Pulsed-beam propagation in lossless dispersive media. II. A numerical example

In Part I of this two-part investigation we presented a theory for propagation of pulsed-beam wave packets in a homogeneous lossless dispersive medium with the generic dispersion relation k(ω). Emphasis was placed on the paraxial regime, and detailed studies were performed to parameterize the effect of dispersion in terms of specific physical footprints associated with the PB field and with properties of the k(ω) dispersion surface. Moreover, critical nondimensional combinations of these footprints were defined to ascertain the space–time range of applicability of the paraxial approximation. This was done by recourse to simple saddle-point asymptotics in the Fourier inversion integral from the frequency domain, with restrictions to the fully dispersive regime sufficiently far behind the wave front. Here we extend these studies by addressing the dispersive-tonondispersive transition as the observer moves toward the wave front. It is now necessary to adopt a model for the dispersive properties to correct the nondispersive high-frequency limit (formula presented) with higher-order terms in (1/ω). A simple Lorentz model has been chosen for this purpose that allows construction of a simple uniform transition function which connects smoothly onto the near-wave-front-reduced generic k(ω) profile. This model is also used for assessing the accuracy of the various analytic parameterizations and estimates in part I through comparison with numerically generated reference solutions. It is found that both the asymptotics for the pulsed-beam field and the nondimensional estimators perform remarkably well, thereby lending confidence to the notion that the critical parameter combinations are well matched to the space–time wave dynamics.