Abstract

Adiabatic elimination is a standard procedure applied to the Maxwell-Bloch laser equations when one variable or more is slaved to the remaining variables. An important case in point is a laser with an extremely large gain bandwidth satisfying the condition γ⊥ ≫ γ_||, k where γ⊥ is the polarization dephasing rate, γ_|| the de-energization rate and k the cavity damping constant. For example, color center gain media satisfy this criterion and support hundreds of thousands of longitudinal modes in synchronous pumped mode-locking operation. For simple single mode plane wave models the crude adiabatic elimination step of setting the derivative of the polarization variable to zero can be avoided by using center manifold techniques [1]. In this general class of singular perturbation problem, the idea is to coordinatize the problem using linear stability analysis about some known solution and then to construct an approximation to the center manifold on which the (possibly dynamic) solution remains for all time. This procedure has been successfully applied to the Maxwell-Bloch equations describing a single mode homogenously broadened ring laser [2]. Extension of the procedure to nonlinear partial differential equations is very difficult in general as the resulting center manifold may be an infinite dimensional object. When transverse (or additional longitudinal) degrees of freedom are introduced in the Maxwell-Bloch equations in order to investigate spatial pattern formation (or mode-locking dynamics) we find that a crude adiabatic elimination (henceforth referred to as standard adiabatic elimination SAE) leads to nonphysical high transverse (or longitudinal) spatial wavenumber instabilities [3]. Recent attempts to apply the center manifold technique to the transverse problem have met with mixed success [4]. In fact the high transverse wavenumber instability shows an even stronger divergence than the SAE case for positive sign of the laser-atom detuning. Moreover, the analysis becomes unwieldy even in situations when the center manifold approach appears to work.

© 1992 Optical Society of America