# Analytical Solutions for the Propagation of UltraShort and UltraSharp Pulses in Dispersive Media
## 1 Introduction  
Ultrashort pulses are ubiquitous in numerous applications including in bio- and medical-imaging (see, for example, [1; 2; 3]), optical communications (see, for example, [4; 5]), and microscopy (see, for example, Ref. [6; 7]), etc. One of the clear shortcomings of short pulses is the consequences of their wide spectrum. When the spectrum is spectrally wide, the absorption variations within the spectrum increases. However, even if the absorption is approximately homogenous within the pulse spectral range, the refraction index is wavelength-dependent and therefore the pulse experiences dispersion effects. Dispersion occurs whenever each one of the pulse's spectral components propagates with a different velocity. Consequently, the pulse experiences shape distortions. The temporally shorter the pulse, the larger the distortions. Moreover, while the dispersion effects are proportional to the medium's length, it is inversely proportional to the square of the pulse's temporal width; therefore, ultra-short pulses are prone to being affected by dispersion even in relatively short media [8; 9; 10]. Clearly, most of the dispersion mitigation techniques were implemented in the optical communication arena, where dispersion can have a destructive effect on the transmitting signals. It has been shown that via ordinary smf28 fiber (the most ubiquitous fiber type) it is impossible to decode data, which is transmitted at a 10 Gb/s rate, beyond a distance of 160 km (see Ref. [11] and in the quantum analogy see Ref. [12]). Clearly, in the presence of noise, this distance shrinks substantially. If the data is carried by picosecond pulses (which is only 1% of the temporal width of the 10 Gb/s signal's individual pulses), this maximum distance is reduced to merely 16 m, and the distortion effects will be felt within less than a meter. In the femtosecond domain, the dispersion effects are, using theterminology and example of Ref. [13], more dramatic: A 10 fs laser pulse in the 800 nm spectral regime, which propagates through 4 mm of BK7 glass, will be temporally broadened to 50 fs. It is well known that dispersion is governed by the dispersion equation, which can easily be solved numerically. However, to evaluate the effects of dispersion and to predict them, some intuition is required. In most textbooks, Gaussian pulses are used as a litmus tool to predict the dispersion effect [8; 13; 14]. Gaussian pulses are very useful since, on the one hand, they are localized pulses with a well-defined width and energy, and on the other hand, the shape of their spectrum is mathematically similar to the dispersion effect. Therefore, the propagation of Gaussian pulses in dispersion medium has a relatively simple analytical solution. This simple pulse model can be used to qualitatively evaluate some fundamental limitations. However, it fails in quantifying their values (see Ref. [11] and the references therein). In fact, the Gaussian model predicts that the pulse keeps its Gaussian shape, and its width is proportional to the square root of the elapsed period. This conduct resembles the spatial spreading of a Gaussian distribution of particles in a diffusion process [15]. Since the \"particle\" realization of the dispersion process is common in the popular description of chromatic dispersion, the similarity to diffusion may be misleading, since, in fact, this similarity appears only in the Gaussian scenario. Whenever the initial pulse profile deviates from the Gaussian's one, the differences between diffusion and dispersion are clearly shown. In general, the two processes tend to smooth out the signal. However, the processes are fundamentally different, and the main differences appear after short distances. This discrepancy clearly appears in most optical communication protocols, where the pulses never have Gaussian profiles. Consequently, the signal distortion is completely different from the Gaussian analysis predictions. To overcome this problem, super-Gaussian pulses are commonly used to model real communications scenarios [16; 17; 18].