Fundamental obstacles to self-pulsations in low-intensity lasers
- Turaev, Dmitry
2010 Mathematics Subject Classification
- 78A60 34E15 37D10 34C29
- distributed feedback semiconductor laser, singular perturbation, averaging, invariant manifold, normal form, rate equations
We investigate most general properties of possible laser equations in the case where optics is linear. Exploiting the presence of a natural small parameter (the ratio of the photon lifetime in the laser device to the relaxation time of the population density) we establish the existence of an exponentially attracting invariant manifold which contains all bounded orbits, and show that only a small number of electromagnetic modes is sufficient to describe accurately the dynamics of the system. We give a general form of the reduced few-mode systems. We analyze the behavior of single-mode models and a double-mode model with a single optical frequency. We show that in the case where only one electromagnetic mode is excited, the rate equations are close to integrable ones, so the dynamics in this case can be understood by analytic means (by averaging method). In particular, it is shown that a non-stationary (periodic) output is possible only in relatively small (of order of some fractional powers of the small parameter) regions in the space of parameters of the system near some specially chosen parameter constellations. Estimates on the size of these regions and on the frequency of periodic self-pulsations are given for different situations.