Effect of Multi-Resonance Subband Structure on the Kerr Nonlinearity of Quantum-Cascade Lasers

This work focuses on the investigation of the optical Kerr lensing effect in quantum-cascade lasers with multiple resonance levels. The Kerr refractive index $n_{2}$ is obtained through the third-order susceptibility at the fundamental frequency \textit{$\chi $}$^{(3)}$(\textit{ $\omega $}). Resonant two-photon processes are found to have almost equal contributions to \textit{$\chi $}$^{(3)}$(\textit{$\omega $}) as the single-photon processes, which result in the predicted enhancement of the positive $n_{2}$, and thus may enhance mode-locking of quantum-cascade lasers. Moreover, an isospectral optimization strategy for further improving $n_{2}$ through the band-structure design is also demonstrated, in order to boost the multimode performance of quantum-cascade lasers. Simulation results show that the optimized stepwise multiple-quantum-well structure has a twofold enhancement on $n_{2}$ over the original flat quantum-well structure. This leads to a refractive-index change $\Delta n$ of about 0.01, which is at the upper bound of those reported for typical Kerr medium. This stronger Kerr refractive index may be important for quantum-cascade lasers ultimately to demonstrate self-mode-locking.