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\title{{\bf WALK-OFF AND PATTERN SELECTION
IN OPTICAL PARAMETRIC OSCILLATORS}}
\author{{\it Marco Santagiustina, Pere Colet, Maxi San Miguel, Daniel Walgraef
\cite{Daniel}}}
\address{Instituto Mediterraneo de Estudios Avanzados, IMEDEA (CSIC-UIB) 
\cite{www},\\
E-07071 Palma de Mallorca, Spain}

\maketitle

%\vskip 1.5cm

\begin{abstract}
The effect of the walk-off in pattern selection in optical parametric
oscillators is theoretically examined. We show that a dynamical mechanism
allows to observe the formation of structures also for positive signal
detunings. In this regime the generated pattern is a periodic array of kinks 
which separate regions where alternatively one of two stable steady-states
is selected. This structure can be regarded as a train of dark 
stripe solitons because the two solutions have opposite signs.
The wavelength of the selected pattern is theoretically predicted and the
prediction agrees with the results of the numerical solutions of the equations 
governing the device.
\end{abstract}

\narrowtext
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\vskip 1.5cm

Optical parametric oscillators (OPO's) are one of the nonlinear optical
systems where transverse pattern formation has been predicted 
\cite{stal92}.
Recent experimental results on spatial 
modulational instability in bulk quadratic media \cite{fuer97} are very 
encouraging about the possibility of observing similar pattern formation
in a nonlinear resonator.
The OPO's have been also investigated because of their properties as generators
of squeezed and other non-classical states of the light \cite{kimb92},
which make them a paradigmatic example of the interplay among quantum and 
macroscopic effects in optical systems \cite{lugi96}.
The foreseen applications of these systems are also very valuable,
including low-noise measurements and detection \cite{kimb92} and optical
storage elements with 2-dimensional localized structures \cite{stal96}.
\newline
Pattern formation was predicted in the OPO's when the pump amplitude is above 
the signal generation threshold and the detuning between 
the cavity resonance and the signal frequency is negative \cite{stal92}.
Under this condition the diffraction can compensate the detuning and
the homogeneous state becomes unstable for plane-waves which have a small
wavevector component in the transversal direction.
For positive detunings the steady-state is unstable for homogeneous
perturbations and thus no pattern with a characteristic periodicity is 
predicted. Two homogeneous stable solutions with equal amplitude but
opposite phase also exist and then for positive detunings domains can be
generated where one of the two solutions is selected. The walls separating the 
domains are dark soliton stripes as shown in ref. \cite{tril97}.
\newline
Here, we study the effect of the walk-off on the pattern formation processes.
This phenomenon was considered for soliton and pulse propagation in quadratic 
media \cite{smit95} but only recently its influence has started to 
be taken into account for OPO's \cite{pre}.
It stems from the birefringence of the nonlinear crystal which is 
exploited to phase-match the down-conversion process. 
Considering a Type I phase-matching scheme, the fundamental and second harmonic waves (FH,SH) are linearly and orthogonally polarized and
therefore, because of the anisotropy, their Pointying vectors are misaligned
as soon as the propagation is not parallel to an optical axis, i.e. they
walk-off one each other.
\newline
Walk-off causes the pattern to drift and forces the orientation of the stripes
to be orthogonal to the drift direction; this is due to the 
fact that the mode orthogonal to the drift motion is the most rapidly spreading 
in the system \cite{pre}.
The walk-off also increases the threshold of the absolute instability and a parameter region exists where the instability is convective.
In the latter regime the drift motion overwhelms the instability growth and a 
pattern can be observed only if noise is present \cite{pre,prl}.
Here, we demonstrate that the presence of the walk-off dynamically induces
the growth of a spatial pattern also for positive signal detunings. In this case
the structure is a periodic array of kinks and the walls separating different 
domains are dark soliton stripes. We also predict the wavelength of the
selected pattern and a very good agreement is found with the results
of the numerical solutions of the equations modeling the device. 
\newline
The equations describing the time evolution of the transverse
SH ($A_0(x,y,t)$) and FH ($A_1(x,y,t)$) electric fields in a Type I,
phase-matched, degenerate OPO in the mean-field approximation are
\cite{stal92,pre}:
\begin{eqnarray}
\label{master}
\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_1^2] \\
\label{master2}
\partial_t A_1 &=& \gamma_1 [ -(1+i \Delta_1) A_1 + \rho_1 \partial_y A_1 + 
i a_1 \nabla^2 A_1 + i K_0 A_1^* A_0]
\end{eqnarray}
where the coefficients $\gamma_{0,1}$, $\Delta_{0,1}$, $a_{0,1}$, $K_0$,
represent respectively the cavity decay rates, the detunings, the diffraction
and the nonlinearity; other parameters are the injected
pump $E_0$ and the walk-off coefficient $\rho_1$.
\newline 
When $\rho_1=0$, a linear stability analysis of the solution 
$A_0=E_0/(1+i\Delta_0)$, $A_1=0$ shows that for negative signal detunings
$\Delta_1$ the most unstable modes for $A_1$ are travelling waves
$\exp[i(q_x x + q_y y) + \lambda ({\vec q}) t]$ with wavevector
modulus $|{\vec q}|=q_c=\sqrt{-\Delta_1/a_1}$. They all become unstable 
when $E_0$ is such that $\Re[\lambda]=0$ but only two conjugate waves of this 
ring of unstable modes will be eventually selected and a stripe structure will
arise in the FH intensity field \cite{stal92}.
The final orientation of the standing wave is random, depending only on the
initial conditions.
For positive detunings the steady-state is unstable for homogeneous 
perturbations (${\vec q}=0$) and the system can evolve to any of the
two stable non-zero homogeneous solutions of eqs. (\ref{master},\ref{master2})
\cite{stal96,tril97}:
\begin{equation}
\label{hom}
A^{\pm}_1 = \pm \left( {I\over 2} \right)^{1/2} \left( \left( 1+{\Delta_0 + \Delta_1 \over
K_0E_0} \right)^{1/2} +i \left( 1-{\Delta_0 + \Delta_1 \over K_0E_0} 
\right)^{1/2} \right)
%A_1^{\pm}=\pm \left[ \frac{i K_0 E_0 I}{2 K_0^2 I+(1+i \Delta_0) (1+i 
%\Delta_1)} \right]^{1/2}
\end{equation}
where the intensity $I=|A_1^{\pm}|^2$ is given by:
\begin{equation}
\label{intens}
I=\frac{1}{2 K_0^2} \left( \Delta_0 \Delta_1 - 1 + \sqrt{K_0^2 E_0^2 -
(\Delta_0 + \Delta_1)^2} \right)
\end{equation}
From random initial conditions one observes the growth of spatial domains
where the field is either $A_1^+$ or $A_1^-$.
The walls separating these domains are stripes of phase defects,
because the amplitude of the field crosses zero.
These structures can be also considered as dark solitary waves; their existence
and properties have been reported and discussed in ref. \cite{tril97}.
\newline
As we show here, this scenario drastically changes in presence of the walk-off
which induces the creation of a pattern as a dynamical effect.
This behaviour is analogous to that of 1-dimensional physical systems
in which front propagation into an unstable state creates a pattern behind the
front \cite{sarl88}.
In the OPO's the injected pump is a beam of finite size and the FH drifts
at a given speed $\rho_1$. Then, the steady-state $A_1=0$ becomes unstable by
crossing the boundary where the pump $E_0$ reaches the signal generation
threshold.
The calculation to determine the selected wavevector of the structure
is formally the same of refs. \cite{sarl88}. It is based on a linear
analysis in which the wavevector is extended to the complex space.
In particular, we first determine the wavevector of the most unstable modes by calculating the complex saddle point ${\vec k_s}=(k^s_x,k^s_y)$
\begin{eqnarray}
\label{saddle0}
\nabla_{\vec k} \lambda({\vec k}) |_{\vec k_s} =0 \;, \;
\Re[\nabla_{\vec k}^2 \lambda({\vec k}) |_{\vec k_s}] \geq 0
\end{eqnarray}
where $\lambda({\vec k})$, the eigenvalue which results from the
linearization of eqs. (\ref{master},\ref{master2}) around the steady-state,
below threshold, can be written as:
\begin{equation}
\label{eigen2}
\lambda({\vec k})=-1+ \rho_1 k_y + \sqrt{F^2 - a_1^2 (q_c^2 + k_x^2 + k_y^2)^2}
\end{equation}
and $F=K_0 E_0/(1+i\Delta_0)$. Note that we have made the formal substitution
$i{\vec q} \rightarrow {\vec k}$ and thus the wavevector of the plane wave is 
represented in this notation by $\Im[{\vec k}]$. 
Equations (\ref{saddle0},\ref{eigen2}) yield $\Im[k^s_x]=0$ so that the pattern
is expected to be always oriented orthogonally to the drift 
direction. The $y$ component of the wavevector of the generated pattern, say 
$Q_y$, can be calculated as follows. The linearly unstable
wavevector is $\Im[k^s_y]$ which oscillates with frequency
$\Im[\lambda({\vec k_s})]$ and drifts at velocity $\rho_1$.
By imposing the conservation of the flux of the field nodes from the linearly
unstable regime to the stable state we have \cite{sarl88,maxi93}:
\begin{equation}
\label{wvec}
Q_y=\frac{\Im[\lambda({\vec k_s})]}{\rho_1}
\end{equation}
In fig. 1 we present the predicted $Q_y$ as a function of the signal
detuning calculated for the value of $E_0$ at the absolute instability 
threshold, 
which is determined by the additional condition $\Re[\lambda({\vec k_s})]=0$
\cite{pre}; for comparison, we also include $q_c$.
Note that for large negative detunings the selected wavevector
depends on the walk-off only slightly. For small negative detunings
the difference increases and the pattern for $\Delta_1=0$ has a non-zero 
wavelength as soon as $\rho_1 \neq 0$.
The predicted $Q_y$ decays to zero only for 
$\Delta_1 \ge \Delta_c(\rho_1) > 0$; thus for positive detunings a pattern can
form too. In this case the field alternatively assumes the values $A_1^{\pm}$
and a novel kind of optical pattern, i.e. a periodic array of kinks, is
predicted.
\newline
To confirm this, we have solved numerically eqs. (\ref{master},\ref{master2}) 
with initial random conditions and a flat top pump beam $E_0$. Initially
we observe the random formation of domains separated by dark stripes and rings
\cite{tril97}. Later, these structures drift outside the pumping
region and are replaced by an ordered array of stripes such for example that
of fig. 2.
The wavevector selected in the numerical solutions is displayed in fig. 1.
The $x$ component is zero (see fig. 2) and the $y$ component is in a very
good agreement with the theory for any $\Delta_1$.
In fig. 3 we show a transverse section at $x=0$ which exhibits the
ordered periodic array of dark solitons.
As the detuning increses the regions of constant amplitude get larger and
larger and finally, above $\Delta_c$, the pattern becomes
homogeneous.
\newline
In conclusion, we have demonstrated that the walk-off effect in OPO's induces
a dynamical selection of the pattern likewise the problem of front propagation
into an unstable state. In the OPO the front of the instability 
is spatially fixed by the injected pump beam and the generated fundamental
harmonic field drifts, because of the walk-off. The strength of the walk-off
sets the wavelength of the generated pattern and the critical value of
the signal detuning $\Delta_c$ beyond which no pattern forms. 
Through this mechanism an array of kinks can be dynamically generated
for positive detunings. The characteristic
wavevector observed in the numerical solutions is in agreement with the theory 
for the whole range of parameters.
The experimental observation of this phenomenon should be possible with the
available technology \cite{tril97,pre}.
\newline
Finally, note that at a fixed position in space, the OPO is generating a 
temporal train of dark solitons. Then, possible applications of this dynamical 
pattern formation can be forseen in the field of optical communications where 
dark soliton trains have been proposed as supporting structures to reduce 
soliton jitter in optical fiber links \cite{akhm97}.
\newline
\newline
This work is supported by QSTRUCT (Project ERB FMRX-CT96-0077). Financial
support from DGICYT (Spain) Project PB94-1167 is also acknowledged.

\begin{thebibliography}{99}

\bibitem[+]{Daniel} Permanent address: Center for Nonlinear Phenomena and
Complex Systems, Universit\'e Libre de Bruxelles, Campus Plaine, Blv. du
Triomphe B.P 231, 1050 Bruxelles.

\bibitem[*]{www} http://www.imedea.uib.es/PhysDept

\bibitem{stal92} K. Staliunas, Opt. Comm. {\bf 91}, 82 (1992); J. Mod. Opt.
{\bf 42}, 1261 (1995); G-L. Oppo, M. Brambilla, L.A. Lugiato, Phys. Rev. A,
{\bf 49}, 2028 (1994); G-L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, L.A. 
Lugiato, J. Mod. Opt. {\bf 41}, 1151 (1994); S. Longhi, J. Mod. Opt. {\bf 43}, 
1089 (1996); J. Mod. Opt. {\bf 43}, 1569 (1996);  Phys. Rev. A, {\bf 53}, 4488 
(1996); S. Longhi, A. Geraci, Phys. Rev. A, {\bf 54}, 4581 (1996); G.J. de 
Varcarcel, K. Staliunas, E. Roldan, V.J. Sanchez-Morcillo, Phys. Rev. A,
{\bf 54}, 1609 (1996); Phys. Rev. A, {\bf 56}, 3237 (1997).

%\bibitem{oppo94} .

%\bibitem{long96} .

%\bibitem{valc96}.

\bibitem{fuer97} R.A. Fuerst, D-M Baboiu, B.Lawrence, W.E Torruelas,
G.I. Stegeman, S. Trillo, Phys. Rev. Lett. {\bf 78}, 2756 (1997);

\bibitem{kimb92} H.J. Kimble, Fundamental systems in quantum optics
(J. Dalibard, J.M. Raimond, J. Zinn-Justine, Elsevier Sc., Amsterdam) 
545 (1992).

%\bibitem{kimb87} L.A. Wu, H.J. Kimble, J. Hall, H. Wu, Phys. Rev. Lett.
%{\bf 57}, 2520 (1986); L.A. Wu, M. Xiao, H.J. Kimble, J. Opt. Soc. Am. B
%{\bf 4}, 1465 (1987).

%\bibitem{reid88} M.P. Reid, P.D. Drummond, Phys. Rev. Lett. {\bf 60}, 
%2731 (1988); J. Mertz, T. Debuisschert, A. Heidmann, C. Fabre, E. Giacobino,
%Opt. Lett. {\bf 16}, 1234 (1991); P.G. Kwiat, K. Mattle, H. Weinfurter,
%A. Zeilinger, A.V. Sergienko, Y. Shih,  Phys. Rev. Lett. {\bf 75}, 4337 (1995).

\bibitem{lugi96} L.A. Lugiato, A. Gatti, H. Wiedemann, Quantum Fluctuations (S. 
Reynaud, E. Giacobino, J. Zinn-Justine Eds., Elsevier Sc., Amsterdam), 431 (1997).

\bibitem{stal96} K. Staliunas, V.J. Sanchez-Morcillo, Opt. Comm., {\bf 139},
306 (1997), Phys. Rev. A {\bf 57}, 1454;  M. Tlidi, P. Mandel, M. Haelterman,
Phys. Rev. E, {\bf 56}, 6524 (1998).

%\bibitem{lede97} C. Etrich, V. Peschel, F. Lederer, Phys. Rev. Letts.,
%{\bf 79}, 2454 (1997)

%\bibitem{mand98}

\bibitem{tril97} S. Trillo, M. Haelterman, A. Sheppard, Opt. Lett. {\bf 22}, 970
(1997).

\bibitem{smit95} W.E. Torruelas, Z. Wang, D.J. Hagan, E.W. Van Stryland,
G.I. Stegeman, L. Torner, C.R. Menyuk, Phys. Rev. Lett., {\bf 74}, 5036 (1995); 
A.V. Smith, W.J. Alford, T.R. Raymond, M.S. Bowers, J. Opt. Soc. Am. B
{\bf 12}, 2253 (1995).

\bibitem{pre} M. Santagiustina, P. Colet, M. San Miguel, D. Walgraef,
submitted (1998).

\bibitem{prl} M. Santagiustina, P. Colet, M. San Miguel, D. Walgraef, Phys. Rev.
Lett., {\bf 79}, 3363 (1997).

\bibitem{sarl88} G.T. Dee, W. Van Sarloos, Phys. Rev. Letts., {\bf 60}, 2641
(1988).

\bibitem{maxi93} R. Montagne, A. Amengual, E. Hernandez-Garcia, M. San Miguel,
Phys. Rev. E, {\bf 50}, 377 (1994); M. San Miguel, R. Montagne, A. Amengual,
E. Hernandez-Garcia, Instabilities and Nonequilibrium Structures V, 85
(E. Tirapegui, W. Zeller Eds., Kluwer Ac., Amsterdam) (1996).

\bibitem{akhm97} P.D. Miller, N.N. Akhmediev, A. Ankiewicz, Opt.Lett. {\bf 21},
1132 (1996).

\end{thebibliography}

\newpage

Figure Captions:
\newline

Figure 1: The predicted wavevector of the pattern is presented in solid
line for $\rho_1=0.15$. The asterisks represent the wavevector selected in the 
numerical solutions and the dashed line the predicted wavevector for $\rho_1=0$.
Numerical solutions are calculated for $E_0$ always $1\%$ above the absolute
instability threshold (see text) and $\Delta_0=0$. We also set $\gamma_{0,1}=1$, $a_0=a_1/2=0.125$, $K_0=1$.

Figure 3: The transverse intensity pattern at $t=2000$ is shown; the parameters
used are $\Delta_0=0$, $\Delta_1=0.3$, $\rho_1=0.15$, $E_0=1.12736$, 
$\gamma_{0,1}=1$, $a_0=a_1/2=0.125$, $K_0=1$.

Figure 3: The real (solid line) and imaginary (dashed line) parts of the
electric field envelope in function of the space variable $y$ are given
for $x=0$. The value of the uniform regions agree with the solutions predicted
by eq. (\ref{intens}), $A_1^{\pm}=\pm(0.166+i0.126)$. Other parameters are as
in fig. 2.

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