Analytical expression for Risken-Nummedal-Graham-Haken instability threshold in quantum cascade lasers

Abstract

We have obtained a closed-form expression for the threshold of Risken-Nummedal-Graham-Haken (RNGH) multimode instability in a Fabry-Pérot (FP) cavity quantum cascade laser (QCL). This simple analytical expression is a versatile tool that can easily be applied in practical situations which require analysis of QCL dynamic behavior and estimation of its RNGH multimode instability threshold. Our model for a FP cavity laser accounts for the carrier coherence grating and carrier population grating as well as their relaxation due to carrier diffusion. In the model, the RNGH instability threshold is analyzed using a second-order bi-orthogonal perturbation theory and we confirm our analytical solution by a comparison with the numerical simulations. In particular, the model predicts a low RNGH instability threshold in QCLs. This agrees very well with experimental data available in the literature.

Dynamics of a broad-band quantum cascade laser: from chaos to coherent dynamics and mode-locking

“…In particular we are interested in the role played by the mode separation and the LEF. The instability of the TW solutions in QCL models based on MBEs for 2-level systems in case of FP and ring configurations has been linked to the RNGH instability, associated with the parametric gain of the cavity modes in resonance with the Rabi frequency [13,20]. In our case, though, the phase-amplitude coupling points towards a different character, namely the Benjamin-Feir instability [38], that appears to be peculiar of the multimode s.c. laser dynamics as assessed in [24,39] and is triggered by the growth of modes having wavevectors larger than a critical value that depends on boundary conditions.…”

“…In literature, different models have been proposed to theoretically interpret the experimental evidences [13]- [21]. A first class of models are essentially based on Maxwell-Bloch equations (MBEs) (in2 or 3-level medium approximation) [13,14,16,17,[19][20][21], where in some cases additional effects (e.g. cubic nonlinearity, saturable absorption, cavity dispersion) have been phenomenologically introduced to provide a proper interpretation of the experimental findings.…”

“…A common understanding emerging thereof was the role of Spatial Hole Burning (SHB) (introduced to describe the effect of counter-propagating fields in a Fabry-Perot (FP) scheme of the laser cavity) in reducing the instability threshold leading to the mode-locking (see e.g. [20]). In other approaches adopting a closer semiconductor (s.c.) optical response [15,18], a standard adiabatic elimination of the macroscopic semiconductor polarization has been introduced, leading to the commonly termed rate equation model.…”

“…In the model, we consider a unidirectional ring cavity configuration; this hypothesis allows us to show that phase-amplitude coupling (associated to the LEF) and mode competition can provide, even in absence of SHB typical of Fabry-Perot (FP) configurations considered e.g. in [13,20], a radically lower threshold for coherent multimode dynamics than what predicted by the Risken-Nummedal-Graham-Haken (RNGH) instability [28][29][30], which is typical of two-levels laser. In this regard, we observe that unidirectional lasing in ring cavities in QCLs has been demonstrated in both a monolithically integrated cavity and an external one [31,32].…”

Tailoring Risken-Nummedal-Graham-Haken instability in quantum cascade lasers

“…Thus in Refs. [6,7] the second threshold reduction in QCLs is attributed to the mode coupling through the induced gratings of population inversion and coherences (medium polarization). The following expression for the second threshold was obtained for the long-cavity QCLs [6]: Following along these lines, the mechanism of the RNGH instability switching due to a change of the gain recovery time is illustrated by Figure 1(a), where the behavior of the second threshold (1) in the range of typical carrier lifetimes for MIR QCLs is plotted.…”