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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
/ 0Strong 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 / 30 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
/ , / 1Energy 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 SS S S S
uS 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 / 30
The slow mode energy flux remains finite
implying conversion of the incident light to the frozen modewith 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 atstationary inflection point and the case of tota
.
l reflection at regula
rpho
.T dS u W
tonic band edge.
k
ω
d
Degenerate band edge
11
( )( )
( )( )( )
x
y
x
y
E zE z
zH zH z
Incident waveΨI
Reflected wave
ΨR
Lossless semi-infinite photonic slab
Transmitted slow mode
ΨT
0
1/ 30
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.
13Summary 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 Ws
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 Ws
k k
NW Ws
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 cul l lc
l D u Dl 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 iE 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 zE z
z M JA A A J J JH zH 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 mediax
yz
x
y
z
E zE z
z i M z z z M JMJH zcH 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 JT
is
Explicit expressions for the transfer matrices of individualhomogeneous 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 Uk k k k k k k k
T U T Uk k k k k k
3 4,k k
34
11
22
3
4
1
0
0
0
1
0 0 00 0 00 0 00 0 0
, : 0 0 10 0 00 0 00
Regular : Band edge
S.I.P. :
0 0
0
0 00 1 00
0 10 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 00 1 0
, D.B.E. :0 0 10 0 0
LT
0 1
0
0
0
30 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
iek k
36
0
0 0
0Consider 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