Self-consistent Maxwell-Bloch model of quantum-dot photonic-crystal-cavity lasers

We present a powerful computational approach to simulate the threshold behavior of photonic-crystal quantum-dot (QD) lasers. Using a finite-difference time-domain (FDTD) technique, Maxwell-Bloch equations representing a system of thousands of statistically independent and randomly positioned two-level emitters are solved numerically. Phenomenological pure dephasing and incoherent pumping is added to the optical Bloch equations to allow for a dynamical lasing regime, but the cavity-mediated radiative dynamics and gain coupling of each QD dipole (artificial atom) is contained self-consistently within the model. These Maxwell-Bloch equations are implemented by using Lumerical's flexible material plug-in tool, which allows a user to define additional equations of motion for the nonlinear polarization. We implement the gain ensemble within triangular-lattice photonic-crystal cavities of various length N (where N refers to the number of missing holes), and investigate the cavity mode characteristics and the threshold regime as a function of cavity length. We develop effective two-dimensional model simulations which are derived after studying the full three-dimensional passive material structures by matching the cavity quality factors and resonance properties. We also demonstrate how to obtain the correct point-dipole radiative decay rate from Fermi's golden rule, which is captured naturally by the FDTD method. Our numerical simulations predict that the pump threshold plateaus around cavity lengths greater than N = 9, which we identify as a consequence of the complex spatial dynamics and gain coupling from the inhomogeneous QD ensemble. This behavior is not expected from simple rate-equation analysis commonly adopted in the literature, but is in qualitative agreement with recent experiments. Single-mode to multimode lasing is also observed, depending on the spectral peak frequency of the QD ensemble. Using a statistical modal analysis of the average decay rates, we also show how the average radiative decay rate decreases as a function of cavity size. In addition, we investigate the role of structural disorder on both the passive cavity and active lasers, where the latter show a general increase in the pump threshold for cavity lengths greater than N = 7, and a reduction in the nominal cavity mode volume for increasing amounts of disorder.