1 slow light and frozen mode regime in photonic crystals april, 2007 alex figotin and ilya vitebskiy...
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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.