Radiative local density of states in three-dimensional photonic band-gap crystals to interpret time-resolved emission

We investigate the spontaneous emission of light in three-dimensional (3D) photonic crystals through theoretical calculations and simulations. It is well known that spontaneous emission depends on the radiative local density of states (RLDOS). Photonic band-gap crystals radically modulate the RLDOS, thereby controlling spontaneous emission. We compare two different methods to calculate the RLDOS: the plane-wave expansion (PWE) method and the finite-difference time-domain (FDTD) method. The PWE method directly calculates the RLDOS of an infinite photonic crystal, whereas the FDTD method simulates the RLDOS through the power emitted by a dipole in a finite photonic crystal. We demonstrate that the methods yield similar frequency-dependent trends in the RLDOS, with relative differences of less than 12% that originate from the different boundary conditions. We employ the plane-wave expansion method to compute distributions of emission rates that are relevant to many optical experiments where quantum emitters are distributed within a crystal. Such distributions of emission rates enable us to compute and directly interpret the time-resolved decay as observed in experiments. We expect that our results promote the RLDOS to the realm of optical design and products.