Slow and fast light associated with polariton interference
B. Gu,
1
N. H. Kwong,
2
R. Binder,
1,2
and Arthur L. Smirl
3
1
Department of Physics, University of Arizona, Tucson, Arizona 85721, USA
2
College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA
3
Laboratory for Photonics and Quantum Electronics, University of Iowa, Iowa City, Iowa 52242, USA
Received 26 May 2010; published 22 July 2010
Propagation times of optical pulses through a medium near an absorptive resonance with and without spatial
dispersion are studied and contrasted. When spatial dispersion is not present, a light pulse is expected to
traverse a medium in a time inversely proportional to its group velocity. In a medium with spatial dispersion,
where two polariton modes exist here, bulk GaAs as an example, a similar description is obtained if the losses
are such that light propagates primarily in one mode. However, we show that, when the broadening of the
resonance dephasing rate is below a critical value, a frequency range exists near resonance where the transit
times are determined by interference between copropagating polaritons and deviate strongly from expectations
based on the group velocities of the polariton branches. When the interference is constructive at the samples
end face, the transit times are determined by the average of the inverse group velocities; when it is destructive,
we find abrupt transitions between very slow long positive and very fast large negative transit times. We
present quantitative criteria for the resolution of these features and for distortion-free propagation in the
spectral vicinity of them. Our analysis puts the well-known slow- and fast-light effects in systems without
spatial dispersion into a broader context by illustrating that they are a limiting case of systems with spatial
dispersion.
DOI: 10.1103/PhysRevB.82.035313 PACS numbers: 71.36.+c, 42.25.Bs, 78.40.Fy
I. INTRODUCTION
Over the past decade, interest in controlling the velocity
of light pulses has been renewed, in part because of potential
applications in telecommunications, spectroscopy, and so-
called microwave photonics for recent introductions and re-
views see Refs. 1 and 2. Studies have shown that the inter-
action of light with matter can lead to extreme changes in the
effective or apparent velocities of pulses of light: for ex-
ample, pulse envelopes that travel a few meters per second,
3
that appear to exit the material before the peaks of the pulses
enter it,
4–10
and that appear to travel backward in the
material
8
have been reported. These phenomena have been
investigated in media ranging from atomic vapors
3,11–13
to
solid materials, such as doped crystals,
14
optical fibers,
15,16
and semiconductors doped,
4
bulk,
17
and quantum wells
18
.
Most recent schemes for producing slow, fast or backward
traveling light take advantage of sharp resonances in nonlin-
ear processes, such as electromagnetically induced
transparency,
3
stimulated Brillouin scattering,
15,16
or coher-
ent population oscillations.
8,14,19–21
The simplest demonstration of slow subluminal and fast
superluminal light propagation, however, is the linear in-
teraction of a light pulse with an absorptive medium consist-
ing of identical, localized dipole oscillators. This topic has
been considered for almost a century
22
and is commonly ad-
dressed in text books.
23
In the absence of interactions be-
tween the two and in the absence of losses, the light and
oscillator have independent dispersion relations frequency
vs wave vector given by k = ck and k = E
x
/ , respec-
tively, as sketched in Fig. 1a. As we discuss in the next
section, for pulses that have a sufficiently narrow spectral
width, the pulse envelope is transmitted undistorted with a
group velocity v
g
.
Most conventional descriptions of light propagating near
a resonance neglect direct coupling between the dipole oscil-
lators i.e., they are only coupled indirectly through their
interaction with the same light field. If present, electronic
coupling will allow the light-induced optical polarization
exciton to move through the system. This motion can be
included through the kinetic energy of the optical polariza-
tion,
2
k
2
/ 2M
x
, where M
x
is the effective mass of the polar-
ization. Figure 1b shows a sketch of the uncoupled material
and light dispersions for M
x
. The case of vanishing elec-
(a)
Light
x
Light
NSED
(b)
Light
SED
x
Re k
FIG. 1. Color online Sketch of the uncoupled and unbroad-
ened exciton and photon-dispersion relations with real and real
k for two exciton masses: a M
x
= , labeled NSED for no spatial
exciton dispersion and b M
x
labeled SED for spatial exciton
dispersion.
PHYSICAL REVIEW B 82, 035313 2010
1098-0121/2010/823/03531310 ©2010 The American Physical Society 035313-1