Our web: Educational introductions. All
our papers on line, including for general audience. Press coverages.
This page is generated by

**Ad
Lagendijk**, physicist and member of the physics
group

**Complex
Photonic Systems**.

**Comment on the paper by Ad L**.

The following paper describes in an exciting way the developments
in the field of localization of light, during 1990 and 1991,
when we had discovered the effect of slowing down of light
(and other classical waves) in strongly scattering media -
M.P. van Albada, B.A. van Tiggelen, A. Lagendijk, and A. Tip,
*Speed of propagation of classical waves in strongly scattering
media, *Phys. Rev. Lett. **66**, 3132-3135 (1991). At
that time there was quite some opposition against our ideas.
Now, six years later, our arguments indeed turned out to be
correct and have been accepted by the community. They are,
for instance, discussed in great detail in the following book:
P. Sheng, *Introduction to Wave Scattering, Localization,
and Mesoscopic Phenomena* (Academic, San Diego 1995) Chapter
4.

Light
travels more slowly through strongly scattering materials

BY BARBARA GOSS LEVI

*Taken from Physics Today June 1991 page 17-19 *

Search and Discovery section

We expect light to travel through a medium at a rate given
by the speed of light in a vacuum divided by the average index
of refraction of the material. But a recent experiment performed
in the Netherlands indicates that in some strongly scattering
materials it may actually take light five to ten times longer
to traverse a sample than one would expect from this simple
formula for the phase velocity. It appears that the light
resonates for a while with the dielectric microspheres of
which the material is made so the light takes longer to travel
through the sample. This result challenges some assumptions
made in the analysis of data from light scattering experiments,
especially searches for localized light.

**Localization of light **

Interest in light multiply scattered from random dielectrics
has been piqued by the analogous phenomenon of electron scattering
in disordered materials. Philip Anderson discovered in 1958
that in a sufficiently disordered material electrons can be
localized: The electrons are scattered so strongly that they
become confined to small regions and their transport is effectively
halted. Once this behavior was recognized as essentially a
wave phenomenon, researchers began to look for its occurrence
in electromagnetic and even acoustic waves. (See the article
by Sajeev John in PHYSICS TODAY, May, page 32.) In the case
of photons, the role of the disordered material is played
by a medium with a sharply and randomly varying dielectric
constant, which can strongly scatter electromagnetic waves.

Localization is caused by the interference between waves
scattered by the medium. As waves are scattered through the
material they interfere with waves traveling other paths.
When the average distance the photon travels between scattering
events, the mean free path, becomes comparable to a wavelength,
the interference effects can affect the transport of light.
Under the condition that the mean free path is about equal
to the wavelength (divided by 2p
), a rule known as the Ioffe-Regel condition, the extended
normal modes can become localized. The electromagnetic energy
is then concentrated in standing rather than traveling waves.

Although localization comes directly out of Maxwell's equations,
it has only recently been appreciated. Previously, multiple
scattering was either ignored or avoided because of the horrendous
complications of keeping track of which waves interfere. Now,
however, physicists are creating rather than shunning materials
with random gradations in the index of refraction. Typically
these are micron-sized spheres of one material embedded within
another material. Because light localization is predicted
to occur only within a certain range of frequencies, it is
very difficult to achieve. To approach localization within
real materials one must make the elastic scattering cross
section large without correspondingly increasing the absorption.
Experimenters must thus select just the right combination
of composite materials, microstructures and packing density.

The hallmark of electron localization is the disappearance
of the transport diffusion coefficient. Searches for photon
localization have also focused on studies of the diffusion
constant. According to the Boltzmann expression, the diffusion
constant D equals 1/3 *v* *l* , where *v* is
the transport velocity and *l* is the mean-free path.
Traditionally, the transport velocity was expected to be closely
approximated by the phase velocity c/n, where n is the average
index of refraction. Thus a small value of D was thought to
imply a small mean free path. Near the localization regime
the diffusion constant gets renormalized by a factor that
depends on the sample size, coherence length and absorption
length.

The message of the Dutch experiment, however, is to inject
some caution into how experimenters interpret the size of
the diffusion constant: Because the transport velocity can
be much less than the phase velocity, a small diffusion coefficient
may imply a small *1*, a small *v*, or both. The
difference between the two velocities is expected to be especially
great near resonant scattering - but that is just the strong
scattering regime where many of the localization searches
have concentrated.

One of the past experiments whose interpretation may be affected
by the new results is a 1989 study by Michael Drake (Exxon
Research and Engineering) and Azriel Genack (Queens College
of the City University of New York) of the transmission of
light through a collection of small spheres of titanium oxide
in air.[1] They reported a value for the diffusion coefficient
that was half the size expected, and they inferred a correspondingly
small value for the mean free path.

**Trapped by resonant scattering**

The Dutch transport velocity experiment was done by Meint
P. van Albada of the University of Amsterdam, Bart A. van
Tiggelen and Adriaan Tip of the FOM Institute for Atomic and
Molecular Physics, Amsterdam, and Ad Lagendijk, who has a
joint appointment at both institutions.[2] Lagendijk discussed
the results at the March APS meeting in Cincinnati. His group
studied the passage of 633-nm light through samples of what
was essentially white paint-particles of titanium oxide, whose
index of refraction is 2.7, surrounded by air. The size distribution
of the particles was centered at 220 nm, and the particles
filled 36% of the volume.

The experimenters made two types of measurements. In one
type, the researchers measured both the backscattered and
transmitted light under static conditions. Because these steady-state
measurements should not depend at all on the transport speed,
they were used in two ways to infer the mean free path.

In the other type of measurements, the FOM-Amsterdam group
varied the frequency of the laser light and measured the changes
in the speckle intensity of the transmitted and reflected
light. The speckles are dark and light areas resulting from
the patterns of constructive and destructive interference.
The intensity-intensity correlations as a function of frequency
gave a measure of the time t between
scattering events. Using the time determined from these dynamic
measurements and the mean free path from the static measurements,
Lagendijk and his colleagues found that the value of the transport
velocity, defined as *v* *= l/t
, *was (5 ± l) x 10^7 m/sec.
The phase velocity for this material would be about 2.5 x
10^8 m/see, a factor of 5 faster.

This experiment was conducted quite near resonance, with
the wavelength about equal to the diameter of the titanium
oxide spheres. Thus Lagendijk suggests a picture in which
the light, being just about the right length to create a standing
wave within the spheres, bounces around inside them for a
bit before resuming the journey through the sample. Lagendijk's
picture also resonates with some other studies involving scattering
in spheres.

The FOM-Amsterdam group derived a theoretical expression
for the transport velocity and found a value of 4.2 x 10^7
m/sec. As a result of their experiment, the group recommends
that the predictions of theories in which diffusion constants
are calculated from the phase velocity should be reconsidered.
In addition, they assert, one should be very careful in inferring
a mean free path from a dynamic measurement.

Many observers find the result of this experiment to be at
once quite striking and quite reasonable. One classic optics
text notes that both the group velocity and the phase velocity
become unphysical near a resonance but provides no details
about how to treat this case. Lagendijk points out, however,
that a 1960 text by Leon Brillouin *(Wave Propagation and
Group Velocity, *Academic Press) derives an expression
for an energy velocity. Brillouin was concerned with the actual
speed of transport of light through media that have some internal
resonances (such as two-level atoms or classical oscillators).
In that case, as in the recent experiment with standing waves
inside spherical scatterers, the energy is not available for
transport part of the time. Lagendijk considers his group's
treatment to be a generalization of Brillouin's.

The FOM-Amsterdam group has used their microscopic theory
to calculate the phase, group and transport velocities. The
ratios of these velocities to the speed of light are plotted
in the figure presented below as a function of a size parameter
for a certain density of scatters. Both the phase and group
velocities are larger than the speed of light for some values
the size parameter an hence become unphysical, but the transport
velocity remains well behaved.

**Phase, group and transport velocities ***relative
to the speed of light. These ratios were calculated for light
passing through a strongly scattering medium made of dielectric
spheres filled up to 36 vol%. The size parameter is a measure
of the sphere diameter. When the spheres are the right size
for the light to scatter resonantly within them, the phase
velocity (red) and group velocity (black) grow larger than
c. But the speed of energy transport (dark blue) remains well
behaved. The second curve for transport velocity (light blue)
is a more approximate calculation. (Adapted from ref. 2.)*

The scattering behavior is reminiscent of the phenomenon
of self-induced transparency, observed by Erwin Hahn (University
of California a Berkeley) and Samuel McCall (now a Bell Labs)
in 1967. In both cases, the light is held up in some way as
it traverses the material, but the self-induced transparency
involves a no linear effect.

**Refinements**

Most observers believe that the FOM Amsterdam group has uncovered
very real effect. However, the exact determination of such
parameters a mean free path and scattering time involves some
very subtle considerations. Lagendijk and some colleagues
issued a caution [3] sever years ago about inferring the transport
mean free path from static transmission measurements: The
light may suffer some internal reflection the boundary, causing
the condition at the boundary to differ from those in the
interior, where one wants to know.. Based on some systematic
study done with his colleagues at Queens College,[4] Genack
has found that the effect might cause the inferred value of
the mean free path to differ from its true value in some systems,
but he admits that the effect may be small in the recent FOM-Amsterdam
experiment. Lagendijk contends that internal reflectivity
is not an important factor in his group's experiment.

Paul Fleury (AT&T Bell Labs) has some reservations about
deriving the mean time between scattering even from dynamic
measurements because the quantitative relation between the
speckle pattern and the mean free time is a very sensitive
function of such things as absorption, boundary conditions
and launch conditions.[5] Genack, however, defends the dynamic
measurement citing experiments in which he h compared measurements
in both t time and frequency domains a found good agreement.[6]

The theoretical treatment of the subject by the FOM-Amsterdam
group has raised some questions well, because at first glance
it appears to violate a conservation law that known to hold
for electrons. Th conservation law, the Ward identity translates
for electrons into a requirement that the square of the wavefunction
remain constant, By analogy theorists analyzing the scattering
light have required that the square the amplitude of the classical
wave conserved. Past theories have considered only the electric
field amplitude. However, Lagendijk and his colleagues point
out that this approach is only approximate: The conserved
quantity is the energy density, which involves both the electric
and magnetic fields.

Ping Sheng of Exxon Research and Engineering was excited
about the Dutch result because he and his colleagues have
seen what they feel is an additional example of resonant behavior
in experiments on acoustical transmission through dense colloidal
systems.[7] In particular they have observed an unexpected
low-frequency mode of propagation in addition to the expected,
higher-frequency mode. The velocity of this new mode, like
the transport velocity in the recent optical experiment, is
very much lower than normal. The researchers believe that
it results because tails of resonant waves in adjoining scatterers
couple together. While the FOM-Amsterdam group finds that
light is bouncing around inside spheres, which interact *incoherently,
*the Exxon team finds a *coherent *coupling between
the scatterers.

**The search for localization**

While the interpretation of the 1989 experiment by Drake and
Genack is affected by the recent Dutch result, Genack and
his Queens College colleague, Narciso Garcia, have recently
done an experiment to which the new caution does not apply.[8]
They have studied the scattering of microwaves in random mixtures
of aluminum and Teflon spheres. (See the May cover of PHYSICS
TODAY.) The radiation does not penetrate the aluminum spheres,
so the phase and transport velocities are expected to be equal.

The absorption present in this system prevents the experimenters
from seeing the diffusion coefficient go to zero. Thus Genack
and Garcia do not claim to have reached localization in the
sense of a vanishing diffusion constant. However, they do
report several types of behavior that are predicted for localized
systems. For one, the transmission falls off as 1/L^2, where
L is the thickness of the sample. Such behavior is predicted
by the scaling theory of localization to occur at the threshold
of localization. The experimenters saw this scaling behavior
only when the spheres occupied a certain fraction (about 30%)
of the volume.

Genack feels that their experiment satisfies the Ioffe-Regel
criterion for localization as well as a criterion formulated
by David Thouless (University of Washington), which requires
essentially that the width of energy levels within the scattering
medium be small compared with the spacing between levels.
However, because the absorption in the microwave system complicates
the data analysis, observers would still like to see additional
confirmation of localization.

A collaboration between the University of California, at
San Diego, and AT&T Bell Labs has produced some intriguing
maps of the electromagnetic power distributions in a two-dimensional
system, in which localization is far easier to achieve. The
team consists of Sheldon Schultz, Rachia Dalichaouch, John
Armstrong and David Smith of UCSD and McCall and Philip Platzman
at Bell Labs. In their setup, microwaves scatter from a square
grid of dielectric cylinders, with the electric field vector
of the microwaves parallel to the z axis of the cylinders.
At the APS March meeting McCall discussed their observation
of a photonic bandgap with this arrangement. (Eli Yablonovich
and his colleagues at Bellcore have solved the more difficult
problem of fabricating a material with a complete microwave
bandgap in *three* dimensions, as pictured in John's
article in PHYSICS TODAY.)

Recently the San Diego-Bell experimenters made the regular
array irregular by removing half of the 36 x 27 array of dielectric
cylinders in a random pattern. They probed the resulting electromagnetic
energy distribution at frequencies where they; anticipated
that the system was strongly localized. The distribution clearly
shows the electric energy density of the localized mode. (See
the figure below.) There may be little surprise in finding
a localized state in two dimensions, but there is great satisfaction
in visualizing it.

**Microwave electric energy density ***in a
two-dimensional system enables one to visualize a localized
state. Microwaves entering from the left traverse a 36X 27
array of dielectric cylinders, half of which have been removed
at random. The peaks near the center are part of one localized
mode, unrelated to the neighboring mode seen at the upper
left. The experiment was performed by a group from the University
of California, at San Diego, and from AT&T Bell Labs.
They showed a similar figure at the March APS meeting.*

References:

1. J. Drake, A. Genack, Phys. Rev. Lett.
**63, **259 (1989).

2. B. A. van Tiggelen, A. Tip, A. Lagendijk,
Phys. Rev. Lett., to be published.

3. A. Lagendijk, R. Vreeker, P. de Vries,
Phys. Lett. **A136**, 81 (1989).

4. A. Genack, N. Garcia, W. Polkosnik, Waves
in Random Media 1, to appear in July 1991.

5. G. H. Watson, S. L. McCall, P. A. Fleury,
K. B. Lyons, Phys. Rev. **B41**, 10947 (1990).

6. A. Z. Genack, N. Garcia, J. Li, W. Polkosnik,
J. M. Drake, Physica A **168**, 387 (1990).

7. X. Jing, P. Sheng, M. Zhou, Phys. Rev.
Lett. **66**, 1240 (1991). J. Liu, L. Ye, D. A. Weitz,
P. Sheng, Phys. Rev. Lett. **65, **2602 (1990).

8. A. Z. Genack, N. Garcia, Phys. Rev. Lett. **66**, 2064
(1991).

back to top