Joint Time-Frequency and FDTD Analysis of Precursor Fields in Dispersive Media

Superluminal group velocities, defined as group velocities exceeding the speed of light in vacuum, c, have been theoretically predicted and experimentally observed in various types of dispersive media, such as passive and active Lorentzian media, one-dimensional photonic crystals, and undersized waveguides. Though the group velocity was found to be superluminal in these media, it has been suggested that the pulse “front” and associated transient field oscillations, known as the precursors or forerunners, will never travel faster than c, and hence relativistic causality is always preserved. Until now, few rigorous studies of these transient fields in structures exhibiting superluminal group velocities has been performed. In this paper, we present the dynamic evolution of these earliest field oscillations in one dimensional photonic crystals (1DPC), using finite-difference time-domain (FDTD) techniques in conjunction with joint time-frequency analysis (JTFA). Our study clearly shows that the precursor fields associated with superluminal pulse propagation travel at subluminal speeds, and thus, the arrival of these precursor fields must be associated with the arrival of “genuine information”. Our study demonstrates the expected result that abnormal group velocities do not contradict Einstein causality. More surprisingly, and in contrast to previously held beliefs, we also show that the speed of the “front” is less than, and not equal to c. This work also shows that FDTD analysis and JTFA can be combined to study the dynamic evolution of the transient and steady state pulse propagation in dispersive media.