  Fast saturable absorbers (FSAs) play a critical role in stabilizing many
passively modelocked lasers. The most commonly used averaged model to study
these lasers is the Haus modelocking equation (HME) that includes a third-order
nonlinear FSA. However, it predicts a narrow region of stability that is
inconsistent with experiments. To better replicate the laser physics, averaged
laser models that include FSAs with higher-than-third-order nonlinearities have
been introduced. Here, we compare three common FSA models to each other and to
the HME using the recently-developed boundary tracking algorithms. The three
FSA models are the cubic-quintic model, the sinusoidal model, and the algebraic
model. We find that all three models predict the existence of a stable
high-energy solution that is not present in the HME and have a much larger
stable operating region. We also find that all three models predict
qualitatively similar stability diagrams. We conclude that averaged laser
models that include FSAs with higher-than-third-order nonlinearity should be
used when studying the stability of passively modelocked lasers.
