Analytic time-domain expression for transients in a periodic structure

The time-domain propagation of a wave packet in a periodic waveguide is analyzed. Firstly, a general representation of a propagating wave packet is given, and then we consider the first TM mode only. Taking advantage of the periodicity of the dispersion relation, the latter may be approximated by the first two terms of the Fourier representation enabling us to develop a general analytic expression for a transient. As an example, a simulation of the propagation of a Gaussian pulse deformed by dispersion is illustrated in comparison with the propagation in free space. It is shown that the amount of energy that keeps up with the front of the pulse decays exponentially in time, and that this energy is smaller for a mode of narrower pass-band.