Time-Frequency and Pulse Propagation in Dispersive Media

Abstract

Abstract : We apply the methods of time-frequency analysis to pulse propagation in a dispersive medium. We give exact expressions for conditional moments and in particular for the contraction and spreading of the pulse, the covariance between position and wave number and other relevant physical quantities. For the contraction time it is shown that the important quantity is the initial correlation between position and group velocity. A simple physical and mathematical model is presented that explains why sometimes pulses contract before eventually expanding. In addition, we give formulas for the calculation of the instantaneous frequency of a pulse at a given position which show how dispersion effects instantaneous frequency. Also, we apply the Wigner distribution to study pulse propagation in dispersive media. Doing so provides insight and simplifies the view of pulse propagation and further leads to simple approximation methods that evolve a pulse in time. In addition, we derive representations that jointly involve the four variables of time, frequency, space, and spatial frequency, and we apply these representations to wave propagation. A number of examples are used to illustrate the formulas derived.