\documentstyle[pre,preprint,aps,epsfig]{revtex}

\tighten

\begin{document} 

 
\title{Fluctuations and Correlations in the Polarization Patterns of a Kerr
 medium} 

\author{Miguel Hoyuelos, Pere Colet and Maxi San Miguel}

\address{Instituto Mediterr\'aneo de Estudios Avanzados, IMEDEA 
(CSIC-UIB),\\ Campus Universitat Illes Balears, E-07071 Palma de Mallorca, 
Spain.\cite{www}}

\date{December 22,1997}

\maketitle 


\begin{abstract}
We study correlations among different components of the spectrum of the light 
intensity field
close to a pattern forming instability  associated with the polarization 
of the light field. In particular we find
strong correlations between opposite wavevectors of critical
wavenumber and anticorrelations of the zero wavenumber and the critical 
one. These anticorrelations are a
manifestation of nonlinear critical fluctuations of the polarization of light.
\end{abstract}

\pacs{PACS numbers: 42.65.-k, 42.65.Sf, 47.54.+r}

\narrowtext

\section{Introduction}

It is well known that close to an instability point fluctuations become
large, as they do for example close to a critical point in an equilibrium
phase transition. In an instability leading to pattern formation the
spectrum of fluctuations below the instability threshold is peaked around
a wave number associated with the spatial periodicity of the pattern which
is formed above threshold. In this way the spectrum of fluctuations is a
noisy precursor \cite{Wiesenfeld} which identifies a preferred wave number
and anticipates the above threshold pattern. This situation has been
studied in detail in experiments in fluid dynamics \cite{fluids} where
thermal fluctuations are observed below the onset of thermal convection.
The fluctuating power spectrum has been used to characterize the selected
wave number and for a quantitative measurement of the strength of thermal
fluctuations.

A first basic wavevector selection process is of linear nature and, for an
isotropic system, determines the modulus of the wavevector of
fastest growth. This is reflected in a ring of maximum power in the below
threshold spectrum. The selection of a discrete set of wavevectors
within this ring is a nonlinear process in which the correlations among
the wavevectors with same modulus play a definite role. The
discrete set of selected modes determine the pattern which is formed above
threshold. Such correlations among the linearly selected wavevectors, 
which were not previously analyzed in fluid systems, have been
recently considered \cite{quantumimages} in pattern forming nonlinear optical 
systems 
\cite{optpatt,grynberg}. Correlations can be here understood in terms of
process involving the simultaneous emission of twin photons. In addition,
these systems present some important peculiarities which make the study of
these correlations particularly interesting. First the power spectrum of
fluctuations below threshold is easily observed as the far field intensity
pattern. Second the fluctuations have contributions of quantum origin (quantum 
noise) and
the observed correlations can encode specific features of quantum
statistics. Thirdly, patterns here are spatial structures of a light
field, and light has a vector character which very naturally leads to the
study of vectorial correlations. This vector character is associated with
the polarization of light which is an additional degree of freedom leading
to polarization patterns.


In this paper we analyze such correlations close to an instability leading
to a polarization pattern. Specifically we consider an optical cavity
filled with a nonlinear optical material, in this case an isotropic Kerr
medium. The cavity is pumped with an external linearly polarized input
field (which we take as $\hat x$-polarized). This situation is described by a
vectorial version of the the scalar model of Lugiato and Lefever
\cite{lugi,firth} in which a Turing optical instability was described.
The generalization to account for the vectorial degree of freedom of 
light was described by Geddes et al \cite{gedd}. For a selfdefocusing
medium no pattern forming instability occurs when neglecting the vector
character of the field \cite{footnote}. However, in the vector model a roll 
$\hat 
y$-polarized
pattern emerges, beyond an instability, on the top of an $\hat x$-polarized
homogenous background \cite{gedd}. We find two types of strong
correlations which can be physically understood in terms of individual fourwave
mixing processes: a) Among the wavevectors with linearly selected
wavenumber a maximum correlation between opposite vectors occurs. 
This is a linear phenomenon that can be found outside the critical region of 
fluctuations.
It is related to symmetry breaking by the pump field. b) An anticorrelation
is found between the zero wavevector of the spectrum of fluctuations
(associated with the $\hat x$-polarized component) and the ring of linearly 
selected wavevectors (associated with the $\hat y$-polarized component). This 
anticorrelation
is a manifestation of nonlinear critical fluctuations and, therefore,
 only observed very close to
threshold. We also study these two types of correlations above threshold, 
showing 
that indeed
the correlations below threshold anticipate
properties found in the pattern that emerges above threshold.

Our results here are obtained within a semiclassical approach in which
specific features of quantum statistics are neglected. The obtained
anticorrelations between the two polarization components of the light
field in the pattern formation process opens the way to the search for
quantum noise aspects of these anticorrelations \cite{alicia}


\section{Description of the model}

The dynamics of the electric field inside an optical cavity with a Kerr medium 
can 
be 
described, in the mean field approximation, by two equations for the independent 
components of the scaled slowly
varying amplitude of the field \cite{gedd,hoyuelos},
\begin{equation}
\frac{\partial E_\pm}{\partial t} = -(1 + i\eta \theta)
E_\pm + i a \nabla^2 E_\pm + E_0 + i\eta [ A
|E_\pm|^2 + (A+B) |E_\mp|^2] E_\pm,
\label{emasmenos}
\end{equation}
where $E_\pm$ are the circularly right and left polarized components of
the field, $E_0$ is an $\hat x$-linearly polarized input field, $\eta=+1 \ (-1)$
corresponds to a self-focusing (self-defocusing) medium, $\theta$ is the
cavity detuning, $a$ represents the strength of diffraction and $\nabla^2$
is the transverse Laplacian. $A$ and $B$ are parameters related to the
components of the susceptibility tensor $\chi$. We consider an isotropic
medium for which $A + B/2 = 1$ ($B \le 2$). The circularly polarized 
components of the
field are expressed in terms of the $\hat x$ and $\hat y$ linearly polarized 
components
$E_x$ and $E_y$ as 
\begin{equation}
E_\pm = (E_x \pm iE_y)/\sqrt{2}. 
\end{equation}

Eq. (\ref{emasmenos}) has an $\hat x$ polarized homogeneous symmetric solution 
in which 
$E_{s+}=E_{s-}=E_s$,
\begin{equation}
E_0 = E_s (1 - i\eta (2I_s - \theta)),
\label{estationary}
\end{equation}
where $I_s= |E_s|^2$. It is well known that the homogeneous solution 
(\ref{estationary}) presents
bistability for $\theta > \sqrt{3}$. We will restrict ourselves here to
the non-bistable regime $\theta < \sqrt{3}$. We note that Eq. (\ref{emasmenos})
has also an asymmetric solution ($E_{s+} \neq E_{s-}$). We will not 
consider the asymmetric solution in this work because it is only relevant 
for high values of the input field, above the first instability threshold 
of the symmetric solution \cite{hoyuelos}. We are here concerned with 
correlations close to this instability threshold.

To analyze the stability of the homogeneous steady state $E_s$, as well as 
the effect of the fluctuations in this state, we consider perturbations of 
the form
\begin{equation}
E_\pm = E_s + \psi_\pm
\label{perturba}
\end{equation}
in Eq. (\ref{emasmenos}). The linear stability analysis
\cite{gedd,hoyuelos} shows that there are  two modes that can become
unstable: a symmetric mode (S) with $\psi_+ = \psi_-$, and an asymmetric
mode (A) with $\psi_+ = -\psi_-$.  In the selffocusing case the mode that
becomes unstable is the symmetric one. As the unstable mode has the same
polarization than the input field this situation can be described with
a scalar model \cite{lugi,firth}. In two transverse dimensional systems
this instability leads to an hexagonal stationary pattern. We are here
interested in the instability of the asymmetric mode which occurs in the
selfdefocusing case. This instability leads to the growth of the component
of the field orthogonally polarized to the driving field
\cite{gedd,hoyuelos}. In the linear regime one finds,
\begin{equation}
E_\pm = E_s \pm \psi,
\label{6}
\end{equation}
and,
\begin{eqnarray}
E_x &=& \sqrt{2} E_s \nonumber \\
E_y &=& -i \sqrt{2} \psi,
\label{exey}
\end{eqnarray}
The instability threshold is located at $I_s^c = 1/B$ and the instability 
occurs at a critical wavenumber $k_c$, 
\begin{eqnarray}
k_c = \sqrt{(\theta + 1 - 2/B)/a}, \;\;\mbox{for } \theta > 2/B - 1 \nonumber \\
k_c = 0, \;\;\mbox{for } \theta \leq 2/B - 1.
\label{k_c}
\end{eqnarray}
For $\theta > 2/B - 1$ a stationary stripe pattern emerges in the 
$\hat y$-polarized component, while the $\hat x$-component remains homogeneous. 
Above this instability we find, therefore, for the total field an elliptically
polarized pattern with spatially periodic ellipticity. From an
experimental point of view an interesting aspect of this situation is
that it is possible to separate the finite wavenumber component of the
field (pattern) from the homogeneous part by simply using a polarizer.


\section{Correlations below threshold}

In order to study correlations below the threshold of pattern formation we need 
to introduce sources of noise in our description. We follow here the approach 
already used to study fluctuations in the one-dimensional and scalar version of 
this problem \cite{aguado}: In a semiclassical description of the problem we can 
represent different sources of fluctuations by adding Gaussian white noise 
$\xi_\pm(\vec x,t)$  with zero mean and correlations to the right hand side of
Eq. (\ref{emasmenos}),
\begin{eqnarray}
\langle \xi_i(\vec x,t) \xi_j^*(\vec x',t') \rangle & = & 2 \epsilon^2 
\delta_{ij} \delta(\vec x - \vec x') \delta(t-t') ,\nonumber \\
\langle \xi_i(\vec x,t) \xi_j(\vec x',t') \rangle & = & 0 ,
\label{correlations}
\end{eqnarray}
where the subindices $i$, $j$ stand for the circularly polarized components 
$\pm$. 

Slightly below threshold noise excites fluctuations of all wavenumbers.
However, the wavevector $k_c$ which is going to become unstable at
threshold is much less damped than the rest of the modes and, as a
consequence, is singled out in the power spectrum of the field. Because a
pattern is not yet formed, the excited wavevectors have random 
directions. In the far field (Fourier transform of the near field) there
is a ring of higher intensity with a radius equal to $k_c$. In Fig.
\ref{far_near} we show the intensities $I_x$, $I_y$ of the far field of
the two components of the field $E_x$, and $E_y$, slightly below the threshold
of instability of the A mode. We also show a snapshot of the spatial 
configurations of
the two components of the electric field (near field)
\cite{algorithm}. The far field of the $\hat x$-polarized component has a single
peak at zero wavenumber, while the $\hat y$-polarized component displays the
characteristic ring. The near field of the $\hat x$-component shows a noisy
homogeneous pattern. The $\hat y$-component in the near field is also 
homogeneous
if a long time-average is taken. A configuration at a fixed time shows a 
disordered structure. 
This structure reveals a characteristic less stable wavenumber $k_c$ when 
averaged, for noise reduction,
over a short
time interval. 

We next analyze the correlations among the wavevectors in the ring
of the $\hat y$-polarized far field. We fix one of these vectors by choosing the
point $\vec k' = (k_c,0)$ in the plot of Fig. \ref{far_near}, and
calculate its correlation with any other wavevector. This is given
by the correlation function,
\begin{equation}
C_1(\vec k) = \langle \delta I_y(\vec k') \delta I_y(\vec k) \rangle,
	\label{8}
\end{equation}
where $\delta I_y(\vec k)$ is the fluctuation of the intensity,  $\delta 
I_y(\vec k) = I_y(\vec k) - \langle I_y(\vec k) \rangle $, and $I_y(\vec k) = 
|E_y(\vec k)|^2$ (equivalent expresions hold for the $\hat x$-component). 
Brackets 
$\langle ... \rangle$ stand for time average.

In Fig. \ref{C1}(a) we plot $C_1(\vec k)$ for a distance to threshold $I_s = 
0.98 I_s^c$ (same parameters as in
Fig. \ref{far_near}). We can see a selfcorrelation at $\vec k = \vec k'$
and another peak at $ \vec k = -\vec k'$. We also calculate the
correlation $C_1$ along the ring as a function of the angle $\alpha$ 
between $\vec k$ and $\vec k'$, taking $|\vec k|=|\vec k'|=k_c$. In this case we 
can select different vectors $\vec k'$ on the ring, calculate the correlation 
$C_1$ for each one of these vectors, and obtain an average. In Fig. \ref{C1}(b) 
we plot $C_1$ versus $\alpha$ and the correlation at $\alpha = \pi$ is clearly 
displayed.
As we get closer to 
the threshold we can expect an enhancement of the correlation since fluctuations 
become larger.  This is, indeed, the case as we can see in Fig. \ref{C1}(c), 
where 
we plot $C_1$ as a function of $\alpha$ for a distance to threshold $I_s = 0.999 
I_s^c$.


Since we are considering correlations below threshold, we may think that
the system is not far from the homogeneous solution and fluctuations could be 
described within a
linear analysis. If we linearize Eq. (\ref{emasmenos}) around that solution, 
we find for the 
$\hat x$ and $\hat y$ components of the fluctuations $\delta E_x$ and $\delta 
E_y$,
\begin{eqnarray}
\frac{\partial \delta E_x}{\partial t} &=& - [ 1 + i\eta (\theta - 2I_s) - 
ia \nabla^2 ] \delta E_x + i\eta 2I_s (\delta E_x + \delta E_x^*) +  \xi_x(\vec 
x,t)
\nonumber \\
\frac{\partial \delta E_y}{\partial t} &=& - [ 1 + i\eta (\theta - 2I_s) - 
ia \nabla^2 ] \delta E_y + i\eta I_s B (\delta E_y^* - \delta E_y) +  \xi_y(\vec 
x,t).
	\label{xy}
\end{eqnarray}
where $\delta E_x = E_x - \sqrt{2} E_s$ and $\delta E_y = E_y$.
These equations show that $\delta E_x$ and $\delta E_y$ are linearly uncoupled.
The correlations in the far field of $E_y$ described above can be understood 
considering
Eq. (\ref{xy}) for the $\hat y$ component in Fourier space,
\begin{equation}
\frac{\partial \delta E_y(\vec k)}{\partial t} = - [ 1 + i\eta (\theta - 2I_s) + 
ia k^2 ] \delta E_y(\vec k) + i\eta I_s B (\delta E_y^*(-\vec k) - \delta 
E_y(\vec k)) +  \xi_y(\vec k,t).
	\label{yk}
\end{equation}
Eq. (\ref{yk}) shows a linear correlation between wavevectors $\vec k$ and 
$-\vec 
k$ 
as found in Fig.  
\ref{C1}(a) and (b). It is important to understand the origin of such 
correlation:
Mathematically it comes from the 
term  $i\eta I_s B (\delta E_y^*(c) - \delta E_y(\vec k))$ which breaks the 
phase
invariance
of the equation for the complex amplitude $\delta E_y$.
This symmetry breaking term is proportional
to $I_s$ which is nonzero because of the pump field $E_0$. The origin of the 
correlations
in the far field of $E_y$ is
then traced back to symmetry breaking caused by the pump field, which also 
breaks 
the global
phase symmetry of Eq. (\ref{emasmenos}).

At a microscopic level the correlation between $\vec k$ and $-\vec k$
can be interpreted as a manifestation
of the individual four wave mixing process in which there is simultaneous
emission of two photons that conserve transverse momentum. In this
interpretation these two photons are $\hat y$-polarized and originate in the
annihilation of two $\hat x$-polarized photons of the pump field which have zero
 transverse wavenumber.
It is then natural to expect an anticorrelation between the homogeneous
$\hat x$-polarized component of the field and the $\hat y$-polarized component.
However, Eqs.(\ref{xy}) do not account for this anticorrelation since 
$\delta E_x$ and $\delta E_y$ are uncoupled.

In order to investigate this possible anticorrelation in which an increase of 
$I_y$
should be accompanied by a decrease of $I_x$, we calculate a second
correlation function:
\begin{equation}
C_2(k) = \langle \delta I_y(k) \delta I_x(0) \rangle,
	\label{9}
\end{equation}
$C_2(k)$ does not take into account the angle of $\vec k$, but it only depends 
on
its modulus $k$. $\delta I_x(0)$ represents the fluctuations of the
homogeneous part of the $\hat x$-component of the field. $\delta I_y(k)$ is the 
average 
over orientations of $\vec k$ of the fluctuations of $I_y$ in Fourier space.
Specifically, taking 
$\vec k = (k \cos \phi, k \sin \phi)$, we define 
$\delta I_y(k)={1\over 2\pi}\int_0^{2\pi}{\delta I(k\cos\phi,k\sin\phi)d\phi}$.  
$\delta I_y(k)$ at 
$k = k_c$ corresponds to the fluctuations of the $\hat y$-polarized precursor of 
the 
pattern.
If we calculate $C_2$ for the same parameters as in Fig. \ref{far_near}, where 
the distance to threshold is $I_s = 0.98 I_s^c$, we do not obtain any 
anticorrelation.  From our analysis of Eq. (\ref{xy}), we know that if these 
anticorrelations exist they have to originate in nonlinear terms. Nonlinearities
become important close to the critical point where fluctuations are enhanced. 
In Fig. \ref{C2}(a) 
we plot $C_2(k)$ in a situation much closer to threshold, $I_s = 0.999 I_s^c$.
For these parameters we enter the critical nonlinear regime and, in agreement 
with our previous reasoning, we do now find  an anticorrelation at $k = k_c$.

In summary, below the threshold for pattern formation there is a linear 
correlation between the fluctuations of $\vec k$ and $-\vec k$ in the $\hat 
y$-polarized far field.  These fluctuations and correlations become larger in 
the critical region. In addition there is a critical nonlinear anticorrelation 
between the fluctuations of the $\hat x$ and $\hat y$-polarized fields which has 
the same physical origin.



\section{Correlations above threshold}

The results discussed in the previous section are for pumping intensity below 
the threshold for pattern formation. We now consider the situation 
above this threshold. We show that the same type of correlations are found. This 
shows that correlations among the fluctuations below threshold anticipate the 
features to be observed in the coherent pattern that emerges above threshold.
There are two additional reasons for this above threshold analysis. 
First, experimental measurements of far field correlations have been already 
done above threshold in pattern formation in nonlinear optical systems 
\cite{grynberg}.
Second, as already mentioned, the correlations studied here between the 
two polarization components of the field in the pattern formation process open
the way to search for quantum noise aspects of these correlations. 
Quantum noise aspects close to an instability have been studied either below or 
above the instability threshold \cite{quantumimages,castelli}. 

Above threshold a stripe structure is developed in the $\hat y$-component of the
field, see Fig. \ref{above_thr}. In the $\hat x$-component also appears a stripe
structure, produced by nonlinear couplings with the $\hat y$-component, but of
smaller amplitude; the main contribution to the far field (Fourier
transform) is still the homogeneous mode. The far field of the 
$\hat y$-component has two dots that indicate the arbitrary direction chosen by 
the system to develop the stripes.

To calculate $C_1$ above threshold we choose the vector $\vec k'$
corresponding to one of the dots in $I_y(\vec k)$, as shown in Fig.
\ref{above_thr}. As in the previous section, we can calculate the
correlation $C_1$ between $\delta I_y(\vec k')$ and $\delta I_y(\vec k)$
in the whole plane or along a ring of radius $k_c$ (we do not average
over different positions of $\vec k'$ in the circle, since it is fixed in
one of the bright dots of $I_y(\vec k)$). Both results are shown in Fig.
\ref{C1_above} with similar results as those obtained for the case below 
threshold.
In Fig. \ref{C1_above}(a) we see a peak showing the
selfcorrelation at $\vec k = \vec k'$, and another peak at $\vec k = -\vec
k'$. In Fig. \ref{C1_above}(b) we plot $C_1$ as a function of the angle
$\alpha$ in the ring of radius $k_c$, and we find, as expected, a strong
correlation at $\alpha = \pi$. The main difference between Figs. \ref{C1}
and \ref{C1_above} is that we now have a coherent pattern so that fluctuations 
in the ring of radius $k_c$ are largely suppressed in comparison with the mean 
value of $I_y$ in the two peaks of the power spectrum.  These peaks  
characterize the pattern  with a well defined orientation (see Fig. 
\ref{far_near}).

We have also calculated the anticorrelation between $I_x(0)$ and
$I_y(k_c)$  above threshold. In this case we calculate $C_2$ in the direction
determined by
the two bright dots in the far field of the $\hat y$ polarized intensity 
$I_y(\vec k)$ (see Fig. \ref{above_thr}). The
result, shown in Fig. \ref{C2}(b), displays the anticipated strong 
anticorrelation. 


\section{Acknowledgements}

Financial support from the European Union TMR network QSTRUCT (Project 
FMRX-CT96-0077) and from DGICYT Project PB94-1167 (Spain) is acknowleged.
We also acknowledge helpful discussions with L. Lugiato.
M. H. wants to acknowledge financial support from the
FOMEC project 290, Dep. de Fisica FCEyN, Universidad Nacional de Mar del
Plata, Argentina.



\begin{figure}
\vspace*{10cm}
\psfig{figure=figure1.ps,width=14cm}
\caption{From left to right and from top to bottom: $I_x(\vec x)$ (near field),
$I_x(\vec k)$ (far field), $I_y(\vec x)$ (near field), $I_y(\vec k)$ (far 
field).
Parameter values: $I_s = 0.98 I_s^c$ (below threshold), $a=1$, $\theta=1$, 
$\eta=-1$, $\epsilon^2=10^{-5}$ and $B=3/2$. $I_x(\vec x)$ and $I_y(\vec x)$ 
were 
averaged over 100 samples separated $\Delta t = 0.02$ to reduce noise. $I_y(\vec 
k)$ was averaged over 40000 samples separated $\Delta t = 0.5$.}
\label{far_near}
\end{figure}

{\psfig{figure=figure2a.ps,width=12cm} \vspace*{-10cm}}
\psfig{figure=figure2b.ps,width=12cm}

\pagebreak

\begin{figure}
\psfig{figure=figure2c.ps,width=12cm} 
\vspace*{-10cm}
\caption{Correlation $C_1$, see Eq. (\protect\ref{8}). (a) $C_1$ shown as 
a function of $k_x$ and $k_y$ for $I_s = 0.98 I_s^c$; (b) $C_1$ calculated along 
the ring $k=k_c$ as a function of $\alpha$ for $I_s = 0.98 I_s^c$; and (c) 
$C_1$ versus $\alpha$ for $I_s = 0.999 I_s^c$.  The rest of the parameters are 
the same as in Fig. \protect\ref{far_near}. For these parameters $k_c=0.82$.
Results obtained after averaging over 40000 samples separated $\Delta t = 
0.5$}
\label{C1}
\end{figure}

\pagebreak

\begin{figure}
\psfig{figure=figure3.ps,width=12cm}
\caption{Correlation $C_2(k)$, see Eq. (\protect\ref{9}). Same parameters
as in Fig. \protect\ref{far_near} except $I_s$. In (a), $I_s = 0.999 I_s^c$. In 
(b), $I_s = 1.05 I_s^c$ (above threshold). Results obtained after averaging
over 10000 samples separated $\Delta t = 0.5$}
\label{C2}
\end{figure}

\begin{figure}
\vspace*{10cm}
\psfig{figure=figure4.ps,width=14cm}
\vspace*{-3cm}
\caption{Results above threshold. From left to right and from top to
bottom: $I_x(\vec x)$ (near field), $I_x(\vec k)$ (far field), $I_y(\vec x)$ 
(near field), $I_y(\vec k)$ (far field). Same parameters as in Fig. 
\protect\ref{far_near} except $I_s = 1.05 I_s^c$.  It is not here
necessary to take an average of the near field intensities over a short time 
since the amplitude of the pattern is much greater than the noise amplitude.  
The far field intensities are also clearly seen without averaging. Extreme 
values of the intensities in the near field are: $I_x(\vec x) \rightarrow 
(1.326,1.356), \; I_y(\vec x) \rightarrow (0,0.113)$}
\label{above_thr}
\end{figure}

\begin{figure}
\psfig{figure=figure5.ps,width=12cm}
\caption{Correlation $C_1$ above threshold (see Eq. (\protect\ref{8}))
shown as a function of $k_x$ and $k_y$ (a), and calculated along the ring 
$k=k_c$
(b). Same parameters as in Fig. \protect\ref{far_near}
except $I_s = 1.05 I_s^c$. Results obtained after averaging over 3000 samples
separated $\Delta t = 0.5$}
\label{C1_above}
\end{figure}


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\bibitem{hoyuelos} M. Hoyuelos, P. Colet, M. San Miguel and D. Walgraef,
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\bibitem{aguado} M. Aguado, R. F. Rodriguez, and M. San Miguel, Phys. Rev.
A {\bf 39}, 5686 (1989).

\bibitem{algorithm} These results are obtained by a direct numerical
integration of the stochastic Langevin partial differential equations as
explained for example in M San Miguel and R. Toral, {\it Stochastic
Effects in Physical Systems}, in {\it Instabilities and Nonequilibrium
Structures VI}, Kluwer Academic (1997). Lattices of size $128 \times 128$
were used, the integration steps in time and space were $dt = 0.005$ and
$dx = 0.75$. 


\bibitem{castelli} L.A. Lugiato and F. Castelli, Phys. Rev. Lett. {\bf 68}, 
3284 (1992); F. Castelli and P. Dotti, Opt. Comm. {\bf 113}, 237 (1994).

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