\documentstyle[osa,manuscript,epsfig,doublespace]{revtex}
\begin{document} 


\title{{\bf Pattern formation in presence of walk-off for a Type-II 
optical parametric oscillator}} 
\author{{\it Gonzalo Iz\'{u}s, Marco Santagiustina, Maxi San Miguel, and 
Pere Colet}} 
\address{Instituto Mediterr\'aneo de Estudios Avanzados, IMEDEA (CSIC-UIB) 
\cite{http},\\ 
E-07071 Palma de Mallorca, Spain} 
 
\maketitle 


\begin{abstract} 
 
 
We show the relevance of walk-off effects in pattern formation in a
type-II optical parametric oscillator at frequency degeneracy. Neglecting
walk-off, only phase patterns are formed and the intensity
distribution is homogeneous. Walk-off changes the instability from absolute
to convective for some parameter range.
In the absolutely unstable regime, it induces for each polarization component 
of light a competition between two phase 
stripe patterns (traveling waves) with different wavelength. 
Phase stripe patterns at each of the wavelengths are equally likely to be
selected and after a transient regime of coexistence, one of them takes over.
In the convectively unstable regime we show the existence of intensity patterns
sustained by noise. The patterns arise form the interference between
traveling waves which are excited by noise and dynamically amplified.
 
\end{abstract} 
\pacs{OCIS codes: 190.4410,190.4420,190.4970,270.2500} 
\newpage
 
Pattern formation in many nonlinear optical systems has been an active 
field of investigation \cite{lugi98}. New experiments on spatial
modulational instability in bulk quadratic media \cite{fuer97} clearly indicate
that pattern formation in optical parametric oscillators (OPO's), theoretically
predicted \cite{stal92}, could be experimentally observed.
Beyond pattern formation and
transverse effects in general \cite{fabr98}, other interesting applications of
these devices include the generation of squeezed and other non-classical states
of the light for low-noise measurements and detection \cite{kimb92} and might
include the storage of information in 2-dimensional optical localized
structures \cite{stal96,ol}. Finally, the interplay among quantum and
macroscopic effects in optical systems was also investigated in OPO's
\cite{lugi98} which are then a valuable test bed for fundamental physics
investigation. 

An important characteristic of OPO's is that the crystal birefringence is 
exploited to phase-match the down-conversion process. The anisotropy can cause 
the Poynting vectors of the extraordinary polarized components to walk-off the
ordinary ones (double-refraction phenomenon) in the plane transverse to the
propagation direction. Basically two main situations can be found: in type-I
OPO's the twin down-converted photons are polarized in the same direction and 
always orthogonally to the pump while in type-II OPO's the down-converted
beams are polarized along both the ordinary and extraordinary axes, i.e. are
polarization non-degenerate. The walk-off effects on the pattern formation
processes in type-I OPO's were studied in refs. \cite{ol,Marco,oe,taki98}.
Pattern formation in a type-II OPO, neglecting the walk-off, has been studied in
ref. \cite{Longhi} and only phase patterns were found in each polarization
component. Here, we study the effects introduced by the walk-off in the pattern
formation in a type-II OPO. Two main new phenomena are found for each component
of the polarization:
a) the competition and pattern selection between two equally preferred traveling
waves of different wavelength; b) in the convective regime, described below,
the existence of stripe intensity patterns, induced by the walk-off and
sustained by noise.

The equations describing the time evolution of the normalized slowly varying
envelopes of the transverse second harmonic (SH) $A_0(\vec r,t)$ and the
fundamental harmonics (FHs) $A_{1,2}(\vec r,t)$ electric fields for a type-II,
phase-matched, ring, frequency non-degenerate OPO (NDOPO) in the mean-field
approximation can be easily derived as an extension of those of ref.
\cite{Marco}: 
\begin{eqnarray} 
\label{master1} 
\partial_t A_0 &=& \gamma_0 [ -(1+i \Delta_0) A_0 + E_0 + i a_0 \nabla^2 A_0 
+ 2 i K_0 A_1 A_2] +\sqrt{\epsilon_0} \; \xi_0(\vec r,t) \\ 
\label{master2} 
\partial_t A_1 &=& \gamma_1 [ -(1+i \Delta_1) A_1 + 
i a_1 \nabla^2 A_1 + i K_0 A_2^* A_0]+\sqrt{\epsilon_1} \; \xi_1(\vec r,t) \\ 
\label{master3} 
\partial_t A_2 &=& \gamma_2 [ -(1+i \Delta_2) A_2 + \rho \; \partial_y A_2 + 
i a_2 \nabla^2 A_2 + i K_0 A_1^* A_0] +\sqrt{\epsilon_2} \; \xi_2(\vec r,t) 
\end{eqnarray} 
where $A_{0,1}$ are ordinary polarized beams and $A_2$ is extraordinary 
polarized; $\vec r=(x,y)$ is the transversal spatial vector, $t$ is the
time and $\gamma_j$ , $\Delta_j$ and $a_j$ are respectively the cavity decay
rate, detuning and diffraction for the field $A_j$. The nonlinear coefficient
is $K_0$, $E_0$ is the injected pump and $\rho$ is the walk-off coefficient. 
Finally, the last term of each
equation is an independent complex Gaussian white noise with zero mean value 
and delta-correlated in space and time. These terms describe quantum noise in 
the Wigner representation for the linearized version of eqs. 
(\ref{master1}-\ref{master3}) \cite{noise}, but they can account for thermal 
or input fluctuations as well. In ref. \cite{Longhi} similar equations for
generic NDOPO have been considered but no walk-off or noise were taken
into account while in ref. \cite{taki98} the study of the walk-off was
restricted to transverse perturbation dynamics effects in a Type-I NDOPO and
in a parameter region where no pattern is expected when walk-off is neglected.

Eqs. (\ref{master1}-\ref{master3}) have the uniform steady state solution $A_0
=E_0/(1+i \Delta_0)$, $A_1=A_2=0$. A linear stability analysis is easily 
performed for the complex vector $(A_0, A_1, A_2^*)$.
The field $A_0$ is always stable while $A_1$ and $A_2^*$
become unstable if the normalized pump intensity $F=K_0 E_0/(1+i\Delta_0)$ is
such that: 
\begin{equation} 
\label{pump} 
|F|^2 \ge 1+\left[ \frac{q^2 \tilde a + \tilde \Delta + \gamma_2 \rho q_y} 
{\gamma_1+\gamma_2} \right]^2 
\end{equation} 
The critical threshold value $|F_c|=1$ corresponds to traveling waves $A_1,
A_2^* \simeq \exp [i \vec q \vec r + \lambda (\vec q ) \, t ]$, whose
two-dimensional real wave-vector $\vec q$ belongs to a circle centered at $\vec
q_0= (0, q_{0,y})$ with radius $R=|\vec q-\vec q_0|=\sqrt{
q_{0,y}^2 -\tilde \Delta/\tilde a}$, where $q_{0,y}=-\gamma_2 \rho / 2 \tilde
a$, $\tilde a=\gamma_1 a_1+ \gamma_2 a_2 $, and $\tilde \Delta=\gamma_1
\Delta_1+\gamma_2 \Delta_2$. In fig. 1 the rings of the most unstable modes
for $A_1$ and $A_2$ at $|F_c|=1$ are shown. Without
walk-off rings would overlap and, for $|F|>|F_c|$, any $\vec q$ mode on the
ring, and the opposite for the orthogonal component, can be selected by
spontaneous symmetry breaking. Hence a phase pattern appears in $A_1$ and
$A_2$ while the intensity in both polarizations remains homogeneous 
\cite{Longhi}.

As a result of the linear stability analysis, we can highlight several new 
effects induced by the walk-off. Pattern formation requires $R>0$
which only occurs for $\tilde \Delta < 0$ when $\rho=0$.
The walk-off allows pattern formation even for positive detunings, as it
also happens for a type I OPO \cite{ol}. The walk-off
shifts the center of the rings along the direction of the advection term in
opposite directions for $A_1$ and $A_2$ and also increases their radius $R$.
Note that, although the walk-off is present only in one polarization component,
the shift appears on both. This stems from the conservation of photon momentum
which has no transverse component for the input SH. Finally, the walk-off
induces the pattern to drift and breaks the rotational symmetry so that the
selected wave-vector orientation is parallel to the walk-off direction. 

The transversal drift induced by the walk-off can overcome the spreading
velocity of the unstable modes that form the pattern, so that the pattern is
effectively washed out of the system.
As in other nonlinear optical systems \cite{Marco,taki98,prl} this is the
convectively unstable regime, which is found for
pump amplitudes up to the absolutely instability threshold $|F_a|$. This
threshold can be determined by extending the linear analysis to complex 
wave-vectors $\vec k \;$ \cite{Marco,prl,deis85} (formally equivalent to the
substitution $i{\vec q} \rightarrow {\vec k}$). Hereafter, we
particularize our analysis to the frequency degenerate type-II OPO (DOPO) by
setting $\gamma_{1,2}=\gamma$, $\Delta_{1,2}=\Delta$ and $a_{1,2}=a$. 
We first determine the wave-vector of the most unstable modes by
calculating the complex saddle point ${\vec k_s}=(k^s_x,k^s_y)$. 
\begin{equation} 
\label{saddle0} 
\nabla_{\vec k} \lambda({\vec k}) |_{\vec k_s} =0 \;, \; 
Re[\nabla_{\vec k}^2 \lambda({\vec k}) |_{\vec k_s}] > 0 
\end{equation} 
where $\lambda({\vec k})$ is the eigenvalue which results from the 
linearization of eqs. (\ref{master1}-\ref{master3}) around the 
steady-state, i.e. 
\begin{equation} 
\label{eigen2} 
\lambda({\vec k})=2 \, \gamma \left[ -1+ \frac{\rho k_y}{2} + 
\sqrt{|F|^2 - a^2 \left[R^2 + k_x^2 + (k_y+ i \frac{\rho}{ 4 a})^2 
\right]^2 } \right] 
\end{equation} 
Eqs.\ (\ref{saddle0}) have been numerically solved for a complex $\vec k_s$
together with the equation $Re[\lambda(\vec k_s)]=0$ to calculate
$|F_a|$. In Fig. 1 we show the absolute instability threshold $|F_a|$ as a 
function of the detuning $\Delta$ for several values of $\rho$. There are 
two modes for which $A_1$ and $A_2^*$ become absolutely unstable at 
$|F|=|F_a|$, both have the same value $k_x^s=0$ and different values for
$k_y^s$. These correspond to two equally likely to be selected phase patterns
with different wavelength and oriented perpendicularly to the $y$-axis.
The wavelength of each pattern can be calculated along the line of
ref. \cite{ol}.
However, for negative detunings, these wavelengths are very close to that of 
the modes on the ring in the inset of fig. 1. In the far field, there are 
only two possible selections of twin beams which conserve momentum,
one indicated by the pair of diamonds and the other by the pair of squares.

For large enough systems, and starting from random initial conditions there is
a transient regime in which a competition between the two
absolutely unstable wave-vectors takes place as shown in fig. 2a. The
$Re(A_1)$ shows stripes perpendicular to the drift direction while 
the intensity $|A_1|^2$ is homogeneous in each of the regions dominated by
a given wavelength. The front between these regions 
displays a set of vertically ordered defects (fig. 2b).
The field $A_2^*$ displays a pattern identical to that shown for $A_1$. 
For long time one of the two wave-vectors is selected, as shown in
fig. 2c and 2d. 

In the convectively unstable regime noise-sustained
structures are expected. These structures are locally sustained
because, although advected away, they are continuously regenerated by noise 
which at each space point excites
all the unstable modes of each polarization \cite{Marco}.
In particular the most rapidly spreading modes can interfere generating
intensity stripes as shown in the numerical solution of fig. 3a.
We remark that the existence of patterns in the intensity is a completely new 
feature for a type-II DOPO, due to the interplay of the convective nature of
the instability (i.e. the walk-off) and the presence of noise.
At the bottom of the pattern, the generation from noise of randomly oriented
stripes can be appreciated. This is a typical feature of 
noise-sustained structures in two transverse dimensions \cite{Marco}. These
spatial modes are clearly visible in the far field, displayed in Fig. 3b. 

In conclusion, in the absolutely unstable regime of a type-II DOPO phase stripes
have been observed, the wavelength being randomly selected out of two possible
different values. 
The difference in the wave-vector stems directly from the walk-off. An initial 
regime of wave-vector competition is also observed. 
In the convectively unstable regime it has been demonstrated that
noise-sustained structures of intensity can be observed. In such a regime
noise forces continuously the most unstable traveling waves in both
polarization components and thus creates a structure in the intensity by
interference like in a type-I interaction.

This work is supported by the European Commission through the Project 
QSTRUCT (ERB FMRX-CT96-0077). Financial 
support from DGESIC (Spain) Projects PB94-1167 and PB97-0141-02-C02 is also 
acknowledged. MSM acknowledges useful discussions with P. Glorieux and 
M. Taki.
 
 
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\begin{figure}
\centerline{\psfig{figure=fig1.ps}}
\vspace{1cm}
\caption{Absolute threshold $|F_a|$ vs detuning (dashed curves) for several
values of $\rho$. The convective threshold $|F_c|$ is the solid curve.
The shadowed region is the convective unstable regime for $\rho=0.15$.
In the inset we show the most unstable modes for both polarization components.} 
\end{figure}

\begin{figure}
\centerline{\psfig{figure=fig2.ps}}
\vspace{5cm}
\caption{Transient competition between the most unstable modes a) $Re(A_1)$,
b) $|A_1|^2$; c) and d) show $Re(A_1)$ for the final state where the largest
or shortest wavelength structure has been selected.
Parameters are $2a_0=a=0.25, \gamma=1, \Delta_0=0, \Delta=-0.15, K_0=1,
\rho=0.15$; $E_0$ is a super-Gaussian beam of maximum amplitude $E_{max}=1.05$.}
\end{figure}

\begin{figure}
\centerline{\psfig{figure=fig3.ps}}
\vspace{2cm}
\caption{Intensity of polarization component $A_1$ in the a) near field and 
b) far field in the convectively unstable regime. Parameters as in fig. 2 
except for $\epsilon_0=\epsilon_1=\epsilon_2=2 \times 10^{-13}$,
$\Delta=-0.2$ and $E_{max}=1.009$.}
\end{figure}

\end{document}