\documentstyle[12pt,times,aps]{revtex} \begin{document} \title{{\bf Two-Dimensional Noise-Sustained Structures \\ in Optical Parametric Oscillators}} \author{Marco Santagiustina, Pere Colet, Maxi San Miguel and Daniel Walgraef \cite{Daniel}} \address{Instituto Mediterraneo de Estudios Avanzados, IMEDEA \cite{www} (CSIC-UIB), \\ E-07071 Palma de Mallorca, Spain} \maketitle \vskip 0.5in \begin{abstract} The problem of two-dimensional (2D), transverse, noise-sustained pattern formation is theoretically and numerically studied, in the case of an optical parametric oscillator for negative signal detuning. This gives a complete analysis of a 2D, convective, pattern forming system which is also relevant to more general 2D physical systems. For the optical parametric oscillator the transversal walk-off due to the nonlinear crystal birefringence, exploited to phase-match the frequency down-conversion process, turns the instability to convective up to a certain threshold. In this regime noise-sustained patterns can be observed. These structures are a macroscopic manifestation of amplified microscopic noise which, in the context of optics, can be of quantum nature. Directly observable properties of the near- and far-field as well as statistical properties of the spectral intensity help to distinguish noise- from dynamics-sustained structures. Moreover, the analysis indicates that the walk-off term breaks the rotational symmetry of the 2D model. This causes a preferential selection of the stripe orientation, which would be otherwise random, being the modulus of the wave-vector the only restricted value. At the convective threshold an entire set of spatial modes becomes unstable; whereas the threshold of absolute instability depends on the relative orientation of the mode. Beyond the threshold for absolute instability this causes the co-existence, in the linear regime of evolution, of modes that are absolutely unstable and others that are only convectively unstable. The numerical solutions of the dynamical equations of the system under study confirm the analytical predictions for the value of the instability thresholds and the kind of pattern selected. Moreover, they allow to investigate the nonlinear regime showing qualitatively the co-existence of modes with different type of instability and giving a quantitative characterization of the transition from noise-sustained to dynamics-sustained structures. \end{abstract} \pacs{PACS numbers: 46.65.Sf, 42.50.-p, 42.65.Yj, 42.65.-k} \newpage \section{Introduction} Pattern formation in nonlinear optical systems is the object of an intense investigation, both theoretical and experimental \cite{abra90}. Spontaneous pattern formation has been observed or predicted in many different nonlinear optical systems such as nonlinear Kerr media in resonators \cite{lugi87,firt92}, Kerr slices in single mirror feedback configuration \cite{firthale,tamb93,gryn94,arec96}, two-level atoms \cite{lugi88}, atomic and molecular gas media \cite{gius93,lang95}, 1 second harmonic generation \cite{lede97} and optical parametric oscillators \cite{stal92,oppo94}. The range of observable patterns and parameter range of observation are highly enhanced when considering the extra degree of freedom associated with the polarization of light \cite{gedd94,mait94,auma97,miguel}. Stripes, squares, hexagons and more complicated stationary and dynamical patterns have been observed and characterized in the publications. A description of many of these phenomena in terms of universal amplitude equations gives the link which interconnects the optical patterns with those similarly observed in other systems \cite{crho93}. \newline Nowadays, the optical parametric oscillators (OPO's) are among the most attracting optical devices where pattern formation is studied \cite{stal92,oppo94,long96,valc96} and the recent experimental observation \cite{fuer97} of spatial patterns in quadratic media is very encouraging. This interest stems from the foreseen applications in the field of all-optical processing and storage of information \cite{abra90} as well as from the fact that OPO's can generate squeezed \cite{kimb87} and other non-classical states of light \cite{reid88}. For the latter reason they are thought to be a paradigmatic example of the interface between classical and macroscopic quantum patterns \cite{lugi96}. On this regard, there are recent studies which focalize mostly onto the quantum correlations associated with a pattern forming instability \cite{lugi96,lugi95,gatt95,marz95,lugi97,lugiPRL97}. There, a particular attention is paid to the noisy precursors observed below the threshold of the instability because they are spatially organized manifestations of quantum fluctuations. This situation is, from the classical viewpoint, conceptually equivalent to the convection experiments of ref. \cite{Ahlers}, aimed to characterize thermal fluctuations. In both cases spatial correlation are investigated very close to the instability threshold where fluctuations are weakly damped. In the classical case features in the correlation functions such as noise squeezing under the minimum classical level are obviously absent. The analogy with other experiments in classical fluid dynamics \cite{babc91} has suggested \cite{prl} that nonlinear optical systems can present macroscopic, noise-sustained structures, above the threshold of the instability in the so-called convectively unstable regime \cite{deis85}. This phenomenon refers to the situation when local perturbations of the steady-state can be advected much rapidly than their rate of spreading. In this case macroscopic patterns can arise and be observed only if noise is continuously applied, the pattern being now regenerated at any time, hence the name of noise-sustained structures. These structures are the result of noise amplification, with magnification factors of several order of magnitudes. They are thus interesting candidates for the study of quantum correlations in spatially structured systems. This situation is distinct from that of the noisy precursors, where noise is selectively enhanced, but not amplified, by the nonlinear spatial filtering effect. Noise sustained structures were predicted and characterized in optical passive cavities filled with a nonlinear Kerr medium and pumped by a tilted external beam \cite{prl}. In this situation the advection motion is induced by the input pump beam tilting \cite{halt92}. Here, we show that the convectively unstable regime and thus noise-sustained structures are also found in type I degenerate OPO's for positive signal detunings too. The present extension includes new important features which are either peculiar of the OPO or of wider interest in the study of convective instabilities in other physical systems. \newline In this article a transversal two-dimensional (2D) model is considered for the OPO. Note that the 1D model of \cite{prl} was justified by both theoretical \cite{lugi87,halt92} and experimental \cite{gryn94} results on the drift instability in nonlinear Kerr devices similar to the one we considered. For the OPO this simplification is generally avoided and the investigation is directed towards 2D transverse devices \cite{stal92,oppo94,long96,valc96}. We want to stress that, in spite of the particular choice of the OPO, the features that are found belong very generally to the whole class of 2D, convective, pattern forming systems. To the best of our knowledge the generation of noise-sustained structures was never studied before in a 2D system, though the theory of convective and absolute instability in higher dimensional systems was developed in plasma physics \cite{infe90,greg90}. We will show that pattern formation is highly affected by the advection term in the convectively unstable regime and also, and this is an important result, in the absolutely unstable regime. In fact, the advection breaks the rotational symmetry of the system and this causes a preferential orientation of the stripes, which can be explained only if the analysis is carried out taking into account the possible convective nature of the instability. \newline It is also worth to note that in the specific case of OPO's the convective regime can hardly be neglected because birefringent crystals are used to phase-match the down-conversion process. As known, the anisotropy of the medium implies that a transverse walk-off effect among the beams, due to the misalignment of the Poynting vectors \cite{bloe65,akhm66,shen84}, can occur. This means that an advection term is present in the governing equation, even when the pump beam is aligned with the cavity. Moreover the typical walk-off angle can be often larger than a few mrad and thus this effect is expected to overcome finite-size locking effects too \cite{lang97}. \newline The paper is organized as follows. In section II we present the model used to represent a type I degenerate OPO. The following section is dedicated to the linear stability analysis, which takes into account the possible convective nature of the instability. In section IV we analyze the generation of the noise-sustained convective structures and characterize the typical features of the pattern. A discussion aboutthe possible the experimental observation of such structures and the conclusions are presented in section V. \section{Governing equations} In this section we introduce the set of semi-classical equations we use to model the OPO. Such device may consist, for example, of a ring resonator, with flat mirrors, filled with a quadratic ($\chi^{(2)}$) nonlinear medium and pumped by an external laser source of frequency $\omega_p$. The output mirror allows a fraction of the internal field to leak out, for detection, in the near-field (NF, i.e. close to the output mirror) and far field (FF, far from the device) configurations. In this system the FF is the spatial Fourier transform of the NF. Under suitable experimental conditions (which we will soon specify) together with the residual input beam at $\omega_p$ two additional frequency components $\omega_s, \omega_i$ (usually called the signal and the idler) can be detected at the output port. They are generated by the nonlinear interaction which takes place inside the crystal: light quanta of the pump beam are down-converted to $\omega_s$ (signal) and $\omega_i$ (idler) photons via the parametric process, which is sketched in figure 1. This process is highly efficient when two conditions are satisfied \cite{bloe65,shen84}: \begin{eqnarray} \label{conservation1} \omega_p &=& \omega_s+\omega_i \\ \label{conservation2} \kappa_p(\omega_p) &=& \kappa_s(\omega_s)+\kappa_i(\omega_i) \end{eqnarray} They represent respectively the conservation of the energy and momentum \cite{notek} of the photons involved in the interaction. In the remaining of the paper we will assume the down-conversion process to be frequency degenerated, i.e. $\omega_s=\omega_i=\omega_0$ (hereafter the fundamental harmonic, FH). Hence eq. (\ref{conservation1}) implies that $\omega_p=2 \omega_0$ (hereafter the second harmonic, SH, or simply the pump). \newline As for eq. (\ref{conservation2}), it means that the process requires a phase-matching condition for the wave-vectors which can be rewritten (taking into account the frequency degeneracy) in the form: \begin{equation} \label{phase-match} n_p(2 \omega_0) = \frac{n_s(\omega_0) + n_i(\omega_0)}{2} \end{equation} where $n_{p,s,i}(\omega)$ are the refractive indexes of the pump, the signal and the idler, which in general depend on both the frequency and the polarization of the fields. The phase-matching condition (\ref{phase-match}) cannot be trivially satisfied for an arbitrary choice of the frequency and of the nonlinear medium. Quite often birefringence is exploited since it allows to compensate the index frequency dispersion by means of orthogonally polarized waves \cite{bloe65,shen84}. In fact, in a uniaxial birefringent crystal there exist two preferential orthogonal linear polarizations, propagating independently with a different refractive index (for a discussion of the normal modes of a birefringent medium see for example \cite{phot}). Thus, there are two types of phase-matching conditions \cite{bloe65,shen84}: one involves polarization non-degenerate, down-converted beams, i.e. the polarization of $\omega_s$ is orthogonal to that of $\omega_i$ and is usually referred to as the type II matching. The type I phase-matching, on the contrary, involves polarization degenerate output photons, i.e. the signal and the idler have the same polarization which is orthogonal to that of the pump in order to satisfy (\ref{phase-match}). We will consider only this second case because type I phase matching is a common experimental set-up in second harmonic generation (SHG) and OPO's \cite{bloe65,shen84,hall93} and because this reduces the number of equations which govern the phenomenon. In this case \begin{equation} \label{indexes} n_p(2 \omega_0)=n_s(\omega_0)=n_i(\omega_0)=n \end{equation} The aim of reviewing these well known features of the parametric down-conversion is to stress that this process is commonly obtained by means of birefringent media and orthogonally polarized beams, a fact that brings important consequences. In fact, except for the particular case in which the propagation takes place along an optical axis (non-critical phase matching), one wave is no longer polarized in a normal mode. It can be demonstrated that for this beam, called extraordinary, the Poynting vector is not parallel to the propagation direction (see again \cite{phot} for details). On the contrary, the other orthogonally polarized beam still has its Poynting vector parallel to the propagation direction and it is thus defined as an ordinary ray. Therefore, the ordinary and extraordinary beams go slightly misaligned during propagation and they walk off one from each other \cite{bloe65,akhm66,shen84,phot}. \newline The walk-off effect, though considered for modeling pulse generation in OPO's \cite{smit95} and solitary wave propagation in SHG \cite{torn95}, has been neglected in the previous 2 on transverse structures formation in the OPO's. In principle it seems logical that this term could be neglected for its smallness in the non-critical or quasi non-critical phase-matching (propagation direction close to an optical axis). However, in the more general case and despite of its smallness this term can be of a fundamental importance as soon as the presence of noise is considered. The effect of the transverse walk-off is in fact equivalent, as it appears in the dynamical equations, to the pump beam tilting of the Kerr case \cite{halt92}. As demonstrated in ref. \cite{prl} a convection term, like the one introduced by beam tilting or walk-off effects, gives rise to a convectively unstable regime, where noise-sustained structures can be observed. Moreover, we will show that in a 2D system this term causes a preferential pattern orientation in the absolutely unstable regime. This effect can be explained only by taking into account the mode selection mechanism introduced by the convection-like term. \newline The equations governing the time evolution of the SH and FH field envelopes in the resonator can be obtained in two steps. First, we can derive in the slowly varying envelope approximation (SVEA) the propagation equations in the nonlinear medium, for example by means of a multiple scales expansion (see also ref. \cite{shen84}). Then, by following the guidelines of \cite{lugi88} we can find, in the mean-field limit (MFL), the time evolution equations when the medium fills a ring resonator. In a ring cavity, photons are generated in the crystal and moves transversally with a walk-off angle that can be typically of the order of 0.1 mrad. So, the actual displacement attained after the propagation in the crystal is very small. Outside the crystal the waves are parallel and thus the signal generated is re-injected collinearly with the pump at the next pass, but slightly displaced in the walk-off direction. The mean field approximation is thus averaging out the effect of this continuous space displacement of photons. We have performed all steps and finally obtained the following equations for the envelopes of the SH ($A_0(x,y,t)$, ordinary polarized) and FH ($A_1(x,y,t)$, extraordinary polarized) (see also \cite{stal92,oppo94,long96,drum80}): \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] + \sqrt{\epsilon_0} \xi_0(x,y,t) \\ \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] + \sqrt{\epsilon_1} \xi_1(x,y,t) \end{eqnarray} where $x,y$ are the transversal spatial dimensions, $t$ the time and the coefficients are defined as follows. The decay rates of the SH and FH fields in the cavity are $\gamma_{0,1}=v_{0,1} T_{0,1}/L$ where $v_{0,1}$ are their group velocities, $T_0 = T_1 =T \ll 1$ are the output mirror transmission coefficients and $L$ is the cavity length. The scaled cavity detunings are: \begin{eqnarray} \label{detun} \Delta_0&=&(\omega_{c0}-2 \omega_0)/\gamma_0 - \Delta \kappa L/T_0 \\ \label{detun2} \Delta_1&=&(\omega_{c1}- \omega_0)/\gamma_1 \end{eqnarray} where $\omega_{c0,c1}$ are the cavity resonances closest to the SH and FH frequencies. The second term in eq. (\ref{detun}) takes into account a possible phase-mismatch $\Delta \kappa$ of the parametric interaction. In order to be consistent with the MFL this term must be small, i.e. $\Delta \kappa \ll 1/L$ so it can be included in the detuning parameter as shown. Note also that the bandwidth of phase-matching is much larger that the cavity modes and thus, if present, the mismatch can be considered constant. Therefore, the limits of validity of this model are those set by the SVEA-MFL approximation (see \cite{lugi88} for a detailed definition) plus the additional condition of a small phase-mismatch. Here, for the sake of simplicity, we will suppose that eq. (\ref{indexes}) holds and thus $\Delta \kappa=0$. The coefficients $a_{0,1}=L/(2 \kappa_{0,1} T_{0,1})$, where $\kappa_{0,1}=\kappa_{p,s}$, represent diffraction. They are related by $a_0=a_1/2$ since $\kappa_0=2 \kappa_1$ as a consequence of eq. (\ref{indexes}) and the frequency degeneracy, and because we have previously set $T_0 = T_1 =T$. The coefficient $\rho_1=L \tan(\alpha_1)/T_1$, where $\alpha_1$ is the angle between the SH and FH Poynting vectors induced by the birefringence \cite{bloe65,shen84,phot,torn95}, is the walk-off parameter. The term $E_0(x,y,t)$ is the input SH pump and $K_0= \omega_0 \chi_{eff}^{(2)} L /( n c T)$ the nonlinear coefficient, where $\chi_{eff}^{(2)}$ is the effective (quadratic) susceptibility and $n$ is defined by (\ref{indexes}). \newline Finally, the last terms in the equations are complex, Gaussian white noise with zero mean ($<\xi_{0,1}(x,y,t)>=0$) and correlations $<\xi_i(x,y,t) \; \xi_j^*(x',y',t')>=2 \delta_{i,j} \delta(t-t') \delta(x-x') \delta(y-y') \; i,j=0,1$. In a linearized version of eqs. (\ref{master},\ref{master2}) they describe quantum noise in the Wigner representation, as shown in \cite{lugi97}. Here, they can account for thermal and input field fluctuations too. \section{Convective and absolute linear instability analysis} This section presents the linear instability analysis of the steady state solution of eqs. (\ref{master},\ref{master2}) which corresponds to the OPO below the threshold of signal generation. This will be carried out according to the theory presented in \cite{deis85,infe90,hall68} in order to be able to discriminate convective from absolute instabilities. \newline For the sake of clarity, we first recall the definition of the convective instability for a 2D system. In general, a steady-state is defined to be absolutely stable (unstable) if a localized perturbation decays (grows) with time. In figure 2a and 2b we show these two situations for a system in which advection is also present along the y-axis. However, there is a third possibility: the perturbation may grow (unstable) but the advection velocity could overwhelm the speed of spreading in the direction of the advection (figure 2c). In this case the perturbation eventually leaves the system which returns to the steady-state. This type of instability is defined to be convective. \newline Clearly, the definition depends on the choice of the system of reference; in fact, by choosing a reference frame moving at the speed of the perturbation peak we would always define a system to be absolutely stable or unstable. However, in finite physical systems, there is always a preferred reference frame, which makes the above definition unambiguous. In the OPO, for example, the pump laser beam is present in a well defined region of space and thus it can be taken as the fixed frame. The FH field moves transversally due to the walk-off effect and thus a perturbation of the FH might leave the pump region at some time. Clearly, outside that portion of space there is no amplification via the down-conversion process and the FH fades because of mirror losses. \newline Thus, we proceed by analyzing the parameter range for which the system of eqs. (\ref{master},\ref{master2}) is stable, convectively unstable or absolutely unstable. When $\epsilon_{0,1}=0$, the equations have the following uniform steady-state: \begin{equation} \label{st-st} A_0=\frac{E_0}{1+i\Delta_0} \; , \; A_1=0 \end{equation} To determine when the latter becomes unstable we can linearize eqs. (\ref{master},\ref{master2}) close to the steady-state and look for solutions of the kind $\exp(i{\vec q} \cdot {\vec r} + \lambda t)$, where ${\vec q}$ is a 2-dimensional vector with real components, ${\vec q}=(q_x, q_y) \in {\cal R}^2$ and ${\vec r}=(x,y)$. It turns out that the steady-state becomes unstable only along the FH component ($A_1$) of the eigenvector and thus the analysis reduces to the study of the linearized form of eq. (\ref{master2}) with $A_0$ given by (\ref{st-st}) \cite{stal92,oppo94,long96,drum80}. The dispersion relation obtained is: \begin{equation} \label{eigenvalue} \lambda_{\pm}({\vec q}) = i q_y \rho_1 -1 \pm \sqrt{F^2 - (\Delta_1 + a_1 (q_x^2+q_y^2))^2} \end{equation} where $F^2=K_0^2 |E_0|^2 /(1+|\Delta_0|^2)$ is a normalized pump intensity. \newline The determination of the nature of the instability entails the estimation of the linearized asymptotic behaviour of a generic perturbation of the FH steady state, say $\psi$, which is given by: \begin{equation} \label{evol} \psi(x,y,t)= \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} dq_x dq_y \tilde{\psi}(q_x,q_y,0) \exp[i(q_x x + q_y y) + \lambda (q_x,q_y) t] \end{equation} where $\tilde{\psi}(q_x,q_y,0)$ is the initial perturbation in the spatial wave-vector space and $\lambda (q_x,q_y)$ is the eigenvalue with largest real part (plus sign). \newline According to the definition given above (figure 2), the system is (absolutely) stable when $|\psi({\vec r_0}+{\vec v}t,t)| \rightarrow 0$ as $t \rightarrow \infty$ for any arbitrary fixed ${\vec r_0}$ and any velocity ${\vec v}$ \cite{deis85}. It is easy to demonstrate that the condition to have absolute stability is $\Re(\lambda(q_x,q_y))<0$ for any $(q_x,q_y) \in {\cal R}^2$, because it yields an integral (\ref{evol}) decaying with time. \newline The modes $(q^m_x,q^m_y)$ with the largest growth rate $\Re(\lambda)$ are those of modulus \begin{equation} \label{qcrit} |(q^m_x,q^m_y)|=q_c=\sqrt{-\frac{\Delta_1}{a_1}} \end{equation} if $\Delta_1<0$ and $(q^m_x,q^m_y)=(0,0)$ otherwise. In the following we treat the case $\Delta_1<0$ where the maximum of $\Re(\lambda)$ is reached on a ring of modes in the wave-vector space \cite{foot}. The threshold for parametric down-conversion takes place when $\Re(\lambda)=0$, i.e. when $F^2=1$, and above threshold formation of a stripe pattern is expected \cite{stal92,oppo94,long96,valc96}. \newline When $\Re(\lambda(q_x,q_y))>0$ for some wave-vector $(q_x,q_y)$, there can be two situations. We can either have that $|\psi({\vec r},t)| \rightarrow \infty$ for any arbitrary value of ${\vec r}$ (absolutely unstable) or that $|\psi({\vec r_0},t)| \rightarrow 0$ and $|\psi({\vec r_0}+{\vec v}t,t)| \rightarrow \infty$ for some velocity ${\vec v}$ and any arbitrary ${\vec r_0}$ (convectively unstable). Note that for determining the asymptotic behaviour of the perturbation we only need to evaluate the integral (\ref{evol}) for those values of $(q_x,q_y)$ for which $\Re(\lambda(q_x,q_y)) \geq 0$, the other modes giving no contribution (they are surely decaying). The extrema of the surface of integration are thus identified by the loci in the real plane $(q_x,q_y)$ where $\Re(\lambda)=0$. In order to illustrate the integration technique we first consider the analysis with only one spatial dimension ${\vec q} \rightarrow q$. The complex function of real variable $\lambda(q)$ can be analytically continued over the complex plane $k$ in the form $\lambda(k/i)$. The previous real domain of definition corresponds to a purely imaginary $k$. Then using the Cauchy theorem, we can equivalently evaluate (\ref{evol}) along any contour in the complex wave-vector plane connecting the extrema of integration, provided that the integrand has no singularities in the area bounded by the original and the new contour. In particular we can evaluate it for a contour which crosses a saddle point $k_s$ in the complex plane, along the direction of the steepest descent. The asymptotic behaviour of the integral is then given by the value of the exponential part of the integrand calculated at the saddle point. Eventually, if $\Re(\lambda(k_s)) > 0$ this term diverges and the instability is absolute (see an example in 1D, Taylor-Couette flows in ref. \cite{tagg90}). In two dimensions we can proceed in a similar way and the integral yields an absolutely unstable solution if the following conditions are satisfied: \begin{eqnarray} \label{marginal} \Re(\lambda(k^s_x,k^s_y))>0 \\ \label{marginal2} \Re(\nabla_{\vec k}^2 \lambda(k_x,k_y) |_{k^s_x,k^s_y}) \geq 0 \end{eqnarray} where $(k^s_x,k^s_y)$, defined by \begin{equation} \label{saddle0} \nabla_{\vec k} \lambda(k_x,k_y) |_{k^s_x,k^s_y} + i \frac{{\vec r}}{t}=0 \end{equation} is the saddle point in the ${\vec k}$ complex space \cite{infe90}. Taking ${\vec r}={\vec r_0}+{\vec v}t$, for $t \rightarrow \infty$ the saddle is given by $\nabla_{\vec k} \lambda(k_x,k_y) |_{k^s_x,k^s_y} + i {\vec v}=0$. In the fixed reference frame of the pump beam ${\vec v}=0$ and so: \begin{equation} \label{saddle} \nabla_{\vec k} \lambda(k_x,k_y) |_{k^s_x,k^s_y} =0 \end{equation} To summarize: if $\Re(\lambda(q^m_x,q^m_y))<0$ where $(q^m_x,q^m_y)$ is the loci (in ${\cal R}^2$) of the maxima of $\Re(\lambda)$ the system is absolutely stable; if $\Re(\lambda(q^m_x,q^m_y))>0$ and $\Re(\lambda(k^s_x,k^s_y))<0$ where $(k^s_x,k^s_y)$ is the saddle point (in ${\cal C}^2$), it is convectively unstable and it is absolutely unstable otherwise ($\Re(\lambda(k^s_x,k^s_y))>0$). \newline For the OPO, by replacing $(q_x,q_y)$ with $(k_x,k_y)/i$ and taking into account eq. (\ref{qcrit}) we can rewrite $\lambda_+$ of eq. (\ref{eigenvalue}) as \begin{equation} \label{eigen2} \lambda(k_x,k_y)=-1+ \rho_1 k_y + \sqrt{F^2 - a_1^2 (q_c^2 + k_x^2 + k_y^2)^2} \end{equation} and thus the saddle is found through eq. (\ref{saddle}) at \begin{equation} \label{thresh0} \frac{2 a_1^2 (q_c^2 + (k^s_x)^2+(k^s_y)^2) k^s_x}{\sqrt{F^2 - a_1^2 (q_c^2 + (k^s_x)^2+(k^s_y)^2)^2}}=0 \; , \; \rho_1 - \frac{2 a_1^2 (q_c^2 + (k^s_x)^2+(k^s_y)^2) k^s_y}{\sqrt{F^2 - a_1^2 (q_c^2 + (k^s_x)^2+(k^s_y)^2)^2}}=0 \end{equation} For $\rho_1=0$ the saddle solution of (\ref{thresh0}) coincides with (\ref{qcrit}) and therefore the criterion of convective and absolute instability coincides. For $\rho_1 \neq 0$ the saddle is given by \begin{equation} \label{thresh} k^s_x=0 \; , \; \rho_1 - \frac{2 a_1^2 (q_c^2 + (k^s_y)^2) k^s_y}{\sqrt{F^2 - a_1^2 (q_c^2 + (k^s_y)^2)^2}}=0 \end{equation} The second of (\ref{thresh}) was numerically resolved for a complex $k^s_y$ together with $\Re(\lambda(0,k^s_y))=0$ to calculate the value of $F$ at the absolute instability threshold. The result is shown, as a function of the FH detuning $\Delta_1$, in figure 3. \newline When the OPO is at threshold ($F=1$) and $\rho_1 \neq 0$, all unstable modes are convective; if the pump amplitude is increased, the first mode to become absolutely unstable satisfies $k_x=0$ (first of eqs. (\ref{thresh})). The advection term breaks the rotational symmetry and stripes parallel to the x-axis are always the selected mode. In fact, from the value of the saddle given by eq. (\ref{thresh}) the selected wave-vector can be determined \cite{dee83}: in particular we get the condition $q_x=0$. Note that this asymmetry cannot be ascribed to a change in the growth rate of the most unstable modes which is equal for all the modes on the ring (\ref{qcrit}). The mechanism which creates this selective action is related to the advection-spreading balance peculiar of the convective regime, as we mention in the previous section. Moreover, as at the absolute threshold the only absolutely unstable modes are those close to $q_x=0$, it is reasonable to assume that the remaining modes should be still convectively unstable. It is then interesting to address the question of the co-existence of absolutely unstable and convectively unstable modes. In the next paragraphs we investigate, through a simple approximated analysis, this phenomenon. The predictions are qualitatively confirmed by numerical solutions which are presented in section IV (see figure 6). \newline First, note that the position of the saddle $(k^s_x,k^s_y)$, and thus of the threshold of the absolute instability, depends on the coefficient of the advection term $\rho_1$ (second of eqs. (\ref{thresh}) and figure 3). In particular, the fact that the faster the advection the larger the absolute instability threshold agrees with the intuitive definition we gave of the convective regime (see fig. 2). The fact that the instability turns to be absolute when the rate of spreading of a perturbation is larger than the advection can be shown by means of a simplified mathematical analysis. Let consider eq. (\ref{evol}) again and make a Taylor expansion of the eigenvalue (\ref{eigenvalue}) around a particular 1 of the ring given by (\ref{qcrit}): $q^m_x=q_c \cos \theta$, $q^m_y= q_c \sin \theta$, where the variable $0 < \theta \le \pi/2$ represents the angle between $(q^m_x,q^m_y)$ and the axis $(q_x,0)$. If the expansion is truncated at the second order in the differences $q_x-q^m_x$ and $q_y-q^m_y$ the integral (\ref{evol}) can be solved analytically (see the 1D examples in \cite{hall68}) and the condition for the absolute instability becomes \begin{equation} \label{mode-th0} \Re(\lambda(q^m_x,q^m_y)) +\frac{(y_0+\rho_1 t)^2}{2 \Re(\lambda_{yy} (q^m_x,q^m_y))t^2} > 0 \end{equation} where $\lambda_{yy}=d^2 \lambda/ d q_y^2 |_{q^m_x,q^m_y}$. Finally, upon substitution of $\lambda_{yy}$ calculated through (\ref{eigenvalue}) and as $t \rightarrow \infty$, we obtain \begin{equation} \label{mode-th} -1 + F \left( 1 - \frac{\rho_1^2}{8 a_1^2 q_c^2 \sin^2 \theta} \right) >0 \end{equation} Equation (\ref{mode-th}) shows clearly that the larger the group velocity $\rho_1$ the larger the threshold value of $F$ for the absolute instability. This approximated solution, valid when the Taylor expansion can be truncated at the second order and for $\theta$ not too close to 0, is in agreement within a few percent, with the exact resolution given in figure 3 for $\theta=\pi/2$. Moreover, it confirms that the mode $\theta=\pi/2$ ($0,q_y$) has the lowest threshold for absolute instability and indicates that for a normalized pump amplitude $F$ above the threshold of absolute instability the modes for which \begin{equation} \label{theta} \theta>\theta_c=\arcsin \left( \frac{\rho_1}{2 a_1 q_c \sqrt{2}} \sqrt{\frac{F}{F-1}} \right) \end{equation} are absolutely unstable, while the others are still convectively unstable. \newline The quantity $\Re(\lambda_{yy}(q^m_x,q^m_y))t$ can be interpreted \cite{hall68} as the mean-square spatial spread of the perturbation in the advection direction. It is then clear that the mode with $\theta=\pi/2$ spreads faster than the modes with smaller $\theta$ and hence, although all modes on the ring have the same growth rate $\Re(\lambda(q^m_x,q^m_y))$, there are some modes which are preferred through a spreading selection mechanism. Eventually, the rotational symmetry is broken. \newline To summarize, the results of this section are the following. In the OPO's the presence of the walk-off term induces the existence of a convectively unstable regime, just above the OPO threshold of signal generation. In this regime perturbations are advected faster than they spread. The walk-off term does not change the growth rate of modes but breaks the rotational symmetry because the critical modes $(q^m_x,q^m_y)$ have different spreading velocities in the direction of the advection. This causes the modes with $q_x=0$ to be preferentially selected both in the convectively and absolutely unstable regime. Moreover, the symmetry breaking implies the co-existence of absolutely and convectively unstable modes in a certain region of parameters in the absolutely unstable regime. \newline Finally, we want to stress that these results, obtained in the specific case of an OPO, are actually very general and could be extended to pattern formation in similar 2D, convective system. To our knowledge, this is the first example of a complete study of the formation of convective structures in 2D systems. \section{Characterization of the noise-sustained structures} Noise-sustained convective structures, as well as features associated to the 2D, convective model are presented in this section as they result from the numerical solutions of the dynamical equations (\ref{master},\ref{master2}). Noise-sustained patterns can be expected in the regime of convective instability where no dynamics-sustained structure can be observed because of the overwhelming advection. In practice, the noise acts as a perturbation which, being continuously applied, regenerates a new pattern at any time. \newline We show the integration of eqs. (\ref{master},\ref{master2}) with a noise intensity $\epsilon_{0,1}=1.5 \times 10^{-11}$ and with the same integration scheme of \cite{prl}. The system size was $320 \times 320$ scaled spatial units and the pump was a super-Gaussian beam: $E_0(x,y,t)=E_m \exp(-((x^2+y^2)/\sigma_0^2)^{m}/2)$, with $m=5$ and $\sigma_0=112$; the grid was of $512 \times 512$ points and the time step $0.01$ normalized units. The super-Gaussian beam is flat top and this allows to apply the results of the linear stability analysis which are strictly valid only for a uniform steady-state. We also set: $\gamma_0=\gamma_1=1$, $\Delta_0=1, \Delta_1=-0.25$, $a_0=a_1/2=0.125$, $K_0=1$, $\rho_1=0.15$. This is equivalent to have scaled: the time with the decay rate; the space with a multiple of the diffraction length; the amplitude with the nonlinear coefficient. Under this, or similar, scaling the equations become dimensionless; thus, all quantities in the figures are dimensionless as well. Note that the parameters of the simulations were chosen in order to make simpler the visualization of the results, but they are not critical. Noise-sustained structures were found also with smaller beams, larger and smaller advection terms, different SH and FH detunings and diffraction coefficients, close to the one presented. \newline In the figures we do not show the whole integrating window, but only the central parts. For the NF the spatial region where the generated FH is appreciably intense ($|x|<103.75, -97.50$ the trivial steady-state is unstable for homogeneous perturbations and the field evolves to a a uniform solution so no pattern is formed. However, the walk-off can affect the selection mechanism and an inhomogeneous pattern can be formed also for small positive signal detunigs. This process is not related to the convective instability phenomenon and the noise-sustained structures and it is described in \cite{ol}. \bibitem{ol} M. Santagiustina, P. Colet, M. San Miguel, D. Walgraef, "Walk-off and pattern selection in optical parametric oscillators", to be published, Optics Letters. \bibitem{tagg90} R. Tagg, W.S. Edwards, H.L. Swinney, Phys. Rev. A, {\bf 42} 831 (1990). \bibitem{dee83} G. Dee, J. Langer, Phys. Rev. Lett., {\bf 50}, 383 (1983); W. Van Saarloos, Phys. Rev. A, {\bf 37} 211 (1988). \bibitem{opos} J-Y Zhang, J.Y. Huang, Y.R. Shen, Laser Science and Technology, {\bf 19}, (Harwood Ac.) (1995); C.L. Tang, L.K. Cheng, Laser Science and Technology, {\bf 20} (Harwood Ac.) (1995). \bibitem{note} Recently, birefringent materials with large nonlinearities and no walk-off ($LiNbO_3, KTA$) have been made available. The walk-off effect disappears when the propagation in the crystal takes place along the principal axes. However, even in this case a weak walk-off, due to experimental crystal misalignment, should be present and this is enough to obtain a convective regime, as shown. \bibitem{note2} For the example given the oscillation frequency would be about 25 MHz. \bibitem{sant97} F. Castaldo, D. Paparo, E. Santamato, Opt. Commun., {\bf 143}, 57 (1997). \end{references} \newpage {\bf Figure Captions:} \newline Figure 1: Schematic representation of the frequency down-conversion process \newline Figure 2: Pictorial definition of the stable and unstable regimes in a two-dimensional system with a transversal walk-off. The left (right) column is a lateral (top) view of a perturbation of the steady-state. Dashed (solid) curves represent the $t=0$ ($t>0$) conditions for: a) the stable, b) the absolutely unstable and c) the convectively unstable regimes. At the fixed position ${\vec r_0}=(x_0,y_0)$ the field amplitude a) decays, b) grows, c) decays due to the strong advection compared to spread. \newline Figure 3: Stability diagram as a function of the FH detuning for the steady-state solution (\ref{st-st}). In the region below the solid line the solution is always stable; the dashed curves represent the absolute instability thresholds for different values of the walk-off parameter: a) $\rho_1=0.25$, b) $\rho_1=0.2$ and c) $\rho_1=0.15$. The region between the solid and the dashed line is the domain of convective instability for a given value of $\rho_1$. In particular we shadowed the region referring to $\rho_1=0.15$ (other parameters: $\gamma_0=\gamma_1=1$, $a_1=0.25$). The star (*) and the plus (+) signs indicate the parameters used for the numerical solutions of figure 4 and figure 5 respectively. \newline Figure 4: The a) near-field and b) far field intensity images of the pattern in the absolutely unstable regime are shown ($t=1500$) in a linear gray-scale. The initial condition is zero for the FH and the steady-state solution for the pump beam; random noise of intensity $\epsilon_0=\epsilon_1=1.5 \times 10^{-11}$ is continuously applied. The other parameters as in figure 3 (*) and $\Delta_0=1$. \newline Figure 5: Same than figure 4 for the convectively unstable regime (+ of figure 3). The gray-scale of a) is the same of figure 4a); the far field in figure b) is scaled to its maximum to enhance the visibility. \newline Figure 6: Intensity distribution for the different spatial modes as a function of the angle $\theta$ and the time as it results from a numerical integration. The intensity content of the modes with $\theta$ closer to zero tends to decrease because of the overwhelming advection, i.e. these modes are convectively unstable. On the contrary the most rapidly spreading modes, with larger $\theta$, are absolutely unstable and can grow. The parameters are as for figure 3 (*) except for $\epsilon_0=\epsilon_1=0$ (no noise); the FH field was initially set to a small intensity, random condition. \newline Figure 7: Spectral intensity of the field amplitude at a fixed spatial position ($0,6.25$) for $F \simeq 1.0465$ (solid curve) and $F \simeq 1.025$ (dashed curve). Noise level was the same of figures 4 and 5. The dashed intensity spectrum has been multiplied 3 times in order to make it visible in the scale of the solid one. \newline Figure 8: Variance $\sigma_{ij}^2$ of the spectra detected in different positions $(x_i,y_j); \; i=1,2,3 \; j=1,2$ with $x_1=-25, x_2=0, x_3=25, y_1=37.5, y_2=68.75$ and their average (solid line). The positions correspond to the symbols: ($x_1,y_1$) filled triangle; ($x_2,y_1$) star; ($x_3,y_1$) triangle; ($x_1,y_2$) filled square; ($x_2,y_2$) cross; ($x_3,y_2$) square. The dotted vertical line indicates the threshold predicted theoretically (see fig. 3) for the walk-off parameter used ($\rho_1=0.15$). \end{document}