Superluminal Group
Velocities
(a.k.a. Fast Light)
Dan Gauthier
Duke University
Department of Physics,
The Fitzpatrick Center for Photonics
and Communication Systems
SCUWP
January 17, 2010
Information on Optical Pulses
Modern Optical Telecommunication Systems:
Transmitting information encoded on optical fields
http://www.picosecond.com/objects/AN-12.pdf
1
0
1
1
0
RZ data
clock
Where is the information on the waveform?
How fast does it travel?
Slow Light
Controllably adjust the speed of an optical pulse
propagating through a dispersive optical material
Slow light:
 g  c (ng  1)
Slow-light medium
control
Motivation for Using “Slow” Light
Optical buffers and all-optical tunable delays
for routers and data synchronization.
data packets
router
router
Outline
• Introduction to “Slow" and "Fast" Light
• Fast and backward light
• Reconcile with the Special Theory of Relativity
Pulse Propagation in
Dispersive Materials
Propagation through glass
Propagation Through Dispersive Materials
Q: How fast does a pulse of light propagate through a
a dispersive material?
A: There is no single velocity that describes how light
propagates through a dispersive material
dispersive
media
A pulse disperses (becomes distorted) upon propagation
An infinite number of velocities!
Propagating Electromagnetic Waves: Phase Velocity
monochromatic plane wave
E
phase velocity
z
E( z, t )  Ae
phase
i ( kz t )
 c. c
  kz  t
Dz  c
p   
Dt k n
Points of constant phase move
a distance Dz in a time Dt
Dispersive Material: n = n()
Linear Pulse Propagation: Group Velocity
Lowest-order statement
of propagation without
distortion
d
0
d
different
p
group velocity
c
c
g 

dn ng
n 
d
Control group velocity: metamaterials, highly dispersive materials
Variation in vg with dispersion
Vg
c
4
3
2
slow light
1
4
3
2
1
1
1
2
3
4
fast light
2
3
dn
d
Pulse Propagation: Slow Light
(Group velocity approximation)
Achieving Slow Light
Boyd and Gauthier, in Progress of Optics 43, 497-530 (2002)
Boyd and Gauthier, Science 306, 1074 (2009)
When is the dispersion large?
2-level system
absorption
1
Absorption
coefficient
0.5
0
|2>
n-1
0.5
laser
field
Index of
refraction
0
-0.5
|1>
Group
index
ng - 1
0
-10
-20
-4
-2
0
2
4
frequency (a.u.)
Electromagnetically-Induced
Transparency (EIT)
absorption
1
3-level system
0
0.5
laser field
|3>
n-1
|2>
control
field
Absorption
coefficient
0.5
Index of
refraction
0
-0.5
|1>
ng - 1
80
Group
index
40
0
-40
-4
S. Harris, etc.
-2
0
2
4
frequency (a.u.)
EIT: Slowlight
Hau, Harris, Dutton, and Behroozi, Nature 397, 594 (1999)
Group velocities as low as 17 m/s observed!
Fast-Light
 g  c or  g  0
Fast light theory, Gaussian pulses:
C. G. B. Garrett, D. E. McCumber, Phys. Rev. A 1, 305 (1970).
Fast light experiments, resonant absorbers:
S. Chu, S. Wong, Phys. Rev. Lett. 48, 738 (1982).
B. Ségard and B. Macke, Phys. Lett. 109, 213 (1985).
A. M. Akulshin, A. Cimmino, G. I. Opat, Quantum Electron. 32, 567 (2002).
M. S. Bigelow, N. N. Lepeshkin, R. W. Boyd, Science 301, 200 (2003)
Pulse Propagation: Fast Light
(Group velocity approximation)
Fast-light via a gain doublet
Steingberg and Chiao, PRA 49, 2071 (1994)
(Wang, Kuzmich, and Dogariu, Nature 406, 277 (2000))
Achieve a gain doublet using stimulated Raman
scattering with a bichromatic pump field
Wang, Kuzmich, and Dogariu, Nature 406, 277 (2000)
Fast light in a laser driven potassium vapor
dd+
o
AOM
K
vapor
K
vapor
d-
waveform
generator
8
22.3 MHz
egl=1,097
7
gain coefficient, gL
d+
6
large anomalous
dispersion
5
4
3
egl=7.4
2
1
0
190
200
210
220
230
probe frequency (MHz)
240
250
12
10
power (W)
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
tadv=27.4 ns
8
6
advanced
vacuum
4
2
0
-300
-200
-100
0
100
200
power (W)
Observation of large pulse advancement
300
time (ns)
tp = 263 ns
A = 10.4%
vg = -0.051c
ng = -19.6
M.D. Stenner, D.J. Gauthier, and M.A. Neifeld, Nature 425, 695 (2003).
Reconcile with the
Special Theory of Relativity
Problems with superluminal information transfer
a)
b)
c)
t
t
t
A
B
x
x
event
event
x
event
C
D
Light cone
Minimum requirements of the optical field
L. Brillouin, Wave Propagation and Group Velocity, (Academic, New York, 1960).
(compendium of work by A. Sommerfeld and L. Brillouin from 1907-1914)
A. Sommerfeld
A "signal" is an electromagnetic wave
that is zero initially.
front
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Sommerfeld.html
Primary Finding of Sommerfeld
(assumes a Lorentz-model dielectric with a single resonance)
The front travels at c
regardless of the details of the dielectric
Physical interpretation: it takes a finite time for the
polarization of the medium to build up; the first part of the
field passes straight through!
Generalization of Sommerfeld and Brillouin's work
P
point of non-analyticity
t
knowledge of the leading part of the pulse cannot be used
to infer knowledge after the point of non-analyticity
new information is available because of the "surprise"
Chiao and Steinberg find point of non-analyticity
travels at c. Therefore, they associate it with the
information velocity.
Implications for fast-light
vacuum
transmitter
receiver
transmitter
receiver
with dispersive material
receiver
transmitter
information still available at c!
A
1.5
Y Data
optical pulse amplitude (a.u.)
1.0
0.5
advanced
advanced
"0"
vacuum
0.0
-300
-200
-100
0
100
200
time (ns)
1.2
Send the symbols
through our fast-light
medium
"1"
B
300
1.8
advanced
1.6
1.0
1.4
0.8
1.2
0.6
1.0
0.4
0.2
0.8
vacuum
0.6
-60
-40
-20
time (ns)
0
 i ,adv  (0.4  0.5)c
Fast light, backward light and the light cone
The pulse peak can do weird things, but can't go
beyond the pulse front (outside the light cone)
Summary
•
Slow and fast light allows control of the speed of optical
pulses
•
Amazing results using atomic systems
•
Transition research to applications using existing
telecommunications technologies
•
Fast light gives rise to unusual behavior
•
Interesting problem in E&M to reconcile with the special
theory of relativity
Collaborators
Duke
Rochester
R. Boyd, J. Howell
Cornell
A. Gaeta
UCSC
A. Willner
UCSB
D. Blumenthal
http://www.phy.duke.edu/
U of Arizona
M. Neifeld
Sir Hamilton 1839
G.G. Stokes 1876
J.S. Russell 1844
Lord Rayleigh 1877
A beam with two frequencies: The group
velocity
E (z, t )  2 A cos(k L z   Lt )  2 A cos(k H z   H t )
nL L  nH H
 L  H
nL L  nH H
F
G
H
 4 A cos
z
2
2
IJ F
KG
H
t sin
2
z
E t
2
1.5
1
0.5
0.5
1
1.5
2
20
40
Photos from: http://www-gap.dcs.st-and.ac.uk/~history/l
60
80
z
 L  H
2
t
IJ
K
Speed of the envelope in dispersive
materials
(n  n )
H
L