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\begin{document}

\title{Order parameter description of walk-off effect on pattern selection in
degenerate optical parametric oscillators}
\author{Majid Taki}
\address{Laboratoire de Physique des Laseres, Atomes et Mol\'{e}cules, \\
CNRS UMR 8523, Centre d'\'{E}tudes et de Recherches Lasers et Applications,\\
Universit\'{e} des Sciences et Technologies de Lille, \\
UFR de Physique B\^{a}t. P5, F-59655 Villeneuve d'Ascq Cedex, France}

\vskip 1cm

\author{Maxi San Miguel, Marco Santagiustina}
\address{Instituto Mediterr\'aneo de Estudios Avanzados, IMEDEA\\
(Consejo Superior de Investigaciones Cientificas -\\
Universitat de les Illes Balears) \cite{http},\\
E-07071 Palma de Mallorca, Spain}
%\date{}
\maketitle

\vskip 2cm

\begin{abstract}
Degenerate optical parametric oscillators can exhibit both uniformly
translating fronts and non-uniformly translating envelope fronts under the
walk-off effect. The nonlinear dynamics near threshold is shown to be
described by a real convective Swift-Hohenberg equation which provides the
main characteristics of the walk-off effect on pattern selection. The
predictions of the selected wave-vector and the absolute instability
threshold are in very good quantitative agreement with numerical solutions
found from the equations describing the optical parametric oscillator.
\end{abstract}

\vskip 2cm

PACS numbers: 46.65.Sf, 42.50.-p, 42.65.Yj, 42.65.-k


\newpage

In the last few years, the interest in pattern formation in nonlinear optics
has increased considerably \cite{chaos1}. This stems from the fact that
nonlinear optical systems show the complexity of other spatially extended
systems \cite{Hohenberg} in a context which presents additional
peculiarities such as the light polarization \cite{Hoyu} and the macroscopic
manifestations of quantum properties \cite{lugi98}. Among all the possible
devices, optical parametric oscillators (OPO's) have recently appeared as
one of the most promising not only for the richness in nonlinear dynamics 
\cite{REFSOPO} but also for their potential applications \cite{Piskarskas},
including low-noise measurements and detection \cite{Kimble} and the
possible storage of information \cite{Tlidi,stal}.

In OPO's, the nonlinear quadratic down-conversion process is most efficient
when the phase matching condition is fulfilled. This can be achieved by
taking advantage of the crystal birefringence to compensate for the
unavoidable material chromatic dispersion. As a result ordinary and
extraordinary rays may have different Poynting vector directions and
therefore the rays may diverge, i.e. walk-off, from one another as they
propagate through the nonlinear medium. Very recently convective
instabilities and noise-sustained patterns have been shown to appear due to
the walk-off \cite{SanM1,Taki}. For degenerate (D) OPO's, an interesting
recent result \cite{SanM} is the continuous generation of patterns of dark
solitary waves \cite{Trillo1} for positive signal detuning (where only a
homogeneous state can develop in the absence of walk-off). The pattern is
created behind a propagating front leading to non-uniformly translating
envelope fronts. However for DOPO's, the authors of Ref.\cite{Taki}
predicted, based on an amplitude equation description, that only uniformly
translating fronts (where no pattern can be formed) exist for large \textit{%
finite} positive signal detunings. Therefore, DOPO's exhibit both types of
fronts and a universal description of the walk-off effect on pattern
selection is worth to be addressed.

In this report we present a theoretical study of the dynamical transition
from uniformly translating fronts, leading behind them a homogeneous state,
to non-uniformly translating envelope fronts, giving rise to the formation
of a nonlinear pattern. The exact analytical expression of the transition
curve in terms of signal detuning and walk-off parameters is found. It
reveals the existence of a scaling law governing the transition. Using this
scaling we have performed an order parameter description, near threshold,
which leads to a real convective Swift-Hohenberg equation. This equation
provides the generic description of the walk-off effect on transverse
pattern formation. Based on this reduced model we have obtained analytical
expressions for the wave-vector selection rules and the threshold of the
absolute instability. The predictions are in very good agreement, both
qualitatively and quantitatively, with numerical simulations of the DOPO's
original equations.

We start from the standard model of a type I phase-matched degenerate OPO in
the mean-field approximation \cite{SanM1,Taki}

%\begin{mathletters}
\begin{eqnarray}  \label{eqdopo}
\partial _{t}A_{0} &=&\gamma _{0}[-(1+i\Delta _{0})A_{0}+ia_{0}\triangle
_{\bot }A_{0}-A_{1}^{2}+E(x,y)] \\
\partial _{t}A_{1} &=&\gamma _{1}[-(1+i\Delta _{1})A_{1}+ia_{1}\triangle
_{\bot }A_{1}+A_{0}A_{1}^{*}+\rho_{1}\partial _{y}A_{1}]
\end{eqnarray}

where $A_{0}$ and $A_{1}$ are the normalized slowly varying envelopes for
pump and signal fields respectively. The parameters $\Delta _{0,1}$, $\gamma
_{0,1}$, and $a_{0,1}$ are the detunings, the cavity decay rates and the
diffraction, respectively. The remaining parameters are the normalized
external pump $E$ and the signal walk-off coefficient $\rho_{1}$. For a
detailed discussion of the linear stability analysis (LSA) of the steady
state $A_0=E_0/(1+i\Delta_0), A_1=0$ including the walk-off we refer to
previous published papers \cite{SanM1,Taki}. Here we just recall that when $%
\rho _{1}\neq 0$ the absolute instability threshold, at which a dynamically
self-sustained structure can develop, is such that Re($\lambda (\mathbf{k}%
^{s})$)$=0$ where $\mathbf{k} ^{s}=(k_{x}^{s},k_{y}^{s})$ is a complex
saddle point satisfying $\nabla _{\mathbf{k}}\lambda (\mathbf{k)=0}$ and Re$%
[\nabla _{\mathbf{k}}^{2}\lambda (\mathbf{k)}]\geq 0$ where $\lambda (%
\mathbf{k)}$ is the linearized eigenvalue \cite{SanM1,Taki}. The use of a
complex wave-vector to express the linearized solutions $exp(\mathbf{k}
\cdot \mathbf{r} + \lambda t)$ is not surprising in this context \cite
{SanM1,Taki,SanM}: Its imaginary part accounts for the mode oscillations and
the real part for the envelope front. The gradient condition is satisfied
for $k_{x}^{s}=0$, which means that the stripes of the selected pattern are
oriented orthogonally to the walk-off \cite{SanM1,Taki,SanM}. At the
transition from nonzero wavelength patterns to homogeneous states, Im($%
k_{y}^{s}$) vanishes which leads to a \textit{real} wave vector $\mathbf{k}%
^{s}=($ $k_{x}^{s}=0,k_{y}^{s}={Re}(k_{y}^{s}))$. This simplifies
considerably the complex expression of $\lambda (\mathbf{k)}$ which reduces
to a real function of $\mathbf{k}$. Thus we obtain for the most unstable
mode (Re($\lambda (\mathbf{k}^{s})$)$=0$) : 
\begin{equation}
2a_{1}^{2}(k_{y}^{s})^{3}-(2a_{1}\Delta _{1}-\rho _{1}^{2}) k_{y}^{s} -\rho
_{1}=0.
\end{equation}
Real solutions $k_{y}^{s}$ are possible only if the signal detuning $\Delta
_{1}$ satisfies 
\begin{equation}
\Delta _{1}\geq \Delta _{1}^{c}(\rho _{1})=\frac{3}{2(2a_{1})^{\frac{1}{3}}}
\rho _{1}^{\frac{2}{3}}+\frac{1}{2a_{1}}\rho _{1}^{2}.
\end{equation}

Note the dependence of $\Delta _{1}^{c}(\rho _{1})$ on the diffraction
parameter $a_{1}$ leading to high values of critical signal detunings for
small values of $a_{1}$. The critical curve at transition $\Delta _{1}=$ $%
\Delta _{1}^{c}(\rho _{1})$ is shown in Fig. 1.

Since the walk-off parameter $\rho_1$ is small, the above relation shows
that pattern formation is possible only for small detunings, i.e. such that $%
\Delta _{1}<\Delta _{1}^{c}(\rho _{1})$. This is in agreement with the fact
that for $\Delta _{1}$ of order unity only nonlinear homogeneous states can
develop. In this case nonlinear dynamics is well described by a real
Ginzburg-Landau equation (GLE) \cite{Taki}. However it is clear that GLE
fails in the parameter range of pattern formation. A universal description,
valid in both regimes, can be derived (for $a_{1}$ fixed) from the scaling
law $\Delta _{1}^{c}(\rho _{1})\sim \rho _{1}^{\frac{2}{3}}$. To this
purpose, we have performed a weakly nonlinear analysis of the full DOPO
equations (\ref{eqdopo}) around threshold for signal generation by setting $%
\Delta _{1}=\varepsilon \Delta $ which leads to $\rho _{1}=\varepsilon ^{%
\frac{2}{3}}\rho $ with $\Delta $, $\rho$ of order one. The small parameter $%
\varepsilon $ measures the distance from threshold ($\mu ^{2}-1=\varepsilon
^{2}$) in the order parameter description approach developed in a previous
work \cite{Taki}. The evolution equations for the field amplitude variables
are obtained by expanding the solution $(A_{0},A_{1})$ in power series of $%
\varepsilon :$ $(A_{0},A_{1})=\varepsilon $ $(\psi _{0},\psi
_{1})+\varepsilon ^{2}(\psi _{0}^{(2)},\psi _{1}^{(2)})+$ $...$

With these scalings the following convective Swift-Hohenberg equation (SHE)
is obtained for the signal : 
\begin{equation}
\partial _{\tau }\psi _{1}\;=\frac{1}{2}(\mu ^{2}\;-\;1)\psi _{1}-\frac{1}{2}
\left( \Delta _{1}-a_{1}\triangle _{\bot }^{2}\right) ^{2}\psi _{1}\;+\rho
_{1}\partial _{y}\psi _{1}-\psi _{1}^{3}/(1+\Delta _{0}^{2})
\end{equation}
where we have set $\tau =\gamma _{1}t$ and $\psi_1 $ is a real function as a
consequence of the fact that the OPO is degenerate. The description of
DOPO's in terms of a SHE was also the result of Refs. \cite{Long1,Valc};
here we also account for the walk-off with the correct scaling relations
presented above. 
Our main aim is to examine the emerging dynamically
self-sustained patterns so that we restrict the analytical study to the
occurrence of the absolute instability for which the linearization of the
convective SHE around $\psi_1 =0$ gives: 
\begin{equation}
\lambda (\mathbf{k)=}\frac{1}{2}(\mu ^{2}-1)-\frac{1}{2}\left( \Delta
_{1}-a_{1}k^{2}\right) ^{2}+\rho _{1}k_{y}
\end{equation}
where $\mathbf{k}$ is a complex wave-vector satisfying the saddle point
condition : 
\begin{eqnarray}  \label{saddle}
\frac{\partial \lambda }{\partial k_{x}} &=&2a_{1}\left( \Delta
_{1}-a_{1}k^{2}\right) k_{x}=0  \nonumber \\
\frac{\partial \lambda }{\partial k_{y}} &=&2a_{1}\left( \Delta
_{1}-a_{1}k^{2}\right) k_{y}+\rho _{1}=0
\end{eqnarray}
From the system (\ref{saddle}) we recover the condition $k_{x}^{s}=0$ while
the $k_{y}^{s}$ component verifies: 
\begin{equation}  \label{ky}
2a_{1}^{2}(k_{y}^{s})^{3}-2a_{1}\Delta _{1}(k_{y}^{s})-\rho _{1}=0.
\end{equation}
Note that this equation is more tractable mathematically than the original
one and it also allows to recover the transition curve $\Delta _{1}^{c}(\rho
_{1})$ to the leading order $\rho _{1}$. From Eq. (\ref{ky}), the formation
of patterns corresponding to non-uniformly translating envelope fronts is
possible if $\Delta _{1}<\Delta _{1}^{c}(\rho _{1})$ and their wave-vectors
are 
\begin{eqnarray}  \label{re}
{Im}(k_{y}^{s}) &=&\frac{\sqrt{3}}{2}\left( \frac{\rho _{1}}{4a_{1}^{2}}
\right) ^{1/3}\left[ \xi _{+}^{1/3}-\xi _{-}^{1/3}\right] \\
{Re}(k_{y}^{s}) &=&-\frac{1}{2}\left( \frac{\rho _{1}}{4a_{1}^{2}} \right)
^{1/3}\left[ \xi _{+}^{1/3}+\xi _{-}^{1/3}\right]
\end{eqnarray}
where $\xi _{\pm}=1\pm \sqrt{1-\left( \Delta _{1}/\Delta _{1}^{c}\right)
^{3} }$. The most unstable modes are those for which ${Re}[\lambda
(k_{y}^{s})]=0$ and they oscillate with frequency ${Im}[\lambda (k_{s}) ] $.
Since they drift at velocity $\rho _{1}$ and, as far as no nodes disappeared
or are created in the nonlinear region behind the front, the conservation of
the linear flux of the field nodes can be applied\cite{SanM} ; the wave
number of the generated pattern is then defined as 
\begin{eqnarray}  \label{wavevector}
Q_{y} &\equiv &{Im}[\lambda (k_{s})]\mathbf{/}\rho _{1}  \nonumber \\
&=&{Im}(k_{y}^{s})\left\{ 1+2a_{1}\rho _{1}^{-1}{Re}(k_{y}^{s}) \left[
\Delta _{1}-a_{1}\left( {Re}(k_{y}^{s})^{2}-{Im} (k_{y}^{s})^{2}\right)
\right] \right\} .
\end{eqnarray}

Notice that, apart from the case of uniformly translating fronts (${Im}%
(k_{y}^{s})=$ $Q_{y}=0$), ${Im}(k_{y}^{s})$ and $Q_{y}$ are generally
different. The predicted wave number $Q_{y}$ as a function of the signal
detuning is reproduced in Fig. 2 together with the wave number selected in
the exact numerical solutions of (\ref{eqdopo}) and that were predicted by
the LSA of the same equations. As it can be seen from the figure the
agreement between the analytical predictions and the exact, numerically found
values is excellent, especially for small $\rho_1$ and $\Delta_1$.

The threshold of absolute instability can be calculated analytically from
the condition ${Re}[\lambda (k_{y}^{s})]=0$ for both types of fronts. After
straightforward calculations we obtain 
\begin{eqnarray}  \label{threshold}
\mu _{A}^{2} &=&1+4a_{1}^{2}{Re}(k_{y}^{s})^{2}\left[ 3{Re}(k_{y}^{s})^{2}+ {%
Im}(k_{y}^{s})^{2}\right] \quad \thinspace for \quad {Im}(k_{y}^{s})\neq 0 
\nonumber \\
\mu _{A}^{2} &=&1+\left( \Delta _{1}-a_{1}{Re}(k_{y}^{s})^{2}\right) ^{2}-2
\rho _{1}{Re}(k_{y}^{s})\qquad for \quad {Im} (k_{y}^{s})=0
\end{eqnarray}

A closed analytical form of the absolute threshold as a function of
parameters can be obtained by substituting $Im(k_{y}^{s})$ and $Re(k_{y}^{s})
$ by their expression in Eqs. (\ref{re}) respectively. We have
reproduced the absolute threshold curves Eqs. (\ref{threshold}) versus the
signal detuning $\Delta _{1}$ for various $\rho _{1}$ and the curves compare
very well with the corresponding numerical results from the LSA, as shown in
Fig. 3. We have also showed (dashed curves) the thresholds obtained with the
real Ginzburg-Landau approximation and from the figure it is clear that this
approximation must be discarded in the parameter range of small signal
detunings. 
Similar approximations for $\Delta _{1}>0$ of order unity have been proposed,
leading to GLE either without walk-off \cite{muss} or with walk-off \cite{kutz}.
In particular, for the DOPO, a Fisher-Kolmogorov equation (FKE, a form of the
GLE with real coefficients) has been derived \cite{kutz}. It is interesting to
note that the FKE with a walk-off term does not show the phenomenon of pattern
formation behind a propagating front studied in \cite{SanM}. However, a
correction of the type $\nabla^4$ (in the spirit of a SHE) is known to account
for such phenomenon \cite{dee}.
 

In summary we have shown that DOPO's, in presence of walk-off, can exhibit
either uniformly translating fronts or non-uniformly translating envelope
fronts. The latter allows the formation of patterns even for positive
detunings giving rise to a non trivial critical transition curve between
pattern forming and homogeneous states. We also demonstrated that the
critical transition curve can be obtained analytical and provides a scaling
law which leads to a unified description of the dynamics through an equation
of Swift-Hohenberg type. The predictions obtained with this equation,
regarding the wave-vector selection and the absolute instability threshold,
are compared with the numerical simulations of original DOPO's equations and
the agreement is very good.

\textbf{Acknowledgements : }One of the authors (M. T.) acknowldges helpfull
discussions with P. Glorieux, N. Ouarzazi and H. Ward.
The work of M.S. is supported by the European Commission (QSTRUCT Project ERB 
FMRX-CT96-0077).

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\newpage 

%\newline
%\noindent \textbf{Figure Captions} \vspace{1cm} %\bigskip
%\newline

\begin{figure}
\centerline{\psfig{figure=fig1.ps}}
\vspace{1cm}
\caption{Critical curve (solid line) $
\Delta_{1}^{c}(\rho _{1})$ (Eq. (5)) limiting the new region of positive
signal detunings of the existence of patterns. The thin horizontal solid
line is the limit without walk-off. The diffraction parameter $a_{1}=0.25$.} 
\end{figure}

\begin{figure}
\centerline{\psfig{figure=Figure2.ps}}
\vspace{1cm}
\caption{The solid curves are the predicted wave-vectors
(Eq. (\ref{wavevector})) of the emerging signal pattern from the convective
SHE for: a) $\rho_1=0.05$, b) $\rho_1=0.15$, c) $\rho_1=0.25$. The dotted
curves are the corresponding numerical solutions from the LSA. The crosses
represent the wave-vectors selected in the numerical solutions of the
governing equations (\ref{eqdopo}). The remaining parameters are set to $%
\Delta_0=0$, $\gamma_1=\gamma_0=1$ and $a_0=0.125$.} 
\end{figure}

\begin{figure}
\centerline{\psfig{figure=Figure3.ps}}
\vspace{1cm}
\caption{Absolute threshold $\mu_A$ for the same values
of $\rho_1$ as in Fig. 2. The solid curves are the values predicted by Eqs.
(\ref{threshold}), while the dotted curves are the result of the numerical
solutions from the DOPO's LSA. The dashed curves are the threshold given by
the GLE [11].} 
\end{figure}


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