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We have obtained a closed-form expression for the threshold ofRisken-Nummedal-Graham-Haken (RNGH) multimode instability in a Fabry-P\'erot(FP) cavity quantum cascade laser (QCL). This simple analytical expression is aversatile tool that can easily be applied in practical situations which requireanalysis of QCL dynamic behavior and estimation of its second threshold. Ourmodel for a FP cavity laser accounts for the carrier coherence grating andcarrier population grating as well as their relaxation due to carrierdiffusion. In the model, the RNGH instability threshold is analyzed using asecond-order bi-orthogonal perturbation theory and we confirm our analyticalsolution by a comparison with the numerical simulations. In particular, themodel predicts a low second threshold in QCLs. This agrees very well withexperimental data available in the literature.

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