Purcell effect with extended sources: The role of the cross density of states

We analyze the change in the spontaneous decay rate, or Purcell effect, of an extended quantum emitter in a structured photonic environment. Based on a simple theory, we show that the cross-density of states is the central quantity driving interferences in the emission process. Using numerical simulations in realistic photonic cavity geometries, we demonstrate that a structured cross-density of states can induce subradiance or superradiance, and change subtantially the emission spectrum. Interestingly, the spectral lineshape of the Purcell effect of an extended source cannot be predicted from the sole knowledge of the spectral dependence of the local density of states