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


\tighten

\begin{document}
\title{Polarization patterns in Kerr Media}

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

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

\date{March 18, 1998}

\maketitle

\begin{abstract} 
We study spatiotemporal pattern formation associated with the polarization
degree of freedom of the electric field amplitude in a mean field model
describing a Kerr medium in a cavity with flat mirrors and driven by a coherent
plane-wave field. We consider linearly as well as elliptically polarized
driving field and situations of self-focusing and self-defocusing. For the case
of self-defocusing and linearly polarized driving field there is a stripe
pattern orthogonally polarized to the driving field. Such pattern changes into
an hexagonal pattern for elliptically polarized driving field. The range of
driving intensities for which the pattern is formed shrinks to zero with
increasing ellipticity. For the case of self-focusing, changing the driving
field ellipticity leads from a linearly polarized (for linearly polarized
driving) to a circularly polarized hexagonal pattern (for circularly polarized
driving). Intermediate situations include a modified Hopf bifurcation at finite
wave number leading to a time dependent pattern of deformed hexagons and a
codimension two Turing-Hopf instability resulting in an elliptically
polarized stationary hexagonal pattern. Our numerical observations of different
spatiotemporal structures are described by appropriate model and amplitude
equations. 
\end{abstract}

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

\section{Introduction}

Spatiotemporal patterns in the transverse direction of an optical field have
now been widely studied theoretically and experimentally \cite{Chaossolfract}.
In particular, the non linear optical configuration of a thin slice of Kerr
material with a single feedback mirror analyzed in \cite{Alessandro} is the
basis of many results recently obtained in this field. Studies of optical
pattern formation share a number of aspects and techniques with general
investigations of pattern formation in other physical systems \cite{CrossHoh},
but they also have specific features such as the role of light diffraction. A
special feature of light patterns comes from the vectorial degree of freedom
associated with the polarization of the light electric field amplitude. A
vectorial degree of freedom also appears in recent studies of two-component
Bose-Einstein condensates \cite{busch} modeled by coupled nonlinear 
Schr\"odinger equations. Consideration of this degree of freedom opens the way
to study a rich variety of {\it vectorial} spatiotemporal phenomena. However,
in many studies of optical pattern formation this extra degree of freedom has
not been taken into account. Those studies correspond to situations in which a
linear polarization of light is well stabilized. We will refer to these
situations of frozen polarization as the ``scalar case''. An early study of
polarization dynamical instabilities in nonlinear optics is due to Kitano et
al. \cite{kitano}. For a review on polarization instabilities and
multistability see \cite{zheludev}. Space independent polarization
instabilities have been also studied in lasers
\cite{puccioni,abraham,matlin,serrat}. More recently, vectorial patterns
associated with polarization instabilities have been considered in lasers
\cite{maxiPRL,gil93,toniPRL,sti,CO2Meucci,VCSELs,prati} as well as in non 
linear passive optical media. For this latter case, new type of vectorial
instabilities have been predicted for cavity \cite{geddes1} or single feedback
mirror \cite{Scroggie96} systems. Several experiments in cells with alkaline
vapors have been reported either without cavity \cite{Tam77,Gahl} or with a
single feedback mirror \cite{Grynberg94,Aumann97}. Dynamically evolving 
patterns produced in a cell of rubidium vapor with counterpropagating beams
have also been studied experimentally \cite{Maitre95}. Within this context of
recent experimental studies we address in this paper several aspects of
polarization transverse patterns and pattern dynamics in passive optical
systems presenting a systematic study of a model system.

Pattern formation in non linear cavities for the scalar case was already 
considered in \cite{MoloneyKerr}. A prototype simple model which has been very
useful for the understanding of pattern formation in this case is a mean field
model describing a Kerr medium in a cavity with flat mirrors and driven by a
coherent plane-wave field \cite{lugi,firt}. This model was extended to take
into account the polarization degrees of freedom in \cite{geddes1,corrkerr}.
Even if a Kerr material model does not give a faithful description of alkali
vapors, it shares with it some basic polarization mechanisms of pattern
formation. In addition, the relative simplicity of the model in \cite{geddes1}
makes it worthwhile to study it in depth as a general prototype model for the
basic understanding of vectorial patterns. We undertake here such study going
beyond the situations already considered in \cite{geddes1}. The study in
\cite{geddes1} was limited to the case in which the driving field is linearly
polarized. Allowing for elliptically polarized driving field, as we do here,
gives rise to a rich variety of new phenomena. In addition, the role of
elliptically polarized homogeneous solutions in a number of pattern forming
instabilities is discussed in detail. Those solutions already exist in the case
of linearly polarized driving field. 

Our study involves a combination of linear stability analysis, numerical
simulations and amplitude and model equations. Guided by the results of linear
stability analysis we search numerically for different spatiotemporal
structures. The general features of these structures are then shown to be
described by model and amplitude equations which are justified by general
arguments of symmetry, form of the linear instability and identification of
relevant non linear couplings.

The paper is organized as follows: In Sec. \ref{model} we describe the model
we are considering, its spatially homogeneous solutions and general properties
of the stability analysis of these states. In Sec. \ref{linearpolarization}
we consider the case of linearly polarized driving field, while in Sec.
\ref{ellipticalpolarization} we discuss the case of elliptically polarized
driving field. Two particular situations of this last case are considered in
the two following sections. Section \ref{Blinking Hexagons} is devoted to
describe the deformed dynamical hexagons occurring in a modified Hopf
bifurcation and Sec. \ref{cod2} discusses the Turing-Hopf codimension
two bifurcation. A summary of results and their connection with related
studies is given in Sec. \ref{results}. Finally, some general concluding 
remarks are given in Sec. \ref{conclusions}.



\section{Description of the model, reference steady states and stability
analysis}

\label{model}

The system we consider is a Fabry-P\'erot or ring cavity filled with an
isotropic Kerr medium. The cavity is driven by an external input field of
arbitrary polarization. The situation in which the polarization degree of
freedom of the electromagnetic field is frozen was first considered by L.
A. Lugiato and R. Lefever \cite{lugi,firt}. Geddes et al \cite{geddes1}
generalized the model of \cite{lugi} to allow for the vector nature of the
field. Their description of this system is given by a pair of coupled
equations for the evolution of the two circularly polarized components of
the field envelope $E_+$ and $E_-$, defined by
$$ 
E_\pm = \frac{1}{\sqrt{2}}(E_x \pm iE_y). 
$$ For an isotropic medium, the equations are
\begin{eqnarray}
\frac{\partial E_\pm}{\partial t} &=& -(1 + i\eta \theta)
E_\pm + i a \nabla^2 E_\pm + E_{0\pm} \nonumber \\
&& + i\eta [ A |E_\pm|^2 + (A+B) |E_\mp|^2] E_\pm ,
\label{1}
\end{eqnarray}
where $E_{0\pm}$ are the circularly polarized components of the input
field, $\eta = +1 \ (-1)$ indicates self-focusing (self-defocusing),
$\theta$ 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. As we
are considering here an isotropic medium, $A + B/2 = 1$ ($B \le 2$).

The case in which $E_{0+}=E_{0-}$ was considered in reference \cite{geddes1}
which corresponds to a linearly polarized input field. Here we consider an
input field with arbitrary ellipticity $\chi$ defined as
\begin{eqnarray}
E_{0+} &=& \sqrt{I_0} \cos(\chi/2) , \nonumber \\
E_{0-} &=& \sqrt{I_0} \sin(\chi/2) ,
\label{2}
\end{eqnarray}
where $I_0$ is the intensity of the input field. We consider that $E_{0+}$
and $E_{0-}$ are real. This assumption fixes the main axis of the ellipse
in the X or Y direction. Note that $\chi=\pi/2$ corresponds to linear
polarization along the X axis and that ellipticity increases when the
value of $\chi$ is changed away from this value. $\chi=0$ ($\chi=\pi$)
corresponds to a right-handed (left-handed) circularly polarized input
field. Finally, $\chi=-\pi/2$ corresponds to linear polarization along the
Y axis. The intensities of the circularly polarized components of the
input field are $I_{0+} = I_0 \cos^2(\chi/2)$ and $I_{0-} = I_0
\sin^2(\chi/2)$. We note that the case of scalar field considered in
\cite{lugi,firt} is formally recovered from (\ref{1}) for a circularly
polarized input. The circular component of the field which is not excited
by the input field decays to zero and the equation for the other component
coincides with the one in \cite{lugi} up to a rescaling of the field
amplitude.

The steady state homogeneous solutions of Eq. (\ref{1}) are reference states
from which transverse patterns emerge as they become unstable. These patterns
are described in the following sections for different situations. The steady
state homogeneous solutions $E_{s\pm}$ are given by the implicit equation
\begin{equation}
E_{0\pm} = E_{s\pm} \left\{ 1 - i\eta\left[ \left(1 - {B \over 2}
\right) I_{s\pm} + \left(1 + {B \over 2} \right) I_{s\mp} - \theta
\right] \right\} ,
\label{2.5}
\end{equation}
where $I_{s\pm}= |E_{s\pm}|^2$. For the intensities we have
\begin{equation}
I_{0\pm} = I_{s\pm} \left\{ 1 + \left[ \left(1 - {B \over 2} \right) 
I_{s\pm} + \left(1 + {B \over 2}\right) I_{s\mp} - \theta \right]^2
 \right\}.
\label{3}
\end{equation}
This gives a pair of coupled cubic polynomials in $I_{s+}$ and $I_{s-}$. 
Solving for $I_{s+}$ and $I_{s-}$ leads to a polynomial of degree nine 
from which it is not possible, in principle, to find an analytical
expression. 

For the particular case of linearly polarized input field, Eq. (\ref{3}) 
admits symmetric ($I_{s+}=I_{s-}=I_s$) and asymmetric ($I_{s+} \neq
I_{s-}$) solutions. The symmetric solution corresponds to linearly
polarized output light, while the asymmetric is elliptically polarized.
For the symmetric solution Eq. (\ref{3}) reduces to the single equation
\cite{footnote1}
\begin{equation}
I_0/2 = I_s(1 + (2I_s - \theta)^2),
\label{symmetricsol}
\end{equation}
which gives an implicit formula for $I_s$. As it is well known, Eq. 
(\ref{symmetricsol}) implies bistability for $\theta > \sqrt{3}$. We will 
restrict our analysis to non-bistable regimes, i.e. $\theta <\sqrt{3}$. The
asymmetric solution is obtained from the general Eq. (\ref{3}). This
solution breaks the ($+$,$-$) symmetry of the problem and is
degenerate. There is one solution with $I_{s+}>I_{s-}$ and a second
equivalent solution in which $I_{s+}$ and $I_{s-}$ are interchanged. The
asymmetric solution only exists for values of $I_0$ greater than a
threshold value for which $I_{s+}=I_{s-}=I'$. An example of the symmetric
and asymmetric solutions for linearly polarized input is given in Fig.
\ref{fig1}. The value of $I'$ is given by
\begin{equation}
I' = \frac{\theta(B-2) + \sqrt{\theta^2B^2+4(B-1)}}{4(B-1)}.
\end{equation}

For circularly polarized input light, for example $\chi=0$, Eq. (\ref{3}) 
reduces to
\begin{equation}
I_0 = I_{s+}\left\{1 + \left[ \left(1 - {B \over 2}\right) I_{s+} - 
\theta \right]^2 \right\} ,
\label{circpolarizedsol}
\end{equation}
and $I_{s-}=0$. It is clear from Eqs. (\ref{symmetricsol}) and 
(\ref{circpolarizedsol}) that the solution for circularly polarized input is
the same as the symmetric solution for linearly polarized input, up to a
rescaling of the intensities. 

An elliptically polarized input breaks the ($+$,$-$) symmetry of the system.
The symmetric solution (\ref{symmetricsol}) found for linearly  polarized input
no longer exists. Instead there is a single asymmetric solution  (\ref{3}) which
favors the ellipticity of the input field. When the  ellipticity of the input
field is decreased, this solution approaches the asymmetric solution obtained
for linearly polarized input. An example of this  homogeneous elliptically
polarized solution, obtained from (\ref{3}), is  shown in Fig. \ref{fig3a} for
various values of the ellipticity of the input field. This solution favors
$I_{s+}$ and an equivalent solution favoring  $I_{s-}$ is found when the
ellipticity is changed from $\chi$ to $\pi-\chi$.

Basic features of the stability of the steady state homogeneous solutions
can be analyzed by considering the evolution equations for perturbations
$\psi_{\pm}$ defined by
\begin{equation}
E_{\pm} = E_{s\pm}[1+ \psi_{\pm}] .
\label{perturba}
\end{equation}
From Eqs. (\ref{1}) and (\ref{perturba}) we find
\begin{eqnarray}
\partial_t \psi_{\pm} &=& -\left[1+i\eta \left(\theta - S
\pm {BR\over 2}\right) - ia \nabla^2 \right]\psi_{\pm}\nonumber \\
&& +i\eta {S\over 2}\left[\left(1 - {B \over 2}\right)
(\psi_{\pm}+\psi_{\pm}^*+\vert\psi_{\pm}\vert^2) + \left(1 +{B \over 2}\right) 
(\psi_{\mp}+\psi_{\mp}^*+\vert\psi_{\mp}\vert^2) \right]
(1+\psi_{\pm})\nonumber \\
&&\pm i\eta {R \over 2} \left[\left(1 - {B \over 2}\right) 
(\psi_{\pm}+\psi_{\pm}^*+\vert\psi_{\pm}\vert^2)
-\left(1 + {B \over 2}\right) 
(\psi_{\mp}+\psi_{\mp}^*+\vert\psi_{\mp}\vert^2) \right]
(1+\psi_{\pm}) ,
 \label{psimasmenos}
\end{eqnarray}
with 
\begin{equation}
I_{s\pm}={1\over 2}( S \pm R ) .
\label{imasmenos}
\end{equation}
The parameter $R$ measures the deviation from a symmetric solution
vanishing for linearly polarized solutions.

It is convenient to make a change of variables to the
following basis \cite{geddes1}:
\begin{equation}
\Sigma = \left \{ \begin{matrix}
{\sigma_1\cr \sigma_2\cr \sigma_3\cr \sigma_4}\end{matrix}
\right \} =
\left \{ \begin{matrix}
{\Re(\psi_+ + \psi_-)\cr
\Im(\psi_+ + \psi_-)\cr
\Re(\psi_+ - \psi_-)\cr
\Im(\psi_+ - \psi_-)}\end{matrix}
\right \} .
\end{equation}
In this basis, which emphasizes the role of symmetric ($\psi_+=\psi_-$)
and anti-symmetric ($\psi_+=-\psi_-$) modes, Eq. (\ref{psimasmenos}) may
be written as:
\begin{equation} \label{vectevol}
\partial_t \Sigma = L \Sigma + N_2(\Sigma\vert\Sigma) + 
N_3(\Sigma\vert\Sigma\vert\Sigma) ,
\end{equation}
where the linear matrix (in Fourier space) is
\begin{equation}
L = \left( \begin{array}{cccc}
-1 & -\eta (S-\theta_k) & 0 & \eta BR/2 \\
\eta(3S-\theta_k) & -1 & -\eta(B/2-2)R & 0 \\
0 & \eta BR/2 & -1 & -\eta(S-\theta_k) \\
-3\eta BR/2 & 0 & \eta(S(1-B)-\theta_k) & -1
\end{array} \right) ,
\label{linmatrix}
\end{equation}
with 
\begin{equation}
\theta_k=\theta+\eta ak^2. 
\end{equation}
The structure of the linear matrix is particularly simple for a symmetric 
solution $R=0$ \cite{geddes1}: $L$ becomes a matrix with $2\times 2$ 
blocks in which the symmetric and antisymmetric modes are decoupled. As a
consequence, the linear instabilities lead to the growth of either a
symmetric or an antisymmetric mode (see Sec. \ref{linearpolarization}).
The eigenvalues $\lambda$ of $L$ are
\begin{eqnarray}
\lambda_{1,2} &=& -1 \pm \sqrt{(\theta_k-3S)(S-\theta_k)} ,
\nonumber\\
\lambda_{3,4} &=& -1 \pm \sqrt{(\theta_k+(B-1)S)(S-\theta_k)} .
\label{lambdasymmetric}
\end{eqnarray}
In the general case of elliptically polarized input, the linear unstable
modes are not purely symmetric or antisymmetric. The general expression
for the 4 independent eigenvalues of $L$ are
\begin{eqnarray}
&&\lambda_{1,2,3,4} = -1 \pm {1 \over 2}\sqrt{f_1 \pm \sqrt{f_2}} ,
\nonumber \\
&&f_1 = (2B-8)S^2 - 2\theta_k(B-6)S - 2B(B-1)R^2 -
4\theta_k^2 , \nonumber \\
&&f_2 = 4(B+2)^2S^2(S-\theta_k)^2 - 4B(5B^2-16B+20)R^2S^2
\nonumber \\
& & \; \; \; \; \; \; + 12\theta_kB(B-2)(B-6)R^2S + B^2(B+2)^2R^4 
\nonumber \\
& & \; \; \; \; \; \; + 32 B\theta_k^2(B-2)R^2.
\label{10}
\end{eqnarray}
The different eigenvalues correspond to the four different combinations of
plus and minus signs in the square roots. Replacing the values of $R$ and
$S$, the eigenvalues are given as functions of the steady state
intensities $I_{s+}$ and $I_{s-}$. Their dependence on $\eta$ is implicit
in $\theta_k$. A given homogeneous steady state solution
($I_{s+}$,$I_{s-}$) becomes unstable when the real part of one eigenvalue
becomes positive. These instabilities are described in detail in Sec. 
\ref{linearpolarization} for linearly polarized input and in Sec. 
\ref{ellipticalpolarization} for the general case of elliptically
polarized input.

The non linearities in (\ref{vectevol}) include quadratic 
$N_2(\Sigma\vert\Sigma)$ and cubic terms $N_3(\Sigma\vert\Sigma\vert\Sigma)$.
The structure of these terms also gives some general information on the
nature of the instabilities. In particular, if the quadratic non linearity
$N_2(\Sigma\vert\Sigma)$ does not vanish, one expects the formation of
hexagonal patterns instead of stripes. In addition, a stationary
instability (which corresponds to a purely real eigenvalue becoming positive)
is expected to be subcritical.

For the symmetric solution ($R=0$), the quadratic non linearity is given by
\begin{equation}
N_2^S(\Sigma\vert\Sigma) = {\eta S\over 2}\left \{ \begin{matrix}
{B \sigma_3 \sigma_4 - 2\sigma_1 \sigma_2\cr
3\sigma_1 \sigma_1 + \sigma_2 \sigma_2 +(1-B) \sigma_3
\sigma_3 + \sigma_4 \sigma_4 \cr
B\sigma_2 \sigma_3 - 2\sigma_1 \sigma_4\cr
2(1-B)\sigma_1 \sigma_3 - B\sigma_2\sigma_4}
\end{matrix} \right \} \,.
\label{n1symmetric}
\end{equation}
Inspection of $N_2^S(\Sigma\vert\Sigma)$ identifies that quadratic non
linearities, and therefore hexagonal pattern formation, are only expected for
an instability of the symmetric mode. In this case, the critical modes are
linear combinations of the modes $\sigma_1$ and $\sigma_2$, and
$N_2^S(\Sigma\vert\Sigma)$ plays an important role since the first two
components contains products of two unstable modes. Alternatively, if an
asymmetric mode becomes unstable, the critical modes are linear combinations of
$\sigma_3$ and $\sigma_4$. The third and fourth components of the vector
$N_2^S(\Sigma\vert\Sigma)$ contain products of one stable and one unstable
mode, but no products of two unstable modes. The adiabatic elimination of the
stable modes yields quadratic terms involving two unstable modes, but these are
terms of higher order and can be neglected.

For an elliptically polarized solution, there is an additional 
contribution to the quadratic non linearity $N_2$; we find
\begin{equation}
N_2(\Sigma\vert\Sigma) = N_2^S(\Sigma\vert\Sigma) + 
{\eta R \over 4} M_2(\Sigma\vert\Sigma),
\label{n1} 
\end{equation}
where
\begin{equation}
M_2(\Sigma\vert\Sigma) = \left \{ \begin{matrix}
{2B\sigma_1 \sigma_4 - 4\sigma_2 \sigma_3 \cr
2(4-B)\sigma_1 \sigma_3 +4\sigma_2 \sigma_4 \cr
2B\sigma_1 \sigma_2 - 4\sigma_3 \sigma_4 \cr
-3B\sigma_1 \sigma_1 +(4-B)\sigma_3 \sigma_3
-B(\sigma_2 \sigma_2 +\sigma_4 \sigma_4)}
\end{matrix} \right \} \,.
\label{m1}
\end{equation}
Therefore, even for a purely asymmetric unstable mode there are
important quadratic contributions which involve the unstable modes 
($\sigma_3 \sigma_3$, $\sigma_3 \sigma_4$ and $\sigma_4 \sigma_4$), and
hexagonal pattern formation is generally expected.



\section{Linearly polarized input field}
\label{linearpolarization}

In this section we discuss transverse polarization patterns in the case of
linearly polarized input field, which we take to be X-polarized. We have found
(see Fig. \ref{fig1}) two types of homogeneous steady state solutions: a
symmetric solution that is also X-polarized and an asymmetric one. The marginal
stability curve for each solution is obtained from the eigenvalues of the
matrix $L$ given in Eqs. (\ref{lambdasymmetric}) and (\ref{10}).

We first consider the stability properties of the solutions
with respect to homogeneous perturbations. The symmetric solution becomes
unstable for a zero wave number perturbation ($k=0$) for $I_s = I'$. This
is the point where the asymmetric solution appears. For $I_s>I'$, the
asymmetric solution is stable with respect to homogeneous perturbations.

Finite wave number perturbations destabilize the symmetric solution for
$I_s < I'$ and the asymmetric solution for $I_s > I'$. In Fig. \ref{fig2}
we plot marginal stability curves for $\theta=1$ as a function of $\eta
ak^2$, so that positive values of this parameter correspond to
self-focusing and negative values to self-defocusing. Figure \ref{fig2}(a)
shows the marginal stability curve for the symmetric solution
\cite{geddes1}. For the symmetric solution, the shape of the marginal
stability curves is, in fact, the same for any value of the detuning $\theta$. 
This is because the eigenvalues $\lambda_i$ given by Eq. (\ref{lambdasymmetric})
depend only on $\theta_k$ and $S=2I_s$, so a change in the value of
$\theta$ is equivalent to a displacement of the origin of $\eta ak^2$
(vertical dashed line) by the same amount. The vertical dashed line 
separating the self-focusing and self-defocusing cases moves to the right
if the detuning $\theta$ is increased, and it intersects the left corner of 
region I for $\theta=\sqrt{3}$ (the value of $\theta$
beyond which there is bistability). Since in this paper we are only
considering the non-bistable regime, the vertical dashed line is always
situated to the left of region I. 

Figure \ref{fig2}(b) shows marginal stability curves for the asymmetric 
solution which merge continuously with the marginal curves of the 
symmetric solution for $I_s<I'$. In this case, the eigenvalues 
$\lambda_i$ given by Eq. (\ref{10}) depend on $\theta_k$, $S$ and also on
$R$. As $S$ and $R$ are related by the steady state solution (\ref{3}) 
which depends explicitly on $\theta$, the shape of the marginal stability
curves changes slightly for different values of $\theta$. 

For values of $\theta$ in the range $2/B-1<\theta<\sqrt{3}$, the situation
is similar to the case shown in Figs. \ref{fig2} (a) and (b). The
homogeneous symmetric (linearly polarized) solution is stable for low
values of the input or the field intensity and it becomes unstable to
finite wave number perturbations for values of $I_s <I'$. The marginal
stability curves for the symmetric solution intersect the $k=0$ vertical
axis at $I_s = I'$ in agreement with the analysis of homogeneous
perturbations mentioned above. This analysis also implies that there is
no instability along the vertical axis $k=0$ for the asymmetric solution
Fig. (\ref{fig2}(b)). The analysis of the instability thresholds that arise by
increasing the input intensity is the same in both Figs. \ref{fig2} (a)
and (b), with the threshold being lower for self-focusing than for the
self-defocusing case. However, as shown in Fig. \ref{fig2}(b), for the
self-defocusing case the region of instability IV is now reduced to an
island, so that further above this threshold an homogeneous elliptically
polarized solution is stable. Given that there is no instability along
the vertical axis $k=0$ in Fig. \ref{fig2}(b), the island IV extends around its
minima until it touches tangentially the vertical axis at $k=0$.
If $\theta$ is decreased, the vertical axis moves to the left and the size of
the island IV becomes smaller until it disappears for $\theta=2/B-1$.

No regions of stability are found above $I'$ in Fig. \ref{fig2}(b) for the
self-focusing case. The different instability islands and
tongues found here will turn out to be a useful guide for analyzing the 
stability of the homogeneous solution for elliptically
polarized input. We recall that such a solution appears continuously from
the asymmetric solution analyzed here. In particular the instability
tongue in the middle, labeled as II, is associated with an eigenvalue
with non-vanishing imaginary part and therefore it identifies a possible
Hopf bifurcation at finite wave number.

It is possible to do a weakly non linear analysis to predict the kind of
pattern that emerges at threshold. The structure of the amplitude 
equations may be easily obtained either in the self-focusing or 
self-defocusing case. In the self-defocusing case (negative part of the
horizontal axis in Fig. \ref{fig2}), the homogeneous symmetric solution
becomes unstable for $I_{s\pm}^c=1/B$ and the critical wave number is 
given by $ak_c^2=\theta+1-2/B$. The instability is stationary,
supercritical and it comes from the $\sigma_3,\sigma_4$ box of the linear
matrix $L$ in Eq. (\ref{linmatrix}), so that the critical mode is an
antisymmetric mode of zero frequency. As discussed in Sec. \ref{model}, 
this implies that quadratic non linearities are not important and stripe
patterns are expected. The amplitude equation of the stripe pattern has been 
presented in \cite{geddes1}. Given the antisymmetric nature of the unstable 
mode, the
X-polarized component of the field is stable and remains homogeneous,
while the stripe pattern appears in the Y-polarized component, which has
zero value below the instability. Overall, the electric field displays an
elliptically polarized spatial structure. We remark that such an instability
is of pure vectorial nature with no analogue when the polarization degree of
freedom is frozen. In fact, no pattern formation instability occurs in
this case for a self-defocusing medium.

In Fig. \ref{fig4} we give an example of this polarization pattern 
instability \cite{num_method} showing a sequence of plots of $|E_+|^2$ for
increasing values of the input field. The first plot corresponds to a
situation close to threshold where the stripe pattern emerges. The
snapshots shown correspond to long lived transient states that evolve to
an ordered stripe pattern by defect evolution and annihilation. In the 
last plot, the input intensity is such that the homogeneous asymmetric
state is stable since we are outside the island of instability in Fig.
\ref{fig2}(b). The system segregates into two phases which correspond to the
two equivalent homogeneous elliptically polarized solutions. The evolution
of the system at later times is dominated by the motion of the interfaces
separating the two stable phases.

For the self-focusing case (positive part of the horizontal axis in Fig.
\ref{fig2}), the homogeneous symmetric solution becomes unstable for
$I_{s\pm}^c = 1/2$ with a critical wave number given by $ak_c^2=2-\theta$.
The instability is also stationary, but it comes now from the
$\sigma_1,\sigma_2$ box of the linear matrix $L$ in Eq. (\ref{linmatrix})
\cite{geddes1}. The critical mode is therefore a symmetric mode. In this
case, quadratic non linearities are present, and as discussed in Sec.
\ref{model}, one then expects formation of hexagonal patterns via a
subcritical bifurcation. Such hexagonal patterns are shown in Fig.
\ref{fig5}. The situation corresponds to the case discussed in
\cite{firt}, in which the polarization degree of freedom is not taken
into account. The instability leads to an X-polarized pattern while the
Y-polarized component of the field continues to be zero. If the intensity
is further increased, the hexagonal structure is disarranged
\cite{firthsolfract}. In this case, the state of the system is far away
from the steady state solution, so the marginal stability plots of Fig.
\ref{fig2} are no longer useful. We observe a spatiotemporal dynamics in
which the intensity of the hexagonal peaks grow, leading to high intensity 
localized structures placed randomly. These structures eventually burst,
producing circular waves that propagate in the transverse plane and
dissipate away (see Fig. \ref{fig5}). In the two dimensional self-focusing
non linear Schr\"odinger equation, there is a phenomenon of wave collapse 
\cite{NLS}. Collapse is known to be prevented by dissipation
\cite{goldman80} or by a saturation nonlinearity \cite{NLS}. In our
problem, we have dissipation and a driving field. We have checked that the
same phenomenon appears when our cubic nonlinearity is replaced by a
saturating nonlinearity. What we then observe is then a strong effect of 
self-focusing in a situation with no collapse.

We finally consider the range of detuning values $\theta \le 2/B-1$. In
this range, the vertical dashed line $k=0$ is at the left of the minimum
of region IV in Fig. \ref{fig2} (a). In the equivalent of Fig. \ref{fig2}
(b) for these values of $\theta$, the island IV appears to the right of
the vertical axis $k=0$ and extends around its minimum,
located at $ak_c^2=-\theta-1+2/B$, until it touches tangentially the
vertical axis $k=0$. According to this picture, in the self-defocusing
case the linearly polarized homogeneous solution becomes unstable at
$I_s=I'$ to $k=0$ perturbations, leading to an elliptically polarized
homogeneous solution with no pattern being formed. In the self-focusing
case, the situation is very similar to the one described before for
$2/B-1<\theta<\sqrt{3}$. Although region IV is now located in the
self-focusing region, it does not contribute to the instability scenario,
since its minimum value ($I_{s\pm}^c = 1/B$) is above the minima of
region I ($I_{s\pm}^c = 1/2$). So, as the input field intensity is
increased, the homogeneous symmetric solution becomes unstable for
$I_{s\pm}^c = 1/2$ with a critical wave number given by $ak_c^2=2-\theta$
and a hexagonal pattern is formed via a subcritical bifurcation. For the
particular limiting value $B=2$, the minimum of the region IV is located 
at the same value as the minimum of region I. These two instability
regions are associated to real eigenvalues, so they identify two
stationary (Turinglike) bifurcations. So, for $B=2$ and $\theta \le
2/B-1=0$, starting from the linearly polarized homogeneous solution, as
the input field is increased, the system crosses the two instability
thresholds simultaneously. This is a codimension 2 bifurcation involving
two stationary modes. The critical mode associated to region I is
symmetric and has a critical wave number $ak_c^2=2-\theta$, while a
critical mode associated to region IV is asymmetric and has a critical
wave number $ak_c^2=-\theta$. 


\section{Elliptically polarized input field}
\label{ellipticalpolarization}

In this section we present a general description of the stability analysis
of the stationary state when the input field is elliptically polarized.
Throughout this section, we are considering a detuning $\theta =1$.

Fig. \ref{fig3b} shows, for different input field ellipticities $\chi$, 
the marginal stability curve obtained introducing the stationary solution
(\ref{3}) in Eq. (\ref{10}) and solving $\Re(\lambda)=0$. 

For small input field intensity, when the ellipticity is small, the 
homogeneous solution is close to the symmetric solution for linearly 
polarized input field so that $R<<S$. The eigenvalues $\lambda$ of
the linear evolution matrix given in Eq. (\ref{10}) can be expanded as a 
series in $R^{2n}$ in the form
\begin{equation}
\lambda_i = \lambda^0_i +\lambda^1_i R^2 ,
\end{equation}
where $\lambda^0_i$ are the eigenvalues for the case of the purely
linearly polarized input field $\chi=\pi/2$. Thus, the instability 
thresholds displayed in Fig. \ref{fig3b}(a) for $\chi=87^\circ$ are very 
similar to the ones of Fig. \ref{fig2}(a) corresponding to the symmetric 
solution of the linearly polarized input field. For larger input field and
small ellipticity, the homogeneous solution is close to the asymmetric 
solution for linearly polarized input field, and the marginal stability
curves shown in Fig. \ref{fig3b}(a) are very similar to the ones of Fig.
\ref{fig2}(b).

In the self-defocusing case (negative part of the horizontal axis of Fig. 
\ref{fig3b}), the size of the instability region of the homogeneous
solution is decreased as ellipticity is increased. For large enough
ellipticity, this instability island disappears and the elliptically
polarized homogeneous solution is always stable (see Fig. \ref{fig3b}(b)
and (c)). For small ellipticity, the critical modes are basically
combinations of the antisymmetric modes $\sigma_3$ and $\sigma_4$. Despite
the correction in the eigenvalues and eigenvectors being small, the
non linear terms in the amplitude equation are modified. The quadratic
terms $N_2(\Sigma\vert\Sigma)$ now contain products of the two unstable 
modes through $M_2(\Sigma\vert\Sigma)$
(see Eqs. (\ref{n1}) and (\ref{m1})). As discussed in Sec. \ref{model},
these terms should induce the formation of hexagonal rather than stripe
patterns (as was the case for linear polarization of the input field), at
least close to threshold \cite{CrossHoh,geddes2}. The formation of
hexagonal patterns rather than stripes due to the presence of small
ellipticities in the input field has been experimentally observed in
rubidium vapor \cite{Maitre95}. Since the quadratic terms are proportional
to $R$, the range of stability of hexagonal planforms should be
proportional to $R^2$ \cite{CrossHoh,shoka}. In principle, in this
situation, on increasing the bifurcation parameter (input field
intensity), hexagons should become unstable and stripes should be observed
\cite{shoka}. However, due to the small size of the island where the
homogeneous solution is unstable, when increasing the input field
intensity, we never found in our simulations the stripe pattern. Instead,
we have a transition back to the elliptically polarized homogeneous state
which is stable. In Fig. \ref{fig6} we present a sequence of plots of
$|E_+|^2$ for increasing values of the input intensity. Near threshold we
have hexagons, and for a large enough intensity, the homogeneous solution is
restored. Different from the case of linearly polarized input, here there
is no competition between two phases with dominant $E_+$ or $E_-$, because
the small ellipticity that we have introduced makes the system choose
the solution with $I_{s+}>I_{s-}$. Since the quadratic non linearities are
proportional to $R$, the sign of $R$ should determine if the hexagons are
of the $0$- or $\pi$-type \cite{Aumann97,shoka,Ackeman95}. As the
dynamical evolution of the perturbations $\psi_+$ and $\psi_-$ is
associated with $\pm R$ (Eq. (\ref{psimasmenos})), we find the opposite
type of hexagons for $|E_+|^2$ and $|E_-|^2$. Changing the ellipticity
$\chi$ to $\pi-\chi$ induces a transition, for a given circularly
polarized component of the field, from one type of hexagons to the other,
similarly to what has been reported in Ref. \cite{Aumann97}. We also note
that, near threshold, the hexagonal pattern looks different if we consider
the $X$ or $Y$ components of the field. In Fig. \ref{fig6b} we plot
$|E_x|^2$ and $|E_y|^2$, for which we find hexagonal pattern of the
``black eyes'' type. Similar patterns have been observed in chemical
systems \cite{ouyang}. Here they arise because of the superposition of
$E_+$ and $E_-$.

In the self-focusing case (positive part of the horizontal axis of Fig. 
\ref{fig3b}), and for small ellipticity, the islands and tongues of
instability of the homogeneous solution are obtained continuously from the
ones for the asymmetric solution of linearly polarized input field (Fig.
\ref{fig2} (b)). The island I in Fig. \ref{fig3b} (a)) corresponds to a
stationary instability of the modes which by continuity go to the
symmetric modes when $R\rightarrow 0$. As there are quadratic terms in the
amplitude equation, an hexagonal pattern is formed, similarly to the case
of linearly polarized input field. The tongues II and III are far away
from the threshold for instability of the homogeneous solution and are
plotted in this figure only to display how they move as we increase the
ellipticity. As in the case of linear polarization of the input field, the
tongue II is associated with a Hopf bifurcation and the tongue III with
a stationary instability.

Figure \ref{fig3b}(b) shows the marginal stability curve for
$\chi=78^\circ$. When the input field intensity is increased starting from
zero, we have, as usual, a first instability, of the Turing type, where an
hexagonal structure emerges. The instability island I is now smaller and
there is a window for $I_{s+}$ around 2, where the elliptically polarized
homogeneous solution is stable. By further increasing the input field
intensity, a second instability appears when the value of $I_{s+}$
crosses the instability threshold of the tongue II. The corresponding
eigenvalues of the linear evolution matrix have non zero imaginary parts
and cross the imaginary axis at finite wave number, so that this
instability is a Hopf bifurcation with broken space translational
symmetry. This situation is discussed in detail in Sec. \ref{Blinking
Hexagons}.

As we can see from the sequence of plots in Fig. \ref{fig3b}, the tongue
II moves upwards and the tongue III downwards as the ellipticity of the
input field is increased ($\chi$ is decreased). Beyond the island of
instability I, the patterns that are expected to form depend crucially on
the relative position of the Hopf instability (tongue II) and the
stationary instability of tongue III. When the stationary instability is
the first to appear on increasing the bifurcation parameter, steady
spatial structures may be expected. If the Hopf bifurcation is the
first to appear, however, one should obtain wavy spatiotemporal structures. 
If both instabilities are at the same level, one has a codimension 2 
situation, as shown in Fig. \ref{fig3b}(c) for $\chi=73^\circ$. With
respect to the situation of Fig. \ref{fig3b}(b), the instability tongue II
has moved upwards and to the right, while tongue III has moved downwards
until the instabilities associated with each of the two tongues take place
at the same value of $I_{s+}$. There is now a large range of values of
$I_{s+}$ between the island I and the two tongues for which the elliptically
polarized homogeneous solution is stable. Increasing $I_{s+}$ from a value
in this range, the homogeneous state has a codimension 2 bifurcation
where steady and wavy modes should interact, ending with pure Turing, pure
Hopf or mixed modes, according to their non linear interaction. This case
is discussed in more detail in Sec. \ref{cod2}.

For $\chi \rightarrow 0$, tongue II disappears, and tongue III is the only
remaining region of instability. In Fig. \ref{fig3b}(d) we plot the
marginal stability curve for a right-handed circularly polarized input 
field ($\chi=0$). As discussed in Sec. \ref{model}, this
case is equivalent to the scalar case, already described in
\cite{lugi,firt}. The steady state solution given by Eq. 
(\ref{circpolarizedsol}) is the
same as the symmetric solution for linearly polarized input except for a
rescaling of the intensities. After this rescaling, in the self-focusing
case, the marginal stability curve is also the same as the one for
linearly polarized input, and the same patterns are observed above
threshold. In the self-defocusing case, however, we do not have any
instability of the homogeneous solution. As stated in Sec.
\ref{linearpolarization}, for linearly polarized input field the
self-defocusing instability involves the asymmetric modes. Here, there is
only one relevant component of the field, and there are not enough degrees
of freedom for such an instability to occur.


\section{Modified Hopf Bifurcation: Deformed Dynamical Hexagons}
\label{Blinking Hexagons}

In Fig. \ref{fig7d}, we plot the squared absolute value of $E_+$ and $E_-$ 
in the near and the far field for an input field intensity such that the
value of $I_{s+}$ is slightly above the threshold of the Hopf instability
(region II) shown in Fig. \ref{fig3b}(b). We can see that a distorted 
hexagonal structure appears for
$E_+$ and $E_-$. The component $E_-$ is correlated with $E_+$ but has a
lower intensity because $\chi < 90^\circ$ gives preference to $E_+$. The
structure has a dynamical evolution as shown in Fig. \ref{fig7a}, where we
plot four configurations of $|E_+|^2$ at different times. Inspection of
the numerical results for the far field indicates that $E_+$ is dominated
by a triad of 3 wave vectors with $|\vec k| = k_h$, while $E_-$ is
dominated by the triad with opposite wave vectors. In addition, we observe
that these 3 wave vectors are not equivalent, since two of them, which form
an angle close to 90$^\circ$, carry a higher spectral power than the third
one. The modes of $E_+$ with highest intensity are identified in Fig.
\ref{fig7e:hexagons_modes}. The two equivalent wave vectors are labeled
$\vec k_2$ and $\vec k_3$, and the third dominant wave vector is labeled
$\vec k_1$. A basic feature of the dynamical evolution of the pattern can
be understood considering the time evolution of each of these modes. The
amplitude of any of these Hopf modes $|\vec k| = k_h$ evolves with a fixed
frequency. The frequency has the same absolute value for all these modes
but different signs, as indicated in Fig. \ref{fig7e:hexagons_modes}. In
Fig. \ref{fig7b} we display, as an example, the phase of the mode $\vec
k_2$. The frequency obtained from this plot, $\omega = 1.63$, coincides
with the imaginary part of the critical eigenvalue associated with the
Hopf instability.

Since the numerical results are obtained slightly above the instability
threshold, we may hopefully interpret them in the framework of reduced
dynamics and amplitude equations for the unstable modes. Let us first
recall that we are dealing with a Hopf instability with broken
translational symmetry. The real part of the most unstable eigenvalues,
$\lambda_1$ and $\lambda_2$, is plotted in Fig. \ref{fig7c}. These
eigenvalues are complex conjugated for $k \simeq k_h$ and $\lambda_{1,2}
(k_h) = \pm i\omega$.

The field $E_\pm$ can be projected on the eigenvectors of the linear
evolution matrix $L$ and using Eq. (\ref{perturba}) may be written as
\begin{eqnarray}
\left( \begin{array}{c} E_+ \\ E_+^* \\ E_- \\ E_-^* \end{array} \right) =
\left( \begin{array}{c} E_{s+} \\ E_{s+}^* \\ E_{s-} \\ E_{s-}^* \end{array}
\right) + \left( \begin{array}{c} E_{s+} \psi_+ \\ E_{s+}^* \psi_+^* \\
E_{s-} \psi_- \\ E_{s-}^* \psi_-^* \end{array} \right) =
\left( \begin{array}{c} E_{s+} \\ E_{s+}^* \\ E_{s-} \\ E_{s-}^* \end{array}
\right) \nonumber\\
+ \sum_{\vec k} e^{i\vec k \vec r} \left( V_1 \sigma_{1,\vec k} + V_2
\sigma_{2,\vec k} + V_3 \sigma_{3,\vec k} + V_4 \sigma_{4,\vec k} \right),
\end{eqnarray}
where $V_i$ are the four eigenvectors, in Fourier space, and 
$\sigma_{i,\vec k}$ are the corresponding amplitudes. 
The usual procedure of reducing the dynamics to the dynamics of the unstable 
modes only, leads to the adiabatic elimination of the modes $V_3 
\sigma_{3,k}$ and $V_4 \sigma_{4,k}$ because $\lambda_3$ and $\lambda_4$ are 
stable eigenvalues.
Taking into account that the eigenvalues $\lambda_1$ and
$\lambda_2$ are most unstable for $\vec k = \vec k_h$ with $|\vec k_h| =
k_h$, we may write, close to the instability \cite{CrossHoh}
\begin{eqnarray}\label{redfield}
&&\left( \begin{array}{c} E_+ \\ E_+^* \\ E_- \\ E_-^* \end{array} \right) =
\left( \begin{array}{c} E_{s+} \\ E_{s+}^* \\ E_{s-} \\ E_{s-}^* \end{array}
\right) \nonumber \\
&&+ \sum_{\vec k_h}
\left( V_1 \sigma_{1,\vec k_h} e^{i \vec k_h \vec r + i \omega t} + V_2
\sigma_{2,\vec k_h} e^{i \vec k_h \vec r - i \omega t} \right) + \dots .
\end{eqnarray}
The amplitudes, $\sigma_{1,\vec k_h}=\sigma_{1,\vec k_h}(\vec X,T)$
and $\sigma_{2,\vec k_h}=\sigma_{2,\vec k_h}(\vec X,T)$, only depend on
the slow variables $\vec X=\varepsilon^{1/2}\vec r$ and
$T=\varepsilon^{-1}t$, where $\varepsilon= {I_{s+}-I_{s+}^c \over I_{s+}^c}$ 
is the reduced distance to the instability threshold (we are using $I_{s+}$ 
as the bifurcation parameter and $I_{s+}^c$ is the critical value at the Hopf 
bifurcation). For each particular pattern,
their evolution equations, or amplitude equations, may be derived with
standard procedures \cite{daniel}. However, it is often convenient to
study instead order parameter equations of the Swift-Hohenberg type.
These equations reduce to the correct amplitude equations near the onset of
instability but take care of the orientational degeneracy of the unstable
wavectors, preserve the correct symmetries of the problem, allow the
description of transitions between patterns of different symmetries and
contain rapid spatiotemporal variations which may be important for
pattern selection or transient dynamics \cite{CrossHoh,shoka,manneville}.
In the present case, we consider a model order parameter dynamics of the
type
\begin{eqnarray}\label{SHOsc}
\partial_t\sigma_{\pm} &=&
[\varepsilon - \xi_h^2(k_h^2 +\nabla^2 )^2 \pm i\omega ]\sigma_{\pm} +
v\sigma_{\mp}^2 \nonumber \\
&& - (1 \pm i\beta)\sigma_{\pm}^2\sigma_{\mp} ,
\end{eqnarray}
where the subscript $+$ and $-$ refer to the sign of the frequency, so
that the complex $\sigma_{\pm}$ are proportional to the wave packet 
$\sigma_{1(2),\vec q}\exp (i\vec k \vec r \pm i\omega t)$, with 
$\vert \vec k\vert \simeq k_h$. These equations contain quadratic non 
linearities,
as discussed in Sec. \ref{model}, and are thus equivalent to the equations
describing oscillatory convection in hydrodynamic systems with no
``up-down" symmetry \cite{brand}. In this case, the authors obtain
monoperiodic regular states of hexagonal symmetry that correspond to a
two-dimensional standing wave. We will call these states pulsating hexagons.

Our system has, however, the following originality. The eigenvalue $\lambda_2$ 
is real and only slightly negative for the modes $\sigma_{0}$ with a wavevector
such that $\vert \vec k\vert = k_s \simeq \sqrt 2 k_h$ (see Fig. \ref{fig7c}).
As $k_s \simeq \sqrt 2 k_h$, the modes $\sigma_{0}$ may be coupled with pairs
of Hopf modes $\sigma_{\pm}$ with orthogonal wave vectors via quadratic
resonances. Since the modes $\sigma_{0}$ are only slightly damped, one should
incorporate them in the order parameter dynamics, which then becomes
\begin{eqnarray}\label{SHgen}
\partial_t\sigma_{\pm} &=&
[\varepsilon - \xi_h^2(k_h^2 +\nabla^2 )^2 \pm i\omega ]\sigma_{\pm}
+ v_0\sigma_{\mp}^2 + v_1\sigma_{\pm}\sigma_{0} \nonumber\\
&& + v_2 \sigma_{0}^2 - (1 \pm i\beta)\sigma^2_{\pm}\sigma_{\mp} 
-( \gamma \pm i\delta)\sigma^2_{0}\sigma_{\pm}\nonumber\\
\partial_t\sigma_{0} &=&
[\mu - \xi_s^2(k_s^2 +\nabla^2 )^2 ]\sigma_{0} + \bar v_1
\sigma_{+}\sigma_{-} +\bar v_2 \sigma_{0}^2- \sigma_{0}^3 \nonumber\\
& & - u\sigma_{0}\sigma_{+}\sigma_{-} ,
\end{eqnarray}
where $\mu<0$ is the linear damping of the homogeneous mode ($\mu \propto
\lambda_2(k_s)$). The kinetic coefficients could be obtained numerically
from Eq. (\ref{perturba}). However, this formidable task may be avoided,
since we are mainly interested in generic dynamical behaviors, which
essentially depend on their signs and orders of magnitude.

Let us look for the possible asymptotic solutions of this system. Systems
described by the dynamics (\ref{SHOsc}) have been mainly studied in
one-dimensional geometries where the resulting patterns correspond to
traveling or standing waves \cite{CrossHoh,haken,lomdahl}. These solutions
are recovered here. Effectively, the uniform amplitude equations for
critical unidirectional counterpropagating traveling waves corresponding
to the modes $\sigma_0=0$, $\sigma_+ = L\exp (i\vec k_h\vec r +i\omega t)
+ R^*\exp (-i\vec k_h\vec r + i\omega t)$ and $\sigma_-= L^*\exp (-i\vec
k_h\vec r - i\omega t) + R\exp (i\vec k_h\vec r - i\omega t)$ ($\vert \vec
k_h\vert = k_h$) may be deduced from (\ref{SHgen}), and are
\begin{eqnarray}\label{TW}
\dot L &=& \varepsilon L - (1+i\beta )L(\vert L\vert^2
+2 \vert R\vert^2 )\nonumber\\
\dot R &=& \varepsilon R - (1-i\beta )R(\vert R\vert^2 +2 \vert L\vert^2) .
\end{eqnarray}
The non linear cross-couplings coefficients of the field equations
are twice the self-coupling coefficients. In this situation, like in 
reaction-diffusion systems with scalar couplings \cite{shoka}, traveling 
waves are stable structures whereas standing waves are unstable.

As we are considering two-dimensional systems, we have to
study the stability of such waves versus modulations with wave vectors
pointing in other directions. In the absence of coupling between
$\sigma_{\pm}$ and $\sigma_0$, one should obtain pulsating hexagons, as in
\cite{brand}. The coupling between oscillatory $\sigma_{\pm}$ and steady
$\sigma_0$ modes may modify this picture, however. Effectively, let us 
consider the linear stability
of a traveling wave defined by $\sigma_-=A \exp \left( i
k_h{x+y\over\sqrt{2}} - i\omega_0 t \right) $, $\sigma_+= A^* \exp \left(
-i k_h{x+y\over\sqrt{2}} + i \omega_0 t \right) $, with $\vert
A\vert^2=\varepsilon$ and $\omega_0 =\omega -\beta\varepsilon$ which
corresponds to a solution of (\ref{TW}) with $L=0$ and $R=A e^{i\beta
\varepsilon t}$. (The wave vector direction is arbitrary, and we may choose
this particular one, anticipating results of the following discussion. Based on
the wave vector definition of Fig. \ref{fig7e:hexagons_modes}, $x$ and
$y$ being the spatial coordinates in the plane, the right traveling wave
just described corresponds to mode $A$.) Mode $A$ is quadratically coupled
to the modes $B \exp \left( i k_h{x-y\over\sqrt{2}}- i \omega_0 t \right)
$ and $D e^{ik_sy}$ and their complex conjugates. Taking $\sigma_-=A \exp
\left( i k_h{x+y\over\sqrt{2}} - i \omega_0 t \right) + B \exp \left( i
k_h{x-y\over\sqrt{2}}- i \omega_0 t \right)$, $\sigma_+= A^* \exp \left(
-i k_h{x+y\over\sqrt{2}} + i \omega_0 t \right) + B^* \exp \left( -i
k_h{x-y\over\sqrt{2}} + i \omega_0 t \right) $, and $\sigma_0 = D
e^{ik_sy} + D^* e^{-ik_sy}$, the corresponding linearized amplitude
equations are of the form
\begin{eqnarray}
\dot B &=& -\varepsilon B + v_1 A D^* - \dots \nonumber\\
\dot D &=& (\mu - u\varepsilon) D + \bar v_1 A B^* - \dots .
\end{eqnarray}
The characteristic equation of the corresponding evolution matrix is
\begin{equation}
s^2 - s((1+u)\varepsilon -\mu )+\varepsilon (u\varepsilon -\mu)
- v_1\bar v_1\varepsilon = 0 ,
\end{equation}
and the traveling waves are thus unstable for
\begin{equation}
\varepsilon < \varepsilon_c = {1\over u }(v_1\bar v_1 +\mu ) .
\end{equation}
If $v_1\bar v_1$ is negative, $\varepsilon_c < 0$ and unidirectional
traveling waves are stable. This is, for example, the case in systems
where the order parameter dynamics contains non linear couplings between
the gradients of the field \cite{proctorjones}. Alternatively, if
$v_1\bar v_1$ is positive, and we may suppose that this is the case here,
$\varepsilon_c $ is positive when $\mu $ has sufficiently small absolute
value (we recall that $\mu<0$). Hence, unidirectional traveling waves are
unstable versus two-dimensional spatiotemporal patterns in the range
$0<\varepsilon < \varepsilon_c$.

We will now try to identify these patterns, taking into account the fact
that the dynamics favors propagating waves and contains quadratic non
linearities. Hence, we consider that the dynamics may be reduced to the
dynamics of the modes A, B, C, D and their respective complex conjugates, 
as defined in Fig. \ref{fig7e:hexagons_modes}. Starting with mode A, modes 
B and D should be included in the description because A is unstable versus 
B and D. Mode C should be also included because of the coupling between 
A and B. The coupling between A and B also generates higher harmonics with
frequency $-2\omega$ (vector $\vec k_5$ in Fig. \ref{fig7e:hexagons_modes}) 
which can be observed in the far field shown in Fig. \ref{fig7d}. However, 
since the dynamics of these harmonics is slaved to the dynamics of A and B,
they are not considered here. The amplitude equations for modes A, B, C and D 
obtained from (\ref{SHgen}) taking into account quadratic non 
linearities between wavy modes are
\begin{eqnarray}
\partial_t A &=& \varepsilon A+4\xi_h^2(\vec k_2.\vec\nabla )^2A + v_1 DB +
v_0C^*B^* e^{i(3\omega t +\kappa x)}\nonumber\\
&-&(1+i\beta)A(\vert A\vert^2 +
2 (\vert B\vert^2 +\vert C\vert^2))-(\gamma+i\delta)A\vert D\vert^2\nonumber\\
\partial_t B &=&\varepsilon B+ 4\xi_h^2(\vec k_3.\vec\nabla )^2B + v_1 AD^*
+ v_0 C^*A^* e^{i(3\omega t +\kappa x)}\nonumber\\
 &-& (1+i\beta)B(\vert B\vert^2 +2
(\vert A\vert^2 + \vert C\vert^2))-(\gamma+i\delta)B\vert D\vert^2\nonumber\\
\partial_t C &=& \varepsilon C+ 4\xi_h^2(\vec k_1.\vec\nabla )^2C + v_0
A^*B^* e^{i(3\omega t +\kappa x)} \nonumber\\
&-& (1+i\beta)C(\vert C\vert^2 +2
(\vert B\vert^2 + \vert A\vert^2)) -(\gamma+i\delta)C\vert D\vert^2\nonumber\\
\partial_t D&=&\mu D+ 4\xi_s^2(\vec k_4.\vec\nabla )^2D + \bar v_1 AB^* -
\dots ,
\end{eqnarray}
where $\kappa =(1-\sqrt 2 )k_h $. $D$ may be adiabatically eliminated and
$D\simeq -{\bar v_1 AB^*\over \mu}$. The mismatch between the modes A, B 
and C may be absorbed in their phases ($\phi_i \to \bar\phi_i - \kappa
x/3$). Writing $I= R_I\exp \phi_I$ and defining the total phase as
$\Phi = \bar\phi_A +\bar\phi_B+\bar\phi_{C} -3\omega t $, we finally get
the following equations for uniform amplitudes and phases, up to cubic
non linearities:
\begin{eqnarray}\label{dyn}
\partial_t R_A &=& \varepsilon_0 R_A+ v_1 R_BR_{C} \cos \Phi - R_A(R_A^2
+\bar\gamma R_B^2 +2 R_{C}^2)\nonumber\\
\partial_t R_B &=& \varepsilon_0 R_B+ v_1 R_AR_{C} \cos \Phi - R_B(R_B^2
+\bar\gamma R_A^2 +2 R_{C}^2)\nonumber\\
\partial_t R_{C} &=& \varepsilon_1 R_{C} + v_1R_AR_B\cos \Phi - R_{C}(R_C^2 +
2 (R_A^2+R_B^2))\nonumber\\
\partial_t\bar\phi_A &=& - v_0{R_BR_{C}\over R_A}\sin \Phi +
O(\beta,\delta,R^2)\nonumber\\
\partial_t\bar\phi_B &=& - v_0{R_AR_{C}\over R_C}\sin \Phi +
O(\beta,\delta,R^2)\nonumber\\
\partial_t\bar\phi_{C} &=& - v_0{R_AR_B \over R_{C}}\sin \Phi + O
(\beta,\delta,R^2)\nonumber\\
\partial_t\Phi &=& -3\omega - v_1{2R_{C}^2+R_A^2 \over R_{C}}\sin \Phi + O
(\beta,\delta,R^2) ,
\end{eqnarray}
where $\varepsilon_0 = \varepsilon - {2\over 9}\xi_h^2(1-\sqrt 2 )^2k_h^4
\simeq \varepsilon - 0.038\xi_h^2k_h^2$, $\varepsilon_1 = \varepsilon -
{4\over 9}\xi_h^2(1-\sqrt 2 )^2k_h^4 \simeq \varepsilon -
0.075\xi_h^2k_h^2$ and $\bar\gamma = 2 +{v_2\bar v_1 \over \mu}$. From
these equations, it turns out that $A$ and $B$ are equivalent, $R_A = R_B
\ne R_C$ and $\bar\phi_A - \bar\phi_B = \varphi$, where $\varphi$ is an 
arbitrary constant. We may choose $\varphi=0$ for simplicity.

Except for sufficiently small $\omega$, where the system (\ref{dyn}) may 
admit fixed point solutions, this system of equations is expected to generate 
time-periodic
solutions corresponding to pulsating deformed hexagons. Over a period,
the mean value of the amplitude of the modes A and B should be equal
while the mean value of the amplitude of C should be smaller ($\varepsilon_1
<\varepsilon_0$, and $\bar\gamma <\gamma$). In the absence of coupling
with the $D$ mode ($\mu << 0$), one should recover the pulsating hexagons
found by Deissler and Brand \cite{brand}. It is the particularity of this
system to present a resonant interaction with a slowly evolving stable
mode which induces the deformation of the hexagonal pattern. In general
$A$ and $B^*$ modes need not make a particular angle for their quadratic
resonance with a stationary mode. If they make an angle $\psi$ with the
y-axis, they should be coupled with the stable $D$ mode (with $k_s \hat y
= 2k_h\sin \psi \hat y$, where $\hat y$ is the unit vector along the
Y-axis) in such a way that
\begin{equation}
\partial_t D(\psi) = [\mu - 4\xi_s^2(k_s^2- 4\sin^2\psi k_h^2)^2]D(\psi) +
\bar v_1 AB^* +\dots .
\end{equation}
Hence, the adiabatic elimination of this mode leads to a renormalized
coefficient $\bar\gamma$ which is minimum for $\sin \psi = k_s / 2k_h$,
The angle $\psi$ is then half of the selected angle between $A$ and $B^*$ 
wave vectors. Changing the detuning $\theta$, the ratio between $k_s$ 
and $k_h$ can be tuned, which then changes the distortion of the 
hexagonal pattern.

If one now wishes to reconstruct the field from the results of this weakly
non linear analysis, one obtains from equations (\ref{redfield}) and 
(\ref{dyn}), at leading order,
\begin{eqnarray}\label{reconstruct}
\left( \begin{array}{c} E_+ \\ E_+^* \\ E_- \\ E_-^* \end{array} \right) &-&
\left( \begin{array}{c} E_{s+} \\ E_{s+}^* \\ E_{s-} \\ E_{s-}^* \end{array}
\right) = \sum_{\vec k_h}
\left( V_1 \sigma_{1,\vec k_h} e^{i \vec k_h \vec r + i \omega t} + V_2
\sigma_{2,\vec k_h} e^{i \vec k_h \vec r - i \omega t} \right) + \dots
\nonumber\\
&=&
V_1 e^{i \omega t} \left( A^* e^{-i \vec k_2 \vec r} + B^* e^{-i \vec k_3
\vec r } + C^* e^{-i \vec k_1 \vec r} \right) 
+ V_2 e^{-i \omega t} \left( A e^{i \vec k_2 \vec r} + B e^{i \vec k_3 
\vec r} + C e^{i \vec k_1 \vec r } \right) + \dots
\nonumber\\
&=& V_1 e^{ i \omega t } \left( 2R_A \cos {k_h y\over \sqrt 2}
e^{-i {k_h x\over \sqrt 2} - i\bar\phi_A } + R_{C} e^{i k_h x -
i\bar\phi_{C}}\right) \nonumber\\
&& + V_2 e^{ -i \omega t}
\left( 2R_A \cos {k_h y\over \sqrt 2}e^{i {k_h x\over \sqrt 2} + i\bar\phi_A}
+ R_{C} e^{-i k_h x + i\bar\phi_{C}}\right) + \dots .
\end{eqnarray}

From (\ref{dyn}), it may be seen that $\bar\phi_A$ and $\bar\phi_C $ are
functions of $\Phi$. Since $\omega$ is finite, close to the instability
threshold one may expand $\Phi$ around $3\omega t$ \cite{kuramoto}, and
(\ref{reconstruct}) becomes
\begin{eqnarray}\label{reconstructbis}
\left( \begin{array}{c} E_+ \\ E_+^* \\ E_- \\ E_-^* \end{array} \right) -
\left( \begin{array}{c} E_{s+} \\ E_{s+}^* \\ E_{s-} \\ E_{s-}^* \end{array}
\right) & = &
 V_1 e^{ i \omega t } \left( 2R_A \cos {k_h y\over \sqrt 2}
e^{-i {k_h x\over \sqrt 2} } + R_{C} e^{i k_h x } \right) \nonumber\\
&& + V_2 e^{ -i \omega t}
\left( 2R_A \cos {k_h y\over \sqrt 2}e^{i {k_h x\over \sqrt 2} }
+ R_{C} e^{-i k_h x }\right) + \dots
\end{eqnarray}
where $R_A$ and $R_B$ still contain time-dependent contributions of
frequencies $3\omega, 6\omega, \dots $. The corresponding spatiotemporal
patterns are thus different from pulsating hexagons, since, besides their
deformation, they are built on a triad of traveling waves propagating in
the $\vec k_1$, $\vec k_2$ and $\vec k_3$ (or $-\vec k_1$, $-\vec k_2$ and
$-\vec k_3$) directions, leading to what we call deformed dynamical
hexagons. These conclusions are in qualitative agreement with the
numerical results presented in Figs. \ref{fig7d}, \ref{fig7a} and
\ref{fig7b}, which tell us, furthermore, that the eigenvectors $V_1$ and
$V_2$ should have dominant contributions to $E_+$ and $E_-$, respectively.
Effectively, it appears that the dominant contributions to $E_-$ come from
the modes $k_1$, $k_2$ and $k_3$ with frequency $-\omega$, while $E_+$ is
built on the modes $k_1$, $k_2$ and $k_3$ with frequency $\omega$.

We thus think that the main feature which determines the properties of the
patterns presented in Figs. \ref{fig7d} and \ref{fig7a} is the fact that a
constructive coupling occurs between a nearly marginal stationary mode and
unstable oscillatory modes. Couplings between steady and oscillatory modes
have already been shown to be able to induce subharmonic Hopf bifurcations
in one-dimensional reaction-diffusion systems in codimension 2 situations
\cite{Lima96,DeWit96}. Here, this particular coupling is allowed by the
two-dimensional geometry of the system, which induces the bifurcation to
spatiotemporal patterns with deformed hexagonal shape.


\section{Codimension two Hopf and Turinglike instabilities}
\label{cod2}

In this section, we consider the situation shown in Figs. \ref{fig3a}(c)
(steady state) and \ref{fig3b}(c) (marginal stability), corresponding to
an ellipticity $\chi = 73^\circ$ and detuning $\theta=1$. We see from the
marginal stability curve that there are two different wave numbers that
become unstable at nearly the same value of the control parameter
$I_{s+}$. Similar situations can be obtained changing simultaneously the
ellipticity and the detuning. For example, for $\chi = 67^\circ$ and
$\theta=0.6$ the same type of situation occurs (the relation between the
two critical wave numbers changes but the qualitative results are not 
affected). In Fig. \ref{fig8b} we plot the unstable eigenvalues
$\lambda_1$ and $\lambda_2$ as functions of the wave number $k$ near the
instability threshold for $\chi = 67^\circ$ and $\theta=0.6$. In the plot
of $\lambda_2$ we can see that the modes $k_h$ and $k_s$ become 
simultaneously unstable, and, furthermore, $\lambda_2(k_h)$ is complex,
while $\lambda_2(k_s)$ is real. Hence, we have a codimension two 
situation where the oscillatory instability corresponding to a Hopf 
bifurcation with broken spatial symmetry and the stationary Turinglike 
instability are close together. We show in Fig. \ref{fig8a} numerical 
results for the near and the far field of $E_+$ at three different times during
the transient following this codimension two bifurcation. These results
display the competition between the Hopf and the static modes. At short
times the Hopf modes dominate, but at long times a static hexagonal
pattern is formed.

In the vicinity of the codimension two point, the field variables may be 
written as: 
\begin{eqnarray}
\left( \begin{array}{c} E_+ \\ E_+^* \\ E_- \\ E_-^* \end{array} \right) &=&
\left( \begin{array}{c} E_{s+} \\ E_{s+}^* \\ E_{s-} \\ E_{s-}^* \end{array} 
\right) \nonumber \\
&&+ \sum_{\vec k_h}\left(
V_1 \sigma_{1,\vec k_h} e^{i \vec k_h \vec r + i \omega t} + 
V_2 \sigma_{2,\vec k_h} e^{i \vec k_h \vec r - i \omega t} 
\right) \nonumber \\
&&+ \sum_{\vec k_s}V_3 \sigma_{0,k_s} e^{i \vec k_s \vec r } ,
\end{eqnarray}
where $\vert \vec k_h\vert = k_h$ and $\vert \vec k_s \vert = k_s$ are the
critical wave numbers associated with each of the instabilities mentioned
before. This expansion is the generalization of Eq. (\ref{redfield}) to
the codimension 2 situation, where one has to expand the fields on all the
unstable modes of the problem, including here the Turinglike unstable
mode. The amplitudes $\sigma_{1,\vec k_h}(\vec X,T)$,
$\sigma_{2,\vec k_h}(\vec X,T)$ and $\sigma_{0,\vec k_s}(\vec X,T)$ only
depend on the slow variables of the problem $\vec X=\varepsilon^{1/2}\vec
x$ and $T=\varepsilon^{-1}t$. Furthermore, we define $\mu$ as the reduced
distance to the stationary instability threshold, ($\mu = {I_{s+}-I_{s+}^c
\over I_{s+}^c}$, where now $I_{s+}^c$ is the critical value of the
bifurcation parameter at this Turinglike instability; $\mu$ is positive,
contrary to the case discussed in the preceding section).

The structure of the evolution equations for these amplitudes may easily 
be obtained using the symmetries of the problem \cite{daniel}. Contrary to 
the case discussed in Sec. \ref{Blinking Hexagons}, in the present situation 
$k_s < k_h$ so that there should be no quadratic couplings between
oscillatory hexagonal planforms and steady modes. In the absence of such
resonances, oscillatory and stationary modes are first coupled through
cubic non linearities. For example, the dynamics of a pair of unidirectional
counterpropagating traveling waves of amplitudes $A$ and $B$, coupled to
an arbitrary number of steady modes of amplitudes $R_i$
correspond to the following coupled Ginzburg-Landau and Swift-Hohenberg
equations:
\begin{eqnarray}\label{hopfturing}
\partial_t A &=& GL_A(\varepsilon ,A,B) -g
(1+id)A\Sigma_i\vert R_i\vert ^2\nonumber\\
\partial_t B &=& GL_B^*(\varepsilon ,A,B) -g
(1-id)B\Sigma_i\vert R_i\vert ^2\nonumber\\
\partial_t R_i &=& SH_i({R_j}) -w R_i(\vert A\vert ^2 + \vert B\vert ^2)
\end{eqnarray}
where, in the absence of mean flow or group velocity, $GL_A(\varepsilon
,A,B)=\varepsilon A+(1+i\alpha )\partial_x^2A - (1+i\beta )A(\vert
A\vert^2+ \gamma \vert B\vert ^2)$, with $\vec k_h \Vert \hat x$.
$SH_i({R_j})$ represents the generic evolution terms of the
Swift-Hohenberg type close to a Turinglike instability. In the absence of
``up-down" symmetry, as it is the case here, quadratic non linear couplings
between stationary modes are important. The corresponding dynamical
operator may then be written, for an arbitrary triad of modes, as ($i,j =
1,2,3$):
\begin{eqnarray}\label{hex}
SH_1({R_j}) &=&\mu R_1 + (\vec k_{s1}\vec \nabla )^2 R_1+ vR_2^*R_3^*
-\vert R_1\vert ^2 R_1 \nonumber\\
&& - u (\vert R_2\vert ^2 +\vert R_3\vert^2)R_1\nonumber\\
SH_2({R_j}) &=&\mu R_2 + (\vec k_{s2}\vec \nabla )^2 R_2+ vR_1^*R_3^*
-\vert R_2\vert ^2 R_2 \nonumber\\
&& - u (\vert R_1\vert ^2 +\vert R_3\vert^2)R_2\nonumber\\
SH_3({R_j}) &=&\mu R_3 + (\vec k_{s3}\vec \nabla )^2
R_3+ vR_1^*R_3^* -\vert R_3\vert ^2 R_3 \nonumber\\
&& - u (\vert R_2\vert ^2 +\vert R_1\vert^2)R_3 ,
\end{eqnarray}
with $\vec k_{s1}+\vec k_{s2}+\vec k_{s3}=0$. Since we are dealing with a 
system with scalar non linear couplings, $\gamma$, $u$, $g$ and $w$ 
should be larger than one.

Codimension 2 situations have been extensively studied in one-dimensional 
reaction-diffusion systems where Turing and zero-wave number Hopf
instability thresholds are close together \cite{shoka,Lima96,DeWit96}.
According to the non linear couplings between unstable modes, the resulting
patterns may be pure Turing, pure Hopf, or mixed mode patterns. We are
considering here two-dimensional geometries and a Hopf bifurcation with
finite wave vector. The Eqs. (\ref{hopfturing}) and (\ref{hex})
admit as steady states pure stripes of amplitude $\sqrt{\mu}$ or
hexagonal planforms, where $R_i = R_0$ and $A=B=0$, with 
$$
R_0 = {v + \sqrt{v^2 +4(1+2u)\mu}\over 2(1+2u)} .
$$ 
Stripes are unstable versus hexagons for $\mu < v^2/(u-1)^2$, which is 
expected to be the case here since $v$ is finite and $\mu \ll 1$. They also 
admit as asymptotic solutions traveling waves (of amplitude $\vert A\vert
= \vert A_0\vert = \sqrt \varepsilon$, $B=0$, R$=0$ or $\vert B\vert =
\vert B_0\vert = \sqrt \varepsilon$, $A=0$, R$=0$) and mixed modes.

The stability of hexagonal patterns versus wavy modes may be studied with
the following linearized equations:
\begin{eqnarray}
\partial_t A &=& \varepsilon A+(1+i\alpha )\partial_x^2A -3g (1+id)\vert
R_0\vert ^2
A\nonumber\\
\partial_t B &=& \varepsilon B+(1+i\alpha )\partial_x^2B -3g (1+id)\vert
R_0\vert ^2 B .
\end{eqnarray}
The result is that hexagonal planforms are stable for
\begin{equation}
\varepsilon < {3g \over 2(1+2u)}(v + \sqrt{v^2 +4(1+2u)\mu}) ,
\end{equation}
which is the case to be expected here since $\varepsilon$ is small and
$v$ finite.

Since we suppose that $\gamma$ is larger than 1, the pure wavy solutions
of Eq. (\ref{hopfturing}) are traveling waves of amplitude 
$\sqrt{\varepsilon}$. The evolution of the steady modes $R_i$ in the presence 
of pure traveling waves ($|A_0| = \sqrt{\varepsilon}$, $|B_0|=0$ or $|B_0| =
\sqrt{\varepsilon}$, $|A_0|=0$) is given by
\begin{equation}
\partial_t R_i= SH_i({R_j}) -w \varepsilon R_i .
\end{equation}
Traveling waves are then linearly stable if $\mu - w \varepsilon < 0$,
which should be the case here. Hence, for the situation discussed in this
section, hexagonal and wave patterns are expected to be simultaneously
stable.

The condition for the existence of mixed hexagon-traveling wave modes is 
found to be 
\begin{equation}
1+2u > 3g w
\end{equation}
and is not expected to be satisfied in reaction-diffusion dynamics with
scalar non linear couplings.

It is important to note that the Hopf bifurcation is
supercritical while the Turinglike transition to hexagons is subcritical.
As a result of their supercriticality, wavy patterns grow first. Although
these patterns are linearly stable versus stationary hexagonal planforms,
the dynamics of the latter present destabilizing quadratic non linearities.
The result is that the hexagonal patterns grow faster and finally take
over. Since steady hexagons are stable versus waves, they should thus be
the final pattern, although wave patterns may appear as transients during
the first part of the evolution. This is indeed what is observed in the
numerical simulations of Fig. \ref{fig8a}: at time $t=50$ there is a
dominance of Hopf modes with arbitrary orientations with weakly excited
Turing modes, as clearly seen in the far field. At late times ($t=1700$)
the Hopf modes have lost the competition and only Turing modes giving an
hexagonal pattern survive. Complicated dynamical competition occurs at
intermediate times. In Fig. \ref{fig_compet}, we show the integrated power
of Hopf and Turing modes as functions of time. The Hopf modes are seen to
grow supercritically first, but eventually the subcritical Turing modes
grow faster until they overcome the Hopf modes.


\section{Summary and discussion of results}
\label{results}

In previous sections we have reported a rich phenomenology for the broad range
of values of the cavity detuning ($2/B - 1 < \theta < \sqrt{3}$) that we have
explored. We summarize here these results and discuss their connection to
other related studies. We have revisited in Sec. \ref{linearpolarization}
the case of linearly polarized driving field for self-focusing as well as
self-defocusing situations. For the self-defocusing case there is an
instability leading to a stripe stationary pattern which is orthogonally
polarized to the driving field \cite{geddes1}. However, increasing the
intensity of the driving field the pattern disappears leading to a final
homogeneous elliptically polarized pattern. The transient dynamics involves the
spatial coexistence of two equivalent elliptically polarized homogeneous states
separated by moving interfaces, like in a process of phase separation dynamics
\cite{gunton}. Spatial coexistence of domains of different structures has been
reported in \cite{Residori96} for a liquid crystal light valve with rotated
feedback loop, while stationary spatial coexistence of circularly polarized
states has been reported in alkali vapors driven by a linearly polarized field
in a single mirror system \cite{Grynberg94} and in cells without mirrors
\cite{Tam77,Gahl}. For a linearly polarized driving field and a self-focusing
situation, there is an instability leading to an hexagonal pattern with the
same polarization of the driving field \cite{geddes1}. This is the same process
as in a scalar model \cite{lugi,firt}. When the intensity of the driving field 
is increased the hexagonal pattern is destabilized \cite{firthsolfract}. We
have observed that a further increase of the driving field intensity leads to a
complicated spatiotemporal dynamics with bursting spots that create
propagating circular waves. A related phenomenon has been reported in a model
which includes the dynamics of atomic variables \cite{Berre97}.

For an elliptically polarized driving field (Sec. \ref{ellipticalpolarization}) 
and a self-defocusing situation, the stripe
pattern is converted into an hexagonal pattern in each of the two independent
vectorial components of the electric field. A transition from bright to dark
hexagons (or viceversa) in each of the field components is obtained by changing
the ellipticity of the driving field. In addition, the range of parameters
(cavity detuning and input intensity) for which a pattern exists shrinks to
zero as the ellipticity departs from its value for linearly polarized driving
field. Beyond a certain ellipticity, the homogeneous solution, which now is
elliptically polarized, never looses stability. The change from stripes or
squares to hexagons for any finite ellipticity follows from general symmetry 
considerations also made in \cite{geddes2} for a Kerr medium with
counterpropagating beams. It has been predicted \cite{Scroggie96} and observed
\cite{Aumann97} in a Na cell with single feedback mirror. It has also been
invoked as an explanation of the observations in \cite{Maitre95} for a cell of
rubidium vapor in two counterpropagating beams. The transition from bright to
dark hexagons by changing the ellipticity of a driving field has been discussed
in \cite{Ackeman95} and observed in \cite{Aumann97} for Na vapor with a single
feedback mirror.

For an elliptically polarized driving field and a self-focusing situation, the
field ellipticity is a tuning parameter that permits to explore several
situations. For circularly polarized driving field the scalar case is
recovered and a circularly polarized hexagonal pattern emerges. Two
particularly interesting cases for intermediate ellipticity involve dynamical
patterns occurring through a Hopf instability at a finite wave number $k_h$.
The first of these instabilities considered in Sec. \ref{Blinking Hexagons}
leads to a time dependent pattern consisting of deformed dynamical hexagons.
The Hopf bifurcation is modified by a linearly damped stationary mode
$k_s>k_h$. The quadratic resonance of these two modes gives rise to deformed
hexagons. The spatiotemporal pattern can be described by a superposition of a
triad of $k_h$ Hopf modes, each one associated with a traveling wave with the
same frequency, but one of them has an amplitude different from the other two.
A second interesting case considered in Sec. \ref{cod2} is a codimension
two bifurcation in which a stationary Turinglike instability occurs
simultaneously with the Hopf instability. Now the stationary unstable mode is
such that $k_h>k_s$, and we do not find quadratic resonance. The transient
dynamics involves complicated states with strong competition of the Hopf and
stationary modes. The final state is a stationary hexagonal pattern in which
the Hopf modes have died. Dynamical patterns and codimension Turing-Hopf
bifurcations have been considered in related studies. A first case of dynamical
hexagons, in a Kerr medium with counterpropagating beams, is discussed in
\cite{geddes2}, but they arise as a secondary pure Hopf instability. Dynamical
hexagons arising from a codimension two situation, for alkali vapors with a
single feedback mirror, are considered in \cite{LogvinEPL,Logvin96}. The
difference with the situations we have found is twofold. First, the codimension
two situation is different from ours because $k_s>k_h$, so that a mixed
Turing-Hopf mode occurs through a quadratic resonances. Second, the
dynamical pattern of \cite{LogvinEPL,Logvin96} is different from our deformed
dynamical hexagons because the Turinglike mode is linearly unstable and
therefore appears with a large amplitude. In spite of these differences, we
note that the deformation of hexagons in \cite{Logvin96} has an origin similar
to ours. The competition of unstable Hopf modes and unstable Turinglike modes
with $k_s>k_h$ has also been considered experimentally in \cite{Residori96} for
a large system. By appropriate control of parameters, either of the two types
of modes can dominate. Close to the codimension two situation, spatial
coexistence of domains in which one of the two modes dominates is observed
\cite{last_news}. A codimension two bifurcation involving a Turinglike mode and
a Hopf mode, but of zero wave number ($k_h=0$), has also been recently
considered in an optical parametric oscillator with saturable losses
\cite{tlidi}.

\section{Conclusions}
\label{conclusions}

In this paper we have presented a systematic analysis of a prototype vectorial 
model of pattern formation in nonlinear optics describing a Kerr medium in a
cavity with flat mirrors and driven by a coherent plane-wave field. We have
considered linearly as well as elliptically polarized driving fields and
situations of self-focusing and self-defocusing. We have described, by
numerical simulations, amplitude and model equations, a rich variety of
phenomena that illustrate the relevance of the polarization degree of freedom
in optical spatiotemporal dynamics. In particular, we have shown that this
degree of freedom allows for new asymmetric homogeneous solutions, induces new
instabilities, and changes the basic symmetry of the pattern formed beyond an
instability. 

A particular relevant aspect of our results is to show that the ellipticity of
the pump can be used as a tuning parameter, easily accessible to the
experimentalist, that permits to explore different types of pattern forming
instabilities. For example, we have shown that by changing the ellipticity we
find situations which include: i) a modified Hopf bifurcation at finite wave
number leading to a time dependent pattern of deformed hexagons, ii) a
codimension two Turing-Hopf instability resulting in an elliptically
polarized stationary hexagonal pattern. Small changes in the ellipticity also
change the symmetry of the pattern, for example from stripes to hexagons.
Another general relevant aspect of our results is that the information on two
different patterns is encoded in each of the two independent polarization
components which are easily separated by a polarizer. From a fundamental point
of view this opens the possibility to study spatiotemporal correlations
between two different patterns emerging from the same physical system
\cite{sti}. Correlations between different polarization components close to
the onset of pattern formation in the model of Ref. \cite{geddes1} have already
been studied in \cite{corrkerr}.

We finally mention that vectorial degrees of freedom are also important in 
pattern formation in lasers where vectorial topological defects of the field
amplitude can occur if the rotational symetry of the system is not broken by a
pump field \cite{gil93,prati,pismenbook,2dVCGLE}. Corresponding phenomena
should also exist in other fields such as two-component Bose-Einstein
condensates \cite{busch}. This calls for further systematic investigation of
vectorial spatiotemporal phenomena for which a number of experimental results
are becoming available. 



\section{Acknowledgements}

We want to acknowledge interesting discussions with Dr. M. Santagiustina.
We also acknowledge Dr. A. Vicens for a careful reading of the manuscript.
This work is supported by QSTRUCT (Project ERB FMRX-CT96-0077). Financial
support from DGICYT (Spain) Project PB94-1167 is also acknowledged.
M. H. wants to acknowledge financial support from the FOMEC 
project 290, Dep. de F\'{\i}sica FCEyN, Universidad Nacional de Mar del Plata, 
Argentina.


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any of the components of the input field, so in the
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\end{thebibliography}


\begin{figure}
%\psfig{figure=figure1.ps,width=14cm}
\caption{Steady state homogeneous solutions, as a function of the input
field intensity, for linearly polarized light, $\chi=90^\circ$. The solid
line is the symmetric solution. The dashed line corresponds to the
asymmetric solutions. The two branches of the asymmetric solution give the
values of $I_{s+}$ and $I_{s-}$ in one of the two degenerate solutions,
and they meet the symmetric solution for $I_{s+}=I_{s-}=I'$. Parameter
values: $a=1$, $B=1.5$ and $\theta=1$. These parameter values are the same
for all the figures except where otherwise noticed. The quantities plotted
in all the figures are dimensionless.}
\label{fig1}
\end{figure}

\begin{figure}
%\psfig{figure=figure2.ps,width=14cm}
\caption{Steady state solutions for different values of the ellipticity
$\chi$ of the input field. The solid lines correspond to the value of
$I_{s+}$ and the dashed lines to $I_{s-}$. Input field ellipticity values:
(a) $\chi=87^\circ$, (b) $\chi=78^\circ$, (c) $\chi=73^\circ$, (d)
$\chi=0^\circ$. Note that $I_{s-}=0$ in Fig. (d).}
\label{fig3a}
\end{figure}

\begin{figure}
%\psfig{figure=figure3.ps,width=14cm}
%\vspace*{-3cm}
\caption{Marginal stability curves for linearly polarized input field as
described in the text. (a) Stability of the symmetric solution and (b)
stability of symmetric solution (up to $I'$) and asymmetric solution. The
dotted line displays the value of $I'$. The vertical dashed line separates
the self-focusing case (to the right) from the self-defocusing one
(left).}
\label{fig2}
\end{figure}

\begin{figure}
%\vspace*{5cm}
%\psfig{figure=figure4.ps}
%\vspace*{-5cm}
\caption{Plots of $|E_+|^2$ for increasing values of the intensity of the 
linearly polarized input field in the self-defocusing case ($\eta=-1$). 
From left to right and from top to bottom: $I_{s+}=0.8$ ($I_0=1.8$), 
$I_{s+}=1.5$ ($I_0=3.1$), $I_{s+}=2.6$ ($I_0=5.1$) and $I_{s+}=3.0$ 
($I_0=6.0$). Gray scale values: black$=0.1$, white$=3.8$.}
\label{fig4}
\end{figure}

\begin{figure}
%\vspace*{5cm}
%\psfig{figure=figure5.ps}
%\vspace*{-2.5cm}
\caption{Plots of $|E_+|^2$ for increasing values
of the input field in the self-focusing regime ($\eta=1$). In this figure
the gray scale is logarithmic. From top to 
bottom $I_{s+} = 0.48$ ($I_0=0.96$, black$=0.16$, white
$=3.5$), $I_{s+} = 0.55$ ($I_0=1.1$, black$=0.0018$, white$=9.8$) and
$I_{s+} = 1.7$ ($I_0=3.2$, black$=0.0028$, white$=17$).}
\label{fig5}
\end{figure}

\begin{figure}
%\psfig{figure=figure6.ps,width=14cm}
\caption{Marginal stability curves for the same values of the ellipticity
$\chi$ of the input field as in Fig. \protect\ref{fig3a}: (a)
$\chi=87^\circ$, (b) $\chi=78^\circ$, (c) $\chi=73^\circ$, (d)
$\chi=0^\circ$.}
\label{fig3b}
\end{figure}

\begin{figure}
%\vspace*{5cm}
%\psfig{figure=figure7.ps}
%\vspace*{-5cm}
\caption{ $|E_+|^2$ for increasing values of the quasi linearly polarized
input field ($\chi=87^\circ$) in the self-defocusing case ($\eta=-1$).
From left to right and from top to bottom: $I_{s+}=0.8$ ($I_0=1.8$),
$I_{s+}=1.7$ ($I_0=3.1$), $I_{s+}=2.1$ ($I_0=3.8$) and $I_{s+}=2.3$
($I_0=4.2$). Gray scale values: black$=0.13$, white$=2.6$.}
\label{fig6}
\end{figure}

\begin{figure}
%\vspace*{2cm}
%\psfig{figure=figure8.ps}
%\vspace*{-10cm}
\caption{ $|E_x|^2$ (left figure, black$=1.2$, white$=1.5$) and $|E_y|^2$
(right figure, black$=0$, white$=0.58$) for $I_{s+}=0.8$ ($I_0=1.8$).
The values of the other parameters are the same as in Fig. 
\protect \ref{fig6}.}
\label{fig6b}
\end{figure}

\begin{figure}
%\vspace*{10cm}
%\psfig{figure=figure9.ps}
%\vspace*{-5cm}
\caption{From left to right and from top to bottom: 
$|E_+(x,y)|^2$ (near field, black$=1.1$, white$=6.5$), $|E_+(\vec k)|^2$ 
(far field, white$=0$, black$=0.02$), $|E_-(x,y)|^2$ (near field,
black$=0$, white$=1.2$) and $|E_-(\vec k)|^2$ (far field, white$=0$,
black$=0.019$). The far fields are drawn in logarithmic scales. 
The homogeneous mode ($k=0$) is in the center of the far field
plots and has been eliminated in all figures. 
Parameters: $I_0=3.92$, $\chi=78^\circ$ and $\eta=1$.}
\label{fig7d}
\end{figure}

\begin{figure}
%\vspace*{10cm}
%\psfig{figure=figure10.ps}
%\vspace*{-5cm}
\caption{Four configurations of $|E_+|^2$ during one period $T$ of oscilation 
($T=3.86$). From left to right and from top to bottom: $t=0$, 
$t=0.97$, $t=1.93$ and $t=2.90$. The values of the parameters are the same as 
in Fig. \protect\ref{fig7d}. Gray scale values: black$=1.1$, white$=6.5$.}
\label{fig7a}
\end{figure}

\begin{figure}
%\hspace*{1cm}\psfig{figure=figure11.eps,width=14cm}
%\vspace*{1cm}
\caption{Definition of the unstable Hopf and slaved stationary modes
coupled through quadratic non linearities and described by the dynamical
system (\ref{dyn}). For asymptotic times, the modes $k_1$, $k_2$ and $k_3$
dominate and build the dynamical hexagons described in the text. The
intensity range of these modes obtained from simulations (see $|E_+(\vec
k)|^2$ in Fig. \protect \ref{fig7d}) is from 0.0008 (damped static modes)
to 0.019 (brightest Hopf mode).}
\label{fig7e:hexagons_modes}
\end{figure}

\begin{figure}
%\hspace*{-1.5cm}\psfig{figure=figure12.ps}
\caption{Phase of the Hopf mode $k_2$ (see Fig. 
\protect\ref{fig7e:hexagons_modes}) as a function of time. The values of the 
parameters are the same as in Fig. \protect \ref{fig7d}. A period $T=3.86$ is 
obtained from the plot.}
\label{fig7b}
\end{figure}

\begin{figure}
%\psfig{figure=figure13.ps}
\caption{Real part of most unstable eigenvalues $\lambda_1$ and
$\lambda_2$. The values of the parameters are the same as in Fig. 
\protect \ref{fig7d}.}
\label{fig7c}
\end{figure}

\begin{figure}
%\psfig{figure=figure14.ps}
\caption{Real part of unstable eigenvalues $\lambda_1$ and
$\lambda_2$. Parameters: $I_0=10.9$, $\chi=67^\circ$ and $\eta=1$.}
\label{fig8b}
\end{figure}

\begin{figure}
%\vspace*{-2cm}
%\psfig{figure=figure15.ps}
%\vspace*{1cm}
\caption{Near field ($|E_+(x,y)|^2$) and far field ($|E_+(\vec k)|^2$, in
logarithmic scale) at three different times for the Turing-Hopf
competition. From top to bottom: $t=50$ (near field: black$=3.6$,
white$=7.1$; far field: white$=0$, black$=5.3\times 10^{-4}$), $t=700$
(near field: black$=4.6$, white$=5.5$; far field: white$=0$, 
black$=2.1\times 10^{-4}$) and $t=1700$ (near field: black$=0.6$,
white$=16$; far field: white$=0$, black$=0.039$). The values of the 
parameters are the same as in Fig. \protect \ref{fig8b}.}
\label{fig8a}
\end{figure}

\begin{figure}
%\hspace*{-1.5cm}\psfig{figure=figure16.ps}
\caption{Time dependence of the amplitudes $|A_s|^2$ (solid line) and
$|A_h|^2$ (dashed line), corresponding to wave numbers $k_s$ and $k_h$, in
logarithmic scale. The amplitudes were calculated integrating over a
circle of radius $k_s$ or $k_h$ in Fourier space. The plot shows the
exponential growth of the Hopf modes at short times, the competition 
between the modes and the final domination of the static modes.}
\label{fig_compet}
\end{figure}


\end{document}