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Cartwright, Natalie A.

Mathematics

2004

Ph. D.

Ultrawideband pulse propagation through dispersive materials is a topic of central importance in applied mathematics, physics and electrical engineering due to the many practical applications ultrawideband pulses have in the world today. The unit step function modulated sine wave signal is a canonical ultrawideband pulse. An approximation of the dynamic evolution of this waveform through a single-resonance Lorentz model dielectric is given in closed-form based upon uniform asymptotic methods. This closed-form asymptotic approximation may be represented as the sum of three terms: the Sommerfeld precursor, the Brillouin precursor and the pole contribution. A generalization of the uniform asymptotic theory due to Bleistein [Com. Pure and Appl. Math. XIX (1966)] is introduced for signals with carrier frequencies that lie within the absorption band of the material.
Both the Sommerfeld and Brillouin precursors are subtracted from a numerical simulation of the total field, leaving a numerical determination of the pole contribution. This is the first time that the pole contribution has been separated from the rest of the field enabling one to determine the accuracy of the uniform asymptotic pole approximation. The rms error between the asymptotic pole contribution and the numerical pole contribution is given as a function of propagation distance. The rms error is shown to generally decrease with increasing propagation distance for the applied carrier frequencies considered. Associated applications of ultrawideband pulse propagation through dispersive materials are given in the papers attached as appendices.