Uniform asymptotic description of the Brillouin precursor in a single-resonance Lorentz model dielectric

Abstract

It is well known that the dynamical evolution of the Brillouin precursor field in a single-resonance Lorentz model dielectric can be fully explained in terms of a pair of saddle points that evolve in a region of the complex -plane near the origin such that , where is the undamped resonance frequency of the dispersive medium. As time increases at a fixed propagation distance, these two near first-order saddle points first approach each other along the imaginary frequency axis, then coalesce into a second-order saddle point at the time , and finally separate from each other in the lower half-plane, one with an increasing real part and the other with a decreasing real part. The uniform asymptotic description of the Brillouin precursor provides an accurate description of the field evolution about the observation time , at which the saddle-point order changes discontinuously. However, previous approximate expressions for the phase behaviour in the region of the near saddle points have resulted in an inaccurate field evolution around . This inaccuracy is corrected in this paper. Numerical illustrations of the complete precursor evolution for the delta function pulse and the step function modulated signal are provided.