We consider pulse formation dynamics in an actively mode-locked laser. We show that an amplitude-modulated laser is subject to large transient growth and we demonstrate that at threshold the transient growth is precisely the Petermann excess noise factor for a laser governed by a nonnormal operator. We also demonstrate an exact reduction from the governing PDEs to a low-dimensional system of ODEs for the parameters of an evolving pulse. A linearized version of these equations allows us to find analytical expressions for the transient growth below threshold. We also show that the nonlinear system collapses onto an appropriate fixed point, and thus in the absence of noise the ground-mode laser pulse is stable. We demonstrate numerically that, in the presence of a continuous noise source, however, the laser destabilizes and pulses are repeatedly created and annihilated.
We study theoretically the effect of transverse boundary conditions on the traveling waves found in infinitely extended and positively detuned laser systems. We find that for large-aspect-ratio systems, well above threshold and away from the boundaries, the traveling waves persist. Source and sink defects are observed on the boundaries, and in very-large-aspect-ratio systems these defects can also exist away from the boundaries. The transverse size of the sink defect, relative to the size of the transverse domain, is important in determining the final pattern observed, and so, close to threshold, standing waves are always observed.