Photonic Band Gap
Materials
A Sem iconductor
for Light
www.physics.utoronto.ca/~john
Photonic Band Gap Materials
– Two Fundamental Optical Principles
• Localization of Light
– S. John, Phys. Rev. Lett. 53,2169 (1984)
– S. John, Phys. Rev. Lett. 58,2486 (1987)
• Inhibition of Spontaneous Emission
– E. Yablonovitch, Phys. Rev. Lett. 58,
2059 (1987)
! State of research
– Since 1995, the number of scientific and engineering
publications per year has been doubling every 18
months.
History of Localization and Photonic Band Gaps
1958:
1958: Electronic
Electronic Localization
Localization:: All
All electronic
electronic states
states localize
localize
P.W.
Absence of
of Diffusion
Diffusion in
in Certain
Certain Random
Random Lattices
Lattices““
P.W. Anderson,
Anderson, Phys.Rev.
Phys.Rev. 109
109,, 1492
1492 (1958)
(1958) ""Absence
ρ(ω)
ρ(ω)
ω
ω
"Anderson Transition"
ρ(ω)
1983:
1983: Phonon
Phonon Localization
Localization::
Some
states are
are localized
localized
Some states
S.
S. John
John &
& M.J.
M.J. Stephen
Stephen
Phys.Rev.
6358 (1983)
(1983)
Phys.Rev. B28
B28,, 6358
ω
ρ(ω)
1984:
1984: Photon
Photon Localization
Localization::
Any
Any states
states localized?!
localized?!
S.
S. John,
John, PRL
PRL 53
53,, 2169
2169 (1984)
(1984)
P.W.
Mag. B52,505
P.W. Anderson,
Anderson, Phil.
Phil.Mag.
B52,505 (1985)
(1985)
ω
1985:
1985: Weak
Weak Localization
Localization of
of Light
Light (Coherent
(Coherent Backscattering)
Backscattering)
M.P.
M.P. Von
Von Albada
Albada &
& A.
A. Lagendjik
Lagendjik,, PRL
PRL 55
55,, 2692
2692 (1985)
(1985)
P.E.
P.E. Wolf
Wolf &
& G.
G. Maret
Maret,, PRL
PRL 55
55,, 2696
2696 (1985)
(1985)
1987:
Photonic Band
Band Gap
Gap
1987: Prediction
Prediction of
of Photonic
S.
S. John
John and
and E.
E. Yablonovitch
Yablonovitch
1990:
1990: Computational
Computational Demonstration
Demonstration of
of PBG
PBG (band
(band structure)
structure)
K.M.
K.M. Ho,
Ho, C.T.
C.T. Chan,
Chan, C.M.
C.M. Soukoulis
Soukoulis,, PRL
PRL 65,
65, 3152
3152 (1990)
(1990)
1991:
1991: Experimental
Experimental Demonstration
Demonstration of
of Microwave
Microwave Localization
Localization and
and PBG
PBG
2D
2D system:
system: S.
S. Shultz
Shultz et
et al.,
al., Nature
Nature 354,
354, 53
53 (1991)
(1991)
3D
3D system:
system: A.Z.
A.Z. Genack
Genack &
& N.
N. Garcia,
Garcia, PRL
PRL 66,
66, 2064
2064 (1991)
(1991)
E.
E. Yablonovitch
Yablonovitch,, T.J.
T.J. Gmitter
Gmitter,, K.M.
K.M. Leung,
Leung, PRL
PRL 67,
67, 2295
2295 (1991)
(1991)
1990-present:
1990-present: Quantum
Quantum Electrodynamics
Electrodynamics in
in aa PBG
PBG
S.
S. John
John &
& J.
J. Wang,
Wang, PRL
PRL 74,
74, 2418
2418 (1990)
(1990)
…
…
T.
T. Quang
Quang et
et al.,
al., PRL
PRL 79,
79, 5238
5238 (1997)
(1997)
1994:
1994: Experimental
Experimental Observation
Observation of
of “Laser
“Laser Paint”
Paint”
N.M.
N.M. Lawandy
Lawandy et
et al.,
al., Nature
Nature 368,
368, 436
436 (1994)
(1994)
Photon-Atom
Photon-Atom
Bound
Bound State
State
1995:
1995: “Large”
“Large” Scale
Scale 2D
2D PBG
PBG Macroporous
Macroporous Silicon
Silicon
U.
U. Gruning
Gruning,, V.
V. Lehman,
Lehman, C.M.
C.M. Englehardt
Englehardt,,
Applied
Applied Phys.
Phys. Lett
Lett.. 66,
66, 3254
3254 (1995)
(1995)
U.
U. Gruning
Gruning,, V.
V. Lehman,
Lehman, S.
S. Ottow
Ottow,, K.
K. Busch,
Busch,
Applied
Applied Phys.
Phys. Lett
Lett.. 68,
68, 3254
3254 (1996)
(1996)
1997:
1997: Experimental
Experimental Demonstration
Demonstration of
of Light
Light Localization
Localization
D.
D. Wiersma
Wiersma et
et al.,
al., Nature
Nature 360,
360, 671
671 (1997)
(1997) cover
cover story
story
1995
-present: ““Woodpile”
Woodpile”
1995-present:
Structures
Structures
E.
E. Ozbay
Ozbay,, Bilkent
Bilkent University
University
S.
S. Noda,
Noda, Kyoto
Kyoto University
University
S.
S. Lin,
Lin, Sandia
Sandia National
National Lab
Lab
1998-present:
1998-present: Inverted
Inverted Opals
Opals
TiO
TiO22,, CdSe
CdSe,, Ge
Ge,, Si
Si,, GaP
GaP,, …
…
2001: Square Spirals
Si, SiO2
Schrodinger
-Maxwell Analogy
Schrodinger-Maxwell
!
! !
1 ∂B
∇× E = −
c ∂t
!
22
! ! !
1 ∂ B
∇ × ∇ × E = − 22 22
c ∂t
!
! ! 1 ∂E
∇× B =
c ∂t
!
ω 22
⇒
ε ( x) E
22
c
ε00
εfluct
fluct
Dielectric
( x)
Dielectric constant
constant ε ( x) = ε 00 + ε fluct
fluct
! ! ! ! ω 22
! ω 22 !
( x ) E = 22 ε 00E
−∇ E + ∇(∇E ) − 22 ε fluct
fluct
c
c
22
ε 00 > −ε fluct
fluct ( x)
Real,
Real, Positive
Positive Dielectric
Dielectric Constant
Constant
ω → 0
 → Extended States
ω → ∞
Can
Can there
there exist
exist localized
localized (bound)
(bound) states
states at
at energies
energies higher
higher than
than the
the highest
highest potential
potential barrier?
barrier?
Types
Types of
of scattering
scattering::
1.
1. Rayleigh
Rayleigh
2.
2. Geometric
Geometric Optics
Optics
3.
3. Resonance
Resonance "
Microscopic
" Microscopic
"
Macroscopic
" Macroscopic
S.
2169 (1984)
(1984)
S. John,
John, PRL
PRL 53
53,, 2169
Conventional
Conventional
Localization
Localization
Criterion
Criterion
a
lscatt
Ioffe-Regel:
Ioffe-Regel:
""**
22π
≈≈11
λ
l*
Classical
Classical (Elastic)
(Elastic)
Transport
Transport
Mean
Mean Free
Free Path
Path
l*
Rayleigh
l*~λ4
weak disorder
strong disorder
Resonance
a
Geometric ray optics
λ/2π
a
Resonance
Resonance Regime
Regime
Generalized
Generalized Localization
Localization
Criterion
Criterion
S. John
PRL 53, 2169 (1984)
( )
""** ××( phase
phase space
space) ≈≈ 44ππ
22
2
not
not necessarily
necessarily 44ππ kk 2
In the presence of scattering resonances the Photon Density of States ρ(ω) is strongly
modified from the free space value ρ0(ω)=ω2/(π2c3).
Near Photonic Band Gap ρ(ω) << ρ0(ω)
( )
22
*
ππ 22ccρρ((ω
ω)) ""* ≈≈11 ⇒
⇒ Localization with very weak disorder (l* >> λ vacuum length)
S.John, R.Rangarajan, Phys.Rev.B. 38, 10101 (1988)
Photonic Band Gap (PBG) Formation: A synergetic interplay
between Microscopic and Macroscopic Resonances
Illustrative Example: (d=1) Scalar Wave
2
2
ω
ω
ω2
ω2
((xx))EE == 22 εε00EE
−∇
−∇ E −− 22 εε fluct
cc fluct
cc
22
E
ε(x)
L= lattice constant
a = "sphere radius"
ω
ω
Largest
Largest 1-d
1-d gap
gap occurs
occurs when
when single
single
scattering
scattering resonance
resonance and
and Bragg
Bragg
resonance
resonance conditions
conditions coincide
coincide..
ππ/L
/L
2π
/L
2π/L
Macroscopic
Macroscopic Bragg
Bragg Resonance
Resonance
ω m
mπ
==
cc
LL
kk
Microscopic
Mie” Resonance
Microscopic ““Mie”
Resonance
transmission resonance
maximum reflection λ/4=2a
ω
ππ
22ππcc
ω
λ
=
⇒
=
λ
=
⇒
=
Refractive index n
ω
cc 22nn(2
ωnn
(2aa))
PBG Formation (continued)
Choose a and L so that microscopic and macroscopic resonances occur at the
same frequency: volume filling fraction f = 2a/L = 1/2n
∆ω/ω0
1/2n
f
Detailed Band Structure Calculation:
n > 2 Diamond lattice of spheres.
3D
K. Ho, C.T. Chan, C.Soukoulis PRL
65, 3152 (1990)
Topology of Dielectric Microstructure
Cermet (nsphere > nbackground)
• Low velocity (high index)
component is not connected.
•Favoured by scalar and elastic
waves
Network
•Favoured by EM waves.
PhotoElectrochemical Etching of 2-d
Photo-Electrochemical
and 3-d Silicon Photonic Crystals
(collaboration
(collaboration with
with Max-Planck-Institute
Max-Planck-Institute of
of Microstructure
Microstructure Physics,
Physics, Germany)
Germany)
2-d
Si Photonic
Photonic Crystal
Crystal
2-d Si
(1,0,0)
(1,0,0) surface
surface
pores
pores etched
etched along
along (1,0,0)
(1,0,0) direction
direction
3-d
Si PBG
PBG Material
Material
3-d Si
(1,1,1)
(1,1,1) surface
surface
criss
-crossing pores
criss-crossing
pores along
along (1,1,3)
(1,1,3) direction
direction
Micro-Circuitry in a 2-D Photonic Crystal
Group Velocity Dispersion
in 2D Silicon PBG ε=1
ε=11.9
r/a=0.48, ri=0.4
ε=4.2
GVD
(psec/nm/km)
Group
Velocity
Configurable WDM add
drop filters
F2
F1
F1, F2, …
Holey Fibre
Silica Opal Templates
#
#
#
#
#
#
#
#
Mono-disperse
Mono-disperse silica
silica spheres,
spheres, 2-5
2-5 %
% variation
variation in
in diameter
diameter
Self-assembled
Self-assembled into
into fcc
fcc lattice
lattice
Sintered
Sintered to
to induce
induce necking
necking between
between spheres
spheres
Control
Control of
of infiltration
infiltration and
and etching,
etching, mechanical
mechanical and
and photonic
photonic properties
properties
Si Inverted Opal
Optical Reflection Spectrum in the Γ- L Direction
Density of States for the FCC Lattice
air
macroporous) silicon,
air voids
voids in
in ((macroporous)
silicon, closed
closed packed
packed
1.0
DOS
DOS
(arbitrary
(arbitrary
units)
units)
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ω
a/2πc
ωa/2πc
Kurt
Kurt Busch
Busch &
& S.
S. John,
John, Phys.Rev.
Phys.Rev. EE 58
58,, 3896
3896 (1998)
(1998)
0.9
|E| at Dielectric Band Edge
(3-d Si Inverse Opal)
|E| at Air Band Edge
(3-d Si Inverse Opal)
Liquid
Liquid Crystal
Crystal Photonic
Photonic Band
Band Gap
Gap Materials:
Materials:
The
The Tunable
Tunable Electromagnetic
Electromagnetic Vacuum
Vacuum
Cross-sectional
Cross-sectional view
view through
through the
the inverse
inverse opal
opal backbone
backbone (blue)
(blue) resulting
resulting from
from incomplete
incomplete
infiltration
infiltration of
of silicon
silicon in
in the
the air
air voids
voids of
of an
an artificial
artificial opal.
opal. A
A tunable
tunable PBG
PBG is
is obtained
obtained by
by
infiltrating
infiltrating this
this backbone
backbone with
with nematic
nematic liquid
liquid crystal
crystal (green)
(green) which
which wets
wets the
the inner
inner surface
surface
of
sphere (only
(only one
one is
is shown
shown in
in the
the figure).
figure).
of each
each sphere
Kurt
Kurt Busch
Busch &
& Sajeev
Sajeev John,
John, PRL
PRL 83
83,, 967
967 (1999)
(1999)
Infiltrated Inverted Opal ((Si,
Si, fSiSi =24.5%)
εyy=1.96,
=36.8%
Liquid
Liquid crystal
crystal (BEHA):
(BEHA): εεxx==ε
=1.96, εεzz=2.56,
=2.56, ffBEHA
BEHA=36.8%
1.0
1.0
Total
Total DOS
DOS
(arbitrary
(arbitrary units)
units)
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
ω
a/2πc
ωa/2πc
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
Infiltrated Inverted Opal (Si, fSi =24.5%)
Change in Total Photon Density of States
0.03
0.03
ϑ
ϑ == 00
ϑ
/8
ϑ == ππ/8
ϑ
π/16
ϑ == 55π/16
Total
Total DOS
DOS
(arbitrary
(arbitrary units)
units)
0.02
0.02
0.01
0.01
0.00
0.00
0.76
0.76
0.77
0.77
ω
a/2πc
ωa/2πc
0.78
0.78
0.79
0.79
Square Spirals Structure
S. John and O. Toader
Square Spirals (continued)
Direct structure
15 %
Density of States for
Inverted Structure with
24%
3-D PBG
Inverted structure
24 %
SUMMARY
Light Localization occurs in carefully engineered dielectrics
without the presence of “classical” turning points
Photonic Band Gap formation is a synergetic interplay
between microscopic and macroscopic resonances
2-D photonic crystal micro-fabrication is well developed
3-D PBG materials: inverse diamond structure, woodpile structure
inverse opals (fcc), square spiral (tetragonal)
Optical Micro-circuitry: Band Gap Engineering
Point and Line Defects (sub-gap)
Tunable Photonic Band Gaps
Photonic Band Gap
Materials
A N ew Frontier in
Q uantum and
N onlinear O ptics
www.physics.utoronto.ca/~john
Two Fundamental Optical Principles
Localization of Light
S.
Lett. 53
S. John,
John, Phys.Rev.
Phys.Rev.Lett.
53,, 2169
2169 (1984)
(1984)
S.
Lett. 58
S. John,
John, Phys.Rev.
Phys.Rev.Lett.
58,, 2486
2486 (1987)
(1987)
Inhibition of Spontaneous Emission
E.
Lett. 58
E. Yablonovitch
Yablonovitch,, Phys.Rev.
Phys.Rev.Lett.
58,, 2059
2059 (1987)
(1987)
Consequences
• Photon-atom
Photon-atom bound
bound states
states
• Low
Low threshold
threshold band-edge
band-edge lasing
lasing without
without aa cavity
cavity mode
mode
• New
New quantum
quantum states
states of
of light
light
• Low
Low threshold
threshold and
and other
other anomalous
anomalous Nonlinear
Nonlinear Optical
Optical Response
Response
• Coherent
Coherent control:
control: single
single atom
atom optical
optical memory
memory
• Optical
Optical switching
switching and
and low
low threshold
threshold All-optical
All-optical Transistor
Transistor action
action
• Classical
Classical and
and Quantum
Quantum Gap
Gap Solitons
Quantum Electrodynamics and Collective
Phenomena in a Photonic Band Gap
Photon-Atom
Photon-Atom Bound
Bound State
State
ω
ωcc
ω
ωvv
S.
S. John
John &
& J.
J. Wang,
Wang, PRL
PRL 64
64,, 2418
2418 (1990)
(1990)
impurity
impurity atoms
atoms
ω
ωvv << ω
ω00 << ω
ωcc
excited
excited
ω00
hhω
ground
ground state
state
No
No propagating
propagating modes
modes in
in aa PBG,
PBG, so
so ordinary
ordinary spontaneous
spontaneous emission
emission of
of light
light is
is eliminated.
eliminated.
But
into classically
classically forbidden
forbidden gap
gap..
But photon
photon can
can tunnel
tunnel into
Solve
H Ψ
Solve Schrodinger
Schrodinger equation
equation H
Ψ == EE Ψ
Ψ ;;
Case
Case (i)
(i) Atom
Atom in
in Vacuum
Vacuum
ck
ω
ck
ω
kk
Complex
Complex E-plane
E-plane
scattering
scattering states
states
ih/τ
X
X EE11++ih/τ
Resonance
Resonance Fluorescence
Fluorescence
ττ == spont
spont.. emission
emission time
time
∞
∞
Variational
Variational
( n)
Ψ
Ψ == ∑ φφnn nn ++ ∑∑ ϕϕλλ( n ) λλ;;nn
nn==11
Case
Case (ii)
(ii) Atom
Atom in
in PBG
PBG
ω
ω
kk00
∞
∞
nn==00 λλ
photon
photon in
in
mode
mode λλ
atom
atom in
in
th
th
nn level
level
kk
Complex
Complex E-plane
E-plane
ωvv
hhω
X
X
ωcc
hhω
Real
Real solution:
solution: Photon-atom
Photon-atom bound
bound state
state
Model
Model Hamiltonian
Hamiltonian for
for Quantum
Quantum Optics
Optics in
in aa PBG
PBG material
material
Resonance
Resonance Two-level
Two-level Atom
Atom
ω
ωAA
bb
External
External Laser
Laser Field
Field
ω
ω
kk00
aa
kk
Rabi
Rabi frequency
frequency Ω
Ω
11
i (ω t +φ ) −
− i (ω t +φ ) +
+
+ −
+
H
H == ##ω
Ω eei (ωLL t +φ )σσ − −− ee − i (ωLL t +φ )σσ + 
ω AAσσ zz ++ ∑ ##ω
ωλλaaλλ+aaλλ ++ ii##∑ ggλλ aaλλ+σσ − −−σσ +aaλλ  ++ ii##Ω
22
λλ
λλ
1.
1. Rotating
Rotating Wave
Wave Approximation
Approximation
2.
2. Neglect
Neglect external
external field
field for
for now,
now,
3.
ωAA::
3. Simplify
Simplify by
by going
going to
to aa new
new “rotating
“rotating frame”
frame” with
with frequency
frequency ω
Define
Define Ψ
R((tt)) Ψ
Ψ == R
ΨRR((tt))

 

++
Unitary
a
a
2
σ
+
Unitary Operator
Operator R
R((tt)) ≡≡ exp
a
a
exp −−iiω
2
ωAA  σ zz + ∑ λλ λλ tt

λλ

 
H$ → H = ∑ #∆ λλaλ+λ+aλλ + i#∑ gλλ  aλ+λ+σ −− − σ ++aλλ 
λλ
∆∆λλ == ω
ωλλ −− ω
ωΑΑ
detuning
detuning frequency
frequency
λλ
coupling
coupling ggλλ ==
##
! !
ω
ωAAdd21
21
(ee!λλ ⋅⋅µµ!λλ )
##
22εε00ω
ωλλVV
Solve
Solve Time-dependent
Time-dependent Schrodinger
Schrodinger equation
equation projected
projected onto
onto 1-photon
1-photon sector:
sector:
− i∆ t
Ψ
ΨRR == bb22((tt)) 2,{0}
2,{0} ++ ∑ bb1,1,λλ ((tt)) 1,{
1,{λλ}} ee−i∆λλ t
“interaction
“interaction picture”
picture”
λλ
dd
bb22((tt)) == −−∑ ggλλbb1,1,λλ ((tt))ee−−ii∆∆λλtt
dt
dt
λλ
(1)
(1)
dd
bb1,1,λλ ((tt)) == ggλλbb22((tt))eeii∆∆λλtt
dt
dt
(2)
(2)
tt
ii∆∆λλττ
Formal solution of (2) bb1,1,λλ ((tt)) == ggλλ ∫ ddττ bb22((ττ ))ee
00
2 − i∆ (t −τ )
Memory Kernel G
G((tt −−ττ )) ≡≡ ∑ ggλλ2ee−i∆λλ (t −τ )
t
⇒
⇒
t
dd
bb22((tt)) == −−∫ ddττ bb22((ττ ))G
G((tt −−ττ ))
dt
dt
0
((3)
3)
0
b2(0)=1 $ atom initially excited
λλ
In free space, we obtain exponential decay.
For ω =ck, this is like the integral representation of a δ-function.
γγ spsp
%
G
t
(
)
−
τ
δδ ((tt −−ττ )) →
G (t − τ ) %
→
22
22
t
−−γγsp
sp t
bb22((tt)) %% ee
22
ω 3A3Ad 21
21
γ spsp =
3πε 00#c33
In a PBG material: S. John & T. Quang, Phys. Rev. A 50, 1764 (1994).
Non Markovian
Radiative Dynamics
Photon
Localization
free space
spontaneous
emission rate
PBG
PBG Model
Model Dispersion
Dispersion Relations
Relations
ω
ω
ω
ωcc
Isotropic
Isotropic model
model
2
ω
ωkk %% ω
ωcc ++ AA((kk −− kk00)) 2
!
VV
33 !
d k
→
∑→
33 ∫ d k
(2
)
π
(2
)
π
kk
ρ(ω)
ρ(ω)
kk00
kk
{
}
2
dk
dk kk 2
2
G
exp
G((tt −−ττ )) %% ∫
iA((kk −− kk00))2((tt −−ττ )) ++ iiδδ ((tt −−ττ ))
exp −−iA
ω
ωkk
δδ == ω
ωΑΑ −− ω
ωcc
ω
ωcc
kk0202
ππ
eeiiδδ ((tt−−ττ))

→

→
ω
iA((tt −−ττ ))
ωcc iA
larg
largee t-t-ττ
stationary
stationary phase
phase
approximation
approximation
33//22
i
(
)
β
i
(
)
β
$$((ss)) == −−
,, ββ == ω
Laplace
G
ωAA γγ spsp //ω
ωAA
Laplace transform
transform G
1/
1/22
((ss −− iiδδ ))
(
Anisotropic
Anisotropic model
model
ω
ωk!k!
!! !! 2
%% ω
ωcc ++ AA((kk −− kk00)) 2
)
Power
Power Law
Law
Decaying
Decaying Memory
Memory
22//33
For a physical anisotropic dispersion relation, the Band Edge is
associated with a single point k0 (rather than the sphere |k| = k0)
{
}
2
dq
dq qq 2
2
G
exp
−−iiAq
GAA((tt −−ττ )) %% ∫
Aq 2((tt −−ττ )) ++ iiδδ ((tt −−ττ ))
exp
ω
ωq!q!
eeiiδδ((tt−−ττ))

→

→
((tt −−ττ ))33//22
large
large tt--ττ
ω
ω
Non
Non Markovian
Markovian
Memory
Memory Kernel
Kernel
ρ(ω)
ρ(ω)
ω
ωcc
ω
ω
Non
Non Markovian
Markovian Radiative
Radiative Dynamics
Dynamics has
has direct
direct implications
implications on
on Atomic
Atomic Line
Line Shape
Shape
Define
Define Emission
Emission Spectrum
Spectrum
∞
∞
− i (ω −ω ) t
∗
SS((ω )) ≡≡ ∫ dt
dt ee −i (ω −ωAA )t bb22∗((tt))bb22(0)
(0) ++ cc..cc == 22Re
Rebb$$22 ( −−ii((ω −−ω AA)))
00
∞
∞
where
where bb$$22((ss)) ≡≡ ∫ ee−−ststbb22((tt))dt
dt
integration
integration by
by parts
parts
00
convolution
convolution theorem
theorem
From
From Equation
Equation of
of Motion,
Motion, −−bb2 (0)
+ sb$$ ( s) = −b$$ ( s)G$$ ( s) →
→ bb$$22((ss)) ==
2 (0) + sb22 ( s ) = −b22 ( s )G22 ( s )
$$ ((ss)) ==
where
where G
G
∑
22
λλ
ggλ2λ2
∞
∞
∫ dtdt ee
0
−−sstt −−ii∆∆λλtt
ee
0
== ∑
λλ
ggλ2λ2
11
ss ++ ii((ω
ωkk −−ω
ωAA))
11
$$ ((ss))
ss ++ G
G
22
γγ sspp
→
+
→ iiδδLam
Lambb +
22
in free
space
(Wigner-Weisskopf approx.)
3-level
-configuration
Λ-configuration
3-level Atom
Atom in
in Λ
|3>
|2>
|1>
≅ ω (band edge)
ω
ω31
31 ≅ ωcc (band edge)
Let
and γ 32 be
Let δδ32
be the
the Lamb
Lamb shift
shift
32 and γ32
and
and the
the spontaneous
spontaneous emission
emission rate
rate
for
(far from PBG)
for transition
transition ω
ω32
32 (far from PBG)
Markovian
Markovian
b$22(s) =
1
(iβ )
3/
3/22
s + iδ32
+γ 32
−
32
32
s − iδ
Atomic
Atomic Population
Population
on
on excited
excited states
states |3>
|3>
of
of 3-level
3-level Atom
Atom
Atomic
Atomic Lineshape
Lineshape
near
near Band
Band Edge
Edge
δδ == −β
−β
P(t)
P(t)
δ=ω
−ω
δ=ω31
31−ωcc
ωλλ))
S(
S(ω
55
1.0
1.0
0.8
0.8
Λ
Λ configuration
configuration
|3>
|3>
44
γγ32
=0.0β
32=0.0β
|2>
|2>
33
0.6
0.6
γγ32
=0.1β
32=0.1β
0.4
0.4
22
γγ32
=0.2β
32=0.2β
0.2
0.2
γγ32
=0.5β
32=0.5β
0.0
0.0
00
33
66
99
ββtt
|1>
|1>
δ=2β
δ=2β
δ=−0.5β
δ=−0.5β
11
δ=0
δ=0
3-level
3-level atom
atom
=β
γγ32
32=β
00
12
12
15
15
−−66
−−44
−−22
00
22
)/β
((ω
ωλλ−ω
−ω32
32)/β
44
66
Main conclusions
Radiative
Radiative dynamics
dynamics near
near aa PBG
PBG is
is different
different than
than in
in ordinary
ordinary vacuum.
vacuum. ItIt is
is
much
much richer
richer than
than simply
simply “no
“no spontaneous
spontaneous emission
emission takes
takes place”.
place”.
(1)
Markovian radiative
(1) NonNon-Markovian
radiative Decay
Decay
Photon
Photon Localization
Localization
(2)
(2) Vacuum
Vacuum Rabi
Rabi Splitting
Splitting ++ Fractionalized
Fractionalized Steady
Steady State
State Inversion
Inversion
(3)
(3) Collective
Collective Enhancement
Enhancement of
of Radiative
Radiative Dynamics
Dynamics Near
Near PBG
PBG Edge
Edge
(4)
(4) Lasing
Lasing near
near aa Photonic
Photonic Band
Band Edge
Edge (even
(even though
though particular
particular superradiance
superradiance
may
may be
be hard
hard to
to observe,
observe, similar
similar physics
physics occurs
occurs in
in Laser
Laser action)
action)
Collective Spontaneous Emission
NN
(k )
N
N atoms
atoms Define
Define JJijij == ∑ σσijij( k );;
kk==11
σσij(ij(kk)) ≡≡ ii k k jj
k k
(
+
+
H
− J 21aaλλ
H == ∑ ##∆
∆λλaaλλ+aaλλ ++ ii##∑ ggλλ aaλλ+JJ12
12 − J21
λλ
λλ
)
atomic
atomic states
states of
of kkthth atom
atom
11
−− JJ ≤≤ M
( JJ22 −− JJ11 )
M ≤≤ JJ
22 22 11
11
22
JJ22 == ( JJ21
J
J
J
J
+
+
)
J
J
J
J
+
+
12
12
21
3
3
22 21 12 12 21
JJ33 ==
Initial
Initial State
State:: Single
Single atomic
atomic excitation
excitation
in
superradiant) state
in symmetrical
symmetrical ((superradiant)
state
Ψ
M ==11−− JJ
(0) == JJ,, M
Ψ(0)
In
Markovian)
In ordinary
ordinary vacuum
vacuum ((Markovian)
P (t ) ≡ Ψ (0) Ψ (t )
22
ρ(ω)
ρ(ω)
= e−−NNγγ2121tt
N
Collective
Collective Scale
Scale Factor
Factor ττ−−11∼∼N
ω
ω
ω
ω21
21
Near
Markovian)
Photonic Band
Band Edge
Edge ττ-1-1~N
~NΦΦ (non(non-Markovian)
Near Photonic
Anomalous
Φ determined
Anomalous exponent
exponent Φ
determined by
by band
band edge
edge singularity
singularity
ρ(ω)
ρ(ω)
−1/2
∼(ω−
ωcc))−1/2
∼(ω−ω
1/2
∼(ω−
ωcc))1/2
∼(ω−ω
ω
ω
ω
ω
Isotropic
=2/3
Φ=2/3
Isotropic PBG:
PBG: Φ
ρ(ω)
ρ(ω)
ρ(ω)
ρ(ω)
2D
=1
Φ=1
2D PBG:
PBG: Φ
ω
ω
3D
anisotropic) PBG:
=2
Φ=2
3D ((anisotropic)
PBG: Φ
Superradiance
Superradiance and
and Lasing
Lasing without
without aa Cavity
Cavity Mode
Mode
Consider
Consider an
an initial
initial state
state with
with population
population inversion
inversion and
and infinitesimal
infinitesimal initial
initial polarization
polarization
NN
(
Ψ
(0) NN == ∏
Ψ(0)
kk==11
rr 11 ++ 11−− rr 22
)
−6
rr ~~ 10
10−6
kk
Solve
Heisenberg Equation
Equation of
of Motion
Motion for
for Collective
Collective Atomic
Atomic Operators
Operators
Solve Heisenberg
t
t
dd
JJ12
(t ) = dτ G (t − τ ) J (t ) J 12((ττ ))
12 (t ) = ∫ dτ G (t − τ ) J33 (t ) J12
dt
dt
00
t
t
dd
JJ33((tt)) == −−22∫ ddττ G
(t ) J 12((ττ )) ++ cc..cc..
G((tt −−ττ )) JJ21
21 (t ) J12
dt
dt
00
JJ12
JJ
12
,, yy((tt)) ≡≡ 33
N
N
N
N
Define
Define xx((tt)) ≡≡
and
and perform
perform mean-field
mean-field factorization
factorization
t
t
dx
dx
== Ny
Ny((tt))∫ ddττ G
G((tt −−ττ ))xx((ττ ))
dt
dt
00
t
t
dy
dy
∗∗
== −−22Nx
Nx ((tt))∫ ddττ G
G((tt −−ττ ))xx((ττ )) ++ cc..cc..
dt
dt
00
Non
Non Markovian
Markovian memory
memory kernel
kernel provides
provides feedback
feedback for
for
system
system to
to self-organize
self-organize into
into coherent
coherent (localized)
(localized) state.
state.
Localization
Localization of
of Superradiance
Superradiance (Lasing
(Lasing without
without aa Cavity)
Cavity)
Initial
Initial State
State:: N
N atoms
atoms with
with overall
overall population
population inversion
inversion infinitesimal
infinitesimal initial
initial polarization
polarization
Numerical
Numerical Solution
Solution of
of
S.
3419 (1995)
(1995)
S. John
John &
& T.
T. Quang
Quang,, PRL
PRL 74
74,, 3419
Heisenberg
Heisenberg Equation
Equation of
of Motion
Motion
Spontaneous
Spontaneous Symmetry
Symmetry Breaking
Breaking:: macroscopic
macroscopic atomic
atomic polarization
polarization develops
develops in
in the
the
steady
ϖ
steady state
state limit
limit tt "
"ϖ
+1=N33 (d=3)
Peak
of Superradiant
Superradiant Emission
Emission ~~ N
NΦΦ+1
=N (d=3)
Peak Intensity
Intensity of
(i)
(i) Collective
Collective Emission
Emission
Dynamics
Dynamics
<J12
(t)>|
||<J
12(t)>|
N
N
Average
Average
Behaviour
Behaviour
1.0
1.0
Macroscopic
Macroscopic
Atomic
Atomic Polarization
Polarization
0.5
0.5
0.0
0.0
Atomic
Atomic Inversion
Inversion
0.5
−−0.5
1.0
−−1.0
00
55
10
10
15
15
2/3tt
ββN
N2/3
(ii)
(ii) Fluctuations
Fluctuations
N.Vats
N.Vats &
& S.
S. John,
John, Phys.Rev.
Phys.Rev. A
A 58
58,, 468
468 (1998)
(1998)
Optical
Optical pumping
pumping ++ damping
damping effects
effects "
Band edge
edge microlaser
microlaser
" Band
20
20
25
Atomic
Atomic inversion
inversion near
near
anisotropic
anisotropic PBG
PBG
Atomic
Atomic polarization
polarization amplitude
amplitude
near
near anisotropic
anisotropic PBG
PBG
<<J
(t)>
J12
12(t)>
N
N
δδcc == ω
-ω
ω21
21 - ωcc
0.6
0.6
δδcc == −−0.3
0.3
δδcc == 00
0.4
0.4
δδcc == ω
-ω
ω21
21 - ωcc
0.5
0.5
δδcc == −−0.3
0.3
0.5
0.5
>
<<J
J33(t)
(t)>
N
N
0.0
0.0
0.3
0.3
δδcc == 00
−−0.5
0.5
0.2
0.2
−−1.0
1.0
0.1
0.1
−6
rr == 10
10−6
δδcc == 0.1
0.1
−−1.5
1.5
0.0
0.0
00
22
44
66
N
N22ββ33tt
88
10
10
12
12
δδcc == 0.1
0.1
−6
rr == 10
10−6
00
22
44
66
N
N22ββ33tt
88
10
10
12
12
Atomic
Atomic polarization
polarization distribution
distribution for
for aa system
system of
of 100
100 atoms
atoms at
at an
an isotropic
isotropic
band
band edge,
edge, subject
subject to
to quantum
quantum fluctuations
fluctuations at
at early
early times
times
δδcc == 00
t =5
Im <J12>
Im <J12>
t = t0PBG
Re <J12>
Re <J12>
Steady
state
Im <J12>
Im <J12>
t = 11
2D Photonic Band Edge Laser
SURFACE
SURFACE EMITTING
EMITTING REGION
REGION
ELECTRODE
ELECTRODE
p-InP
p-InP CLAD
CLAD
SCH-MQW
SCH-MQW ACTIVE
ACTIVE
n-InP
n-InP CLAD
CLAD
n-InP
n-InP CLAD
CLAD with
with
TRIANGULAR-LATTICE
TRIANGULAR-LATTICE
STRUCTURE
STRUCTURE
WAFER
WAFER A
A
FUSION
FUSION
n-InP
n-InP SUBSTRATE
SUBSTRATE
WAFER
WAFER B
B
ΓΓ-XX
ΓΓ-JJ
S. Noda et. al.
0.
0.462µm
462µm
Collective Switching and All-Optical Transistor action
in a Doped Photonic Band Gap Material
• Waveguide channels carrying
“pump” and “probe” laser beams
intersect in a region doped with
impurity atoms.
•Frequency of atomic resonance
occurs near an abrupt change
(discontinuity) in the local photon
density of states of the host photonic
crystal.
Coherent Amplification of Weak Probe Beam by Pump Laser
Probe Laser Beam
Pump Laser Beam
Weak
ω,, εεPP
Weak Probe
Probe Field:
Field: ω
ii((ω
−−ii((ω
ω −−ω
ωL ))tt
ω −−ω
ωL )) tt 

##
L ++σσ ee
L
H
HP == εεP σσ12 ee

21
P 22 P  12
21



44
Atomic Excitation by a Coherent Laser Field
N
N22
Average
ω=
ω00
Average Incident
Incident
ω=ω
Energy
W
Energy density
density W
N
N11
N
N22+N
+N11=N
=N
N
N two-level
two-level atoms
atoms
N
N22/N
/N
Steady
Steady State
State solution
solution of
of
Einstein
Einstein Rate
Rate Equation
Equation
N
W
N22
W
==
N
N ##ωρ
W
ωρ((ω
ω)) ++ 22W
0.5
0.5
W
W
Vacuum
ρ(ω)=ω22/π
/π22cc33
Vacuum Density
Density of
of States
States ρ(ω)=ω
Dressed
Dressed atom
atom picture
picture n=#
n=# of
of photons
photons in
in laser
laser mode
mode >>1
>>1
|2,n-1>
|2,n-1>
ε
|1,n>
~
h
breaks
degeneracy
of
<2,n-1|H
|1,n>
~
hε
breaks
degeneracy
of
<2,n-1|Hint
int
|1,n>
|1,n>
{
Mollow splitting Ω
Ω ≡≡
<2,n+1|
<2,n+1|
<1,n+2|
<1,n+2|
(ωω −−ωω00 ) 2 ++εε22
2
Ω
22Ω
Fluorescence
Mollow
Fluorescence Mollow
Spectrum
Spectrum
<2,n|
<2,n|
<2,n-1|
<2,n-1|
<1,n+1|
<1,n+1|
<1,n|
<1,n|
ω00
hhω
ω
ω00−2Ω
−2Ω
ω
ω00 ω
+2Ω
ω00+2Ω
Einstein
Einstein picture
picture requires
requires
that
ρ(ω) is
that ρ(ω)
is smooth
smooth on
on
the
Ω so
the scale
scale of
of Ω
so that
that the
the
rate
rate of
of spontaneous
spontaneous
emission
emission is
is roughly
roughly the
the
same
same in
in the
the Mollow
Mollow
sidebands
sidebands..
Collective
Collective Switching
Switching and
and Inversion
Inversion without
without Fluctuation:
Fluctuation: All-Optical
All-Optical
Transistor
Transistor Effect
Effect in
in aa PBG
PBG Material
Material
Consider
Consider N
N 2-level
2-level atoms
atoms (in
(in aa colored
colored vacuum)
vacuum) interacting
interacting with
with aa Coherent
Coherent Laser
Laser field:
field:
(
)
11
iω t
+
+ e−−iiωωLLttJJ21
H
ωaaJJ33 ++ ∑ ##ω
ωλλaaλλ+aaλλ ++ H
H == ##ω
H11 ++ ##εε eeiωLLtJJ12
12 + e
21
22
λλ
NN
dd
Consider
the
Schrodinger
equation
#
H
i
Ψ
=
Consider the Schrodinger equation H Ψ = i# Ψ
Ψ
dt
dt
Define
R((tt)) Ψ
Ψ((tt)) == R
ΨRR((tt))
Define Rotating
Rotating Frame
Frame Ψ
dd
$
$
#
H
i
⇒
Ψ
=
⇒ H ΨRR = i# Ψ
ΨRR
dt
dt
where
where
(
+
− J 21aaλλ
where
H11 == ii##∑ ggλλ aaλλ+JJ12
where H
12 − J21
λλ
JJijij == ∑ ( ii jj )k
k
kk==11
)
− J 11
;; JJ33 == JJ22
22 − J11

 JJ33

++
where
exp −−iiω
ωLLtt  ++ ∑ aaλλaaλλ 
where RR((tt)) == exp

 22 λλ

dR
++
++ dR
$
$
#
H
R
HR
i
R
=
−
H = R HR − i#R
dt
dt
Dressed
Dressed State
State Basis:
Basis: first
first diagonalize
diagonalize atom
atom ++ external
external field
field part
part of
of H
H
εε 
∆∆a // 22
11 00 
#
for
→
Ω
#
for single
single atom
atom ## a
→
Ω



−∆
−∆aa // 22
00 −−11
 εε
Introduce
Introduce Unitary
Unitary Transf
Transf.. 1$1$ == cos
ϑ 11 −−sin
ϑ 22
cosϑ
sinϑ
(Dressed
(Dressed States)
States)
ππ
2$2$ == sin
ϑ 11 ++ cos
ϑ 22
sinϑ
cosϑ
00 ≤≤ϑ
ϑ ≤≤
22
Define
Rabi”
Define new
new ““Rabi”
22
22
/
2
ε
Ω
=
+
∆
(
)
/
2
Ω
=
+
∆
aa
frequency
frequency

and
sgn(∆∆aa)) 
and sin 22ϑ = 11 1 − sgn(

sin ϑ = 1 −
22
22 
choose
22 
choose
11++ 44εε // ∆∆aa 

N
N
(
Define
Define Dressed
Dressed Collective
Collective Atomic
Atomic Operators
Operators RRijij == ∑ i$i$ $j$j
kk==11
)
kk
22
22
ItIt is
R
R
sin
cos
sin
cos
;
ϑ
ϑ
ϑ
ϑ
=
−
+
R
R
sin
cos
sin
cos
is easily
easily verified
verified that
that JJ12
=
ϑ
ϑ
−
ϑ
+
ϑ RR12
3
21
12
3
21
12 ;
(
)
R
− R11
R33 == R
R22
22 − R11
TT
JJ21
J
=
J
=
12
21
12
2
2
JJ33 == cos
+ R12 )
cos 2ϑ
sin 2ϑ
2sinϑ
cosϑ
ϑ −−sin
ϑ RR33 −− 2sin
ϑ cos
ϑ ( RR21
21 + R12
Hamiltonian
H == H
H00 ++ H
H11
Hamiltonian in
in Dressed
Dressed State
State Basis
Basis H
+
where
H00 == ##Ω
where H
ΩRR33 ++ ∑ ##∆
∆λλaaλλ+aaλλ
λλ
ii
H
H00tt
#
#
e
H
Interaction
Interaction Picture
Picture H$ (t ) ≡ e
i
−− i H
H00tt
#
#
H11e
~~
H(t)
H(t) is
is the
the same
same as
as H
H11 except
except with
with all
all operators
operators replaced
replaced by
by interaction
interaction picture
picture operators
operators
as
as defined
defined by
by dAII i
+
+
= [ H 00, AII ] e.g.
∆λλtt}
e.g. aa$$λλ+((tt)) == aa$$λλ+(0)
(0)exp
exp{ii∆
#
dt
$$ ((tt)) == R
$$ (0)
R
R
R
Ωtt}
exp{22iiΩ
21
21 (0)exp
21
21
We
We wish
wish to
to describe
describe the
the response
response of
of this
this system
system of
of 2-level
2-level atoms
atoms in
in aa Statistical
Statistical Sense
Sense
including
including the
the effects
effects of
of dipole
dipole dephasing
dephasing interactions
interactions with
with environment
environment and
and other
other damping
damping
Consider
Consider Density
Density Operator
Operator XX ((tt)) ≡≡ ∑ ppψψ ψ
ψ ((tt)) ψ
ψ ((tt))
ψ
ψ
ppψψ == probability
ψ >>
probability that
that system
system is
is in
in state
state || ψ
later
later we
we will
will specify
specify these
these to
to be
be
interaction
interaction picture
picture state
state vectors
vectors
Transistor
Transistor Action
Action in
in aa Doped
Doped PBG
PBG Material
Material
ρ(ω)
ρ(ω)
ω
ω00
Density
Density
of
of States
States
ω
ωcc
ω
ω
ω00
hhω
atom
atom
Spontaneous
Spontaneous Emission
Emission Rates
Rates
"
ωcc
ω00 >> ω
for ω
" γγ++ for
"
ω00 << ω
ωcc
for ω
" γγ−− for
Dipole
Dipole dephasing
dephasing rate
rate γγPP due
due to
to phonons
phonons
Markovian regime
For
=0 "
Non-Markovian
regime
For true
true photonic
photonic band
band edge
edge γγ−− =0
" Nonββ
Vacuum
Vacuum Rabi
Rabi splitting
splitting
Photon
Photon localization
localization
External
Colored” Vacuum
External Laser
Laser Field
Field ++ N
N atoms
atoms "
Mollow Splitting
Splitting in
in aa ““Colored”
Vacuum
" Mollow
γγ−−
∆=
ωLL−ω
0
∆=ω
−ω00>>0
Ω
Ω >>
>> ββ
Collective
Collective Switching
Switching from
from aa
Passive
Passive to
to Active
Active Medium
Medium
1.0
1.0
Population
Population
Inversion
Inversion
γγ++
Active
Active
0.5
0.5
Passive
Passive
S.
S. John
John &
& T.
T. Quang
Quang,, PRL
PRL 78
78,, 1888
1888 (1997)
(1997)
Threshold
Threshold
Laser
Laser Intensity
Intensity
Optical Mode Density and Relevant Frequency Scales
ρρ(ω)
(ω)
ωL
ωL−2Ω
ωL+2Ω
ω
ω
ωC
ω
ωCC
ω
ωAA,, ω
ωLL
Ω
Ω
ωA
the
the point
point of
of the
the DOS
DOS discontinuity
discontinuity
the
the laser
laser and
and the
the resonant
resonant atomic
atomic frequency
frequency
the
the generalized
generalized Rabi
Rabi frequency
frequency
∆∆AL
= ω A-- ω
ωLL << 00
AL= ωA
49
Atomic Switching in Photonic Crystals
Collective Atomic Effect
in the Markov Approximation
Single Atom Non-Markovian
Switching near an Anisotropic
Photonic Band Edge
Average Inversion
0.7
0.7
>/N
<J
<J22
22>/N
0.6
0.6
Atomic
Atomic population
population per
per atom
atom on
on the
the
bare
>/N, as a
bare excited
excited state
state <J
<J22
22>/N, as a
function
ε/∆ for
∆=−1,, N=5000,
function of
of ε/∆
for ∆=−1
N=5000,
γγPP/γ
0.5 and
0.3,0.4,0.5.
/γ++==0.5
/γ++==0.3,0.4,0.5.
and γγ−−/γ
0.5
0.5
0.4
0.4
0.2
0.2
Fluctuations in
Atomic Inversion:
Qbb =
22
J 22
22
− J 22
22
22
J 22
22
q-Mandel
as aa function
function
q-Mandel parameter
parameter Q
Qbb as
of
0.001
ε/∆ for
∆=−1,, N=5000,
/γ++==0.001
of ε/∆
for ∆=−1
N=5000, γγ−−/γ
and
0.01,0.1,0.5. Inset
/γ++==0.01,0.1,0.5.
and γγPP/γ
Inset shows
shows an
an
expanded
expanded view
view of
of the
the same
same curves
curves in
in
the
Poissonian statistics
the regime
regime of
of subsub-Poissonian
statistics
of
of the
the excited
excited atoms.
atoms.
γγ−−/γ
0.5,0.4,0.3
/γ++==0.5,0.4,0.3
0.3
0.3
11
500
500
22
ε/∆
ε/∆
44
10
10
Q
Qbb
400
400
55
300
300
00
0.15
0.15
200
200
0.25
0.25
0.35
0.35
γγPP/γ
0.5,0.1,0.01
/γ++==0.5,0.1,0.01
100
100
00
33
0.1
0.1
0.2
0.2
ε/∆
ε/∆
0.3
0.3
0.4
0.4
Photonic Crystal
Weak Probe Absorption Spectrum
ε << εthth
ε ≥ εthth
ε ≤≤εthth
ε >> εthth
If (RED) pump I(t)> I, (BLUE) probe is amplified
If (RED) pump I(t) <I, (BLUE) probe is absorbed
APPLICATIONS:
Optical Micro-Transistor
Optical Wavelength Converter
All-Optical Packet Switch
SUMMARY
Light Localization " Non-Markovian Radiative Dynamics
Photon-Atom Bound State
Fractionalized Steady State Inversion and Vacuum Rabi Splitting
Collective Time Scale Factors near a Photonic Band Edge
Band Edge Lasing (lasing without a conventional cavity)
Inversion of a Two-Level System by Coherent, Resonant Pumping
Collective Atomic Switching " All-Optical Micro-Transistor
Optical Wavelength Converter
Sub-Poissonian Statistics