Electromagnetic waves in disordered dielectric media: Coupled-dipole model

Abstract

An effective approach to multiple scattering of electromagnetic waves by disordered dielectric media is developed. A self-consistent energy-conserving coupled-dipole model is used. Applications to the Anderson localization of electromagnetic waves are presented. A complete set of Maxwell's equations is used to describe the propagation of waves and the vector character of the electromagnetic field is fully taken into account.

Full text

Tr. J. o f Physics
23 (1999) , 887 – 894.
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T¨
UB˙
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Electromagnetic waves in disordered dielectric media:
coupled-dipole model
Arkadiusz OR LOWSKI and Marian RUSEK
Instytut Fizyki, Polska Akademia Nauk
Aleja Lotnik´ow 32/46, 02-668 Warszawa-POLAND
Received 02.02.1999
Abstract
An effective approach to multiple scattering of electromagnetic waves by disor-
dered dielectric media is developed. A self-consistent energy-conserving coupled-
dipole model is used. Applications to the Anderson localization of electromagnetic
waves are presented. A complete set of Maxwell’s equations is used to describe the
propagation of waves and the vector character of the electromagnetic field is fully
taken into account.
1. Introduction
Disordered dielectric media with a high refractive index contrast are a vivid subject
of experimental and theoretical studies. Such environments provide interesting exam-
ples of strongly scattering media where multiple scattering effects play a dominant role.
Transport properties of electromagnetic waves in such media are directly affected by in-
terference and the standard radiative transport theory must be essentially modified to
agree with experiments. It should be noted that electromagnetic waves propagating in
random dielectric structures exhibit many effects that are typical for the behavior of
electrons in disordered semiconductors. Therefore studies of transport properties of light
and microwaves in random dielectrics can benefit from the well-developed theoretical
methods and concepts of solid-state physics. A striking example is the theory of electron
localization in noncrystalline systems such as amorphous semiconductors or disordered
insulators. According to Anderson [1, 2], an entire band of electronic states can be spa-
tially localized in a sufficiently disordered infinite material. So-called strong localization
is achieved when the diffusion constant in the scattering medium becomes zero and the
Anderson transition may be viewed as a transition from particle-like behavior described
by the diffusion equation to wave-like behavior described by the Schr¨odinger equation.
Because interference is the common property of all wave phenomena, we can expect that
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there are some analogs of electron localization for other types of waves. Indeed, many
generalizations of electron localization exist, especially in the realm of electromagnetic
waves [3, 4]. So-called weak localization of electromagnetic waves manifesting itself as
enhanced coherent backscattering is presently relatively well established experimentally
[5, 6, 7] and understood theoretically [8, 9]. The crucial question is whether interfer-
ence effects in random dielectric media can reduce the diffusion constant to zero leading
to strong localization. The essential parameter is the mean free path lwhich should
be rather short. Localization could be experimentally demonstrated if the scale depen-
dence of the diffusion constant in disordered dielectric media would be observed. Such
a demonstration has first been given for two dimensions, where strong localization takes
place for arbitrarily small value of the mean free path (if the medium is sufficiently large).
The strongly scattering medium has been provided by a set of dielectric cylinders ran-
domly placed between two parallel aluminum plates on half the sites of a square lattice
[10]. Only very recently a quite successful experiment dealing with localization of light
in three dimensions has been performed [11]. The strongly scattering dielectric medium
was provided by the semiconductor powder gallium arsenide.
Localization of electromagnetic waves is a very nontrivial effect and proper under-
standing of this phenomenon requires sound theoretical models. To take fully into ac-
count the vector character of electromagnetic fields, such models should be based directly
on the Maxwell equations. They should also be simple enough to provide calculations
without too many approximations. Although there is a temptation to immediately apply
averaging procedures as soon as disorder is introduced into the considered model, such
an approach can be dangerous; averaging of the scattered intensity over some random
variable leads to radiative transport theory which neglects all interference effects [12].
2. General considerations
A possible definition of localized electromagnetic waves which is used in this paper
resembles the definition of localized states in quantum mechanics and makes use of the
analogy between the quantum-mechanical probability density and the energy density of
the field. In general, the electric field ~
E(~r, t) cannot be interpreted as the probability am-
plitude. The correct equivalent of the quantum-mechanical probability density is rather
the energy density of the field and not the squared electric field. Therefore it seems
natural to say that a monochromatic electromagnetic wave
~
E(~r, t)=Ren~
E(~r)e−iωt oand ~
H(~r, t)=Re
n~
H(~r)e−iωt o,(1)
is localized if the time-averaged energy density of the total field tends to zero far from
a region of space which contains the scattering medium. In the following we will use
the fact that for rapidly oscillating monochromatic electromagnetic waves (1) only time
averages are measurable. The total field
~
E(~r)= ~
E(0)(~r)+~
E(1)(~r)and ~
H(~r)= ~
H(0)(~r)+ ~
H(1)(~r),(2)
may be considered as the sum of the incident free field ~
E(0)(~r)and ~
H(0)(~r), which obeys
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OR LOWSKI, RUSEK
the Maxwell equations in vacuum, and waves scattered by various parts of the medium
[13]:
~
E(1)(~r)=~
∇×~
∇× ~
Z(~r)−4π~
P(~r)and ~
H(1)(~r)=−ik ~
∇× ~
Z(~r).(3)
The electric Hertz potential
~
Z(~r)=Zd3r0~
P(~
r0)eik|~r−~
r0|
|~r −~
r0|,(4)
is expressed by the polarization of the dielectric medium ~
P. The above system of equa-
tions determines the electromagnetic field everywhere in space for a given free wave
incident on the system. Analogous relationships between the stationary outgoing and
the stationary incoming wave are known in general scattering theory as the Lippmann-
Schwinger equations [14]. A way of dealing with bound states in the formalism of the
Lippman-Schwinger equation is to solve it as a homogeneous equation, i.e., with the
incoming wave equal to zero.
3. Coupled-dipoles model
Usually localization of light is studied experimentally in microstructures consisting
of dielectric spheres or cylinders with typical dimensions (e.g., diameters) and mutual
distances being comparable to the wavelength. It is well known, however, that in such
a case the theoretical description becomes very complicated. On the other hand, the
theory of multiple scattering of light by dielectric particles is tremendously simplified
in the limit of point scatterers. In principle, this approximation is justified only when
the size of the scattering particles is much smaller than the wavelength. Fortunately,
in practical calculations many multiple-scattering effects including universal conductance
fluctuations, enhanced backscattering, and dependent scattering [15] can be obtained
qualitatively for coupled electrical dipoles. It is reasonable to assume that in many
situations what really counts for localization is the scattering cross section and not the
geometrical shape and real size of the scatterer. Therefore we will represent the dielectric
particles located at the points ~raby single electric dipoles
~
P(~r)=X
a
~paδ(~r −~ra),(5)
with properly adjusted scattering properties. In 2D media it is convenient to introduce
cylindrical coordinates ~r =(~ρ, z) and the dipole model leads to the polarization ~
P(~r)=
Pa~ezpaδ(~ρ −~ρa). Of course we should remember that the dipole model is only an
approximation. Moreover, it is known that several mathematical problems emerge in the
formulation of interactions of point-like dielectric particles with electromagnetic waves
[15, 16]. To safely use the dipole approximation it is essential to use a representation
for the scatterers that rigorously fulfills the optical theorem and conserves energy in the
scattering processes. Therefore, the time-averaged field energy flux integrated over a
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OR LOWSKI, RUSEK
surface surrounding arbitrary part of the medium should vanish:
Zd~s·~
S(~r)= c
4π
1
2Re Zd~s·n~
E(~r)×~
H∗(~r)o=0.(6)
Thus, on average, the energy radiated by the medium must be equal to the energy given to
the medium by the incident wave. Using this energy conservation condition and assuming
that the dielectric particles modeled by the dipoles are spherically symmetrical, we arrive
at the following field-dipole coupling [17]:
2
3ik3~pa=eiφ −1
2~
E0(~ra).(7)
Thus, to provide conservation of energy, the dipole moments ~pamust be coupled to the
electric field of the incident wave ~
E0(~ra) by complex polarizability (eiφ −1)/2. We get for
the 2D model the same kind of coupling, with the only difference being the constant in the
front of the dipole moment: iπk2[18]. To get some insight into the physical meaning of the
parameter φfrom Eq. (7) let us observe that it is directly related to the total scattering
cross-section σof an individual dielectric sphere represented by the single point-like dipole
in 3D. Indeed, within our formalism, the explicit formula for the total scattering cross
section of a dipole reads: 2k2σ=3π(1 −cos φ). A very similar relationship holds also
for the total scattering cross section of a dielectric cylinder modeled by a 2D dipole:
kσ=2(1−cos φ). In both cases φis a function of frequency and the physical parameters
describing the scatterer (dielectric constant and radius of a sphere or a cylinder).
The field acting on the ath dipole,
~
E0(~ra)=~
E(0)(~ra)+X
b6=a
~
Eb(~ra),(8)
is the sum of the free field and waves scattered by all other dipoles:
~
Ea(~r)=˜g(~r −~ra)·~pa.(9)
In 2D the Green function ˜g(~ρ)=2k2K0(−ik|~ρ|) is given by the Bessel function of the
second kind [18]. In 3D ˜gdenotes the proper Green tensor
˜g(~r)=k2eik|~r|
|~r| 3
(k|~r|)2−i3
k|~r|−1~r~r
|~r|2−1
(k|~r|)2−i1
k|~r|−1.(10)
Inserting Eq. (9) into (8), and using (7), it is easy to finally obtain the system of linear
equations:
X
b
Mab·~
E0(~rb)=~
E(0)(~ra),(11)
determining the field ~
E0(~ra) acting on each dipole for a given free field ~
E(0)(~ra). If we solve
it and use again Eqs. (7) and (9), we are able to find the electromagnetic field everywhere
890

OR LOWSKI, RUSEK
in space outside of the dipoles. Nonzero solutions ~
E0(~ra)6= 0 of Eq. (11) for the incoming
wave equal to zero ~
E(0)(~ra)≡0 may be interpreted as localized waves [17]. Let us stress
that perfectly localized waves exist only in infinite systems of dipoles.
5. Towards strong localization
It seems reasonable to expect that each electromagnetic wave localized in a system of
dipoles (5) usually corresponds to a certain curve on the plane {ω, φ}. Nevertheless, in the
case of random and infinite system of dipoles there can exist an entire continuous band
of spatially localized states corresponding to a region in the plane {ω, φ}.Afterchoosing
apoint(ω, φ) from this region a localized wave of frequency (arbitrarily near) ωexists in
almost any random distribution of the dipoles described by the scattering properties φ.
To illustrate this statement we have to study the properties of finite systems for increasing
number of dipoles N(while keeping the density constant). For each distribution of the
dipoles ~raplaced randomly inside a sphere with the uniform density of n=1dipole
per wavelength cubed (in 2D they are placed randomly inside a square with the uniform
density n= 1 dipole per wavelength squared) we have diagonalized numerically the M
matrix from Eq. (11) and obtained the lowest eigenvalue:
Λ(φ)=min
j|λj(φ)|.(12)
The resulting probability distribution Pφ(Λ), calculated from 103different distributions
of Ndipoles, is normalized in the standard way: RdΛPφ(Λ) = 1. Let us now compare
the contour plots of Pφ(Λ) (treated as a function of two variables φand Λ) calculated
for systems consisting of N= 300 three dimensional and two dimensional dipoles. They
are presented in Figs. 1 and 2, respectively. It is seen from inspection of these plots that
there is a region of φvalues (this region strongly depends on dimensionality) for which
the probability distribution Pφ(Λ) is apparently closer to the Λ = 0 axis than for other
values of φ. In 3D this happens only for values of |φ|that are sufficiently greater than
zero but smaller than π. Our numerical investigations indicate that for increasing size
of the system the variances of Pφ(Λ) decrease. In the limit of an infinite medium, the
probability distributions presented in these figures seem to tend to the delta function
lim
N→∞ Pφ(Λ) = δ(Λ),for |φ|≥φcr .(13)
Therefore, for some φthe function Λ(φ) is an example of a self-averaging quantity. This
means that for almost any random distribution of the dipoles ~ra, the equation λj(φ)=0
is satisfied. Thus, as we expected, a localized wave described by the corresponding
eigenvector of the Mmatrix, exists. Similarly, localized electronic states in solids appear
always at discrete energies only. However, in the case of a disordered and unbounded
system a countable set of energies corresponding to localized states becomes dense in
some finite interval. It is a signature that the Anderson localization occurs. However it
is always difficult to distinguish between the allowed energies which may be arbitrarily
close to each other. Therefore, physically speaking, an entire continuous band of spatially
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OR LOWSKI, RUSEK
Figure 1. Contour plot of the probability distribution Pφ(Λ) calculated for 103systems of
N= 300 three dimensional dipoles distributed randomly in a sphere with the density n=1
dipole per wavelength cubed.
Figure 2. Contour plot of the probability distribution Pφ(Λ) calculated for 103systems of
N= 300 two dimensional dipoles distributed randomly in a square with the density n=1dipole
per wavelength squared.
892

OR LOWSKI, RUSEK
localized electronic states exists. In 3D the total scattering cross section of individual
particles σmust exceed some critical value σcr =σ(φcr) before localization will take
place in the limit N→∞. This fact is in perfect agreement with the scaling theory of
localization [19]. Moreover, our preliminary calculations indicate that in three dimensions
the value of k2σcr may decrease with nbut slower than n−2. Within our approach internal
resonances of scatterers can be modeled by |φ|'π. Our calculations do not exclude the
possibility that in infinite three dimensional medium the band of localized waves may
appear in this region of φ. However, in all experiments we can investigate only systems
confined to certain finite regions of space. And, as follows from Fig. 1, in 3D the band
of localized waves appears faster with increasing size of the system when φcr ≤|φ|π,
i.e., when the frequency is not tuned to the internal resonances of individual scatterers.
Let us emphasize that this result is specific for 3D random media. In both one and
two dimensions, macro- and microscopic resonances appear at the same frequencies. To
illustrate this fact, in Fig. 2 we have prepared the plot analogous to Fig. 1 but calculated
for 103configurations of N= 300 dielectric cylinders modeled by 2D dipoles [18]. In this
case, the band of localized waves does appear faster for |φ|'π; in 2D the parameters of
the single scatterers that give the internal and global resonances coincide and matching
the internal resonances helps to establish localization. In our opinion this could be the
main reason that the convincing experimental demonstration of strong localization of
microwave radiation has first been given for two dimensions [10], although also more
practical reasons as, e.g., polydispersity of the actual 3D samples (leading to phase shifts)
play an important role. Results obtained from our model for 2D media (which seem to
agree with experimental results) prove that the surprising features of localization we
observed for 3D random media are not the artifacts produced by the model.
6. Summary
We have developed a quite realistic coupled-dipole model describing multiple scatter-
ing of electromagnetic waves by a disordered dielectric medium. Its relative simplicity
allowed us to discover some new features of the Anderson localization of electromagnetic
waves in 3D dielectric media without using any averaging procedures. Within our theoret-
ical approach one can easily see how localization “sets in” for increasing size of the system.
Very striking universal properties of random matrices describing the scattering from a col-
lection of randomly distributed point-like scatterers have been observed. Self-averaging
of the lowest eigenvalue emerging in the limit of infinite medium has been discovered
numerically and the appearance of the band of localized electromagnetic waves in 3D was
demonstrated. Possible relationships between this phenomenon and the dramatic inhi-
bition of the propagation of electromagnetic waves in spatially random dielectric media
are currently under investigation. It can be understood as a counterpart of Anderson
transition in solid state physics.
Acknowledgments
We are grateful to Edward Kapu´scik and Andrzej Horzela for their kind hospitality
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OR LOWSKI, RUSEK
extended to us in Krak´ow. This work was supported by the Polish Committee for Scien-
tific Research (KBN) under Grant No. 2 P03B 108 12. We thank the Interdisciplinary
Center for Mathematical and Computational Modeling (ICM) of Warsaw University for
providing us with their computer resources.
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