Codimension 2 and 3 situations in a ring cavity with elliptically polarized electromagnetic waves

• We study pattern formation in cavities with elliptically polarized input field.
• The system is well described by two coupled Lugiato–Lefever equations.
• There cannot be more than 3 wavenumbers becoming simultaneously unstable.
• We show a method for finding parameters for codimension 2 and 3 situations.
• Some numerical integration results are shown.

We study pattern formation on the plane transverse to propagation direction, in a ring cavity filled with a Kerr-like medium, subject to an elliptically polarized incoming field, by means of two coupled Lugiato–Lefever equations. We consider a wide range of possible values for the coupling parameter between different polarizations, B̄, as may happen in composite materials. Positive and also negative refraction index materials are considered. Examples of marginal instability diagrams are shown. It is shown that, within the model, instabilities cannot be of codimension higher than 3. A method for finding parameters for which codimension 2 or 3 takes place is given. The method allows us to choose parameters for which unstable wavenumbers fulfill different relations. Numerical integration results where different instabilities coexist and compete are shown.