Spectral method for efficient computation of time-dependent phenomena in complex lasers

Abstract

Studying time-dependent behavior in lasers is analytically difficult due to the saturating nonlinearity inherent in the Maxwell-Bloch equations and numerically demanding because of the computational resources needed to discretize both time and space in conventional FDTD approaches. We describe here an efficient spectral method to overcome these shortcomings in complex lasers of arbitrary shape, gain medium distribution, and pumping profile. We apply this approach to a quasi-degenerate two-mode laser in different dynamical regimes and compare the results in the long-time limit to the Steady State Ab Initio Laser Theory (SALT), which is also built on a spectral method but makes a more specific ansatz about the long-time dynamical evolution of the semiclassical laser equations. Analyzing a parameter regime outside the known domain of validity of the stationary inversion approximation, we find that for only a narrow regime of pump powers the inversion is not stationary, and that this, as pump power is further increased, triggers a synchronization transition upon which the inversion becomes stationary again. We provide a detailed analysis of mode synchronization (aka cooperative frequency locking), revealing interesting dynamical features of such a laser system in the vicinity of the synchronization threshold.Lasers are very rich dynamical systems which exhibit various time-dependent phenomena characteristic of nonlinear systems such as phase and mode locking, selfpulsing and breathing, and generally, spatio-temporal pattern formation and dynamical chaos. Almost all these effects can be understood and quantitatively studied using the semiclassical laser theory in the form of Maxwell-Bloch (MB) equations [1-3], a set of coupled non-linear equations for the space-and time-dependent electric field amplitude E(r, t), and the polarization and inversion of the gain medium P (r, t) and D(r, t). Early work made abundant use of spectral methods, where the field amplitudes entering the MB equations are expanded in a complete basis of spatial modes, reducing MB equations to a set of coupled non-linear ordinary differential equations for time-dependent amplitudes. These early theoretical investigations made a number of simplifying assumptions on the spatial aspects of the problem. The lasing modes were assumed to be simple (uniform, trigonometric, or gaussian) and unmodified from their passive cavity modes, and the openness (optical leakage) was taken into account phenomenologically. While these assumptions are sufficiently general to reproduce qualitatively almost all features of laser dynamics in macroscopic cavities, new laser systems have emerged in the past two decades that raised questions not easily addressable by these spectral approaches.Most novel laser systems are motivated by their deployment as compact and tunable light-sources for on-chip applications [4]. Typically, these lasers feature complex sub-wavelength patterning of the cavity volume to employ light-confinement mechanisms that are based on optical interferen...

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