Download Fast Light, Slow Light

Fast Light, Slow Light
David Jun
3/8/05
Outline:
• Section 1: Introduction
– Definitions
– Recent Studies
– Motivations
• Section 2: Working Principles
– Dispersion
– Wave Velocities (V_phase and V_group)
– Group index of refraction
• Section 3: Controversy/Debate
– Einstein’s Theory of Special Relativity
– Signal Velocity
• Section 4: Conclusions and Reference
Section 1. Introduction
• Definitions:
– Fast light (superluminal light): vg > c or vg is negative
– Slow light (subluminal light): vg << c
– “Stored” or stopped light: vg ~ 0
• Recent Studies:
– A group from Univ. of Rochester used an alexandrite crystal to
reduce the speed of light to 91m/s and minus 800m/s (M. Bigelow
et al. 2003 Science 301 200)
– Connie J. Chang-Hasnain (UC Berkeley), Hailin Wang (Uoreg),
Shun-Lien Chung (UIUC) slowed down the group velocity of light
to about 6 mi/sec in semiconductors (Oct 1, 2004, Optics Letters)
– A group from Harvard University stopped light particle in their
tracks for 10~20 msec in rubidium gas
Section 1. Introduction
• Motivations:
- Motivated by uncovering new physical
phenomena
- Practical applications:
•
•
•
•
•
High performance communications
Controllable optical delay lines
Optical data storage
Optical memories
Devices for quantum information
Section 2: Working Principles - Dispersion
• Dispersion:
– All material media w/ the exception of vacuum is
dispersive; meaning its index of refraction is frequency
dependent
– From Maxwell’s Equations and Wave equation:
c
1
m
v
1
m
– Index of refraction:
c
m
n 
 KeKm
v
m
Section 2: Working Principles - Dispersion
– Assume magnetically “simple” (m~mo):
n  Ke
– n subject to an applied electric field:
• Polarization: (  ) E  P
• Different polarization results depending on the
frequency of the incident electromagnetic wave
Section 2: Working Principles - Dispersion
• Analytic expression for
Dispersion:
– Forced Oscillator
model:
Total force on an electron
due to E(t) = Eo*coswt:
d2X
me 2  qeEo cos wt  (mew 2 X )
dt
Section 2: Working Principles - Dispersion
• Relative displacement between the (-) e-cloud and
the (+) nucleus:
qe / me
x(t ) 
E (t )
2
2
(w  w )
Note:
 w<wo : x(t) and E(t) in phase
 w>wo: x(t) and E(t) out of phase
Section 2: Working Principles - Dispersion
– Density of dipole moment (polarization):
P  qexN  (  ) E
qe 2 N / me
   
(w 2  w 2 )
n  Ke
since
Nqe 
1

n (w )  1 

2
2 
me  w  w 
2
2
(Dispersion Equation)
Section 2: Working Principles - Dispersion
• Complications/Implications/Corrections:
– Multiple natural frequency wo:
2
fj
Nqe
2
n (w )  1 

me j wj 2  w 2
 fj  1
j
– Absorption (damping term added):
fj
Nqe 2
n (w )  1 

me j wj 2  w 2  ijw
2
– Local electric field effect:
fj
n 2  1 Nqe 2


2
2
2
3

me
n 2
j wj  w  ijw
Section 2: Working Principles - Dispersion
• Dispersion Eq:
fj
n 2  1 Nqe 2


2
n  2 3me j wj 2  w 2
 woj^2>> w^2: n gradually
increases w/ frequency
(Normal Dispersion)
 woj^2<< w^2: n gradually
decreases w/ frequency
(Anomalous Dispersion)
Section 2: Working Principles – Wave Velocities
• Wave Velocities:
– Phase Velocity (vp): speed at which any fixed
phase or the shape of the wave is moving
example:
E(t,x) = Eo*cos(kx-wt)
Vp = w/k
Section 2: Working Principles – Wave Velocities
– Group Velocity (vg): speed of the overall shape
(modulation envelop) of the wave’s amplitude
example: consider two harmonic waves
E1(t , x)  E cos( k1x  w1t )
E 2(t , x)  E cos( k 2 x  w 2t )
where k1>k2 and w1>w2
E (t , x)  E1  E 2  2 E cos( kmx  wmt ) cos( k x  wt )
where
1
w  (w1  w 2)
2
1
wm  (w1  w 2)
2
1
k  (k1  k 2)
2
1
km  (k1  k 2)
2
Section 2: Working Principles – Wave Velocities
• Overall wave:
E (t , x)  En( x, t ) cos( k x  wt )
where
En(t , x)  2 E cos( kmx  wmt )
the former = carrier wave
vp _ carrier 
w
k
the latter = modulation envelop
1 (w1  w 2)
w
vp _ envelop 
 2

 vg
1 (k1  k 2)
km
k
2
wm
Section 2: Working Principles – Wave Velocities
If the frequency range w centered about w is
small,
w  dw 
vg 


k  dk w
Section 2: Working Principles – Wave Velocities
The relationship between Vp and Vg in a nondispersive and dispersive system:
In a non-dispersive system (vacuum):
•
Suppose two different traveling harmonic waves with the
same phase velocity (v=v1=v2):
dv
vp=v=w/k
vg  v  k
dk
Phase velocity independent of wavelength (dv/dk=0)
therefore, vp=vg=v1=v2
In a dispersive system:
kc
dw c kc dn
 k dn 
w
vg 
  2
 vp1 

n(k )
dk n n dk
 n dk 
Section 2: Working Principles – Group Index of Refraction
• Group index of refraction (ng):
dw
c
 k dn 
vg 
 vp1 

dk
 n dk  n  n dn
dn
vg = c/ng
where
dn(n )
ng  n(n )  n
dn
Consequences:
If dn/dn is plus, then ng > 1
If dn/dn is minus, then ng < 1
If dn/dv is minus and large, then ng is (-)
vg < c
vg > c
(-)vg
Section 2: Working Principles – Negative Group Velocity
• Negative group velocity:
– The peak of the emerging pulse occurs at an earlier time than the
peak of the incident pulse.
Example: Consider a pulse traversing a medium of length L:
t_traverse in the medium=L/vg
t_traverse in vacuum = L/c
Delay time t = L/vg – L/c = (ng-1)L/c
Negative velocity requires a (-) ng, then t < 0 (i.e. pulse
arrives early)
Section 3: Controversy/Debate
• Einstein’s Theory of Special Relativity:
Nothing can travel faster than the speed of light
(not exactly, examples: vg>c, motion of a spotlight projected on a
distant wall)
– With the Principle of Causality, the relativity theory says
No signal (information) or energy can exceed the speed of
light
 Characteristics of a “signal”:
A train of oscillations that starts from zero at some point and end at some
point. (a pulse, not a simple periodic wave)
Question: what about vg > c?
Section 3: Controversy/Debate
A group velocity greater than c does not contradict
relativity because group velocity is not in general
a signal velocity.
Section 3: Controversy/Debate
In anomalous dispersion:
If vg > c, essentially the entire
transmitted pulse is a
“reconstruction” of a tiny,
early-time tail of the
incidence pulse. No new
information is transferred.
Section 4: Conclusions and References
• Conclusions:
– A light pulse can propagate with a group velocity exceeding or
below “c” due to dispersion
– Faster-than-c group velocities do not violate Einstein’s Theory of
Relativity because the group velocities do not represent the speed
at which real information or energy is moving
• References:
– The speed of information in a “fast-light” optical medium, M.S.
Bigelow et al., Science 301, 200-2 (2003)
– Optics, Eugene Hecht, 4th Edition
– Fast light, Slow light, Raymond Y. Chiao and Peter W. Milonni,
Optics and Photonics News, June 2003
– Gain-assisted superluminal light propagation, L.J. Wang et al.,
Nature, Vol 406, July 2000