Periodic pulse solutions and stability in the fast absorber model

In the theory of passive mode locking with a fast saturable absorber as formulated by Haus [1], the mode-locking equations can be satisfied by periodic solutions which are described in terms of Jacobian elliptic functions. These solutions have been used by Ausschnitt [4] in his theory of transient mode locking. Here we reexamine the Jacobian elliptic dnoidal solutions and operating points, and show how the operating point in the case of overlapping pulses compares to the case of well separated pulses as found by Haus. We establish an exact stability criterion. These solutions are of interest because they provide a convenient description of the mode-locked waveform all the way from the limit of no mode locking (CW operation) to the limit of infinitely separated hyperbolic secant pulses.