UvA-DARE is a service provided by the library of the University of Amsterdam (http
s
://dare.uva.nl)
UvA-DARE (Digital Academic Repository)
Optical transmission through strong scattering and highly polydisperse media,
(article0
Gomez Rivas, J.; Sprik, R.; Soukoulis, C.M.; Busch, K.; Lagendijk, A.
DOI
10.1209/epl/i1999-00108-7
Publication date
1999
Published in
Europhysics Letters
Link to publication
Citation for published version (APA):
Gomez Rivas, J., Sprik, R., Soukoulis, C. M., Busch, K., & Lagendijk, A. (1999). Optical
transmission through strong scattering and highly polydisperse media, (article0.
Europhysics
Letters
,
48
, 22-28. https://doi.org/10.1209/epl/i1999-00108-7
General rights
It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)
and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open
content license (like Creative Commons).
Disclaimer/Complaints regulations
If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please
let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material
inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter
to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You
will be contacted as soon as possible.
Download date:09 Aug 2022
EUROPHYSICS LETTERS 1 October 1999
Europhys. Lett., 48 (1), pp. 22-28 (1999)
Optical transmission through strong scattering
and highly polydisperse media
J. G
´
omez Rivas
1
, R. Sprik
1
, C. M. Soukoulis
2
, K. Busch
3
and A. Lagendijk
1
1
Van der Waals-Zeeman Instituut, Universiteit van Amsterdam
Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
2
Ames Laboratory and Department of Physics and Astronomy
Iowa State University, Ames, Iowa 50011, USA
3
Institute for Theory of Condensed Matter, Department of Physics
University of Karlsruhe - P.O. Box 6980, 76128 Karlsruhe, Germany
(received 21 December 1999; accepted in final form 3 August 1999)
PACS. 42.25Dd – Wave propagation in random media.
PACS. 78.30Ly – Disordered solids.
Abstract. – We present near infrared total transmission measurements through samples of
randomly packed silicon powders. At different wavelengths we analyze in detail the scattering
prop erties and the effects of residual absorption. The lowest value of kl
s
, where k is the wave
vector and l
s
is the scattering mean free path, is 3.2. We also observe that kl
s
is nearly constant
over a wide wavelength range. This phenomenon is associated with the high polydispersity
of the particles. We use the energy density coherent potential approximation to explain our
measurements.
The analogy between the propagation of electron waves and classical waves has led to a
revival in the research of the transport of light in disordered scattering systems [1]. The
final goal of many of these studies has been to observe the optical analogue of Anderson
localization in electronic systems [2]. Anderson localization refers to an inhibition of the wave
propagation in disordered scattering systems due to interference. Localization is essentially a
wave phenomenon and it should hold for all kinds of waves i.e. electrons, electromagnetic and
acoustic waves [3]. For isotropic scatterers Anderson localization is established if kl
s
≤ 1, where
k is the wave vector in the medium and l
s
is the scattering mean free path, or the average
length that the wave propagates in between two elastic collisions. The transition between
extended and localized states occurs when kl
s
' 1. This is known as the Ioffe-Regel criterion
for localization [4]. To approach the Ioffe-Regel criterion, l
s
can be reduced by using scatterers
with a high refractive index, n, and a size such that the scattering cross-section is maximum.
Experimental difficulties in realizing a random medium where the optical absorption is low
enough and the light scattering is efficient enough to induce localization has been the reason
why, for a long time, only microwave localization was realized [5]. In this experiment the
absorption is large and, therefore, complicates the interpretation of the results. Recently, near
c
° EDP Sciences
j. g
´
omez rivas et al.: optical transmission through strong scattering etc. 23
infrared localization in GaAs powders was observed [6]. Nevertheless, the validity of these
measurements has been questioned by the possibility of absorption [6]. It is clear that new
experiments must be carried out in very strong scattering media to distinguish between the
effects of optical absorption and multiple scattering.
In this letter we present near infrared total transmission measurements through samples of
randomly packed silicon powders with particle sizes of the order the wavelength, λ. Silicon is a
semiconductor with the energy band gap at λ
gap
=1.1µm. Therefore, the measurements were
performed at wavelengths, λ>λ
gap
to minimize optical absorption. The high refractive index
of Si (n ' 3.5 in the near infrared) and the size of the Si particles constituting our samples
make the light-matter coupling very strong. We have performed measurements at different
wavelengths to systematically study the influence of the residual band gap absorption and the
scattering properties of the system.
The transition between extended and localized states occurs when kl
s
' 1. If kl
s
À 1,
light propagates by performing a random walk. An enormous simplification in the description
of the transport of light can thus be made by neglecting all interference effects, and the
transport may be described by the diffusion equation [7]. As kl
s
approaches the critical value,
the diffusion approximation may still be used with a renormalized diffusion constant. In the
localization regime the steady-state diffusion breaks down, which means that the diffusion
constant vanished. Defining the transport mean free path, l, as the distance traveled by the
light before its direction of propagation becomes randomized, isotropic scattering implies that l
equals l
s
. If we consider a medium translationally invariant in the x and y directions, the three-
dimensional diffusion equation reduces to the one-dimensional case in the non-translationally
invariant direction, i.e. the z-direction. This is valid for slab-geometry samples in which the
x and y dimensions are much larger than the z-dimension. Then, the energy density in the
stationary state inside the sample, ρ, is given by
∂
2
ρ
∂z
2
−
ρ
L
2
a
= −
1
D
I
0
δ(z − l) , (1)
where L
a
is the absorption length and D the diffusion constant. The diffusion constant is
given by D = v
e
l/3, where v
e
is the energy transport velocity in the medium. In eq. (1), the
incoming energy flux at the boundary z = 0 has been replaced by a source of diffusive radiation
of strength, I
0
, equal to the incident flux and located at the plane z = l [8]. Simulations have
shown that eq. (1) is accurate to within about 1% for slabs sufficiently thick to be opaque [9].
The boundary conditions are determined considering that the diffuse fluxes going into the
sample at z = 0 and L are due to a finite reflectivity at the boundaries [10]. The boundary
conditions can be written as
ρ ∓ z
0
i
∂ρ
∂z
=0,i=
1ifz= 0 (front sample surface) ,
2ifz=L(back sample surface) ,
(2)
where z
0
i
are given by z
0
i
=(2l/3)(1 + R
i
)/(1 − R
i
), and R
i
are the polarization and angular
averaged reflectivities of the boundaries. In the non-absorbing limit (L
a
→∞) eqs.(2) are
equivalent to extrapolate ρ to 0 at a distance z
0
i
outside the sample surface. Therefore z
0
i
are
called the extrapolation lengths. If
R
i
= 0 the diffusion approximation gives z
0
i
=2l/3, very
close to the extrapolation length of 0.7104l given by the Milne solution [7].
The experimentally determined quantities in our experiments are the total transmitted
intensities. The total transmission, defined as the transmitted flux normalized by the incident
24 EUROPHYSICS LETTERS
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.0
Normalized number of particles
r (
µ
m)
1.66
µ
m
Fig. 1
0.9 1.2 1.5 1.8 2.1 2.4
0.0
0.2
0.4
0.6
0.8
1.0
Transmission (a.u.)
λ
(
µ
m)
Fig. 2
Fig. 1. – Normalized histograms of the silicon particle radius considering all the particles as entities,
indep endently of whether they are part of a cluster (solid bars), and considering the clusters as single
particles (dashed bars). The solid and dotted lines are log-normal fits, from which the average particle
radius are calculated. The inset is a scanning electron microscope photograph of the Si particles.
Fig. 2. – Total transmission spectra normalized to their maximum transmissions. Solid line: Total
transmission spectrum of a layer of silicon powder of 57.8 µm thickness. Dotted line: Transmission
sp ectrum of a piece of intrinsic silicon of 1 mm thickness.
flux, is given by [11]
T =
−D(∇ρ)
z=L
I
0
=
sinh(l/L
a
)+(z
0
1
/L
a
) cosh(l/L
a
)
(1 + z
0
1
z
0
2
/L
2
a
) sinh(L/L
a
)+(1/L
a
)(z
0
1
+ z
0
2
) cosh(L/L
a
)
. (3)
If L
a
À L, eq. (3) simplifies to
T =
l + z
0
1
L + z
0
1
+ z
0
2
, (4)
and if L
a
¿ L, it simplifies to
T =
2L
a
(l + z
0
1
)
L
2
a
+(z
0
1
+z
0
2
)L
a
+z
0
1
z
0
2
exp[−L/L
a
] . (5)
In our experiments we measure the total transmission through samples consisting of 99.999%
pure Si particles. These powders are commercially available (Cerac S-1049) containing par-
ticles with sizes ranging from a few hundred nanometers to about 40 µm. To reduce the
polydispersity of the powders we suspended them in spectroscopic chloroform and we let the
Si particles sediment for 5 minutes. Only the particles that did not sediment were used in the
experiments. The polydispersity of the resulting powder was evaluated from scanning electron
microscope photographs like the one showed in the inset of fig. 1. As can be seen in the
photograph, the particles tend to aggregate into clusters. This makes the definition of particle
j. g
´
omez rivas et al.: optical transmission through strong scattering etc. 25
radius difficult. We have evaluated the average particle radius with two different methods:
a) considering all the particles as entities, independently of if they are part of a cluster, and
b) considering the clusters as single particles. Defining the radius of a particle (or a cluster)
as half the maximum distance between parallel tangents to the particle surface, we computed
the radius of the particles. Figure 1 shows the normalized histograms of the particle radius.
The histograms can be fitted with a log-normal function, y = A exp[−ln
2
(r/r
c
)/2W
2
]. The fit
to the histogram obtained with method a) gives A =0.90, r
c
=0.19 µm, W =0.61, and it is
shown by the solid line in fig. 1, while method b) gives A =0.86, r
c
=0.44 µm, W =0.55, and
it is shown by the dotted line in the same figure. This allows to calculate the average particle
radius and its standard deviation: a) r
a
=0.33 ± 0.22 µm, and b) r
a
=0.69 ± 0.41 µm. The
polydispersity defined as the ratio between the standard deviation and r
a
in percentage, is of
67% and 59%, respectively. In other words, our samples are constituted by highly polydisperse
scatterers.
We evaporated the chloroform and made slab geometry samples with the remaining powder
on CaF
2
substrates with the form of a disk of 10 mm diameter. The thicknesses of the powder’s
layers were measured, with a resolution of 1 µm, with an optical microscope. We measured
the thickness of each sample at different points within its central region to be sure that the
powder layer was homogeneous. The resulting sample thicknesses are the average value of
these measurements and they range from 5.9 ± 2 µmto57.8±2µm.
The total transmissions were measured with a Fourier Transform Infrared Spectrometer
(BioRad FTS-60A). A tungsten halogen lamp was the light source. Short wavelengths were
optically filtered and an aperture of 2 mm diameter was placed in front of the sample to measure
the transmission only through the region where the thickness was measured. The power of
light incident on the sample was ' 1 mW. The diffusely transmitted light was collected with a
BaSO
4
-coated integrating sphere and detected with a PbSe photo conductive cell. Before and
after measuring each sample, we measured the transmission through a clean CaF
2
substrate,
which we used as reference to obtain the absolute value of the total transmission through the
Si layer and to check the stability of the set-up. Figure 2 shows a total transmission spectrum
of a sample of thickness L =57.8µm (solid line), and the transmission spectrum of a piece
of intrinsic Si of 1 mm thickness (dotted line) for comparison. Both measurements have b een
normalized by their maximum transmissions.
By weighing the samples, we estimated the Si volume fraction to be φ ' 40%, which gives
rise to a Maxwell-Garnet effective refractive index of the samples of n
e
' 1.5, nearly constant
for wavelengths between 1.4 µm and 2.5 µm. With the value of n
e
, the extrapolation lengths
of the Si-air and Si-CaF
2
interfaces (z
0
1
and z
0
2
, respectively) can be calculated. Due to the
size and irregular shape of the Si particles, we may assume that the scattering is isotropic.
Then, since l = l
s
, the values of the extrapolation lengths are estimated to be z
0
1
' 2.42 l
s
and z
0
2
' 0.78 l
s
; these are taken as fixed parameters in eq. (3) and eq. (4). Figure 3 shows
the total transmission as a function of the sample thickness for λ =2.5µm (squares) and
λ =1.4µm (circles). The error bars are mainly due to intensity fluctuations of the source
and to inhomogeneities in the sample thickness. The total transmission measurements can
be fitted excellently by using classical diffusion theory as it is shown by the solid and dotted
lines in fig. 3. The solid line in fig. 3 is a fit of eq. (4) to the λ =2.5µm measurements with
l = l
s
=0.83 ± 0.08 µm. At this wavelength L
a
À 57.8 µm, thus absorption can b e neglected.
The dotted line is a fit of eq. (3) to the λ =1.4µm measurements with l = l
s
=0.56± 0.06 µm
and L
a
=8.8±1µm.
The wavelength dependence of L
a
is plotted in fig. 4. The increase of absorption for
λ<2.0µm is due to strain in the Si lattice structure. The presence of strain gives rise to a
deformation of the potential which smears the valence and conduction bands. This results in an
