Complex space-time rays and their application to pulse propagation in lossy dispersive media

Asymptotic transient field solutions of the form A(r,t) exp [iS (r, t)], where S is a rapidly and A a slowly varying function of space and time, may be analyzed in terms of wave packets with central frequency ω =-∂S/∂t and central wavenumber k = ∇S. When the (dispersive) medium is lossless, stationary, and homogeneous, wave packets with constant real ω and k move along straightline trajectories called space-time rays. In the presence of dissipation and (or) when the input signal has an exponential amplitude dependence, S is complex. The corresponding wave packets with constant complex ω and k move along complex space-time rays, i.e., along trajectories defined in a complex (r, t) coordinate space. The properties of complex space-time rays and of the fields propagating along them, and their relation to physical fields observed on real (r, t) coordinates, are illustrated for a plane pulse with Gaussian envelope and frequency swept carrier, launched into a lossy environment. Tracking of spatial and temporal maxima is performed by ray techniques, and a paraxial ray regime is defined that permits discussion of a signal velocity. Special attention is given to ray focusing and the associated phenomena of pulse compression. It is shown how a complex input frequency profile can be synthesized so as to achieve optimum compression at a real space-time observation point in a lossy medium. The general results are applied in detail to a cold dissipative plasma, and a representative set of numerical calculations is included.