Transverse patterns in a laser with an injected signal

Pattern formation near threshold in large-aspect-ratio, single-longitudinal-mode, two-level lasers with plane mirrors under the action of a weak injected plane-wave field resonant with the atomic transition frequency and tilted with respect to the laser cavity axis is analytically investigated in terms of the amplitude equations for the system. It is shown that, when the laser emission occurs off-axis, the appearance of resonances among the injected signal field and the unstable emerging tilted waves drastically changes the pattern selection properties of the free-running laser. As a result new transverse patterns, which include singly and doubly periodic stripes and undulating hexagons, may be stabilized by the injected signal. These patterns arise from the superposition of the forced mode with either one, two, or three traveling waves on the critical circle of the free-running laser. This is a distinctive feature from similar structures more commonly found in passive optical systems, where complex patterns always arise as a superposition of standing waves. The stability properties of the various patterns and their selection rules are analytically investigated within the amplitude equations.