Slow Light and Superluminal Propagation
M. A. Bouchene
Laboratoire « Collisions, Agrégats, Réactivité », Université Paul Sabatier, Toulouse, France
Interest in Slow and Fast light
• Fundamental aspect in optical physics• How can we stop (reversibly) the light ?• Can light travels faster than c?
• Applications: optical storage, optical information, telecom (buffering)
Outlook• Group velocity, case of a resonant system
• (Ultra) Slow light: - Electromagnetic Induced Transparency- Coherent Population Oscillations- Zeeman Coherence Oscillations. Coherent control
of the optical response
• Superluminal propagation: - Relativity implications- Different velocities for a light pulse- Backward propagation
• Conclusion
Group Velocity( ) ( )
0
0 0( ) 1 ( )
( )
⎛ ⎞≠ + −⎜ ⎟⎝ ⎠
=
dnn nd
cteω
ω ω ω ωω
α ω
z0( )
0
0
( , ) ( / v ) e
v
i kz tgE t z G E t z
k
ω
ϕ
ω
−= −
=
( ) 1;( ) 0
==
n ωα ω
0 0( )0
00
( , ) ( / ) ei k z tE t z E t z c
kc
ω
ω
−= −
=
0v / ( ) :c nϕ ω=
0
0 0
v :( )
gc
dnnd ω
ω ωω
=⎛ ⎞+ ⎜ ⎟⎝ ⎠
Phase velocity
Group velocity
PROPOGATION WITHOUT DISTORTION UNLESS RESHAPING EFFECTS(higher orders contributions)
c
c vϕ
vg
Group Velocity( ) ( )
0
0 0( ) 1 ( )
( )
⎛ ⎞≠ + −⎜ ⎟⎝ ⎠
=
dnn nd
cteω
ω ω ω ωω
α ω
z0( )
0
0
( , ) ( / v ) e
v
i kz tgE t z G E t z
k
ω
ϕ
ω
−= −
=
( ) 1;( ) 0
==
n ωα ω
0 0( )0
00
( , ) ( / ) ei k z tE t z E t z c
kc
ω
ω
−= −
=
c
c vϕ
vg
Group Velocity( ) ( )
0
0 0( ) 1 ( )
( )
⎛ ⎞≠ + −⎜ ⎟⎝ ⎠
=
dnn nd
cteω
ω ω ω ωω
α ω
z0( )
0
0
( , ) ( / v ) e
v
i kz tgE t z G E t z
k
ω
ϕ
ω
−= −
=
( ) 1;( ) 0
==
n ωα ω
0 0( )0
00
( , ) ( / ) ei k z tE t z E t z c
kc
ω
ω
−= −
=
0v / ( ) :c nϕ ω= Phase velocity
c
c vϕ
vg
Group Velocity( ) ( )
0
0 0( ) 1 ( )
( )
⎛ ⎞≠ + −⎜ ⎟⎝ ⎠
=
dnn nd
cteω
ω ω ω ωω
α ω
z0( )
0
0
( , ) ( / v ) e
v
i kz tgE t z G E t z
k
ω
ϕ
ω
−= −
=
( ) 1;( ) 0
==
n ωα ω
0 0( )0
00
( , ) ( / ) ei k z tE t z E t z c
kc
ω
ω
−= −
=
0v / ( ) :c nϕ ω=
0
0 0
v :( )
gc
dnnd ω
ω ωω
=⎛ ⎞+ ⎜ ⎟⎝ ⎠
Phase velocity
Group velocity
c
c vϕ
vg
Group Velocity( ) ( )
0
0 0( ) 1 ( )
( )
⎛ ⎞≠ + −⎜ ⎟⎝ ⎠
=
dnn nd
cteω
ω ω ω ωω
α ω
z0( )
0
0
( , ) ( / v ) e
v
i kz tgE t z G E t z
k
ω
ϕ
ω
−= −
=
( ) 1;( ) 0
==
n ωα ω
0 0( )0
00
( , ) ( / ) ei k z tE t z E t z c
kc
ω
ω
−= −
=
0v / ( ) :c nϕ ω=
0
0 0
v :( )
gc
dnnd ω
ω ωω
=⎛ ⎞+ ⎜ ⎟⎝ ⎠
Phase velocity
Group velocity
linear-dispersive medium with constant spectral gain/abs: PROPAGATION WITHOUT DISTORTIONHigh orders contributions: RESHAPING EFFECTS
c
c vϕ
vg
( )/ ,i c nL Le eω α
; :g gg
dnn nn d
Group indexcV ωω
= = +
• Dilute medium but vg may be very different from c 1n
; :g gg
dnn nn d
Group indexcV ωω
= = +
• Dilute medium but vg may be very different from c 1n
vg
ωω
dnd-1
Vg=c
0
; :g gg
dnn nn d
Group indexcV ωω
= = +
• Dilute medium but vg may be very different from c 1n
vg
ωω
dnd-1
Vg=c
Vg<0
Backward propagation
Vg>0
SLOW LIGHTVg<c
SUPERLUMINAL
Vg>c
0
; :g gg
dnn nn d
Group indexcV ωω
= = +
NORMAL DISPERSIONANOMALOUS DISPERSION
Forward propagation
• Dilute medium but vg may be very different from c 1n
vg
-1
Vg=c
Vg<0
Backward propagation
Vg>0
SLOW LIGHTVg<c
SUPERLUMINAL
Vg>c
0
; :g gg
dnn nn d
Group indexcV ωω
= = +
NORMAL DISPERSIONANOMALOUS DISPERSION
ωω
dnd
Forward propagation
• Ultra slow light needs w dn/dw >>1How to achieve ??
• vg> c : What about relativity?
• vg< 0 : Propagation of energy?
What happens near a resonant transition?
N 0ωω
0E i n ll c0E e e
ω−α
What happens near a resonant transition?
( )
2
0 2 20
:=− +
absorptioncoefficientγα αω ω γ
N 0ωω
20
00
: , :N absorption at resonance frequency relaxation ratec
μ ωα γε γ
=
0E i n ll c0E e e
ω−α
Increasing ng is possible but absorption also increases!
NORMAL DISPERSION
ANOMALOUS DISPERSION
What happens near a resonant transition?
( )0 0
2 20 0
21 :−= +
− +refracn ction indexα γ ω ω
ω ω ω γ
N 0ωω
20
00
: , :N absorption at resonance frequency relaxation ratec
μ ωα γε γ
=
0E i n ll c0E e e
ω−α
Increasing ng is possible but absorption also increases!
NORMAL DISPERSION
ANOMALOUS DISPERSION
Increasing ng is possible but absorption also increases!Difficult to observe experimentally
NORMAL DISPERSION
ANOMALOUS DISPERSION
I- How to produce slow-fast light?:g n
c
nd
V dωω
=+
Kramers-Kronigs relations2 20
2
2 20
( ')1 ''
4 1 ''
( )
( ')( )
∞
∞
− = ℘−
−= − ℘
−
∫
∫
c d
dc
n
n
α ωω
ωα ω
ωπ ω ωω ωπ ω ω
A narrow spectral hole produce a strong normal dispersion
• Electromagnetic Induced Transparency
• Coherent Population Oscillations
• Zeeman Coherence Oscillations
Slow light
II-a Electromagnetic Induced Transparency
II-a Electromagnetic Induced Transparency
II-a Electromagnetic Induced Transparency
0
1
2
Control field
Propagating fieldC laser intensityΩ ∝
Interpretation
γ
0
1
2
1 2:
2
i teadiabatic states
ω−±± =
Control field
Propagating fieldC laser intensityΩ ∝
Interpretation
γ
0
1
2
1 2:
2
i teadiabatic states
ω−±± =
0
+
−Control field
Propagating field Propagating field
C laser intensityΩ ∝
CΩ
Interpretation
γ
∝ γ
0
1
2
1 2:
2
i teadiabatic states
ω−±± =
0
+
−Control field
Propagating field Propagating field
C laser intensityΩ ∝
CΩ } Autler-TownesSplittingDynamical Starkeffect
Interpretation
γ
∝ γ
0
1
2
1 2:
2
i teadiabatic states
ω−±± =
0
+
−Control field
Propagating field Propagating field
C laser intensityΩ ∝
CΩ
:C γΩ >> Propagating field is non-resonant
} Autler-TownesSplittingDynamical Stark effect
Interpretation
γ
∝ γ
propag. 10ω − ω
AB
SOR
PTIO
N C
OEF
FIC
IEN
T
1 2:
2
i teadiabatic states
ω−±± =
0
+
−
Propagating field
CΩ
Interpretation
Ω ≤C γ
1 2:
2
i teadiabatic states
ω−±± =
0
+
−
Propagating field
CΩ
Interpretation
Ω ≤C γ
Interference between quantum paths 0 and 0→ + → −
RESULT ?
propag. 10ω − ω
AB
SOR
PTIO
N C
OEF
FIC
IEN
T
TOTAL DESTRUCTIF INTERFERENCEpropag. 10ω − ω
AB
SOR
PTIO
N C
OEF
FIC
IEN
T
Coherent Population Trapping
0
1
2
CΩ
PΩ
Bright stateB sin 0 cos 2 :
D sin 2 cos 0 : Dark state
= ϕ + ϕ
= − ϕ + ϕ
p
c
tanΩ
ϕ =Ω
γ
1
BD
No coupling
Coherent Population Trapping
p 10ω = ω
1
2 2P CΩ = Ω + Ω
BD
No coupling
Coherent Population Trapping
p 10ω = ω
1
B
D
Coherent Population Trapping
p 10ω = ω
Coherent Population Trapping
1
B
D
1
B
D
Coherent Population Trapping
1
B
D
Coherent Population Trapping
1
B
D
Coherent Population Trapping
ALL THE ATOMS ARE IMMUNE TO INTERACTION: perfect transparency
CPT
Autler-Townes splitting
NATURE, 397, 594, (1999)
g 7v10 !!
c−=
Vg decreases with CΩ
Vg decreases with CΩ
Is it possible to stop the light ?
Vg decreases with CΩ
Is it possible to stop the light ?
YES
Vg decreases with CΩ
Is it possible to stop the light ?
YESWhat means a stopped light ??
Vg decreases with CΩ
Is it possible to stop the light ?
YESWhat means a stopped light ??
CΩ
E(Z 0)=E(Z L)=
CΩ
E(Z 0)=E(Z L)=
STORAGE
CΩ
E(Z 0)=E(Z L)=
STORAGEWhat kind of
storage?
bcE(z, t)(z, t) cos (t) sin (t)g Nψ = θ − σθ
2 2
(t)cos(t) g NΩ
θ =Ω +
Polariton (mixture field-atom)
2ccos (t) (z, t) 0t z
∂ ∂⎛ ⎞+ θ ψ =⎜ ⎟∂ ∂⎝ ⎠
Shape-preserved propagation with2
gv (t) ccos (t)= θ
a
b
1ω2ω
Bichromatic field :
δ
II-b Coherent Population Oscillations
a
b
1ω2ω
Bichromatic field : ( )1 11 2 22I I I 2 I I cos t= + − ω+ ω
BEATINGδ
II-b Coherent Population Oscillations
a
b
1ω2ω
Bichromatic field : ( )1 21 2 12I I I 2 I I cos t= + − ω+ ω
BEATING
TEMPORAL GRATING
(0) ( ) i t ( ) i tba b a ba ba ban n n n n e n e+ δ − − δ= − = + +
δ
II-b Coherent Population Oscillations
2ω 12ω + δ = ω
δ
Reinforcement of the emission at 1ω
2ω − δ
2ω
II-b Coherent Population Oscillations
2ω12ω + δ = ω
δ
δ0
Abs
orpt
ion
II-b Coherent Population Oscillations
2ω12ω + δ = ω
δ
δ0
Abs
orpt
ion
II-b Coherent Population Oscillations
1POPULATION 1 OSCILLATIONt T t −Δ = ≤ Δ = δ
12T −
11T −
II-c Coherent Zeeman Oscillations
z
x
yprop. axes
π polarized control field
σ polarized probe field
Δ
Control fieldpolarizedπ −
A double two level system
F=½
F=½
a b
c d
mF=½ mF=-½
πProbe fieldpolarizedσ −
Bichromatic field, with strong resonant pulse and detuned pulse
π
σ
π σ
( ) δΦ = Δ −, .r t t k r
( )− Φ= +
,i r tT
Ee e e eE
σπ σ
π
( )Φ ,r t
0 2π π 3 2π 2π
πe
σe
polarization producesa time /space grating
Linear ellipt. L linear ellip. R linear
II-c Coherent Zeeman Oscillations
grating imprints on Zeeman coherences
( )1 2
1
1
δρ ρ ρ ρ=
Δ −
=−
= + = ∑ .( )n
in t k rnZ z z
ne
z2ρ
z1ρ
grating imprints on Zeeman coherences
( ) ( ) 2σ
σ π σρ
ρ θ ρ θ − Φ Γ= − − − −
,Im i r tZ e gi e n n
= =Γ Γ
;dE dEπ σπ σθ θ
z2ρ
z1ρ
( )1 2
1
1
δρ ρ ρ ρ=
Δ −
=−
= + = ∑ .( )n
in t k rnZ z z
ne
grating imprints on Zeeman coherences
Absorption bypopulation
= =Γ Γ
;dE dEπ σπ σθ θ
( ) ( ) 2σ
σ π σρ
ρ θ ρ θ − Φ Γ= − − − −
,Im i r tZ e gi e n n
en
gn
( )1 2
1
1
δρ ρ ρ ρ=
Δ −
=−
= + = ∑ .( )n
in t k rnZ z z
ne
grating imprints on Zeeman coherences
Diffraction of strong fieldby Zeeman coherence
π σ1n =
( ) ( ) 2σ
σ π σρ
ρ θ ρ θ − Φ Γ= − − − −
,Im i r tZ e gi e n n
z2ρ
z1ρ
( )1 2
1
1
δρ ρ ρ ρ=
Δ −
=−
= + = ∑ .( )n
in t k rnZ z z
ne
grating imprints on Zeeman coherences
Diffraction of strong fieldby Zeeman coherence
Absorption bypopulation
Cancel exactly
at
π σ1n =
= 0Δ
( ) ( ) 2σ
σ π σρ
ρ θ ρ θ − Φ Γ= − − − −
,Im i r tZ e gi e n n
z2ρ
z1ρ
( )1 2
1
1
δρ ρ ρ ρ=
Δ −
=−
= + = ∑ .( )n
in t k rnZ z z
ne
F. A. Hashmi and M. A. Bouchene Phy. Rev. A 77, 051803(R) 2008
-2 -1 0 1 2
-0.4
-0.2
0.0
0.2
0.4-2 -1 0 1 2
0.0
0.2
0.4
0.6
0.8
1.0
Sus
cept
ibili
ty (i
n un
its o
f 2α
0k-1
)
Δ (in units of Γ)
Re(χ)
Ωπ= 0 Ωπ= 0.1 Γ Ωπ= 0.2 Γ
Im(χ)
II-c Coherent Zeeman Oscillations
( )2/Width π∝ χ Γ
• Pure Non-linear effect• No dark state• Hybrid properties between CPO and EIT
II-c Coherent Zeeman Oscillations
• Giant Non-linearity:
IId- Coherent Control of the Susceptibility
(2) 2 (3) 3 ...linP E E Eχ χ χ= + + +
If E is strong : ionisation
If resonance : absorption
Increase NL effects increase E or χ
Slow light techniques: reduction of absorption AND resonance
Effects of giant non linearities observable
-2 -1 0 1 2
-0.4
-0.2
0.0
0.2
0.4-2 -1 0 1 2
0.0
0.2
0.4
0.6
0.8
1.0
Sus
cept
ibili
ty (i
n un
its o
f 2α
0k-1
)
Δ (in units of Γ)
Re(χ)
Ωπ= 0 Ωπ= 0.1 Γ Ωπ= 0.2 Γ
Im(χ)
0= :Δ
Optical response canceled in the direction Optical response determined by other emitted waves
kπ
kσ
kδ
kσ2k kπ σ−
II-d Coherent Control of Susceptibility
z
x
yprop. axes
π polarized control field
σ polarized probe field
0Δ
π σ
F=½
F=½
,k kσ π
a b
c d
( ) :Coherent Controlda cbσ σρ ρ ρ ρ ϕ= + =
0 0= =, kδΔ
( , ) .r t t k rδ ϕ ϕΦ = Δ − + =
da cbσρ ρ ρ= +
a b
c d
Absorptionpath
daρ =
II-d Coherent Control of Susceptibility
kσ
da cbσρ ρ ρ= +
a b a b
c d c d
π π+
Absorptionpath
Cross- Kerr type path
daρ =
II-d Coherent Control of Susceptibility
( )k k k kπ σ π σ+ − =kσ
da cbσρ ρ ρ= +
a b a b
c d c d
π π+
Absorptionpath
Cross- Kerr type path
daρ =
II-d Coherent Control of Susceptibility
Cancel each other
( )k k k kπ σ π σ+ − =kσ
0=Δ
da cbσρ ρ ρ= +
a b a b a b
c d c d c d
π π π π+ +
Absorptionpath
Cross- Kerr type path
Phase Conjugate type path
daρ =
II-d Coherent Control of Susceptibility
Cancel each otherDefine the optical response
( ) 2k k k k k
k
π σ π π σ
σ
− − = −
=
( )k k k kπ σ π σ+ − =kσ
0=Δ
0 1if and L ( thin sample ) :π αΩ << Γ <<
II-d Coherent Control of Susceptibility
a b
c d
π π
0 1
i
linearProblem equivalent to propagation of fieif
ltwo leve
de
and L (
in
thin s
an assembly of at
am
l
ple ) :
oms :ϕσ
π αΩ
Ω −
<< Γ <<
II-d Coherent Control of Susceptibility
a b
c d
( ) ( ) ( )2
0 1
i
i i ilin lin
Problem equivalent to propagation of filineartwo lev
if and L ( thin sameld
e in an assembly of atoms :
ple ) :
e
l
e
e
P e
ϕσ
ϕ ϕ ϕσ σ σ
π
σρ χ
α
χ −
Ω
∝ Ω =
Ω < Γ
Ω
< <
−
<
II-d Coherent Control of Susceptibility
a b
c d
( ) ( ) ( )2
0 1
i
i i ilin lin
Problem equivalent to propagation of filineartwo lev
if and L ( thin sameld
e in an assembly of atoms :
ple ) :
e
l
e
e
P e
ϕσ
ϕ ϕ ϕσ σ σ
π
σρ χ
α
χ −
Ω
∝ Ω =
Ω < Γ
Ω
< <
−
<
II-d Coherent Control of Susceptibility
2≈ !ieff line
ϕχ χ
Independent from pump field characteristics!
a b
c d
( ) ( ) ( )2
0 1
i
i i ilin lin
Problem equivalent to propagation of filineartwo lev
if and L ( thin sameld
e in an assembly of atoms :
ple ) :
e
l
e
e
P e
ϕσ
ϕ ϕ ϕσ σ σ
π
σρ χ
α
χ −
Ω
∝ Ω =
Ω < Γ
Ω
< <
−
<
II-d Coherent Control of Susceptibility
2≈ !ieff line
ϕχ χ
-The system behaves as a linear medium with a modified(linear) suceptibility controlled by the phase shift-Giant non-linearity
Independent from pump field characteristics!
a b
c d
F. A. Hashmi and M. A. Bouchene Phys. Rev. Letters 101, 21306 (2008)
2 :ieff line gain dispersion couplingϕχ χ≈ −
-4 -2 0 2 4-0.5
0.0
0.5
1.0
ϕ ϕ
ϕϕ
Re χeff
Im χeff
=0
=3π/4 =π/2
=π/4
Ωπ=100 Ω
σ= 0.1Γ; Γze=2Γd=Γ; α0L=0.2
-4 -2 0 2 4-1.0
-0.5
0.0
0.5
-4 -2 0 2 4-1.0
-0.5
0.0
0.5
χ eff i
n un
its o
f 2α
0k-1
Δ0 in units of Γ
-4 -2 0 2 4-0.5
0.0
0.5
1.0
II-d Coherent Control of Susceptibility
0 0ω ωΔ = −
AbsorberAnomalous dispersion
AmplifierNormal dispersion
Anomalous «dispersion-like»gain
«absorption-like»dispersion
Normal «dispersion-like»gain
«gain-like »dispersion
F. A. Hashmi and M. A. Bouchene, Phys. Rev. Letters 101, 21306 (2008)
Superluminal propagation
CAUSALITY AND RELATIVITY
(1)
( )x, t
(2)
( )x x, t t+ Δ + Δ
(R)
(R’) v
( ) ( )
2
22 2
vt x t uv xct ' 1 ; uc t1 v / c 1 v / c
Δ − Δ Δ Δ⎛ ⎞Δ = = − =⎜ ⎟ Δ⎝ ⎠− −
CAUSALITY t ' t 0Δ Δ ≥ u c≤v c<
u: signal velocity
RELATIVITY (implicit)
v c<
WHAT IS A SIGNAL?
WHAT IS A SIGNAL?
In Optics :Wave packet with sharp front (Sommerfeld 1910)
WHAT IS A SIGNAL?
In Optics :Wave packet with sharp front (Sommerfeld 1910)
z
WHAT IS A SIGNAL?
In Optics :Wave packet with sharp front (Sommerfeld 1910)
C exactly!
z
WHAT IS A SIGNAL?
In Optics :Wave packet with sharp front (Sommerfeld 1910)
C exactly!
z
- Envelope faster than c ! (Vg > c), Vg still meaningful-Pulse distortion occurs if the back part of the pulse catch up the sharp front
Extension: non analyticity (Chiao et al. 2000)
>c or <c
C
Five velocities for a light pulse:Phase velocity
v / ( )c n c or cϕ = > <
v / ( )g gc n c or c= > <
Five velocities for a light pulse:Group velocity
Meaningful in a linear dispersive medium with constant spectral gain
Five velocities for a light pulse:Signal velocity
v :s speed of non analytical point c=
vs
Five velocities for a light pulse:Signal velocity
v :s speed of non analytical point c=
vs
« detectable Information »
3 2 20
30
( ),2 2
FG F
F
r u d r E Br c or c uu d r
εμ
= > < = +∫∫
Five velocities for a light pulse:centroid of field energy
G
Five velocities for a light pulse:centroid of field energy
G
2 20
.0
v ( ),2 2Field En G FE Br c or c u
tε
μ∂
= > < = +∂
Five velocities for a light pulse:centroid of field energy
2 20
.0
v ( ),2 2Field En G FE Br c or c u
tε
μ∂
= > < = +∂
G
Field En.v :" "watched velocity in pulse propagation experiment
Field En g. gv v vv( / ) if dispersivemedium with cte gainα ω= + ∂ ∂ =
Five velocities for a light pulse:centroid of field energy
G
2 20
.0
v ( ),2 2Field En G FE Br c or c u
tε
μ∂
= > < = +∂
Field En.v :" "watched velocity in pulse propagation experiment
Field En g. gv v vv( / ) if dispersivemedium with cte gainα ω= + ∂ ∂ =
Five velocities for a light pulse:centroid of field energy
3
3≠ ∫
∫Field. En.v
F
S d r
u d rG
2 20
.0
v ( ),2 2Field En G FE Br c or c u
tε
μ∂
= > < = +∂
Field En.v :" "watched velocity in pulse propagation experiment
Field En g. gv v vv( / ) if dispersivemedium with cte gainα ω= + ∂ ∂ =
• Can the energy go faster than c ???
• relativity: energy matter and v < c!!
How to solve the paradox ??
≡
Energy has to be defined as the total energy
3 2 20
. 30
v ( ),2 2
FField En F
F
r u d r E Bc or c ut u d r
εμ
∂= > < = +
∂∫∫
Five velocities for a light pulse:centroid of field energy
G
3
3≠ ∫
∫Field. En.v
F
S d r
u d r
« Open » energy
Field En.v :" "watched velocity in pulse propagation experimentField En g. gv v vv( / ) if dispersivemedium with cte gainα ω= + ∂ ∂ =
3 2 20
. . 30
v , ' ( )2 2
t
Tot En
r ud r E B Pu E dt u tt tud r
εμ −∞
∂ ∂= = + + + → −∞
∂ ∂∫ ∫∫
Five velocities for a light pulse:centroid of total energy
G« Closed » energy
3 2 20
. . 30
v , ' ( )2 2
t
Tot En
r ud r E B Pu E dt u tt tud r
εμ −∞
∂ ∂= = + + + → −∞
∂ ∂∫ ∫∫
Five velocities for a light pulse:centroid of total energy
G
. . :<Tot En Subluminal transport of the totalv c energy
3 2 20
. . 30
v , ' ( )2 2
t
Tot En
r ud r E B Pu E dt u tt tud r
εμ −∞
∂ ∂= = + + + → −∞
∂ ∂∫ ∫∫
Five velocities for a light pulse:centroid of total energy
3
. . 3v = ∫
∫Tot En
S d r
ud rG
. . :<Tot En Subluminal transport of the totalv c energy
J. Peatross et al.PRL 84, 2370 (2000)
S. Glasgow et al.PRE 64, 046610 (2001)
NATURE, 406 (2000)
Supraluminal propagation: Backward propagation.PROPAGATION THROUGH NEGATIVE MEDIUM
(ng=-2)
SCIENCE, 312 (2006)
Energy flow in the forward direction
SCIENCE, 312 (2006)
S k∝E
Bg Tot. En.v and v not only differ by their magnitude but also by their
P
aradox:sign!!
Pulse envelope displacement: spatial pulse shaping
Alternative description
amplified absorbed
Any saturable absorber may re-shape (distort) the pulseHere (constant spectral gain), the re-shaping is symmetric!, Vg meaningful
Alternative description
amplified absorbed
Any saturable absorber may re-shape (distort) the pulseHere (constant spectral gain), the re-shaping is symmetric!, Vg meaningful
Construc. interf Destruc. interf
Conclusion• Fast light: better understanding of relativity implications and
light pulse velocities.• Relativity implications on signal and information propagation:
*Signal: non analyticity *transmission of (detectable) Information faster than c possible!
• Relativity implications on energy propagation:*Group velocity characterizes the displacement of the envelope but is not representative of the (total) energy flux!*Group velocity is significative in a constant spectral gain medium: displacement of the EM energy, watched velocity.*Superluminal pulse envelope propagation is the result of both temporal and spatial amplitude pulse shaping by the medium.*Superluminal propagation appears when truncated balance of energy
• Challenge in slow light: extension to short pulses (higher bandwidth): SRS, SBS techniques.
• Development of quantum optical memories