Effect of frequency detunings and finite relaxation rates on laser localized structures

We study, analytically and numerically, the effect of frequency detunings and relaxation processes in laser media on stability and bifurcations of dissipative optical localized structures (DOLS's) in a transversely one-dimensional laser with a saturable absorber. The approximate envelope equation, with an intensity dependent effective coefficient of the diffusion, is derived. Andronov-Hopf bifurcations resulting from frequency detuning and leading to oscillatory DOLS's are analyzed numerically. A numerical and analytical study of bifurcations of transversely motionless DOLS's in a laser with finite relaxation rates of amplifying and absorbing media is performed. New types of DOLS's are found, including those moving with a large transverse velocity and those moving with a periodically oscillating transverse velocity. Hysteresis between different types of DOLS's is demonstrated.