Time delay distribution in Bragg gratings

A layer-by-layer analysis of the time delay of both reflected and transmitted light in one-dimensional photonic band-gap structures is developed and applied to uniform Bragg gratings. An effective Fabry-Perot cavity is associated with every layer along the Bragg grating, multiple paths with a well defined layer traversal time are identified, and the average time is computed, introducing an appropriate weighting factor that accounts for interference between different paths. The analysis presented leads directly to a complex-valued time delay whose real part is shown to be equivalent to the classic phase time delay. Physical meaning is also given to the imaginary part. The local dwell time, interpreted as the average time spent by light in the layer independently of the final (transmitted or reflected) state, is proved analytically to be related to the energy density distribution when small index change gratings are considered. The time delay evolution is derived at different wavelengths and the nonuniform distribution along the grating is discussed. Nonintuitive features such as superluminal transmission time delay for propagation inside the band gap and negative reflection time delay close to transmission resonances are addressed. Finally, the effect of introducing a small perturbation in the structure is shown to be directly related to the local time delay and is proposed as a possible experimental measurement scheme for both its real and imaginary parts.