Analytical Solutions for the Propagation of UltraShort and UltraSharp Pulses in Dispersive Media

Ultrashort pulses are severely distorted even by low dispersive media. While the mathematical analysis of dispersion is well known, the technical literature focuses on pulses, which keep their shape: Gaussian and Airy pulses. However, the cases where the shape of the pulse is unaffected by dispersion is the exception rather than the norm. It is the object of this chapter to present a variety of pulses profiles, which have analytical expressions but can simulate real-physical pulses with greater accuracy. In particular, the dynamics of smooth rectangular pulses, physical Nyquist-Sinc pulses, and slowly rising but sharply decaying ones (and vice-versa) are presented. Besides the usage of this chapter as a handbook of analytical expressions for pulses' propagations in a dispersive medium, there are several new findings. The main ones are: Analytical expressions for the propagation of chirped rectangular pulses, which converge to extremely short pulses; analytical approximation for the propagation of Super-Gaussian pulses; the propagation of Nyquist Sinc Pulse with smooth spectral boundaries and an analytical expression for a physical realization of an attenuation compensating Airy pulse.