Abstract
A minimal model for a laser with delayed feedback is analyzed. The model is motivated by two independent studies of a laser controlled by a fully optical feedback [Otsuka and Chern, Opt. Lett. 16, 1759 (1991)] and a laser with an optoelectronic feedback [Loiko and Samson, Opt. Commun. 93, 66 (1992)]. By reformulating the original laser equations in terms of dimensionless quantities, we obtain a simpler problem which is valid for both lasers. We then investigate the limit of small-amplitude feedback and small damping and determine a bifurcation equation for all periodic solutions. We analyze this condition in terms of increasing values of the delay time and show that each branch of solutions emerges from the basic state and becomes isolated as the delay time is progressively increased. The overlap of bifurcating and isolated branches of solutions explains the coexistence of nearly harmonic and pulsating solutions. Pulsating solutions may change stability through period-doubling bifurcation. We determine a simple approximation for this bifurcation point and study its validity numerically.
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