  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 mechanisms for dynamical
instabilities are identified.
