Floquet analysis, feedback regulation, and observer design for passively Q-switched fiber lasers

We investigate a nonlinear rate-equation model for passively Q-switched fiber lasers from a unified dynamical-systems and control-theoretic perspective. The Q-switched pulse train is interpreted as a self-sustained nonlinear oscillatory regime generated by the interaction between gain recovery, intracavity losses, and saturable absorption. Instead of treating the problem as stabilization of an unstable equilibrium or periodic orbit, the analysis is formulated in terms of orbital regulation and robustness of an intrinsically self-oscillatory laser dynamics. A control-oriented formulation is introduced in which pump modulation acts through the gain channel and provides a physically meaningful mechanism for influencing pulse timing, pulse-energy fluctuations, and robustness under perturbations. The orbital stability of the pulsed regime is analyzed through Floquet exponents and multipliers computed along the periodic pulse train. Numerical pulse characterization is performed using quantities including pulse width, repetition frequency, duty cycle, timing jitter, peak power, and pulse energy. In addition, we formulate an output-based observer system for estimating the internal gain and absorber dynamics from the measured optical power. The observer and feedback mechanisms are treated as distinct but complementary components: the feedback regulates the oscillatory pulse regime through pump actuation, while the observer reconstructs hidden internal variables using output-error injection. The numerical simulations indicate that low-dimensional pump-based feedback can preserve the passive Q-switched regime while reducing timing and pulse-energy variability in the parameter regime considered. The results provide a systematic framework connecting passive Q-switching dynamics, Floquet orbital analysis, nonlinear feedback regulation, and observer design within nonlinear fiber-laser systems.