1

Frozen modes and resonance phenomena in periodic metamaterials

October, 2006

Alex Figotin and Ilya Vitebskiy

University of California at Irvine

Supported by AFOSR

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

1D periodicity 2D periodicity

n1 n2

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

3What are the differences between "conventional" metamaterials and photonic crystals (PC)?

- At frequency range of interest, all PC should be treated as genuinely heterogeneous media. Equivalent uniform media approach usually does not apply (no effective ε and/or μ can be introduced).

- At frequency range of interest, PC of any dimensionality are essentially anisotropic – even those with cubic symmetry.

The reason for this is that at frequencies where a PC behaves as a PC, the spatial dispersion must be very strong.

At the same time, such phenomena as:

- backward waves (those with opposite phase and group velocities) and

- negative refraction

can be found in almost any PC.

4

Different scenarios of "negative refraction" in photonic crystals:

Due to strong structural anisotropy, in either case, the normal component of the transmitted wave phase velocity can be opposite to that of the normal component of its group velocity (possible definition of negative refraction in anisotropic media).

SI

ST

SRSI

ST

SR

5"Zero refraction" scenarios in photonic crystals:

In either case, the normal component of the transmitted wave group velocity vanishes. What happens to the transmitted wave?

SI

ST

SR

a) Total reflection b) Frozen mode regime

Case (a) is analogous to the phenomenon of total internal reflection (this case is typical of uniform materials).

In case (b), the transmitted wave is a grazing mode with huge amplitude and tangential energy flux. This case is impossible in uniform materials.

From now on, PC will be treated as genuinely heterogeneous periodic media.

SI

ST

SR

6

(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 propagating e/ mod .g

k

v k

Electromagnetic dispersion relation in photonic crystals

7

2 1/ 2

3 2 2/3

0 0 0 0

, .

, .

- Regular band edge (RBE):

- Stationary inflection point (SIP):

- Degenerate ba

Examples of stationary points:

g g g g g

g

k k v k k

k k v k k

4 3 3/ 4nd edge (DBE): , . d d g d dk k v k k

EM waves with vanishing group velocity in photonic crystals: stationary points of dispersion relation ω(k)

Each stationary point is associated with slow mode with vg ~ 0.But there are some fundamental differences between these three cases.

g

RBE

wavenumber k

freq

uenc

y

a)

0

SIP

wavenumber k

freq

uenc

y

b)

d

DBE

wavenumber k

freq

uenc

y

c)

8

What happens if the incident wave frequency coincides with that of a 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?

Reflected wave

Incident wave of frequency sTransmitted slow mode

Semi-infinite photonic crystal

A plane wave incident on a semi-infinite photonic crystal

9

Transmission / reflection coefficients

where , , are the energy fluxes of the incident,

transmitted and reflected waves, respectively.

S

, , 1.

Energy conservation considerati n.o

T R

I I

I T R

S S

S S

S S S

low mode energy flux is where .

Therefore, as approaches :

and , un

, 0

0 l 0 ess .

T g g

s

T

S W v v

S W

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

10

2

Regular photonic band edge (generic case)

.

The slow mode mode group velocity is

.

The slow mode energy density at

,

1.

2

2

0bg g g

g

g g g g g

g

k k

v

v k kk

1.

2

The slow mode energy flux vanishes at

as

implying total reflection of the incident wave.

0 ,g

g

T g g gv

W

S W

RBE case

k

ωg

11

Frozen mode regime at SIP

12

300 0

2 2/300 0

0

Stationary inflection point (the frozen mode regime)

The slow mode group

velocity vanishes

while its energy density diverges

.6

,2

2

a

.

s

g

k k

v k kk

2/3

0

0The slow mode energy flux remains finite at

implying conversion of the incident wave to the frozen mode

with huge divergi

.

ng amplitude and nearly zero

gr

p

,

o e

1

u v

T g

W

S v W

locity.

SIP case

k

ω

0

13Steady-state frozen mode regime in ideal lossless semi-infinite photonic slab. Frequency is close (but not equal) to that of SIP.

Electromagnetic field and its propagating and evanescent components inside semi-infinite photonic crystal. The amplitude of the incident wave is unity.Frequency is close (but not equal) to 0 .

As approaches 0 , the frozen mode amplitude diverges as W ~ ( – 0)–2/3.

14Destructive interference of propagating and evanescent components of the frozen mode at photonic crystal boundary at z =

0.

Resulting field at z = 0

Propagating Bloch component at z = 0

Evanescent Bloch component at z = 0

Frequency dependence of the resulting field amplitude (a) and its propagating and evanescent Bloch components (b and c) at the photonic crystal boundary at z = 0. The amplitude of the incident wave is unity.

15

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)

Smoothed profile of the frozen mode at six different frequencies in the vicinity of SIP: (a) ω = ω0 – 10 – 4 c/L, (b) ω = ω0 – 10 – 5 c/L, (c) ω = ω0 , (d) ω = ω0 + 10 – 5 c/L, (e) ω = ω0 + 10 – 4 c/L, (f) ω = ω0 + 10 – 3 c/L. In all cases, the incident wave has the same polarization and unity amplitude. The distance z from the surface of the photonic crystal is expressed in units of L.

Frozen mode profile at different frequencies close to SIP [3 - 4]

16

- Both at normal and oblique versions of the frozen mode regime at SIP, the incident wave is completely converted into the frozen mode with greatly enhanced amplitude and vanishing group velocity.

- At oblique frozen mode regime, one can continuously change the operational frequency within certain range, simply by changing the angle of incidence. There is no need to alter the physical parameters of the periodic structure.

Incident EM wave Transmitted frozen mode with greatly enhanced amplitude and vanishing group velocity

Photonic slab

Frozen mode regime at SIP. Main features. [3 - 4]

17

Frozen mode regime at DBE

18

(4)4

(4)3 3/ 4

Degenerate band edge (the intermidiate case)

The slow mode gro

up velocity is

while its energy density diverges

, 0.24

,6

3.

a

s

dd d d

g

dg d d

d

k k

v

v k kk

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

stationary IP a

.

nd

,

d

T d

W

S u W

the case of total reflection at a regular BE.

DBE case

d

ω

k

19

0 20 40 600

20

40

60

80

100

120

Distance z

Square

d a

mplit

ude

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

mplit

ude d)

0 20 40 600

50

100

150

Distance z

e)

0 20 40 600

10

20

30

40

50

Distance z

f)

Smoothed profile of the frozen mode at six different frequencies in the vicinity of SIP: (a) ω = ωd – 10 – 4 c/L, (b) ω = ωd – 10 – 6 c/L, (c) ω = ωd , (d) ω = ωd + 10 – 6 c/L,(e) ω = ωd + 10 – 5 c/L, (f) ω = ωd + 10 – 4 c/L. In all cases, the incident wave has the same polarization and unity amplitude. The distance z from the surface of the photonic crystal is expressed in units of L.

Frozen mode profile at different frequencies close to DBE [9]

20

Subsurface waves at band gap frequencies close to DBE

21Giant subsurface waves at band gap frequencies close to DBE [9].This phenomenon can be viewed as a particular case of frozen mode regime.

4

d dk k

d

ω

k

2 2 2 2

2 2 2 2

In vacuum (at ), the component of the wave vector is

Surface waves corespond to imaginary value of :

, while

At f

0

req

z x x

z

x x d

z z k

k c k k

k

k k c

uencies close to DBE, the surface wave profile is exactly

the same as that of a DBE related frozen mode.

22Smoothed profile of a regular surface wave at the photonic crystal / air interface.

Smoothed profile of a giant subsurface wave at the photonic crystal / air interface.

In a subsurface wave, the field amplitude decays exponentially in vacuum

(at ). But inside the periodic medium (at ), the field amplitude

increases sharply with the distance and reache

s

0 0

its ma

z z

z

1/ 4 1/ 4

max

ximum value of

at a distance

Then it slowly decays as the distance further increases.

.

d dZ

z

23The following features are unique to giant subsurface waves (GSSW):

1. Although the amplitude of GSSW can be very large, it is value at the surface is relatively small. Therefore, GSSW are much better confined and should be much less leaky, compared to regular surface waves.

2. At any given frequency, there is only one GSSW propagating in a certain direction along the surface of the photonic slab. This direction is a function of frequency.

3. If the layered array with DBE has a defect layer, a GSSW can also be realized as a localized mode with the profile similar to that of the frozen mode. Such a mode can accumulate a lot of energy – much more than a regular localized state with the same amplitude at the maximum.

24

What structures can display band diagrams with RBE, SIP, and DBE?

1. RBE – regular band edge (there is no frozen mode regime here)

2. SIP – stationary inflection point

3. DBE – degenerate band edge

25

2

g gk k k

ωg

1. RBE occurs in any periodic array of any dimensionality

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

LL

Alternating layers A and B can be made of any two different low-loss materials – either isotropic, or anisotropic. L is a unit cell of the periodic structure – in this case it includes two layers.

262. A SIP at normal incidence can occur in periodic arrays of magnetic and misaligned anisotropic layers, with 3 layers in a unit cell L [2].

YY

XX

A1 A2

3

0 0k k

k

ω

0

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

LL

XX

ZZ

YY

A1 and A2 are anisotropic layers with misaligned in-plane anisotropy. The misalignment angle is different from 0 and π/2.

F – layers are magnetized along the z direction.

27

Alternating layers A and B , of which the A layers display off-plane anisotropy (εxz ≠ 0), while the B layers can be isotropic.

A SIP at oblique incidence can occur in periodic arrays of two layers, of which one must display off-plane anisotropy [3, 4].

3

0 0k k

k

ω

0

AA BB AA BB A A B B A A BB

DD

XX

ZZ

YY

28

4

d dk k

d

ω

k

3. A DBE at normal incidence can occur in periodic arrays of with three layers in a unit cell L, of which two must display misaligned in-plane anisotropy [6].

XX

ZZ

YY

LL

AA11 AA22 BB AA11 AA22 BB AA11 AA22 BB

YY

XX

A1 A2

A1 and A2 are anisotropic layers with misaligned in-plane anisotropy. The misalignment angle is different from 0 and π/2.

B – layers can be isotropic.

29

Alternating layers A and B , of which the A layers must display in-plane anisotropy (εxx ≠ εyy , εxz = εyz = 0), while the B layers can be isotropic.

A DBE at oblique incidence can occur in periodic arrays of two layers, of which one must display in-plane anisotropy [9].

4

d dk k AA BB AA BB A A B B A A BB

DD

XX

ZZ

YY

d

ω

k

30

2

1

The presence of anisotropic (birefringent) materials may not be necessary in periodic structures with 2D or 3D periodicity. For example, one can use periodic arrays of identical rods made of isotropic dielectric material to replace uniform but anisotropic A layers. The angles 1 and 2 now define the

orientations of the adjacent arrays of the rods. After the replacement, the entire periodic structure will be a 3D photonic crystal.

31

Summary on what structures can support different versions of frozen mode regime.

The layered structures supporting frozen mode regime at oblique incidence are, generally, simpler, compared to those supporting the normal version of frozen mode regime. For instance, at oblique incidence, a unit cell of the periodic array can include as few as two different layers.

The layered structures supporting DBE are, generally, less complicated and more practical, compared to those supporting SIP. The differences are especially pronounced at optical frequencies.

The use of periodic structures with 2D and 3D periodicity can remove the necessity to employ birefringent constitutive components, which is important. The downside, though, is that layered arrays are, generally, easier to build and much easier to analyze.

32Publications[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] 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, 293-382 (2006).

[9] A. Figotin and I. Vitebskiy. "Frozen light in photonic crystals with degenerate band edge". To be submitted to Phys. Rev. E.

33

Auxiliary slides

34

1. Giant transmission band edge resonance(Fabry-Perrot cavity resonance in the vicinity of DBE)

35

s

Periodic stack is composed of alternating layers and .

Resonant wave lengths:

Resonant wave numbers:

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

, 1, 2,...s g

A B

Ds s

k k s sNL

2

g gk k

Standard arrangement: a transmission band-edge resonance below RBE

k

ωg

AA B A B B A B A B A B A BA B

DD

36Transmission dispersion and resonant field inside the photonic slab: generic case of RBE

Smoothed field intensity distribution at the frequency of first transmission resonance

Stack transmission vs. frequency.ωg is a regular photonic band edge

37

4

d dk k

Giant transmission band edge resonance below DBE [6]

d

ω

k

DD

AA11 AA22 BB AA11 AA22 BB AA11 AA22 BB

At normal incidence, a unit cell of the stack includes two misaligned anisotropic layers A1 and A2, and an isotropic B layer.

AA BB AA BB A A B B A A BB

DD

At oblique incidence, the periodic stack involves one anisotropic layer (A) and one isotropic layer (BB).

38

Field intensity distribution at the frequency of first transmission resonance

Transmission dispersion and resonant field inside the photonic slab: the case of a degenerate photonic band edge

Stack transmission vs. frequency.ωd is a degenerate photonic band edge

39

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

Regular transmission band-edge resonance vs.giant transmission band-edge resonance [6]

A stack of 8 layers with a degenerate photonic BE performs as well as a stack of 64 layers with a regular photonic BE !

d

ω

k

k

ωg

40Regular Band Edge vs. Degenerate Band Edge. Example: photonic slabs with N = 8 and N = 16

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)

41

Additional advantages of the giant transmission resonance at oblique incidence [9]:

- The oblique effect can occur in much simpler periodic array involving just two alternating layers A and B. No misaligned or off-plane anisotropy is required.

- The oblique resonance frequency can be changed continuously within a wide frequency range by adjusting the direction of incidence. There is no need to change any physical parameters of the periodic structure.

- Extreme directional selectivity can be utilized in transmitting and receiving antennas.

42

2. Frozen mode regime derived from the Maxwell equations

43

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

44

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

45

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

46Transfer 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

47

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

48

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 cell

L mm

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

tion:

Symmetric dispersion relation (if ):

for any

, , , , , ,

, , , , , , ,L L

k k k k k k k k

T U T U

k k k k k k k k

49

11

22

3

4

0 0 00 0 0

0 0 00 0 00 . :

0 0 10 0 0

0 00 0 0

: RBE

Jordan normal form of the transfer matrix of the periodic

layered array correspomding to each of the stationary point

s

L La

L

T Tk

T

1 0

0 0

0 0

0 0

0

0 0 0 1 0 0

0 1 0 0 1 0.

0 0 1 0SIP: D

0 1

0 0 0 0 0 0

BE:

a

L LT T

50

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

51

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

52

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