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.

Scattering of ultrashort X-ray pulses (USP) is an important component of the diffraction analysis of matter using modern USP sources. Usually, the specific scattering of such USPs is not taken into account to determine the structure of a substance. Taking into account the specifics of scattering on complex structures will give more accurate results when deciphering complex structures. In this work, it is shown that when X-ray USPs are scattered on diamond with NV centers, it is necessary to take into account the pulse duration. The results obtained can be very different from the widely used theory of diffraction analysis, which confirms the need to take into account the specifics of USP scattering when diagnosing complex structures. It is also shown that scattering spectra are quite sensitive to the concentration of NV centres in the diamond structure and this can be used in diffraction analysis.

X-ray diffraction (XRD) analysis of complex poly-atomic systems, especially of biomolecules, using ultra-short laser pulses (USP), is currently one of the most important fields of modern physics. The basis for interpreting and "deciphering" experimental data is the well-known theory of X-ray scattering, where the main parameter of USPs - its duration - is not taken into account. In the present work it is shown that for scattering of attosecond USPs on DNA and RNA trinucleotides the pulse length is the most important scattering parameter. In this case the diffraction pattern significantly changes with respect to the previously known scattering theory. The results obtained are extremely important in XRD when using attosecond pulses to study trinucleotides of DNA and RNA because using the previously known scattering theory which does not take into account the duration of USPs one can not correctly interpret and therefore "decode" DNA and RNA structures.