Random Resonators and Prelocalized Modes in Disordered Dielectric Films.
V. M. Apalkov and M. E. Raikh
Areal density of disorder-induced resonators with a high quality factor, $Q\gg 1$, in a film with fluctuating refraction index is calculated theoretically. We demonstrate that for a given $kl>1$, where $k$ is the light wave vector, and $l$ is the transport mean free path, when {\em on average} the light propagation is diffusive, the likelihood for finding a random resonator increases dramatically with increasing the correlation radius of the disorder. Parameters of {\em most probable} resonators as functions of $Q$ and $kl$ are found.
Random1.tex
\documentstyle[aps,epsf]{revtex}
%\documentstyle[aps,twocolumn]{revtex}



%\documentstyle[preprint,aps]{revtex}
\tighten
\begin{document}
\draft



\title{Random Resonators and Prelocalized Modes in Disordered 
Dielectric Films.}

%\title{Statistics of Random Resonators in Disordered Dielectric 
%Films}
\author{V. M. Apalkov and M. E. Raikh}
\address{Department of Physics, University of Utah, Salt Lake City,
UT 84112, USA}
\author{B. Shapiro}
\address{Department of Physics, Technion-Israel Institute of
Technology, Haifa 32000, Israel}
\maketitle

\begin{abstract}
Areal density of disorder-induced resonators with a high quality factor,
$Q\gg 1$, in a film with fluctuating refraction index is calculated
theoretically. We demonstrate that for a given $kl>1$, where $k$ is
the light wave vector, and $l$ is the transport mean free path, when
 {\em on average} the light propagation is diffusive, the likelihood 
for finding a random resonator increases dramatically with 
increasing the
correlation radius of the disorder. Parameters of {\em most probable}
resonators as functions of $Q$ and $kl$ are found. 
\end{abstract}
\pacs{PACS numbers: 42.25.Dd, 42.60.Da, 73.20.Fz}





{\em Introduction}. Recent discovery of the {\em coherent} lasing from
various disordered materials adds a new dimension to the conventional
physics of light propagation in multiply scattering media.  In the
experimental works\cite{cao99,cao00,Fro99,Fro99a} it was demonstrated
that, above a certain excitation level, the emission spectra of {\em
ZnO} powders\cite{cao99,cao00}, conjugated polymer films\cite{Fro99}
and dye-infiltrated opals\cite{Fro99a} exhibit a sequence of extremely
narrow peaks, their widths being limited by the spectrometer
resolution.  Until recently, this finite resolution left some room for
doubt as to whether the observed peaks indicated true
lasing\cite{Wiersma2000}. However, the latest experiments on photon
statistics\cite{cao01,Polson01} have unambiguously established the
coherence of the emitted light, thus proving conclusively that the
underlying mechanism of random lasing in
Refs.~\onlinecite{cao99,cao00,Fro99,Fro99a}
%requires 
involves the {\em amplitude} (coherent) 
rather than {\em power} (incoherent) feedback. (The latter is known
to occur via the diffusion process of the light intensity,
as proposed long ago by
Letokhov\cite{letokhov} and observed first by Lawandy\cite{Law94} and
later in other numerous experiments.)
The origin of coherent feedback, responsible for the lasing observed
in Refs. \onlinecite{cao99,cao00,Fro99,Fro99a}, is a subject of 
controversy and debate.
Such a feedback would naturally emerge if the light
were localized. However, the coherent backscattering measurements
\cite{cao99,cao99'',cao99',soest,Polson'01,thesis} 
carried out in parallel with the analysis of the emission spectra  
rule out this possibility. Firstly, the values of $kl$ extracted
from these measurements turn out rather large. Secondly, the onset of
the Anderson localization manifests itself in the rounding of the top
of backscattering cone  
\cite{Schmuurmans99,Berkovits87,edrei90}. No such rounding was observed
in the experiments\cite{cao99,cao99'',cao99',soest,Polson'01,thesis}.



In the early work of Cao {\em et. al.}\cite{cao99,cao99'',cao99'}, 
it was argued that, even in diffusive regime of light propagation,
a photon, scattered off a certain sequence of impurities
(grains of, roughly, $100nm$ size) can move on a closed loop, which
thereby serves as a laser resonator. The picture of such ring cavities
put forward in Refs.\onlinecite{cao99,cao99'',cao99'}
was based on  the experimental evidence\cite{wiersma1}
that the {\em recurrent} scattering processes contribute to
backscattering albedo.
% and get coherently amplified. 
%upon the notion that {\em recurrent scattering} observed in
%Ref. \onlinecite{wiersma1} implies the presence of loops, and argued
%that these loops emulate random cavities.
However, since in each scattering act most of
the energy gets scattered out of the loop, an unrealistically
high gain would be required to achieve the lasing threshold condition
for such a loop. This point was particularly emphasized 
by van Soest\cite{thesis}, who also argued that 
%such 
"impurity loops" are likely to generate
a broad frequency spectrum rather than isolated resonances.
%, instead of isolated resonant peaks.







Certainly the picture of random cavities representing a certain
spatial arrangement of {\em isolated} scatterers is too naive.  This,
however, does not rule out the entire concept of disorder-induced
resonators. Although  sparse, the disorder configurations 
that trap the light
for long enough time can occur in a sample of a large enough size, and
a {\em single} such configuration is already sufficient for lasing to
occur.  Therefore, under the condition $kl \gg 1$, which implies that
{\em overall} scattering is weak, the conclusion about the relevance of
random cavities can be drawn only upon the {\em quantitative}
calculation of their likelihood.
This is the subject of the present paper.

We start with a remark  that the issue of random
resonators for light waves has its counterpart in the electronic 
transport. In particular, the modes with anomalously low losses 
are analogous to the so called {\em prelocalized} electronic states
in diffusive conductors that are responsible for the long-time
asymptotics of the current relaxation.
 Theoretical study of these states  was  launched more 
than a decade ago  (see Ref. \onlinecite{Kravtsov}) and 
later renewed in Ref. \onlinecite{Muzykantskii95}. 
The results obtained to date are summarized in the 
review\cite{mirlin00}. Following this analogy we dub the modes
of  
%high-$Q$ 
random resonators with anomalously high quality factor, $Q$, as 
{\em prelocalized} modes. 



The principal outcome of our study is that for a given $kl \gg 1$, the
probability of formation of a high-$Q$ random resonator depends
crucially on the {\em size} of the scatterers, or, more precisely, on
the correlation radius of the disorder, $R_c $.  
Similarly to the treatment in  
Refs.~\onlinecite{cao00,cao99'} we restrict our consideration to
the two-dimensional case (a disordered film).
Regarding the geometry of a random resonator, we adopt 
the idea  proposed by Karpov in
Ref.~\onlinecite{karpov93} for trapping the  acoustic waves
in three dimensions. According to Ref.~\onlinecite{karpov93}, the 
fluctuations responsible for trapping are the toroidal inclusions
with reduced  sound velocity. 
Correspondingly, in two dimensions, a  random resonator represents 
a ring-shape area  (see Fig.~1a) 
within which the  dielectric constant 
is  enhanced  by some small value $\epsilon _1$ 
(compared to the background value $\epsilon _0$). Then
such a ring can be viewed as a waveguide that supports the modes of
a whispering-gallery type. Due to the azimuthal symmetry, these modes
are characterized by the radial index and the angular momentum,
$m$. Denote by ${\cal N}_m(kl,Q)$ the areal density of resonators
with quality factor $Q$ in the film with a transport mean free path
$l$.  Here $k= \epsilon_0^{1/2} k_0$, where $k_0 = 2\pi /\lambda $,
$\lambda $ stands for the wavelength in vacuum. 
Obviously, in the diffusive
regime, $kl > 1$, the density ${\cal N}_m(kl,Q)$ is exponentially 
small for $Q \gg 1$. In this domain ${\cal N}_m(kl,Q)$ can be
 presented as
\begin{equation}
{\cal N}_m(kl,Q) = {\cal N} _0 e^{-S_m(kl, Q)} ~,
\end{equation} 
 where ${\cal N} _0$ is the prefactor. Now we can ``quantify''
the above statement about the dependence of the resonator
likelihood on $R_c$  
\begin{equation}
\frac{ S_m(k_0R_c > 1)}{ S_m(k_0R_c \ll  1)} =
\frac{ \Phi(\epsilon _1^{1/2} k_0 R_c)}
{\pi ^{1/2} (\epsilon _0 ^{1/2} k_0 R_c)^3}~.   \label{eq1}
\end{equation}
The dimensionless single-parameter function $\Phi $ is 
shown in Fig.~2. It is seen from Eq.~(\ref{eq1}) 
that $S_m$ falls off rapidly with increasing $R_c$.
In the domain $k_0 R_c > 1 $ but $\epsilon _1 ^{1/2} k_0 R_c 
\lesssim 1$ we can set $\Phi \approx 1$, so that we have  
 $S_m \propto (k_0 R_c)^{-3}$. For larger $R_c$, when  
$\epsilon _1 ^{1/2} k_0 R_c \gg 1 $, the function $\Phi $ 
behaves as $\Phi (u) \propto u $. In this domain 
$S_m$ decreases slower with $R_c $: $S_m \propto (k_0 R_c)^{-2}$.
We emphasize again that Eq.~(\ref{eq1}) applies {\em for a given}
$kl$ value, so that the decrease of $S_m$ with $R_c$ {\em leaves 
the backscattering cone unchanged}. Below we will demonstrate 
with rigorous calculations  that the value of 
$S_m(k_0R_c \ll 1)$ is really large
\begin{equation}
S_m (k_0 R_c \ll 1)  
   =  2\left( \frac{\pi ^3 }{3 } \right)^{1/2} kl  |\ln Q |
              ~, \label{eq2}
\end{equation}
which makes 
formation of random resonators with 
$Q\gg 1$ practically impossible for point-like scatterers.   
On the contrary, for $k_0 R_c \gtrsim 2 $ ring-like fluctuations 
of the dielectric constant (see Fig.~1b) make  realistic 
candidates for high-$Q$ random resonators. 

{\em Derivation of Eq.~(\ref{eq1})}. Denote by $P$ the probability
  of fluctuation of dielectric constant, $\delta \epsilon (\bbox{r})$.
Assuming the fluctuations to be Gaussian with r.m.s. $\Delta $, we have 
\begin{equation}
\ln P =
    - \frac{1}{2 \Delta ^2} \int \! \! \! \int d \bbox{r} 
 d\bbox{r}^{\prime }
    \delta \epsilon (\bbox{r}) \delta \epsilon (\bbox{r}^{\prime }) 
 \kappa (\bbox{r}- \bbox{r}^{\prime })  \label{eq3},
\end{equation}
where the kernel $\kappa(\bbox{r})$ is related in the usual way, 
$\int d\bbox{r}^{\prime } \kappa (\bbox{r}- \bbox{r}^{\prime })
 K (\bbox{r} ^{\prime } - \bbox{r}_1) = \delta (\bbox{r}- \bbox{r}_1)$,
to the correlation function, $K(\bbox{r})$, defined as    
\begin{equation}
\langle \delta \epsilon (\bbox{r}) \delta \epsilon 
(\bbox{r} ^{\prime }) \rangle = \Delta ^2
              K (\bbox{r}- \bbox{r}^{\prime }) ~. \label{eq4}
\end{equation}
We search for fluctuations of a type shown in Fig.~1, {\em i.e.},
$\delta \epsilon $ is  azimuthally symmetric (depends only on 
the radius, $\rho $) and is non-zero 
within the relatively narrow ring of width $w \ll \rho _0$, where
$\rho _0$ is the radius corresponding to the middle of the ring
(see Fig.~1a).  
For such a fluctuation the wave  equation for field distribution 
corresponding to the  angular momentum, $m$, reduces to 
\begin{equation}
\hat{L}\chi _m=
\frac{d^2 \chi_m}{d x ^2} + \delta \epsilon ~ k_0^2 \chi_m =
         \epsilon _1 ~k_0^2 \chi _m ~,
\label{eq5}
\end{equation}
where $x=\rho -\rho _0$. The ``eigenvalue'' in the
r.h.s. of Eq.~(\ref{eq5}) is defined as 
%$\epsilon _1 = (m^2 /(k_0 \rho _0 )^2- \epsilon _0 $. 
$\epsilon _1 = \left(\frac{m}{k_0 \rho_0 } \right)^2- \epsilon _0 $.
It is independent of $x$ by virtue of the condition $\rho _0 \gg w$.
The next step is to find the ``most probable'' distribution $\delta
\epsilon (x)$ for a given $\epsilon _1$. This is done within the 
standard optimal fluctuation approach \cite{Halperin66,Zittartz66},
which prescribes minimization of auxiliary functional $\Psi \{\delta
\epsilon \} = \ln P - \lambda (\chi _m \hat{L} \chi _m)$, where
$\lambda $ is the Lagrange multiplier. This yields
\begin{equation}
\delta \epsilon (x ) = \int dx_1  K_0 (x-x_1) \chi _m^2(x_1 )~,
  \label{eq6}
\end{equation}
where we have set $\lambda = 2\pi \rho _0 /\Delta ^2 k_0^2 $. 
The above expression reminds the standard result of optimal 
fluctuation approach for a correlated random potential 
\cite{Halperin66}. The only difference is that,  
due to the angular integration in Eq.~(\ref{eq3}), 
the kernel in Eq.~({\ref{eq6}) is given by a 
function $K_0 $, which is related to the correlator, 
$K(\bbox{r})$, as  $K_0 (x_1-x_2) = \int_{-\infty }^{\infty } dy 
K (\sqrt{ (x_1-x_2)^2 + y^2})$.
A natural $x$-scale for  the eigenmode $\chi _m$ is $(\epsilon_1 ^{1/2}
k_0)^{-1}$. Thus, we present the outcome of the optimal fluctuation 
approach in terms of a dimensionless variable 
$z=\epsilon _1^{1/2} k_0 x$. 
\begin{equation}
S_m = \frac{\pi \rho _0}{\epsilon _1 k_0^2 \Delta ^2} 
\int \! \! \! \int dz_1 dz_2
\chi_m ^2 (z_1) \chi _m^2 (z_2) K_0 (z_1-z_2)~.
   \label{eqP}
\end{equation}
The dimensionless equation for the function $\chi _m$ 
reads
\begin{equation}
\frac{d^2 \chi _m(z)}{dz^2} +\frac{ \chi _m(z)}{\epsilon _1^{3/2} k_0 }
 \int dz _1 K_0(z-z_1) \chi _m^2(z_1)  =
     \chi _m(z) ~.
  \label{eqChi}
\end{equation}
To proceed further we have to specify the correlator, $K$.  
We have chosen the Gaussian form  
$K(\rho ) = \exp \left( - \rho^2 / R_c^2 \right)$. 
The form of the function $\chi _m = A(R_c) \exp[-\gamma (R_c) z^2]$
allows to cover the entire range of correlation radii, from 
``white noise'' ($R_c \rightarrow 0$) 
to the limit of a smooth disorder ($k_0 R_c \gg 1$). Indeed, for 
large $R_c$ this form becomes exact. In the opposite limit,
$R_c \rightarrow 0$, using the above trial function instead of
exact solution $\chi _m \propto 1/\cosh (z)$ leads to 
the overestimate of $S_m$ by a factor $(\pi /3)^{1/2}\approx 1.023$.
The parameter $A(R_c)$ and $\gamma (R_c)$ of the trial
function can be found analytically. Evaluation of  Eq.~(\ref{eqP}) 
reduces to the Gaussian integration which for a given $m \approx 
\epsilon ^{1/2} k_0 \rho _0$ yields 
\begin{equation}
S_m =2^4 3^{-3/2} \pi ^{1/2} m 
\left(\frac{\epsilon _1^3}{\epsilon _0 }\right)^{1/2}
\frac{\Phi (\epsilon _1^{1/2}k_0R_c )}
{(\Delta  k_0 R_c)^2} ~.
   \label{eq7}
\end{equation}
The analytical expression for  the function $\Phi (u)$ introduced 
above in Eq.~(\ref{eq1}), and shown in Fig.~2, is the following 
\begin{equation}
\Phi(u)  =
\frac{3^{3/2}}{16} 
\left( \frac{5+ \sqrt{9+16 u^2}}{3+ \sqrt{9+16 u^2}}\right)^2
   \left[ u^2 + \frac{1}{2} ( 3+ \sqrt{9+16 u^2}) \right]^{1/2} ~.
  \label{eq8}
\end{equation}
Recall that we are interested in the density of random resonators at a
{\em given } value of $kl$. The remaining task is to express the
transport mean free path in terms of $\Delta $ and $R_c $. With
$K(\rho ) = \exp \left( - \rho^2 / R_c^2 \right)$ the expression
simplifies in two limits
%\begin{subequations}
\begin{mathletters}
\begin{eqnarray}
& & kl =  \frac{4 \epsilon _0 }{\pi (k_0 R_c\Delta )^2}  ~~\mbox{for}
~~k_0R_c \ll 1~;   \label{eq9a} \\
& & kl =  \frac{4\epsilon _0^{5/2} k_0R_c}{\pi ^{1/2} \Delta ^2 }
~~\mbox{for}~~~k_0R_c \gg 1 ~\label{eq9b} ~.
\end{eqnarray}
\end{mathletters}
%\end{subequations}
Note, that Eqs.~(\ref{eq9a}), (\ref{eq9b}) can be cast into the 
conventional form $l\propto 1/n\sigma $, where $n$ is the concentration 
of scatterers and $\sigma $ is the transport cross section. In 
particular, Eq.~(\ref{eq9a}) follows from the two-dimensional 
version ($\sigma \propto R_c^4 k_0^3$) of the Rayleigh scattering 
formula for $\sigma $. 
Combining Eq.~(\ref{eq7}) and Eqs.~(\ref{eq9a}), (\ref{eq9b}), 
we arrive at 
 Eq.~(\ref{eq1}). In the limit $\epsilon _1^{1/2} k_0 R_c \gg 1$ 
Eq.~(\ref{eq7}) has a simple interpretation. Namely, in this
limit $S_m $ can be rewritten as 
$S_m \sim (\epsilon _1 /\Delta )^2 (\rho _0 /R_c)$.  
The first factor comes from 
the Gaussian probability to have a fluctuation $\epsilon _1$ within 
a square of an area $\sim R_c ^2$. The second factor accounts 
for the number of  squares needed to cover the ring area 
$A_m \sim \rho _0 R_c$.

{\em $Q$-factor}. It is obvious that at large distances, 
$\rho \gg \rho _0$, the behavior of $\chi _m$ is oscillatory,
$\chi_m \propto \exp (i \epsilon _0^{1/2} k_0 \rho )$, manifesting
that the waveguided mode of a ring, being prelocalized, 
has a finite lifetime.
In other words, the eigenvalue, $\epsilon _1$, in Eq.~(\ref{eq5})
has an imaginary part, $\tilde{\epsilon } _1$, due to 
evanescent leakage. The quality factor is inversely proportional 
to $\tilde{\epsilon } _1$, namely, $Q= \epsilon_0 / 
\tilde{\epsilon } _1$. The leading contribution to
$\tilde{\epsilon } _1$ comes from the region of a 
width, $d$, adjacent to the waveguide (see Fig.~1a). To 
find $d$ we have to take into account that the r.h.s in 
Eq.~(\ref{eq5}) does in fact depend on $x$. This is because the 
precise form of the r.h.s is not $\left( \frac{m}{\rho _0}\right)^2 -
\epsilon _0 k_0 ^2$, but rather $\left( \frac{m}{\rho _0+x}\right)^2 -
\epsilon _0 k_0 ^2$. In the region outside the  waveguide, when
$x \gg 1/\epsilon _1^{1/2}k_0$ (but still $x \ll \rho _0$)
Eq.~(\ref{eq5}) takes the form
\begin{equation}
\frac{d^2\chi_m}{dx^2} = 
\epsilon _1 k_0^2 \left( 1- \frac{x}{d} \right) \chi_m
  \label{eq10}~,
\end{equation}
where the width of the decay region is given by
$d = \epsilon _1 k_0^2 \rho_0^3 /2m^2$.
Equation (\ref{eq10}) is of Airy-type. Semiclassical calculations 
with exponential accuracy yields for the rate of evanescent leakage 
$\tilde{\epsilon}_1 \propto \exp [- (2m/3) 
( \epsilon _1 /\epsilon _0 )^{3/2} ]$, and hence 
\begin{equation}
\ln Q     =
   \frac{2m}{3} \left( \frac{\epsilon _1}{\epsilon _0}
          \right)^{3/2} \label{tt} ~~.
\end{equation}
Substituting $\epsilon _1$ from Eq.~(\ref{tt}) and 
$\Delta $ from Eq.~(\ref{eq9a}) into Eq.~(\ref{eq7}) we arrive at 
Eq.~(\ref{eq2}). The above derivation relied on the assumptions 
$\rho _0 \gg d $ and $d\gg w$. These 
assumptions are justified within the following domain of  the 
quality factors $m \gg \ln Q \gg \max \left\{ 1,~ 
\epsilon _0^{3/4} (k_0 R_c )^{3/2} m^{-1/2} \right\}$. 

{\em Discussion.} Equation (\ref{eq2}) quantifies the effectiveness of 
trapping of light in a random medium with point-like scatterers.
It follows from Eq.~(\ref{eq2}) that the likelihood of high-$Q$ 
cavity  is really small. Indeed, even for rather strong 
disorder,  
$kl = 5$,  the exponent, $S_m$,  in the probability of
having a cavity with a quality factor $Q=50$ is close to $S_m = 120$.
We emphasize that in two dimensional case under consideration, 
this exponent does not depend on $m$ and, thus, on the cavity
radius $\rho _0 = m/\epsilon _0^{1/2} k_0$. To estimate 
the degree to which finite size of scatterers ($\sim R_c$) 
improves the situation we choose $k_0 R_c \approx 2$, which 
already corresponds to the limit 
$k_0 R_c \gg 1$ in Eq.~(\ref{eq9b}), 
but still allows to set $\Phi = 1$. Then for $Q=50$, 
$kl = 5$ we obtain 
$S_m \approx  1.1$, suggesting that the resonators 
with this $Q$ are quite frequent. In the latter estimate 
we have set $\epsilon _0 =4$.

A natural question to address is how large a value of 
 $Q$ can be achieved for a given $kl$ and 
$k_0 R_c$. To address this question we inspect the  argument 
$u = \epsilon _1^{1/2} k_0 R_c $ of the  function $\Phi $. 
With the use of Eq.~(\ref{tt}) it can be presented in the form  
$ u = \epsilon _0^{1/2} k_0 R_c 
\left[ \frac{3\ln Q }{2m} \right]^{1/3}$. Since $\Phi (u) $ 
increases monotonically (see Fig.~2), the latter expression 
suggests that $Q$ can be increased at the expense of larger 
$m$-values. In the example considered above, in order to 
keep $\Phi $ smaller than, say $1.5$, $m$ should be bigger than 
$50$. However, due to slow dependence $u\propto m^{-1/3}$, 
we get rather small value $S_m\approx 3$ for $m$ as small as $m=15$. 
Certainly, the allowed values of $m$ are limited from 
above. This limitation originates from ``vulnerability'' of
waveguiding to the fluctuations of the dielectric constant 
around optimal ring-like distribution. The dangerous fluctuations are
those that enhance the evanescent leakage. Note, that these 
fluctuations do not affect the main exponent $S_m$ in the 
density of resonators. It is obvious that the bigger is the 
area $A_m = 2 \pi \rho _0 d\propto m^{4/3}$, responsible for  
evanescent leakage (see Fig.~1a), the harder it is to ``protect'' 
the waveguiding. Since the fluctuations $\delta \epsilon (\bbox{r})$ 
have the spatial scale $R_c$, the probability that the waveguiding 
``survives'' can be roughly estimated as $\exp (-A_m / R_c^2)$. 
The condition that the exponent $ A_m / R_c^2$ does not 
dominate over the principal exponent $S_m$ can be rewritten 
as $m\lesssim  (\epsilon _0 kl )^{3/5} (\ln Q)^{2/5}$. For the example 
$kl = 5$ and $Q = 50$, addressed above, we get $m\lesssim 10 $. 
Rigorous calculation of the ``survival probability'' is the problem 
of the same complexity as calculation of the prefactor in 
the functional integral \cite{Brezin80}.


  

{\em Conclusion}. 
In the present paper we  provided a quantitative
theory of random resonators that 
 substantiates the intuitive image 
\cite{cao99,cao99''}
of a resonant  cavity as a closed-loop
trajectory of a light wave bouncing between the point-like
scatterers. The  intuitive picture in 
\cite{cao99,cao99''} assumed that light can
propagate along a  loop of scatterers by simply being
scattered from one scatterer to another. Such a picture, however, is
unrealistic due to the scattering out of the loop \cite{thesis}. 
 We have 
demonstrated that the scenario of light traveling along 
closed loops can be remedied. In our picture the "loops", 
i.e. the random resonators, can be envisaged as rings with 
dielectric constant larger than the average value.
On a microscopic level these  resonators
correspond to certain arrangements of scatterers (grains). The main
point, however, is that a resonator acts as a {\em single entity}: 
only
 the coherent multiple scattering of light by {\em all} the scatterers
in the resonator can provide trapping. We also point out that 
correlations in the fluctuating part of the dielectric constant
(due to finite grain size) highly facilitate trapping.



%In the present paper we  provided a quantitative
%theory of random resonators that 
%%%qunatifies 
%substantiates the intuitive image 
%\cite{cao99,cao99''}
%of a resonant  cavity as a closed-loop
%trajectory of a light wave bouncing between the point-like
%scatterers. The drawback of this intuitive picture is the 
%implicit assumption that the scatterers are located far apart
%from each other. As a result, the wave is scattered by each 
%scatterer 
%{\em independently}. Then  the critique of Ref.\cite{thesis}, 
%that, due to the scattering out of the loop, the  light trapping
%is unlikely, applies. As we have 
%demonstrated, the scenario of closed loops can be remedied, 
%if the scatterers constituting a loop are placed close to each other
%(quantitatively, if $k_0R_c \gtrsim 2$). Such a sparce fluctuation,
% $\delta \epsilon (\bbox{r})$, interacts with light 
%{\em as a single entity}, then the possibility of trapping emerges.
%The effectiveness of light trapping is expressed by
%Eq.~(\ref{eq1}).  

The effectiveness of light trapping is expressed by
Eq.~(\ref{eq1}). This expression describes the statistics of 
the quality factors which determines the distribution 
 of the threshold gain for random
lasing. Our consideration pertains to the {\em passive} disordered
films, in the sense, that we neglect the effect of gain on the spatial
distribution of the light intensity
\cite{cao00,cao99',Jiang00,Jiang01}.  Random resonators considered in
the present paper are sparse, so that there is no spatial overlap
between the modes of different resonators, the situation opposite to
that considered in Refs.  \cite{beenakker96,beenakker98}.  We have
also treated scatterers as frequency-independent fluctuations, $\delta
\epsilon (\bbox{r})$, of the dielectric constant. The entirely
different scenario of the collective mode  
formation emerges for {\em resonant}
scatterers \cite{cooperative}.
%The fact that the resonators have sizes much bigger than the 
%wave length indicates that they can support many modes 
%(corresponding to
%different angular momenta, $m$), {\em nearly equidistant in frequency}
%\cite{Polson'01}.



{\em Acknowledgments}. Two of the authors MER and BS
acknowledge the hospitality of the University of K\"{o}ln and of
the Institute for Theoretical Physics at UCSB where the parts of
this work were completed.
This research was supported in part by the National Science
Foundation under Grant No. PHY99-07949.

 
\begin{references}





\bibitem{cao99} H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang,
and R. P. H. Chang,  Phys. Rev. Lett. {\bf 82}, 2278 (1999), 


%\bibitem{cao99} H. Cao {\em et. al.}, Phys. Rev. Lett. {\bf 82}, 2278 (%1999).




\bibitem{cao00} H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho,  
E. W. Seelig,   X. Liu, and R. P. H. Chang, Phys. Rev. Lett. {\bf 84}, 
5584 (2000).

%\bibitem{cao00} H. Cao {\em et. al.}, Phys. Rev. Lett. {\bf 84}, 5584 (%2000).



\bibitem{Fro99} S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov,  
and R. H. Baughman,  Phys. Rev. B {\bf 59}, R5284 (1999).

\bibitem{Fro99a}S. V. Frolov, Z. V. Vardeny, A. A. Zakhidov, and
R. H. Baughman, Opt. Commun. {\bf 162}, 241 (1999).

\bibitem{Wiersma2000}D. Wiersma, Nature {\bf 406}, 132 (2000).
%The smallest random laser
%Cao's experiment is described

\bibitem{cao01} H. Cao, Y. Ling, J. Y. Xu, C. Q. Cao, and P. Kumar,
 Phys. Rev. Lett. {\bf 86}, 4524 (2001).

\bibitem{Polson01}R. C. Polson, A. Chipouline, and Z. V. Vardeny,
Adv. Mater. {\bf 13}, 760 (2001).








\bibitem{letokhov}V. S. Letokhov, Zh. \'{E}ksp. Teor. Fiz. {\bf 53},
1442 (1967) [Sov. Phys. JETP  {\bf 26}, 835 (1968)].

\bibitem{Law94} N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and
E. Sauvain,  Nature (London) {\bf 368}, 436 (1994).
%Isotropic laserlike emission from an optically pumped solution
%of Rhodamine 640 perchlorate with TiO_2 scatterers dispersed
%in a methanol solution

\bibitem{cao99''}H. Cao, Y. G. Zhao, H. C. Ong, and R. P. H. Chang,
Phys. Rev. B {\bf 59}, 15107 (1999).
%Far-field characteristics of random lasers
%highly disordered polycrystalline ZnO films with $l=2.6\lambda$


\bibitem{cao99'}H. Cao, J. Y. Xu, S.-H. Chang, and S. T. Ho,
Phys. Rev. E {\bf 61}, 1985 (2000).
%Transition from amplified spontaneous emission to laser action in
%strongly scattering media
%Not ZnO powder, but ZnO prticles in dye solution +CBS


\bibitem{soest} G. van Soest, F. J. Poelwijk, R. Sprik, and 
A. Lagendijk, Phys. Rev. Lett. {\bf 86}, 1522 (2001).
%Dynamics of a Random Laser above Threshold

\bibitem{Polson'01}R. C. Polson, M. E. Raikh, and Z. V. Vardeny,
 ArXiv: cond-mat 0105360.

\bibitem{thesis}G. van Soest, Ph.D. thesis, Universiteit van Amsterdam,
The Netherlands, 2001 (unpublished).



\bibitem{Schmuurmans99}F. J. P. Schuurmans, M. Megens. 
D.~Vanmaekelbergh, and A. Lagendijk, Phys. Rev. Lett. {\bf 83}, 2183
(1999). 
%Light Scattering near the Localization Transition in Macroporous GaP 
%Networks


\bibitem{Berkovits87} R. Berkovits and M. Kaveh, Phys. Rev. B {\bf 36},
9322 (1987).
%Backscattering of Light near the Optical Anderson Transition}

\bibitem{edrei90} I. Edrei and M. J. Stephen, Phys. Rev. B {\bf 42},
110 (1990).
%Optical coherent backscattering and transmission in diosrdered
%medium near the mobility edge.




\bibitem{wiersma1}D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen,
and A. Lagendijk, Phys. Rev. Lett. {\bf 75}, 1739 (1995).
%Experimental evidence for recurrent multiple scattering of light
%in disordered media

\bibitem{Kravtsov}B. L. Altshuler, V. E. Kravtsov, I. V. Lerner,
in {\em Mesoscopic Phenomena in Solids}, eds. B. L. Altshuler,
P. A. Lee, and R.~A.~ Webb (North Holland, Amsterdam, 1991).
 
\bibitem{Muzykantskii95} B. A. Muzykantskii and D. E. Khmelnitskii,
Phys. Rev. B {\bf 51},  5480 (1995).

%\bibitem{falko95} V. I. Fal'ko, and K. B. Efetov, Phys. Rev. B
%{\bf 50}, R11267 (1994); {\em ibid.} {\bf 52}, 17413 (1995);
%J. Math. Phys. {\bf 37}, 4935 (1996).

\bibitem{mirlin00} A. D. Mirlin, Phys. Rep. {\bf 326}, 259 (2000).




\bibitem{karpov93} V. G. Karpov, Phys. Rev. B {\bf 48}, 12539 (1993).


\bibitem{Halperin66}B. I. Halperin and M. Lax, Phys. Rev. {\bf 148},
722 (1966).

\bibitem{Zittartz66} J. Zittartz and J. S. Langer, Phys. Rev.
{\bf 148}, 741 (1966).

\bibitem{Brezin80}  E. \'{B}rezin and   G. Parisi,
 J. Phys. C {\bf 13} L307  (1980).



\bibitem{Jiang00}X. Jiang and C. M. Soukoulis,  Phys. Rev. Lett. {\bf 85}, 70 (2000).
\bibitem{Jiang01}X. Jiang and C. M. Soukoulis, ArXiv cond-mat/0103042.





\bibitem{beenakker96} C. W. J. Beenakker, J. C. J. Paasschens, 
and P. W. Brouwer, Phys. Rev. Lett. {\bf 76}, 1368  (1996).

\bibitem{beenakker98} T. Sh. Misirpashaev and C. W. J. Beenakker,
 Phys. Rev. A {\bf 57}, 2041 (1998).

\bibitem{cooperative} T. V. Shahbazyan, M. E. Raikh, and Z. V. Vardeny 
 Phys. Rev. B {\bf 61}, 13266 (2000).





%\bibitem{Narimanov2000}see O. A. Starykh, Ph. Jacquod, E. E. Narimanov,
%and A. D. Stone, Phys. Rev. E {\bf 62}, 2078 (2000), and references
%therein.


%\bibitem{Wiersma97}D. S. Wiersma, P. Bartolini, A. Lagendijk,
%and R. Righini, Nature {\bf 390}, 671 (1997).
%%Localization of light in a disordered medium


%\bibitem{Chabanov00}A. A. Chabanov, M. Stoytchev, and A. Z. Genack,
%Nature {\bf 40}, 850 (2000).
%Statistical signatures of photon localization


\end{references}


\begin{figure}
\centerline{
%\epsfxsize=3.2in
\epsfxsize=4.8in
\epsfbox{Fig_1_random.eps}}
\vspace*{0.1in}
\protect\caption[sample]
{\sloppy{(a) The structure of a two-dimensional
resonator is illustrated schematically; only a half of the
ring-shaped waveguide (blank region) is shown. 
(b) Optimal fluctuation of the dielectric constant, $\delta \epsilon 
(\rho )$ (solid line), and the corresponding field distribution 
(dotted line) are shown. Dashed line outside the shaded region 
of a width, $d$, illustrates the evanescent leakage.  
}}
\label{figone}
\end{figure}

\begin{figure}
\centerline{
%\epsfxsize=3.2in
\epsfxsize=4.8in
\epsfbox{Fig_2_random.eps}}
\vspace*{0.1in}
\protect\caption[sample1]
{\sloppy{Dimensionless function $\Phi(u)$ defined in Eq.~(\ref{eq8})
is plotted. Inset: Normalized modulus of the log-density of random
resonators, $\tilde {S}_m = S_m(k_0 R_c)/S_m(0)$,
calculated from Eq.~(\ref{eq1}) for  $\epsilon_0 =4$, $Q=50$, 
and $m=15$ is plotted versus the dimensionless 
correlation radius, $k_0 R_c$. 
}}
\label{figtwo}
\end{figure}


\end{document} 




































The challenge for the
theory is to identify these resonators and to estimate  their number
in a sample of a given size and degree of disorder.
In particular, the quantity to be calculated theoretically is
density of resontors with a {\em given quality factor}, $Q$ (clearly,
this density falls off repidly with $Q$). In the absense of
absorption $Q$ is equal to $\omega t$,
where $t$ is the time during
which the wave of frequency $\omega$ is trapped inside the resonator before
it escapes. Thus, the statistical description of the random resonations 
requires the knowledge of the distribution of the
trapping times, $t$.

 

  The issue of trapped waves, with anomalously long trapping
times, has already been addressed in the context of electronic
conduction in dirty metals [references]. In that case the
objects of study are
delocalized wave functions that take anomalously large values
within some sparse spatial regions  (the "prelocalized states").
The value of the wave function, $|\psi|$, in such a region
determines the trapping time for an electron initially localized
in that region, via the relation $t\propto \vert\psi\vert^2$. The bulk of the work
on disordered metals employed the machinery of the non-linear
sigma-model. Within this approach, averaging over the disorder
is performed at an early stage of the calculation, so that the
tails of the distribution for $|\psi|^2$ emerge without specifying
the actual disorder realizations which are responsible for
the anomalously long trapping times. An attempt to pinpoint
these realizations was made in Ref.x, for the case of a
white-noise random potential. The authors of ref.x, by means of
the instanton approach, 
 obtained, in two dimensions, a trapping
configuration of a shape of a deep circular potential well,
supplemented by a weak periodic modulation at large distances.
In optical terms, this kind of a trapping configuratation would
imply a spatial region with {\em negative} dielectric constant. Indeed,
the scalar wave equation is
\begin{equation}
\label{wave}
\Delta_{\bbox\rho} \Psi + \left[\varepsilon _0 + \varepsilon_1({\bbox\rho})\right]k_0^2\Psi=0,
\end{equation}
where $\varepsilon_0$ and $k_0$ are the dielectric constant and
the wave vector in the averaged medium, and $\varepsilon_1({\bbox\rho})$
is the fluctuating part of the dielectric constant. 
Since $\varepsilon_0$ can be identified with energy in the Schr\"{o}dinger equation, 
a prelocalized state, of the type considered in Ref.x, would
require a negative $\varepsilon_1({\bbox\rho})$, larger in absolute value
than $\varepsilon_0$ - a situation that can hardly occur in dielectric
films.




%{\em Introduction}. Recent discovery of the {\em coherent} lasing from
%various disordered materials adds a new dimension to the conventional
%theory of light propagation in multiply scattering media.  In the
%experimental works\cite{cao99,cao00,Fro99,Fro99a} it was demonstrated
%that, above a certain excitation level, the emission spectra of {\em
%ZnO} powders\cite{cao99,cao00}, conjugated polymer films\cite{Fro99}
%and dye-infiltrated opals\cite{Fro99a} exhibit a sequence of extremely
%narrow peaks, their widths being limited by the spectrometer
%resolution.  Until recently, this finite resolution left opened the
%possibility for the argument that the peaks observed do not indicate
%the true lasing\cite{Wiersma2000}. However, the latest experiments on
%photon statistics\cite{cao01,Polson01} have unambiguously established
%the coherence of the emitted light, thus proving conclusively that the
%underlying mechanism of random lasing in
%Refs. \onlinecite{cao99,cao00,Fro99,Fro99a} is {\em not} a
%conventional diffusive feedback 
%which was 
%proposed by Letokhov\cite{letokhov} and observed first 
%by Lawandy\cite{Law94} and
%then in numerous experimental papers.

%Generation of coherent light requires the {\em amplitude} (coherent) 
%rather than {\em power} (incoherent) feedback. 
%The views expressed in the literature on the origin of the amplitude
%feedback are rather controversial.  
%mute and 
%Such a coherent feedback would naturally emerge if the light was
%localized. However the coherent backscattering 
%measurements\cite{cao99,cao99'',cao99',soest,Polson'01,thesis}
% carried
%out in parallel with spectral 
% measuremets essentially rule out this possibility. 

%First Lagendijk\cite{Wiersma96}
%has mentioned in passing in the theoretical paper on {\em incoherent}
%lasing that {\em random cavities} are possible in disordered
%medium. 

%In early papers\cite{cao99,cao99',cao99''} the picture of the
%coherent feedback was based on the experimental observation of 
%{\em recurrent} scattering processes


%and also in her studies of
%{\em ZnO} nanoparticles in dye solutions\cite{cao99'} has built up
%upon the notion that {\em recurrent scattering} observed in
%Ref. \onlinecite{wiersma1} implies the presence of loops, and argued
%that these loops emulate random cavities. This statement was contested
%by van Soest\cite{thesis}. He argued that a loop-like
%multiple-scattering paths {\em cannot} be regarded as a resonator due
%to a large loss.  In particular van Soest writes:

%``...with such a large loss the frequency spectrum is so broad
%that one cannot speak of a resonant wavelength for which a
%round trip shifts the phase by $n\times 2\pi$. Of course, a rare
%event with smaller loss can exist, but the major characteristic
%of such random laser is unlikelyhood. The essence is not that
%the loops do not exist, it is that they are a scarce and unimportant
%subclass of all possible events.''


%To disprove Cao's results van Soest made experimented on laser dyes
%with $TiO_2$ colloidal scatterers.
 
%In the next PRL\cite{cao00} Cao has attributed her random lasing 
%to the 
%Anderson localization of  light {\em induced} by the presence of 
%a gain.
%Again van Soest has argued that Cao's samples are well away from
%the Anderson localization threshold.  

%Finally, in her latest PRL\cite{cao01} Cao seem to return to the
%picture of random cavities but instead of random cavity modes
%she uses the term ``quasistates'' by which she means the solutions
%of the Maxwell equations within the limited volume.



%Let us note that prior to Refs. {   } the problem of prelocalized
%states was considered by  Karpov (Refs) for the three-dimensional
%case. Karpov has employed the same instanton  approach as in {Smolyarenko}.
%On the basis of analysis of the instanton equation he has concluded
%that along with spherically symmetric traps, there exists a
%qualitatively different class of  solutions, corresponding to traps
%of thoroidal shape. The trapped modes have high angular momentum,
%and their trapping time is much longer than that of zero-angular-momentum 
%modes in a spherical traps




Let us note that, prior to Refs.(all) ,the problem of prelocalized states has been
considered by Karpov (Refs) for the three-dimensional case. 
Karpov has employed the same instanton approach as in (Smolyarenko).
On the basis of analysis of the instanton equation he has concluded that along
with spherically symmetric traps there exists a qualitatively different class
of solutions, corresponding to traps of thoroidal shape.
The trapped modes have high angular momentum,
and their trapping time is much {\em longer} than that
of zero-angular-momentum modes in spherical traps.

%Using the
%instanton approach, Karpov demonstarted that thoroidal-shaped
%traps are more efficient than the spherically symmetric
%once discussed in Ref.x. This kind of traps are also more
%feasible in optics, since they reqire only relatively
%small variation of the dielectric constant.

%In the present work we implement Karpov's idea in  two
%dimensions, for resonators in a disordered dielectric
%film.  We demonstrate that two-dimensional ring- shaped
%resonators are more abundant than those of a circular
%shape. Moreover...

In the present paper we show that the idea of Karpov
that
instantons with
reduced symmetry yield more efficient traps
holds also in two dimnesions. Namely, we demonstrare that
for disordered dielectric film the {\em ring-shape} resonators
are more abundant than those of circular shape. Moreover,
we develop the theory further by establishing that additional
reduction of symmetry that lifts degeneracy of the instanton
(analog of Jahn-Teller effect) leads to even stronger enhancement
of the density of resonators.
%Another conceptual development in the present paper
%is that the value of the instanton action can be strongly
%modified in course of calculation of the prefactor.
%Let us note that prior to Refs. {   } the problem of prelocalized
%states was considered by  Karpov (Refs) for the three-dimensional
%case. Karpov has employed the same instanton  approach as in {Smolyarenko}.
%On the basis of analysis of the instanton equation he has concluded
%that along with spherically symmetric traps, there exists a
%qualitatively different class of  solutions, corresponding to traps
%of thoroidal shape. The trapped modes have high angular momentum,
%and their trapping time is much longer than that of zero-angular-momentum 
%modes in a spherical traps




Another conceptual development in the paper pertains to the role
of fluctuations around the instanton solution. Usually these
fluctuations determine only the prefactor in the functional
integral, leaving the instanton action intact. However, in the
present problem, that deals with the satistics of {\em trapping times},
rather than the average density of states, these fluctuations
lead to the modfication of the {\em instanton action}.


%Another conceptual development in the paper pertains to the role
%of fluctuations around the instanton solution. Usually these
%fluctuations determine only the prefactor in the functional
%integral, leaving the instanton action intact. However, in the
%present problem, that deals with the satistics of trapping times,
%read to the modfication of the instanton action.
%ather than the average density of states, these fluctuations

{\bf DISCUSSION}

The physical picture of the wave trapping is quite transparent. The main part
of the wave energy is stored in the rotational motion, so that a relatively shallow well
of depth $\varepsilon_1\ll \varepsilon_0$ suffices to trap the wave in the radial direction.
Let us denote the width of the well and of the emerging barrier (see Fig. 1) with $w$ and $d$,
respectively. The height of the barrier, $\varepsilon_1$ is related to $d$ as 
$(\varepsilon_1/\varepsilon_0)\sim (d/\rho_c)$, where 
$\rho_c=m/k_0\sqrt{\varepsilon_0}$. The estimate Eq. (x) for the trapping time is then
$\ln t \sim k_0d\sqrt{\varepsilon_1}\sim m(\varepsilon_1/\varepsilon_0)^{3/2}$. Thus to ensure
the prescribed value of $\ln t$, for a given $m$, the depth of the well
should exceed the barrier height $\varepsilon_1 \sim \varepsilon_0 \left(\ln t/m \right)^{2/3}$.
The width of the well, $w$, is different for the cases of a short-range 
and long-range fluctuations. For short-range fluctuations, 
  $w$, determined
from the trapping condition $\sqrt{\varepsilon_1}k_0 w \sim 1$, is much smaller than the
correlation radius, $R_c$. For long-range fluctuations, we have $w\sim R_c$, with a typical
well trapping many radial modes. Thus the probabilty, ${\cal P}$, for the formation of a 
ring-shape fluctuation
with the magnitude $\varepsilon_1$, radius $\rho_c$, and the width $w$, which is given by
$|\ln {\cal P}| \sim \varepsilon_1^2w\rho_c/\Delta^2R_c^2$, takes the value 
${\cal P}_{short} \sim \exp (-\varepsilon_1^{3/2}\rho_c/\Delta^2k_0R_c^2)$ for short-range
disorder, and ${\cal P}_{long} \sim \exp (-\varepsilon_1^2\rho_c/\Delta^2R_c)$ for 
long-range disorder. From the above estimate of the barrier height it follows that 
\begin{equation}
\label{relation}
\varepsilon_1^{3/2}\rho_c \sim \varepsilon_0 k_0^{-1}\ln t,
\end{equation}
 {\em i.e.} the combination
$\varepsilon_1^{3/2}\rho_c$ is independent of the angular momentum. For short-range disorder 
this immediately
leads us to the result $|\ln {\cal P}_{short}|\sim\varepsilon_0\ln t/(\Delta k_0 R_c)^2$,
which coincides with Eq. (x) upon using the "white-noise" expression for the mean free path 
$l^{-1}=2\pi k_0^3R_c^2\Delta^2/\varepsilon_0$.
For the long-range disorder, the  relation (\ref{relation})
yields for ${\cal P}_{long}$ the expression 
$|\ln {\cal P}_{long}|\sim (\varepsilon_0\ln t)^{4/3}/\Delta^2k_0^{4/3}\rho_c^{1/3}R_c$,
which reproduces the  result Eq. (  ) of the analytical calculation.

{\bf CONCLUSION}

The statistics of the quality factors studied in the present paper
determines the distribution of threshold gain for random lasing.

While possible manifestations of prelocalized states in transport are rather limited
(they determine the long-time decay of the response to the voltage pulse (Khmelnitskii)),
in optics, the same prelocalized states  are crucial for random lasing.


The fact that the resonators have sizes much bigger than the wave length indicates
that they can support many modes (corresponding to different angular momenta, m),
{em nearly equidistant in frequency}.

\end{document}
































cond-mat0111047.extracted_bib
\bibitem{cao99} H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang,
and R. P. H. Chang,  Phys. Rev. Lett. {\bf 82}, 2278 (1999), 


%
\bibitem{cao99} H. Cao {\em et. al.}, Phys. Rev. Lett. {\bf 82}, 2278 (%1999).





\bibitem{cao00} H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho,  
E. W. Seelig,   X. Liu, and R. P. H. Chang, Phys. Rev. Lett. {\bf 84}, 
5584 (2000).

%
\bibitem{cao00} H. Cao {\em et. al.}, Phys. Rev. Lett. {\bf 84}, 5584 (%2000).




\bibitem{Fro99} S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov,  
and R. H. Baughman,  Phys. Rev. B {\bf 59}, R5284 (1999).


\bibitem{Fro99a}S. V. Frolov, Z. V. Vardeny, A. A. Zakhidov, and
R. H. Baughman, Opt. Commun. {\bf 162}, 241 (1999).


\bibitem{Wiersma2000}D. Wiersma, Nature {\bf 406}, 132 (2000).
%The smallest random laser
%Cao's experiment is described


\bibitem{cao01} H. Cao, Y. Ling, J. Y. Xu, C. Q. Cao, and P. Kumar,
 Phys. Rev. Lett. {\bf 86}, 4524 (2001).


\bibitem{Polson01}R. C. Polson, A. Chipouline, and Z. V. Vardeny,
Adv. Mater. {\bf 13}, 760 (2001).









\bibitem{letokhov}V. S. Letokhov, Zh. \'{E}ksp. Teor. Fiz. {\bf 53},
1442 (1967) [Sov. Phys. JETP  {\bf 26}, 835 (1968)].


\bibitem{Law94} N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and
E. Sauvain,  Nature (London) {\bf 368}, 436 (1994).
%Isotropic laserlike emission from an optically pumped solution
%of Rhodamine 640 perchlorate with TiO_2 scatterers dispersed
%in a methanol solution


\bibitem{cao99''}H. Cao, Y. G. Zhao, H. C. Ong, and R. P. H. Chang,
Phys. Rev. B {\bf 59}, 15107 (1999).
%Far-field characteristics of random lasers
%highly disordered polycrystalline ZnO films with $l=2.6\lambda$



\bibitem{cao99'}H. Cao, J. Y. Xu, S.-H. Chang, and S. T. Ho,
Phys. Rev. E {\bf 61}, 1985 (2000).
%Transition from amplified spontaneous emission to laser action in
%strongly scattering media
%Not ZnO powder, but ZnO prticles in dye solution +CBS



\bibitem{soest} G. van Soest, F. J. Poelwijk, R. Sprik, and 
A. Lagendijk, Phys. Rev. Lett. {\bf 86}, 1522 (2001).
%Dynamics of a Random Laser above Threshold


\bibitem{Polson'01}R. C. Polson, M. E. Raikh, and Z. V. Vardeny,
 ArXiv: cond-mat 0105360.


\bibitem{thesis}G. van Soest, Ph.D. thesis, Universiteit van Amsterdam,
The Netherlands, 2001 (unpublished).




\bibitem{Schmuurmans99}F. J. P. Schuurmans, M. Megens. 
D.~Vanmaekelbergh, and A. Lagendijk, Phys. Rev. Lett. {\bf 83}, 2183
(1999). 
%Light Scattering near the Localization Transition in Macroporous GaP 
%Networks



\bibitem{Berkovits87} R. Berkovits and M. Kaveh, Phys. Rev. B {\bf 36},
9322 (1987).
%Backscattering of Light near the Optical Anderson Transition}


\bibitem{edrei90} I. Edrei and M. J. Stephen, Phys. Rev. B {\bf 42},
110 (1990).
%Optical coherent backscattering and transmission in diosrdered
%medium near the mobility edge.





\bibitem{wiersma1}D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen,
and A. Lagendijk, Phys. Rev. Lett. {\bf 75}, 1739 (1995).
%Experimental evidence for recurrent multiple scattering of light
%in disordered media


\bibitem{Kravtsov}B. L. Altshuler, V. E. Kravtsov, I. V. Lerner,
in {\em Mesoscopic Phenomena in Solids}, eds. B. L. Altshuler,
P. A. Lee, and R.~A.~ Webb (North Holland, Amsterdam, 1991).
 

\bibitem{Muzykantskii95} B. A. Muzykantskii and D. E. Khmelnitskii,
Phys. Rev. B {\bf 51},  5480 (1995).

%
\bibitem{falko95} V. I. Fal'ko, and K. B. Efetov, Phys. Rev. B
%{\bf 50}, R11267 (1994); {\em ibid.} {\bf 52}, 17413 (1995);
%J. Math. Phys. {\bf 37}, 4935 (1996).


\bibitem{mirlin00} A. D. Mirlin, Phys. Rep. {\bf 326}, 259 (2000).





\bibitem{karpov93} V. G. Karpov, Phys. Rev. B {\bf 48}, 12539 (1993).



\bibitem{Halperin66}B. I. Halperin and M. Lax, Phys. Rev. {\bf 148},
722 (1966).


\bibitem{Zittartz66} J. Zittartz and J. S. Langer, Phys. Rev.
{\bf 148}, 741 (1966).


\bibitem{Brezin80}  E. \'{B}rezin and   G. Parisi,
 J. Phys. C {\bf 13} L307  (1980).




\bibitem{Jiang00}X. Jiang and C. M. Soukoulis,  Phys. Rev. Lett. {\bf 85}, 70 (2000).

\bibitem{Jiang01}X. Jiang and C. M. Soukoulis, ArXiv cond-mat/0103042.






\bibitem{beenakker96} C. W. J. Beenakker, J. C. J. Paasschens, 
and P. W. Brouwer, Phys. Rev. Lett. {\bf 76}, 1368  (1996).


\bibitem{beenakker98} T. Sh. Misirpashaev and C. W. J. Beenakker,
 Phys. Rev. A {\bf 57}, 2041 (1998).


\bibitem{cooperative} T. V. Shahbazyan, M. E. Raikh, and Z. V. Vardeny 
 Phys. Rev. B {\bf 61}, 13266 (2000).





%
\bibitem{Narimanov2000}see O. A. Starykh, Ph. Jacquod, E. E. Narimanov,
%and A. D. Stone, Phys. Rev. E {\bf 62}, 2078 (2000), and references
%therein.


%
\bibitem{Wiersma97}D. S. Wiersma, P. Bartolini, A. Lagendijk,
%and R. Righini, Nature {\bf 390}, 671 (1997).
%%Localization of light in a disordered medium


%
\bibitem{Chabanov00}A. A. Chabanov, M. Stoytchev, and A. Z. Genack,
%Nature {\bf 40}, 850 (2000).
%Statistical signatures of photon localization


