Operating regimes, split-step modeling, and the Haus master mode-locking model

We develop an iterative (averaging) method to characterize the mode-locking dynamics in a laser cavity mode locked with a combination of wave plates and a passive polarizer. The model explicitly accounts for the effects of self- and cross-phase modulation, an arbitrary alignment of the fast- and slow-axes of the fiber with the wave plates and polarizer, fiber birefringence, saturable gain, and chromatic dispersion. The general averaging scheme results in the cubic-quintic Ginzburg-Landau equation at the leading order and the Swift-Hohenberg equation at the next order. An extensive comparison between the full model and the averaged equations shows a quantitative agreement that allows for characterizing the stability and operating regimes of the laser cavity.