Further Examination of Sommerfeld-Brillouin Precursors

Abstract

In this work, the high- and low-frequency precursors discovered by Sommerfeld and analyzed by Sommerfeld and Brillouin are further examined using numerical integration techniques on a modern computer. It builds on the techniques and results found by Jakobsen and Mansuripur, who extended Sommerfeld and Brillouin's work by investigating finite-bandwidth electric fields and analyzing the use of the steepest descent approximation for integration of the Fourier integral. To that method is added two others: the method of stationary phase and an alternate approach to the method of steepest descent using path integrals. The behavior of the saddle-points described in detail by Jakobsen and Mansuripur is reproduced. Then, taking advantage of the numerical techniques to calculate the integral results for over a million points, the original field used by Sommerfeld is examined in greater detail, confirming the results he sketched out for the behavior of the field and expanding upon them to reveal more details of the spectral and intensity behavior of the field. Additionally, two of the limited-bandwidth fields proposed by Jakobsen and Mansuripur are examined, showing the similarities and differences of their behavior compared to those of Sommerfeld's. Finally, since the results for the three different fields were each computed using all three numerical integration methods, the characteristics of the outputs and computation speeds for all three methods are compared, finding advantages and disadvantages of each.