Model of a two–transverse mode laser with injected signal

We derive a simple model for a two transverse mode laser (that considers the TEM00 and TEM10 modes) in which an injected signal with the shape of the TEM10 mode but a frequency close to that of the TEM00 mode is injected. Here we derive a model for a two–transverse mode laser with injected signal. More specifically, we do consider an incoherently pumped homogeneously broadened laser, in a ring cavity configuration in which a particular signal is injected from the outside. We assume that the laser works reasonably close to resonance with the TEM00 mode, but the injected field has not the shape of the TEM00 mode but the shape of the TEM10 mode. The frequency of the injected field is however assumed to be close to that of the TEM00 mode. We first develop a two–mode model which is valid for any value of the decay rates. Later on, we particularize this model to class–A lasers, i.e., lasers in which the material variables decay much faster than the intracavity field. We end up with a single complex equation for the TEM00 mode which is of the Stuart–Landau type.