\documentstyle[aps,a4,epsfig]{revtex} \parindent=0.25in \voffset= -1.1207 true cm \textheight= 22.86 true cm %\hoffset= -1.3831 true cm \hoffset= -1.8 true cm \textwidth= 16.51 true cm %\pagenumbering{0} %\oddsidemargin -0.5cm \begin{document} \draft \tightenlines \title{\bf Two dimensional vectorial localized structures in optical cavities} \author{\bf P. Colet$^1$, R. Gallego$^1$, E. Hern\'andez-Garc\'\i a$^1$, M. Hoyuelos$^1$,\\ G.L. Oppo$^2$, \underline{M. San Miguel$^1$}, and M. Santagiustina $^1$} \address{\it $^1$Instituto Mediterr\'aneo de Estudios Avanzados, IMEDEA (CSIC-UIB),\\ Universitat Illes Balears, E-07071 Palma de Mallorca, Spain. \\ Tel: +34 971 173229, Fax: +34 971 173426, e-mail: maxi@imedea.uib.es, http://www.imedea.uib.es/PhysDept/ \\ $^2$Department of Physics, University of Strathclyde, Glasgow G4 0NG, Scotland, U.K.} \maketitle \vskip 0.5cm Localized structures in lasers and in optical cavities containing nonlinear optical materials are being intensively studied \cite{Rosanov96,Staliunas98,Peschel98,HernandezGarcia99} %Tlidi94,Firth96,Schreiber96,Spinelli98,Weiss99,Oppo99,LeBerre99, %HernandezGarcia99} because of two main reasons. On the one hand complex spatio-temporal dynamics is often dominated by these particle-like objects. Secondly, the ability to control, create and erase these objects opens the way to new technological developments, including parallel information processing. We are here concerned with two dimensional localized objects in the transverse plane of an optical cavity. Generally speaking they are named cavity solitons and they can take the form of vortices or bright or dark dissipative spatial solitons. Most studies of localized structures in optical systems consider light with a fixed polarization. However, the vectorial degree of freedom of light leads to a very interesting phenomenology associated with a space and time dependent polarization \cite{Hoyuelos98}. Such degree of freedom is also relevant from the point of view of information encoding and processing. We address here the question of new types of localized structures emerging from polarization nonlinear dynamics. A visualization of vectorial localized structures can be given in terms of isolated zeroes or peaks of either of the two independent polarization components of the electric field. Situations of hole-hole, peak-hole or peak-peak in a given polarization background can be envisaged. A proper classification \cite{Pismen94} of these objects depends on the reference states preferred by a particular system. We consider here three different systems which give relevant examples of possible vectorial localized structures and of their dynamics. \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent {\large\bf Broad Area Lasers} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \medskip The interaction of the two polarization components of light in rotationally symmetric large aperture lasers can be described, close to threshold, by the Vector Complex Ginzburg Landau Equation (VCGLE) \cite{SanMiguel95}. The VCGLE can be written as coupled equations for the two circularly polarized components $A_\pm$ of the slowly varying part of the vector complex field: \begin{equation} \partial_t A_\pm = A_\pm + (1 + i\alpha) \nabla^2 A_\pm - (1 + i\beta) (|A_\pm|^2 + \gamma |A_\mp|^2) A_\pm. \label{vcgle} \end{equation} \vspace{-0.5cm} \noindent \begin{minipage}[t]{0.48\textwidth} \begin{figure} \centerline{\epsfig{file=fig1.eps,width=\textwidth}} \caption{Frozen field configuration: (a) $|A_+|^2$, (b) $\phi_g$ ($\gamma=0.1$, $\alpha=0.2$, and $\beta=2$).}\label{fig1} \end{figure} \end{minipage} \hfill \begin{minipage}[t]{0.48\textwidth} \begin{figure} \centerline{\epsfig{file=fig2.eps,width=\textwidth}} \caption{Instantaneous configuration for $\gamma=0.8$. (a) $|A_+|^2$, (b) $|A_-|^2$. Bright (dark) spots correspond to maximum (minimum) intensity. }\label{fig2} \end{figure} \end{minipage} \bigskip \noindent The parameters $\alpha$ and $\beta$ are associated with the strength of diffraction and detuning respectively and the condition $1+\alpha\beta> 1$ is always satisfied. For the coupling parameter $\gamma<1$, as considered here, homogeneous linearly polarized solutions in an arbitrary direction are preferred. In this background it is natural to look for localized structures of vortex type as isolated zeros of $A_+$ and/or $A_-$ \cite{HernandezGarcia99}. Charges $n_\pm$ are associated with the change of the phases $\phi_\pm$ of $A_\pm$ around those zeroes. We call {\it vectorial defect} a zero that is present in both components of the field at the same point. A vectorial defect can be of {\it argument} type ($n_+ = n_-=1$) or of {\it director} type ($n_+ = - n_-=1$). They are identified, respectively, by a two-armed spiral or a target pattern of the global phase $\phi_g=\phi_+ + \phi_-$ (Fig. \ref{fig1}). A {\it mixed defect} is a zero present only in one component of the field. Vectorial defects are stable in a range of parameters. Within this range they freeze the dynamics creating exclusion islands with mixed defects concentrated at their borders. Vectorial defects loose their stability either by an unbinding of the zeroes of the two fields or by an instability of the phase spirals. This leads to a dynamical state in which $A_\pm$ are highly anticorrelated. This state is dominated by mixed defects characterized as a localized bound state of a zero of one component and a peak of the opposite circularly polarized component. These localized objects move on a linearly polarized background (Fig. \ref{fig2}). \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent {\large\bf Vectorial self-defocusing Kerr Resonators} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \medskip Consider a ring cavity filled with an isotropic Kerr medium and driven by an external linearly polarized input field. This can be described by coupled driven and damped Non Linear Schr\"odinger Equations for the two circularly polarized components $A_\pm$ of the slowly varying part of the vector complex field. In a mean field approximation they become \cite{Hoyuelos98} \begin{eqnarray} \partial A_\pm &=& -(1 - i \theta) A_\pm + i a \nabla^2 A_\pm + A_0 - i [ \alpha |A_\pm|^2 + (\alpha + \beta) |A_\mp|^2] A_\pm , \label{1} \end{eqnarray} where $A_0$ is the input field, and $\theta$ is the cavity detuning. A stripe pattern orthogonally polarized to the input field occurs close to threshold. For higher values of $A_0$ the system prefers either of two equivalent homogeneous states which are close to circularly polarized states. Polarized localized structures are here visualized as a hole of $A_+$ ($A_-$) in the background of a circularly positive (negative) polarized state together with a peak of $A_-$ ($A_+$) (Figs. \ref{fig3} and \ref{fig4}) . These structures are related to domain walls separating the two equivalent homogeneous circularly polarized states. They are stable for $A_{0,m}