Nonadiabatic semiconductor laser rate equations for the large-signal, rapid-modulation regime

Abstract

The standard multimode rate equation for semiconductor lasers is based on the approximation that modal field shapes (if they change at all) depend on the instantaneous value of the time-dependent dielectric function. This is known as the adiabatic approximation. It will break down if the inverse of the modulation frequency approaches the photon round-trip time in the cavity. In this paper, we derive a criterion for the validity of the adiabatic approximation and find that it also involves the fractional modulation of the dielectric function, not just the photon round-trip time and the modulation frequency. Recognizing that present laser designs are starting to approach the limits of validity as set by our criterion, we study what can be done to get past that limit. We obtain corrections to the equations presently used by rederiving these equations from the time-dependent wave equation without discarding the time derivative of the dielectric function, an approximation made in every other derivation we have encountered. Retaining this time derivative introduces new terms into the usual equations. These terms correctly account for propagation delay-time effects and for nonadiabatic couplings between the modes. Surprisingly, the new terms alter only the source of photons to each mode. The usual rate equations are driven by a spontaneous emission term ${R}_{\ensuremath{\nu}}^{\mathrm{sp}}(t),$ which represents the rate at which photons are emitted spontaneously into the mode \ensuremath{\nu}. In the rate equation derived here, the spontaneous emission term ${R}_{\ensuremath{\nu}}^{\mathrm{sp}}(t)$ is augmented by a term ${\ensuremath{\Xi}}_{\ensuremath{\nu}}(t)$ which counts photons that were earlier emitted spontaneously into other modes \ensuremath{\mu}, accumulated and perhaps amplified there, and are now, because of the breakdown of the adiabatic approximation, leaking into the $\ensuremath{\nu}\mathrm{th}$ mode. Although casting the equations into this form makes sense from a physical point of view, it leads to great computational difficulties in solving the equations because ${\ensuremath{\Xi}}_{\ensuremath{\nu}}(t)$ refers explicitly to the past history of the laser. To overcome this practical problem, we provide an efficient and accurate algorithm for stepping the laser forward in time without having to retain history prior to the start of the present time step. Our method allows the equations to be solved with substantially the same computational effort as is normally expended in solving the conventional rate equations, and, moreover, provides error estimates at each step of the way.