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\begin{document}

\articletitle[Transverse patterns in type-II optical parametric oscillators]
             {Transverse patterns in type-II \\ optical parametric oscillators}

\author{G.~Iz\'us$^{[1,*]}$, M.~Santagiustina$^{[1,2]}$, M.~San Miguel$^{[1]}$,
 and P.~Colet$^{[1]}$}


\affil{$^{[1]}:$ Instituto Mediterr\'aneo de Estudios Avanzados, 
IMEDEA (CSIC-UIB),\\ Universitat Illes Balears, E-07071
Palma de Mallorca, Spain.$^{[+]}$}

\affil{$^{[2]}:$ INFM, Dipartimento di Elettronica ed Informatica, \\
Universit\`a di Padova, via Gradenigo 6/a, 35131 Padova, Italy.}

\begin{abstract}
We study the effects produced by walk-off and cavity induced
polarization coupling in the process of pattern formation in a
type-II optical parametric oscillator (OPO). Walk-off generates a
convectively unstable regime in which intensity patterns sustained
by noise are produced. An absolute unstable regime is also
present. In this regime a competition between two traveling waves
of different wavelength takes place. The presence of a
birefringent and/or dichroic mirror produces homogeneous phase
locked solutions separated by Domain Walls similar to Bloch walls
in magnetic systems. The spontaneous motion of these walls around
point defects produces persistent spatio-temporal states.
\end{abstract}

\section{Introduction}

Pattern formation is an ubiquitous manifestation of nonlinearity
which presents specially interesting features in nonlinear optical
systems \cite{lugi98}. Among these features are phenomena linked
to the vectorial degree of freedom associated with the
polarization of light and to the macroscopic manifestations of
quantum noise (disorder). A paradigm for these studies is the
Optical Parametric Oscillator (OPO) \cite{oppo94} in which
transverse patterns have been recently observed \cite{vaupel99}.



We consider transverse pattern formation in a type-II OPO: The
nonlinear medium inside an optical cavity leads to frequency down
conversion of the photons of a pump beam.  The down-converted
beams are polarized along the ordinary and extraordinary axes,
i.e. are polarization non-degenerate. The anisotropy of the
crystal can cause the extraordinary polarized component to
walk-off the ordinary one in the plane transverse to the beam
propagation direction. The resulting advection-like phenomenon
produces quantum noise amplification \cite{prl}. The
birefringence/dichroism of the mirrors of the cavity leads to an
additional interaction by way of a linear coupling between
ordinary and extraordinary beams. Here, we discuss effects
introduced in the process of pattern formation by: a) 
walk-off \cite{Marco,JOSAB} and b)linear polarization coupling \cite{OL}.



\smallskip



\section{Mean field equations for a type-II OPO}



The equations that describe the time evolution in a phase-matched
ring optical resonator filled with a birefringent, nonlinear
quadratic medium, couple four complex slowly varying envelopes:
the linearly polarized components $A_{0,0'}(\vec r,t)$ of the
intracavity Second Harmonic (SH) field pumped by $E_0$, and the
signal $A_1(\vec r,t)$ (ordinary polarized) and idler $A_2(\vec
r,t)$ (extraordinary polarized) components of the down converted
Fundamental Harmonic (FH) field. We consider a mean-field
approximation in which the longitudinal spatial dependence is
averaged out. For the case of frequency degeneracy between signal
and idler the equations obtained along the lines of the derivation
in ref. \cite{Marco} are



\begin{eqnarray}
\partial_t A_{0'} &=& \gamma_{0'} [ -(1+i \Delta_{0'}) A_{0'}  + i a_{0'} \nabla^2 A_{0'} + c_{0'} \, A_0] +\sqrt{\epsilon_{0'}} \; \xi_{0'}(\vec r,t)
\nonumber \\
\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+c_0 \, A_{0'}] + \nonumber \\
& & \sqrt{\epsilon_0} \; \xi_0(\vec r,t) \nonumber \\
\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+c_1 \, A_2]+  \nonumber \\
& &  \sqrt{\epsilon_1} \; \xi_1(\vec r,t)  \nonumber \\
\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 +c_2 \, A_1]+ \nonumber \\
& & \sqrt{\epsilon_2} \; \xi_2(\vec r,t)
\label{master}
\end{eqnarray}

Here $\vec r=(x,y)$ is the transverse spatial vector, $t$ is the
time and $\gamma_j$, $\Delta_j$ and $a_j$ are respectively the
effective cavity decay rate, effective mode detuning and
diffraction for the field $A_j$. The nonlinear coefficient
coupling is $K_0$ and $\rho$ is the walk-off coefficient. The
linear coupling coefficients $c_{0,0'}, \, c_{1,2}$ are related to
the dichroism and to the birefringence of one of the mirrors of
the ring cavity \cite{OL}. Such linear coupling of the linearly
polarized components can also arise from an intracavity quarter
wave plate in a standing wave cavity configuration \cite{Fabre}.
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 noise terms describe quantum noise in  the Wigner
representation for the linearized version of eqs.(\ref{master})
\cite{noise}, but they can also account for thermal  or input
fluctuations.
\smallskip



\section{Walk-off effects}



We first consider the simpler case in which there is no direct
polarization coupling ($c_{0,0'}, \, c_{1,2}=0$, $A_0'=0$) and study the
effects of walk-off in the pattern forming OPO threshold. In this
case eqs.(\ref{master}) have the uniform steady state solution
$A_0=E_0/(1+i \Delta_0)$, $A_1=A_2=0$. A linear stability analysis
determines that $A_{0}$ is 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}
At the critical threshold  $|F|=|F_c|=1$ traveling waves $A_1,
A_2^* \simeq \exp [i \vec q \vec r + \lambda (\vec q ) \, t ]$
become unstable. The critical two-dimensional real wave-vectors
$\vec q$ are on 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 $\tilde \Delta=\gamma_1
\Delta_1+\gamma_2 \Delta_2$. When there is no walk-off the rings
of the unstable modes for $A_1$ and $A_2$ overlap and, for
$|F|>|F_c|$, any $\vec q$ mode on the ring, and the opposite for
the orthogonal component, are 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}.
\begin{figure}[t]
\begin{minipage}[t]{2.4in}
\epsfig{figure=dib1.eps,width=2.4in,height=2.3in}
{\footnotesize Fig. 1. Transient competition between two
wavelengths associated with the most unstable modes: a)
$\Re(A_1)$, b) $|A_1|^2$; c) and d) show $\Re(A_1)$ for two
possible final states where the largest or shortest wavelength
structure has been selected. Parameters are $2a_0=a_{1,2}=0.25,
\gamma_{0,1,2}=1, \Delta_0=0, \Delta_{1,2}=-0.15, K_0=1,
\rho=0.15$, $\epsilon_{0,1,2}=0$; $E_0$ is a super-Gaussian beam of maximum amplitude
$E_{max}=1.05$.}
\end{minipage}
\hspace{0.1in}
\begin{minipage}[t]{2in}
\centerline{\epsfig{figure=dib2.eps,width=1.2in,height=2.2in}}
{\footnotesize Fig. 2. Intensity of the signal field  $A_1$ in the
a) near field and b) far field, in the convectively unstable
regime. Parameters as in fig. 1 except for
$\epsilon_0=\epsilon_1=\epsilon_2=2 \times 10^{-13}$,
$\Delta_{1,2}=-0.2$ and $E_{max}=1.009$.}
\end{minipage}
\end{figure}
Pattern formation requires an instability at a finite wavenumber,
which only occurs for $\tilde
\Delta < 0$ when $\rho=0$. 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 it also increases their radius $R$. The
shift appears on both rings because of conservation of photon
momentum which has no transverse component in the SH pump.
Walk-off also induces a drift of the pattern and breaks the
rotational symmetry, so that the selected wave-vector orientation
is parallel to the walk-off direction.



The transverse drift induced by the walk-off can overcome the
spreading velocity of the unstable modes that form the pattern, so
that a pattern forming perturbation grows but leaves the system:
this is the convectively unstable regime, which is found for pump
amplitudes larger than $|F_c|$ and up to an absolutely instability
threshold $|F_a|$. This threshold can be determined by extending
the linear analysis to complex wave-vectors $\vec k \,$
\cite{prl,Marco,deis85}.



At the threshold of absolute instability  $|F|=|F_a|$ the same
traveling wave  for $A_1$ and $A_2^*$ is selected. It is oriented
perpendicularly to the $y$-axis \cite{JOSAB}. The wavelength is
locally selected with equal probability between two possible ones.
These two wavelengths correspond to the two wavevectors of the
unstable ring along the two senses of the advection direction.
These are the most unstable modes. For large enough systems a
transient regime is observed in which a competition between these
two wavelengths (phase stripe patterns) takes place as shown in
Fig.1a. $\Re(A_1)$ 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 of different wavelength displays a set of vertically
ordered defects (Fig.1b). The field $A_2^*$ displays a pattern
identical to that shown for $A_1$. For long times one of the two
wavelengths fills the system, as shown in Fig.1c and Fig.1d.



In the convectively unstable regime, i.e., for $1<|F|<|F_a|$,
noise-sustained structures are expected. These structures,
although advected away, are
continuously regenerated by noise which at each space point
excites all the unstable modes \cite{Marco}. The modes of fastest
growth, which are oriented along the advection direction,
interfere generating intensity stripes as shown in the numerical
solution of Fig.2a. The active modes in this regime can be seen in
Fig.2.b (far field). The existence of intensity patterns is a new
feature for a type-II OPO, which results from the interplay of the
convective nature of the instability (i.e. the walk-off) and the
presence of noise \cite{JOSAB}.



\smallskip



\section{Polarization Coupling and Domain Walls}

\begin{figure}
\begin{minipage}[h]{2.3in}
\centerline{\epsfig{figure=dib3.eps,width=2.3in,height=1.6in}}
{\footnotesize Fig.3  A Bloch wall obtained from a numerical
solution of eqs.(\ref{master}) in one spatial dimension. Solid
line represents the real part of $A_1$ and the dotted line the
imaginary one. Parameters are $\gamma_{0,1}=1$,
$\gamma_{0',2}=1.002$, $\Delta_{0,0'}=0$, $\Delta_1=0.01$,
$\Delta_2=0.03$, $a_{0,0'}=0.125$, $a_{1,2}=0.25$, $K_0=1$,
$E_0=1.25$, $c_{0,0'}= 0.01(1+i)$ and $c_{1,2} = 0.01$}
\end{minipage}
\hspace{0.05in}
\begin{minipage}[h]{2.3in}
\centerline{\epsfig{figure=dib4.eps,width=2.3in,height=1.85in}}
{\footnotesize Fig. 4  Velocity of the Bloch walls as a function
of $c_1$ -real- in the regime $\gamma_1 \Delta_1 \neq \gamma_2
\Delta_2$. Here $c_1=c_2$, $\gamma_{0,1}=1.001$,
$\gamma_{0',2}=1$, $\Delta_{0,0'}=0$, $\Delta_1=0.01$,
$\Delta_2=0.03$, $a_{0,0'}=0.125$, $a_{1,2}=0.25$, $K_0=1$,
$E_0=1.25$ and $c_{0,0'}= 0.01(1+i)$.}
\end{minipage}
\end{figure}
%\newline
We now study the effects of  birefringent-dichroic polarization
coupling ($c_{0,0'},c_{1,2} \neq 0$) neglecting for simplicity
walk-off and noise ($\rho=\epsilon_i=0$). A linear stability
analysis shows that the trivial solution $A_{1,2}=0$,
$A_0=(1+i\Delta_{0'})E_0/[1-\Delta_{0}\Delta_{0'}-c_0
c_{0'}+i(\Delta_{0}+\Delta_{0'})]$, $A_{0'}=c_{0'}
E_0/[1-\Delta_{0}\Delta_{0'}-c_0
c_{0'}+i(\Delta_{0}+\Delta_{0'})]$, is stable for $E_0<E_c$ where
$E_c$ can be determined numerically. For $\widetilde{\Delta}>0$
homogeneous perturbations have the largest linear growth rate
beyond threshold $E_0 > E_c$. In this case two homogeneous
solutions $A_{1,2}^+$ and $A_{1,2}^-$ of equal amplitude and
phase-shifted by $\pi$ radiants ($A_{1,2}^+=-A_{1,2}^-$) bifurcate
from $A_1=A_2=0$. They have the same probability to be selected at
threshold. The existence of  two phase-locked solutions is a
consequence of the polarization coupling, since the relative phase
of $A_1$ and $A_2$ is arbitrary when $c_{0,0'}=c_{1,2}=0$.
Numerical integrations of eqs.(\ref{master}) confirm that
stationary and uniform local domains, where $A_{1,2}$ are either
$A_{1,2}^+$ and $A_{1,2}^-$, form spontaneously. These domains of
different but equivalent phase-locked solutions are separated by
Domain Walls.



For small values of $c_{1,2}$ the Domain Walls are of the Bloch
type, while for larger values they are of the Ising type
\cite{Coullet}. In Fig.3 an example of a one-dimensional (1D)
Bloch wall for $A_1$ is shown. The phase can rotate in two
possible senses (chirality) across the interface, clockwise or
counterclockwise in the complex plane. The wall for $A_2$ has the
opposite chirality. The fields $A_{1,2}$ do not vanish at the core
of the wall. On the contrary, on an Ising wall both the real and
imaginary part of the field vanish at the same point in the core
of the wall.
\begin{figure}
\begin{minipage}[t]{4.6in}
\epsfig{figure=dib5.eps,width=4.6in,height=1.53in}
{\footnotesize Fig.5. A snapshot at time t=1600 of the field
$A_1(x,y,t)$ obtained from a numerical integration of
eqs.(\ref{master}) with random initial conditions. a) real part;
b) intensity and c) phase. Parameters are $\gamma_{0}=\gamma_2=1$,
$\gamma_{0'}=\gamma_1=1.002$, $\Delta_{0}=\Delta_{0'}=0$,
$\Delta_1=0.01$, $\Delta_2=0.03$, $E_0=1.25$, $a_{0}=a_{0'}=0.125$,
$a_1=a_2=0.25$, $K_0=1$, $c_0=c_{0'}=0.025 (1+i/2)$ and $c_1=c_2=
0.02 (1+i)$.}
\end{minipage}
\end{figure}
The dynamics of 1D Bloch walls depends critically on the values of
the cavity decay rate and the detuning.  For $\gamma_1
\Delta_1=\gamma_2 \Delta_2$, 1D Bloch walls do not move. However,
for $\gamma_1 \Delta_1 \ne \gamma_2 \Delta_2$ walls of different
chirality move in opposite directions. In Fig.4 a plot of the
velocity of a Bloch wall as a function of polarization coupling is
shown. The point at which this velocity goes to zero identifies
the transition point from Bloch to Ising walls.



In two dimensions (2D) domain walls are lines that locally can
have opposite chirality. The change of chirality takes place in
singular points, where the phase of the field is not defined and
the amplitude goes to zero (defects), as we show in Fig.5.
Observe, in Fig.5a, the domain walls that separate homogeneous
regions of $A_{1,2}^+$ from those of $A_{1,2}^-$; walls of
different chirality are represented respectively by black or white
thick curves. Defects associated with the changes of chirality
appear as black dots in the intensity field (Fig.5b), while the
phase field is shown in Fig.5c.
\begin{figure}[t]
\begin{minipage}[t]{4.6in}
\epsfig{figure=dib6.eps,width=4.6in,height=2.5in}
{\footnotesize Fig.6. Time evolution generated from a Bloch wall
with an array of defects. A succession of Bloch walls of opposite
chirality are emitted: a) t=0; b) t=1000; c) t=1150; d) t=1550.
Above: intensity patterns for $A_1$, below: the corresponding real
part. Parameters are: $\gamma_{0}=\gamma_2=1$,
$\gamma_{0'}=\gamma_1=1.002$, $\Delta_{0}=\Delta_{0'}=0$,
$\Delta_1=0.01$, $\Delta_2=0.03$, $E_0=1.25$, $a_{0}=a_{0'}=0.125$,
$a_1=a_2=0.25$, $K_0=1$, $c_0=c_{0'}=0.01$ and $c_1=c_2 = 0.025$.}
\end{minipage}
\end{figure}
The dynamics of Bloch walls in 2D depends also on the detuning and
damping values, but it is also influenced by the curvature of the
walls. For $\gamma_1 \Delta_1=\gamma_2 \Delta_2$ flat Bloch walls
are stable. In this case the transient dynamics is mainly
controlled by the curvature of the fronts. This leads to the
growth of a phase at the expenses of the other, and to the
annihilation of all the defects. For $\gamma_1 \Delta_1 \ne
\gamma_2 \Delta_2$ walls of different chirality move in opposite
directions. Defects remain notably fixed and Bloch walls of
different chirality tend to spiral around the defects. Fronts of
equal chirality annihilate, when they collide, and new ones are
generated by the defects (Fig.5). This leads to a persistent
dynamical state and the system never reaches the state
corresponding to one of the two homogeneous phase-locked
solutions. An example of such persistent dynamical state, obtained
from an initial condition consisting in a Bloch wall with segments
of different chirality, is shown in Fig. 6.
\smallskip



\section{Summary}

We have studied some polarization effects in the process of
pattern formation in a type-II OPO. We have first considered the
effect of walk-off, which induces the presence of absolutely and
convectively unstable regimes. In the absolutely unstable regime
phase stripes have been numerically observed, the wavelength being
randomly selected out of two possible different values. The
existence of two different values stems directly from the
walk-off. An initial regime of wave-vector competition is also
observed. We have also shown that in the convectively unstable
regime intensity noise-sustained structures exist. These are a
manifestation of quantum noise amplification.

Secondly, we have shown that signal/idler polarization coupling
originated in a birefringent/dichroic cavity mirror leads to two
equivalent phase locked states. Domain walls between these states
can be Bloch walls with defects at points of change of chirality.
The spontaneous motion of these walls sustains a persistent
spatio-temporal pattern.

This work was supported by the European Commission
through the Project QSTRUCT (ERB FMRX-CT96-0077). Financial
support from Ministerio de Ciencia y Tecnolog\'ia (Spain) Projects
PB97-0141-02-C02 and BFM2000-1108 is also acknowledged.

\begin{chapthebibliography}{1}

\bibitem[*] {ggi} Permanent address: Departamento de F\'{\i}sica,
FCEyN, Universidad Nacional de Mar del Plata, Argentina. Member of CONICET
(Argentine).

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

\bibitem{lugi98} Lugiato, L. A., Brambilla, M. and Gatti, A. (1999)
Optical Pattern Formation, {\it Advances in Atomic Molecular and
Optical Physics}, vol. {\bf 40}, 229-306.

\bibitem{oppo94} Oppo, G-L., Brambilla, M., Lugiato, L. A. (1994) Formation and evolution of
roll patterns in optical parametric oscillators, {\it Phys. Rev.
A} {\bf 49}, 2028-2032.

\bibitem{vaupel99} Vaupel, M., Maitre, A., Fabre, C. (1999) Observation of Pattern
Formation in Optical Parametric Oscillators {\it Phys. Rev. Lett.}
{\bf 83}, 5278-5281.

\bibitem{prl} Santagiustina, M., Colet, P., San Miguel, M.  and Walgraef, D. (1997)
Noise-sustained convective structures in nonlinear optics, {\it
Phys. Rev. Lett.} {\bf 79}, 3633-3636.

\bibitem{Marco} Santagiustina, M., Colet, P., San Miguel, M.  and
Walgraef, D. (1998) Two-dimensional noise-sustained structures in
optical parametric oscillators, {\it Phys. Rev. E} {\bf 58},
3843-3853.

\bibitem{JOSAB} Iz\'us, G., Santagiustina, M., San Miguel, M. and
Colet, P. (1999) Pattern formation in the presence of walk-off for
a type II optical parametric oscillator, {\it J. Opt. Soc. Am. B}
{\bf 16}, 1592-1596.

\bibitem{OL} Iz\'us, G., Santagiustina, M.  and San Miguel, M. (2000)
Bloch Domain Walls in Type-II Optical Parametric Oscillators, {\it
Optics Letters} {\bf 25}, 1454-1456.

\bibitem{Fabre} Fabre, C., Mason, E. J. and Wong, N. C. (1999)
Theoretical analysis of self-phase locking in a type-II phase
matched optical parametric oscillator, {\it Optics Communications}
{\bf 170}, 299-307.

\bibitem{noise}
Gatti, A., Wiedemann, H., Lugiato, L. A., Marzoli, I., Oppo, G. L.
and  Barnett, S. M. (1997) Langevin treatment of quantum
fluctuations and optical patterns in optical parametric oscillator
below threshold, {\it Phys. Rev. A} {\bf 56}, 877-897.

\bibitem{Longhi}
Longhi, S. (1996) Traveling-wave states and secondary
instabilities in optical parametric oscillators, {\it Phys. Rev.
A} {\bf 53}, 4488-4499.

\bibitem{deis85}
Deissler,  R. J. (1985)  Noise-sustained structure, intermittency
and the Guinzburg-Landau equation, {\it J. Stat. Phys.} {\bf 40},
371-395; (1989) External noise and the origin and dynamics of
structure in convectively unstable systems, {\it J. Stat. Phys.}
{\bf 54}, 1459-1488.

\bibitem{Coullet} Coullet, P., Lega, J., Houchmanzadeh, B. and
Lajzerowicz, J. (1990) Breaking Chirality in Nonequilibrium
Systems, {\it Phys. Rev. Lett.} {\bf 65}, 1352-1355.

\end{chapthebibliography}

\end{document}