1

SLOW LIGHT AND FROZEN MODE REGIME IN PHOTONIC CRYSTALS

April, 2007

Alex Figotin and Ilya Vitebskiy

University of California at Irvine

Supported by MURI grant (AFOSR)

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

1D periodicity 2D periodicity

1. Each constitutive component is perfectly transparent, while their periodic array may not transmit EM waves of certain frequencies.

2. Strong and controllable spatial dispersion, particularly at λ ~ L.

3. Photonic crystals should be treated as genuinely heterogeneous media – no “effective” homogeneous medium can imitate a photonic crystal.

3

(k)

k

11

22

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

Typical k diagram of a photonic crystal for a given direction of k.

(k)

k

Typical k diagram of an isotropic non-dispersive medium: = v k

(k)

k

* *

Electromagnetic eigenmodes in periodic media are Bloch waves

Propagating Bloch modes: . Evanescent Bloch modes:

-

exp .

dispers

.

r L r ik L

k k k k

k

ion relation ( diagram).

- group velocity energy velocity of a pro/ pagating mode.g

k

v k

Electromagnetic dispersion relation in photonic crystals

4

1. Dramatic increase in density of modes.

2. Qualitative changes in

0, at .

the

EM waves with vanishing group velocity

g s sv kk

2 1/ 2

eigenmode structure (can lead to the frozen mode regime).

- Regular band edge (RBE):

- Stationary inflection

,

poin

.

Examples of stationary points:

g g g g gk k v k k

3 2 2/3

0 0 0 0

4 3 3/ 4

t (SIP):

- Degenerate band edge (DBE):

, .

, .

g

d d g d d

k k v k k

k k v k k

g

RBE

wavenumber k

freq

uenc

y

a)

0

SIP

wavenumber k

freq

uenc

y

b)

d

DBE

wavenumber k

freq

uenc

y

c)

Each stationary point is associated with slow light, but there are some fundamental differences between these three cases.

5

- What happens if the incident wave frequency is equal to that of slow mode with vg = 0 ?- Will the incident wave be converted into the slow mode inside the photonic crystal, or will it be reflected back to space?

Assuming that the incident wave amplitude is unity, let us see what happens if the slow mode is related to (1) RBE, (2) SIP, (3) DBE.

Reflected wave

Incident wave of frequency sTransmitted slow mode

Semi-infinite photonic crystal

What is the frozen mode regime? Example of a plane wave incident on a lossless semi-infinite photonic crystal

6

L

S

e

te

t

ady -state re

,

gime i

,

n semi - i

and be the energy fluxes of the incident,

reflected, and transmitted waves, respec

nfinite photonic crystal :

Energy conservation consideration.

tiveI R TS S S

ly.

Transmission coefficient

Slow mode energy flux is

where .

As

/ .

,

approaches :

0

T I R

I

g g

s

S S S S

S S

S W v v

and , unless0 .0S W

Assuming that the incident wave amplitude is unity, let’s see what happens if the slow mode is related to: (1) a regular band edge, (2) a stationary inflection point, or (3) a degenerate band edge.

7

22

2 2

.

The slow mode mode group veloc

Regular photonic band edge (generic ca

ity is

.

The slow mode energy density at

, 02

2

1

s

e

)

e

.

Th

g g g

g

g g g

g

ak k

v

v a k k ak

W

2

slow mode energy flux vanishes at

as

implying total reflect

2 0

ion of the incident wave.

,g

g

g gvS W a

Regular BE

k

ωg

8

330 0

2 2/3

3 0 0

0

0

The slow mode group velocity vanishes

while its en

.3

,

ergy density di

verges as

Frozen mode regime at stationary inflection point (SIP)

g

ak k

v a k kk

W

2/3

0The slow mode energy flux remains finite at

implying conversion of the incident wave to the frozen mode

with huge diverging amplitude a

.

nd nearly ze

ro group velocity.

1,gS v W

SIP case

k

ω

0

9

In all cases, the incident wave has the same polarization and unity amplitude.

Frozen mode profile at different frequencies close to SIP

2/3 1/3

0 max 0 max 0

20

At : at ,

At : .

I

I

W W z

W z W z

0 20 40 600

50

100

150

200

Distance z

Square

d a

mplit

ude

a)

0 20 40 600

200

400

600

800

Distance z

b)

0 20 40 600

1000

2000

3000

4000

5000

6000

Distance z

c)

0 20 40 600

200

400

600

800

Distance z

Square

d a

mplit

ude

d)

0 20 40 600

50

100

150

200

Distance z

e)

0 20 40 600

10

20

30

40

50

Distance z

f)

SIP

10

44

3 3/ 4

4

The slow mode group veloci

Frozen mode regime at a degen

ty

, 0.4

,

is

while its energy density diverges

era

te band edge (DBE)

as

d d d

g

g d d

d

ak k

v

v a k kk

W

1/ 2

0

1/ 4

The slow mode energy flux vanishes at

implying total reflection of incident wave.

This case is intermediate between the frozen mode regime at

a SIP and the cas

.

e o

,

d

g dS v W

f total reflection at a RBE.

DBE case

d

ω

k

11Frozen mode profile at different frequencies close to DBE

In all cases, the incident wave has the same polarization and unity amplitude.

1/ 2 1/ 4

max max

2

A t : at ,

At :

.

d I d d

d I

W W z

W z W z

(band)

(gap

,

, )

d

d

0 20 40 600

20

40

60

80

100

120

Distance z

Square

d a

mpl

itude

a)

0 20 40 600

200

400

600

800

1000

1200

Distance z

b)

0 20 40 600

2000

4000

6000

8000

10000

Distance z

c)

0 20 40 600

100

200

300

400

Distance z

Square

d a

mpl

itude

d)

0 20 40 600

50

100

150

Distance z

e)

0 20 40 600

10

20

30

40

50

Distance z

f)

DBE

12Summary of the case of a plane wave incident on asemi-infinite photonic crystal supporting a slow mode.

- The case of a regular BE: the incident wave is reflected back to space without producing slow mode in the periodic structure.

- The case of a stationary inflection point: the incident wave can be completely converted into the slow mode with infinitesimal group velocity and huge diverging amplitude.

- The case of a degenerate photonic BE: the incident wave is totally reflected back to space, but not before creating a frozen mode with huge diverging amplitude and vanishing energy flux.

Regular band edge Stationary inflection point Degenerate band edgek

ω

0

k

ωg

d

ω

k

13

In case of transverse electromagnetic waves propagating

ˆˆ ( , ) ; ( , )

in the direction,

the time-harmonic Maxwell equat

Time-harmonic Maxwell equations in layere d mediai i

E r z H r H r z E rc c

z

† † 1

where

, ,

( ) 0 0 0 1

( ) 0 0 1 0( )

ions reduc

, , , ( ) 0 1 0 0

( ) 1 0 0 0

e to

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

14

At any given frequency , the reduced Maxwell equatio

, ,

n

has four solutions which normally can be c

Propagating and evanescent eigenemodes in periodic layered med

ia

z z i M z z M z L M zc

1 2 3 4 1 2

hosen in Bloch form:

Every Bloch eigenmode is either extended or evanescent:

is extended if ,

is evanescent if .

The disp

, 1,2,3,4

Im

ersion relation:

0

Im 0

, , , ,

i

i i

ik Lk k

k

k

z L e z i

z k

z k

k k k k k k

3 4, ,k k

15Transfer matrix formalism

†

0 0

( )

(

The reduced time-harmonic Maxwell equations in layered media

The respective Cauchy problem has a unique solution

where

), ; ( ) ,

( )

( )

(

,

x

y

zx

y

E z

E zz i M z z z M JMJ

H zc

H z

T z

z T z z z

0

† 10 0

is the unitarity transfer matrix

Transfer matrix of a unit cell

, )

, , ,

.

.

,

,0L mm

z J

T z z T z z T z z T JT J

TT T L

16

4 3 23 2 1

where is the Bloch wave number.

The characteristic polyno

Bloch eigenmodes are the eigenvectors

mial

determines the dispersion re

, ,

2 /

de

a

t 1 0

l

L

ikLL k k k

L

T

T z z L z e

k k L

P T I P P P

1 2 3 4 1 2 3 4

1

1 2 3 4 1 2 3 4

tion:

The condition for symmetric dispersion relation:

if , then for any

, , , , , ,

, , , , , , . L L

k k k k k k k k

T T

k k k k k k k k

17

11

22

3

4

Jordan normal form of the transfer matrix of the periodic

layered array correspomding to each of th

0 0 00 0 0

0 0 00 0 00 . :

0 0 10

e stationary point

0 0

00 0 0

: RBE

s

g L La

L

v T T

T

1 0

0 0

0 0

0 0

.

0 0

0 0 0 1 0 0

0 1 0 0 1 0. .

0SI

0 1 0 0 1

0 0 0 0

P: DBE

0

:

0

a

L LT T

18

0

0 0

Consider a Bloch solution

of the reduced Maxwell equation

At the fr

,

, ,

ozen mode frequency defined b

.

,

y

0

The eigenmodes at SIP frequency

ikzk k k k

z k k

k k k

z e z z z L

z i M z z A z L A zc

k

k

0

0 1

2

201 02

there are two propagating Bloch solutions and .

The other two solutions are non-Bloch Floquet eigen

0

modes

,

a n ~d

kk k k

k k

k

z z

z z z z

Dispersion relationwith SIP

k

ω

0

19

Evanescent mode: Im k > 0

Propagating 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

20

k

ω

0

AA11 AA22 FF AA11 AA22 FF AA1 1 AA22 FF AA11 AA22 FF

LL

d

ω

k

LL

AA11 AA22 BB AA11 AA22 BB AA11 AA22 BB

SIP

DBE

What kind of periodic structures can support the frozen mode regime?

21

y

x

z

Anisotropic layer A1 Anisotropic layer A2 Ferromagnetic layer F

M || z

1 2

1 2

0 0 0 0

ˆ ˆ ˆε 0 0 ε 0 ε 0

0 0 0 0 0 0

1 0 0 1 0 0 0

ˆ ˆ ˆ 0 1 0 0 1 0 0

0 0 1 0 0 1 0 0

F

zz zz

F

i

i

i

i

22

2. Frozen mode regime in bounded photonic crystals

So far we have discussed the frozen mode regime in lossless semi-infinite periodic structures. What happens to the frozen mode regime if the photonic crystal has finite dimensions?

23EM wave incident on a finite photonic slab: different possibilities

Different arrangements involving a photonic slab with finite thickness D = N L

Mir

ror

or a

bsor

ber

Photonic slab

Photonic slab

(a) (b)

a) The incident wave is partially transmitted through the photonic slab.

b) There is no transmitted wave if a mirror or an absorber is present.

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

We start with the case (a), involving incident, transmitted, and reflected waves. Then we turn to the case (b), where there is no transmitted wave at all.

24Transmission band edge resonances near a regular photonic band edge (generic case)

Smoothed field intensity distribution at the frequency of first transmission resonance

Finite stack transmission vs. frequency.ωg – regular photonic band edge

1.6 1.65 1.70

1

Frequency

Tra

nsm

itta

nce

g

a) N = 16

1.6 1.65 1.70

1

g

FrequencyT

ran

sm

itta

nce

b) N = 32

1 2 3 4 5 1 2

0 4 8 12 160

20

40

60

80

100

Location z

Square

d a

mplit

ude

a) N = 16, s = 1

0 8 16 24 320

20

40

60

80

100

Location z

Square

d a

mplit

ude

b) N = 32, s = 1

25

Finite stack transmission vs. frequency. ωg – degenerate photonic band edge

Smoothed Field intensity distribution at frequency of first transmission resonance

Giant transmission band edge resonances near a degenerate photonic band edge

0 4 8 12 160

500

1000

1500

2000

2500

Location z

Sq

uare

d am

plitu

de

a) N = 16, s = 1

0 8 16 24 320

10000

20000

30000

40000

Location z

b) N = 32, s = 1

26

22

2

44

4

Regular band edge: :

Degenerate band edge:

2

max

4

max

:

g g

I

d d

I

ak k

NW W

s

ak k

NW W

s

Regular BE vs. degenerate BE

A stack of 10 layers with degenerate photonic BE performs as well as a stack of 100 layers with regular photonic BE !

k

ωg

d

ω

k

27Example: Transmission band edge resonance in periodic stacks of 8 and 16 double layers.

Smoothed electromagnetic energy density distribution inside photonic cavity at frequency of transmission band edge resonance

RBE: Regular photonic band edge (Energy density ~ N2)

DBE: Degenerate photonic band edge (Energy density ~ N4)

28

0 2 4 6 80

10

20

30

40

50

60

70

80

90

100

Location z

Squ

ared

am

plitu

de

N = 8

a) Vacuum - PS - Mirror

0 2 4 6 80

10

20

30

40

50

60

70

80

90

100

Location z

Squ

ared

am

plitu

de

N = 8

b) Vacuum - PS - Vacuum

= 0.01 = 0.1

= 0.01 = 0.1

Frozen mode profile at frequency of a giant transmission band edge resonance: a) with a mirror at the right-hand boundary, b) without the mirror.

, where is dielectric permittivity of the lay s./ eri B

29Publications

[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 magneticphotonic 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 periodicstacks of anisotropic layers. Phys. Rev. E 71, 036612 (2005).

[5] G. Mumcu, K. Sertel, J. L. Volakis, I. Vitebskiy, A. Figotin. RF Propagation in Finite Thickness Nonreciprocal Magnetic Photonic Crystals. IEEE: Transactions on Antennas and Propagation, 53, 4026 (2005)

[6] A. Figotin and I. Vitebskiy. Gigantic transmission band-edge resonance inperiodic stacks of anisotropic layers. Phys. Rev. E72, 036619, (2005).

[7] A. Figotin and I. Vitebskiy. Electromagnetic unidirectionality and frozen modesin magnetic photonic crystals. Journal of Magnetism and Magnetic Materials, 300, 117 (2006).

[8] A. Figotin and I. Vitebskiy. "Slow light in photonic crystals" (Topical review),Waves in Random Media, Vol. 16, No. 3, 293 (2006).

[9] A. Figotin and I. Vitebskiy. "Frozen light in photonic crystals with degenerate band edge". Phys. Rev. E74, 066613 (2006)

30

Auxiliary slides

31

0.8 0.85 0.9 0.95 1 1.050

0.1

0.2

0.3

0.4

0.5

0.6

Frequency

Abs

orpt

ion

d

a

N = 8

a) Vacuum - PS - Mirror

= 0.01 = 0.1

0.8 0.85 0.9 0.95 1 1.050

0.1

0.2

0.3

0.4

0.5

0.6

Frequency A

bsor

ptio

n

d

a

N = 8

b) Vacuum - PS - Vacuum

= 0.01 = 0.1

Fig. 2. Absorption versus frequency of a periodic stack with DBE at ω = ωd : (a) The vacuum – PS – mirror arrangement shown in Fig. 1(a). (b) The vacuum – PS – vacuum arrangement shown in Fig. 1(b). N = 8 is the number of unit cells in the periodic stack. Black and blue curves correspond to two different values of absorption coefficient γ of the isotropic B layers. In either case (a) or (b), larger absorption coefficient (the black curve) gives higher absorption peaks at frequencies of transmission band-edge resonances.

32

0.8 0.85 0.9 0.95 1 1.050

0.5

1

1.5

2

2.5

3

Frequency

Tran

smis

sion

d

a N = 8, = 0, - 0.1, - 0.2

Transmission dispersion of a periodic stack with different values of negative absorption (gain) γ. Solid red curve corresponds to γ = 0. Observe the sharp difference between a regular TBER (just below ωa ) and a giant TBER (just below ωd ).

Frozen mode regime in the presence of negative absorption (one of the constitutive components is a gain medium).

33

0.8 0.85 0.9 0.95 1 1.050

2

4

6

8

10

12

Frequency

Tran

smis

sion

d

a

a) N = 8, = 0, - .1, - .2, -.3

0.8 0.85 0.9 0.95 1 1.050

1

2

3

4

5

6

7

8

Frequency

Ref

lect

ion

d

a

b) N = 8, = 0, - .1, - .2, - .3

Transmission/reflection dispersion of a periodic stack with different values of negative absorption (gain) γ. Compared to the previous slide, the magnitude of negative absorption here is larger. The difference between the regular TBER (just below ωa ) and the giant TBER (just below ωd ) is now extreme.