato, A. Ballato, A. Figotin, and I. Vitebskiy. Frozen light in periodic stacks of anisotropic layers. Phys. Rev. E 71, 036612 (2005).[5] A. Figotin and I. Vitebskiy. Slow light in photonic crystals. Subm. to Waves in Random and Complex Media.(arXiv:physics/0504112 v2 19 Apr 2005).[6] A. Figotin and I. Vitebskiy. Gigantic transmission band edge resonance in periodic stacks of anisotropic layers. Phys. Rev. E 72 (2005).

26

Auxiliary Slides

27

Incident pulse

Photonic crystal

D

D

Passed slow pulse

Photonic crystal

0

0 0

Incident pulse length:

Passed pulse lengt

2 / .

.

1) , ( 2 / ).

2) , ( 2

h:

/ ).

:

l c

ul l l

c

l D u D

l D u D

Two limiting cases

Pulse incident on a finite photonic crystal

28

Regular frequencies:

ω < ωa : 4 ex.

ω > ωg : 4 ev. (gap)

ωa < ω < ωg : 2 ex. + 2 ev.

-------------------------------------

Stationary points:

ω = ωa : 3 ex. + 1 Floq.

ω = ωg : 2 ev. + 1 ex. + 1 Floq.

ω = ω0 : 2 ex. + 2 Floq.

ω = ωd : 4 Floq. (not shown)

kk0

g

a

0

Dispersion relation ω(k)

Eigenmodes composition at different frequencies

29

In case of transverse electromagnetic waves propagating in the direction,

the time-harmonic Maxwell equatio

ˆˆ ( , ) ; ( , )

Time-harmonic Maxwell equations in layered mediai i

E r z H r H r z E rc c

z

† † 1

where

, ,

( ) 0 0 0 1

( ) 0 0 1 0( ) , , ,

( ) 0 1 0 0

(

ns r

) 1

educe t

0 0 0

o

z

x

y

x

y

z i M z zc

E z

E zz M JA A A J J J

H z

H z

30

At any given frequency , the reduced Maxwell equation

has four solutions which normally can be chos

, ,

en

Extended and evanescent eigenemodes in

periodic layered medium

z z i M z z M z L M zc

1 2 3 4 1 2 3

in Bloch form:

Every Bloch eigenmode is either extended or evanescent:

is extended if ,

is evanescent if .

The dispersion

, 1,2,3,4

Im 0

Im 0

, , ,

relation:

, , ,

i

i i

ik Lk k

k

k

z L e z i

z k

z k

k k k k k k k k

4

31Transfer matrix formalism

†

0

The respective Cauchy problem

has a unique solution

( )

( ), ; ( ) ,

( )

( )

, ,

The reduced time-harmonic Maxwell equations in layered media

x

y

zx

y

z

E z

E zz i M z z z M JMJ

H zc

H z

z i M z z zc

0

10 0 0 0

0 0

where is the transfer matri( x, )

, , , , , ,

,

T z z

T z z T z z T z z T z z T z z

z T z z z

32

0 0

0

† 1

The respective Cauchy problem for the transfer matrix is

which implies that is a unitarity matrix

The transfer matrix of an arbitrary stratifi

, , , , ,

( , )

ed medium

z

S

T z z i M z T z z T z z Ic

T z z J

T JT J

T

is

Explicit expressions for the transfer matrices of individual

homogeneous layers are known (they are very cumbersom )

ˆ ˆ, , , ,

e

S mm

m m m m x y

T T

T T k k

33

4 3 23 2 1

Bloch eigenmodes are the eigenvectors

The characteristic polynomial

determines the dispersion re

,

det 1

l

0

a

The transfer matrix of a unit cellL m

m

L

ikLL k k k

L

T T

T

T z z L z e

P T I P P P

1 2 3 4 1 2 3 4

1 1

1 2 3 4 1 2 3 4

1 1

1 2 3 4 1 2

, , , , , ,

, ,

tion:

Symmetric dispersion relation (if ):

for any

Asymmetric dispersion relation (if ):

for any

, , , , ,

, , , , , ,

L L

L L

k k k k k k k k

T U T U

k k k k k k k k

T U T U

k k k k k k

3 4,k k

34

11

22

3

4

1

0

0

0

1

0 0 00 0 0

0 0 00 0 0 , :

0 0 10 0 0

0 0 00

Regular : Band edge

S.I.P. :

0 0

0

0 0

0 1 0

0

0 1

0 0

0

: Jordan normal form of the -mat ix r

L La

a

L

L L

T T

T

T T UT U

0

0

0

0

1 0 0

0 1 0, D.B.E. :

0 0 1

0 0 0

LT

0 1

0

0

0

3

0 1 0 0 1

0

At , the characteristic polynomial is degenerat

0,

e

with triple root relating to the frozen mode

De

,

viati

The vicinity of the frozen mode frequency

ik L ik L

ik Lk

P e e

e z

0 0

1/ 3 1/ 3 2 / 3 2 / 30 0 0

on of form removes the triple degeneracy of

yi

/ 6 , 1, ,

1elding one extended solution ( ) relating to the nearly

frozen mode, and a pair of evanescent solu

i ik k e e

2 / 3

1 2

tions ( )

with infinitesimal m Im . I

ie

k k

36

0

0 0

0

Consider a Bloch solution

of the reduced Maxwell equation

At the frozen mode frequency defined

,

, , .

0,

by

The eigenmodes at

ikzk k k k

z k k

k kk k

z e z z z L

z i M z z A z L A zc

k

k

0

1 0

0

0 0

2

2

01 02 2

there are two extended Bloch solutions and .

The other two solutions are related to the frozen mode

or, explicitely

0,

,

:

k k k

k k

k

k kk k k k

k

z z

z

z z z zk k

37

0 0

0 0 0

0 0

0 0

01 0

2 202

2

2

0

where

are auxiliary Bloch functions (not eigenm

, (~ )

2 , (~ )

,

odes !!!)

At there are only two (not f

k k

k k k

ikz ikzk k k k

k k k k

z z ik z z z

z z iz z z z z

z e z z e zk k

0

1

0

1

201 02

our !!!) Bloch solutions:

1. extended (frozen) mode with and

2. extended mode with and

The other two solutions are the non-Bloch Floquet eigenmode

0

s

0

and ~

k

k

z k k u

z k k u

z z z z

38

Evanescent mode: Im k > 0

Extended mode: Im k = 0

Evanescent mode: Im k < 0

Floquet mode: 01 (z) ~ z

Blo

ch e

igen

mod

esN

on-B

loch

eig

enm

ode