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

Abstract

Abmct-kymptotic transient field mlutions of the form A(r, t ) exp [is@, r)] , where S is a rapidly and A a slowly varying function of space and time, may be analyzed in terms of wave packets with c e n e frequency w = -as/at and central wavenumber f = VS. When the (dis-@e) medium is lodes, stationary, and homogeneous, wave packets with constant real w aud k move dong 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 w and f move dong complex -time rays, ia., dong trajectories defined in a complex (r, r) coordinate space. The properties of complex s p a c e t h e rays and of the fields proprgrting along them, and their relation to physical fields observed on real (r, t ) coordinates, are illustrated for a plane pulse with Guusian envelope and frequency swept carrier, launched into a lossy environment. Tracking of spatial and temporal maxima is performed by ray techniques, and a paraxial my regime is defined that permits discuMion of a signal velocity. Specipr attention is given to my focusing and the associated phenomena of pulse compression. It is shown how a complex input frequency profile can be synthesized 80 as to achieve optimum c o m p r d o n at a real space-time observation point in a lossy medium. The general results are applied in detail to a cold dissipative pLQnr, and a representative set of numerical calculations is included. ray technique, it is appropriate to ask whether a similar pro-Durham under Contract DAHC 04-69-c-0079, and in part by the Joint and quantitative information on, a more g e n e d and prac-Services Electronics Program under Contract F 44620-69-0047. Thh tically important class of wave phenomena. As a first step

A Method for Delivering Spatio-Temporally Focused Energy to a Dynamically Adjustable Target Along a Waveguiding Structure

“…The pulse envelope spectrum , modulation frequency distribution , original and modified propagation relations and , and waveguide transfer function were all computed directly from (41), (14), (1), (17), and (2), respectively, on the sampled values. The spectrum of the waveguide excitation signal at the target was computed as in (15) with discrete convolution using the truncated envelope .…”

“…The derivative of (7) with respect to defines and the derivative of (3) with respect to defines . Applying the chain rule and substituting (3) for yields the modulation frequency distribution (14) Note that, while depends upon spatial location, the dependence cancels out of the relation in (14).…”

“…The spectrum of the signal at the spatial focus is constructed using the modulation frequency distribution given by (14). The spectrum of a cosine-modulated pulse envelope is , so it follows that the spectrum of a continuous distribution of modulated pulse envelopes, weighted with density function , is (15) Note that is real-valued and symmetric because it is the convolution of two real-valued, symmetric functions.…”

“…In the signal construction, the majority of the signal energy is located near , with in (34). While the imaginary part of (37) grows linearly with , the relative amplitude of the signal is inversely proportional to in (14). Hence, the overall signal attenuation can be represented with the attenuation for .…”

“…The overall signal bandwidth covers the region from to approximately one order of magnitude above . The upper bound is due to the modulation frequency distribution in (14), which is inversely proportional to for . (In simulation, an upper frequency limit of produced qualitatively similar results to .)…”