  The time-dependent Bragg diffraction by multilayer gratings working by
reflection or by transmission is investigated. The study is performed by
generalizing the time-dependent coupled-wave theory previously developed for
one-dimensional photonic crystal [Andr{\'e} J-M and Jonnard P, J. Opt. 17,
085609 (2015)] and also by extending the Takagi-Taupin approach of the
dynamical theory of diffraction. The indicial response is calculated. It
presents a time-delay with a transient time that is a function of the
extinction length for reflection geometry and of the extinction length combined
with the thickness of the grating for transmission geometry.
