Stability of passively mode-locked pulses in the fast absorber limit

Abstract

We have reexamined the stability criterion derived by Haus in the analysis of a laser system which is passively mode locked by a fast saturable absorber. In his stability analysis, Haus required perturbations <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\delta\upsilon(t)</tex> of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\delta\upsilon(t) = (sech (t/\tau_{p}))^{\nu}</tex> where τ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> is the steady-state pulsewidth and ν is a parameter which is varied, to be stable and arrived at an approximate stability criterion. Upon numerical evaluation, Haus observed that the result was very close to the locus of apices separating the two branches of solutions, and proposed it as a stability criterion. Here we test the stability of arbitrary small perturbations and show that the result proposed by Haus is exact.