Linearized theory of the fluctuation dynamics in two-dimensional topological lasers

We theoretically study the collective excitation modes of a topological laser device operating in a single-mode steady state with monochromatic emission. We consider a model device based on a two-dimensional photonic Harper-Hofstadter lattice including a broadband gain medium localized on the system edge. Different regimes are considered as a function of the value of the optical nonlinearity and of the gain relaxation time. The dispersion of the excitation modes is calculated via a full two-dimensional Bogoliubov approach and physically interpreted in terms of an effective one-dimensional theory. Depending on the system parameters, various possible physical processes leading to dynamical instabilities are identified and characterized. On this basis, strategies to enforce a stable single-mode topological laser operation are finally pointed out.