Transverse Patterns in Degenerate Optical-Parametric Oscillation and Degenerat Four-wave Mixing

Transverse pattern formation in dcgenerate optical parametric oscillation (DOPO) is a subject of' present interest [ I ] . In this work we carry out an analytical study of pattern formation i n DOPO and degenerate lour-wave mixing (DFWM), which is complemented by numerical integration of the microscopic cquations of both systems. In the limit of small pump and signal detunings both systems are described by the real SwiftHohenberg equation (SHE). This broadens the range of similarities existing between DOPO and DFWM, which comprise both temporal instabilities and quantum properties. On the other hand the derivation of the SHE gives support io the already observed leading role of rolls in DOPO 111. l'he predictions about the stability of these patterns given by the SHE are in good agreement with numerical simulations. Moreover. i t has been recently shown that the SHE contains stable hexagons with an additional zero-mode [ 2 ] . which have been also found by numerical simulation. Contrarily, for small signal and large pump detunings the SHE is modified by the appearance of additional nonlinear tenns, which signals the existence of a nonlinear resonance (NLR). These additional ternis are however different iii DOPO and DFWM, limiting the similarities existing between both systems.