# Analytical Solutions for the Propagation of UltraShort and UltraSharp Pulses in Dispersive Media
## 6 Smooth Pulses
### Smooth Step Function  
In principle, any smooth step function can replace the Heaviside step function; however, there is a clear advantage to choosing the complementary error function for that purpose. The function  
\[\frac{1}{2}erfc\bigg{(}\frac{t}{\Delta}\bigg{)} \tag{54}\]  
is an approximation of the step function in the sense that  
\[\lim_{|t/\Delta|\rightarrow\infty}\frac{1}{2}erfc\bigg{(}\frac{t}{\Delta} \bigg{)}=u(-t) \tag{55}\]  
However, unlike the step function, the transition in Equation (54) occurs within the time scale \(\Delta\).  
Equivalently, a step function signal that passes through a Gaussian low-pass filter whose spectral FWHM is \(\delta\) will have the following form [11]:  
\[\frac{1}{2}erfc\bigg{(}\pi\delta\frac{t}{\sqrt{2\ln 2}}\bigg{)} \tag{56}\]  
Therefore, the relation between the transition time scale \(\Delta\) and the spectral FWHM \(\delta\) is  
\[\delta=\sqrt{2\ln(2)}\theta/\pi\Delta. \tag{57}\]  
Figure 16: Similar to Figure 10, but for the pulse presented by Equation (53). In these plots, \(a=1/\theta\) on both, but \(b=4/\theta\) on the left figure (final distance corresponds to \(\zeta=\beta_{2}z/\theta^{2}=0.1\)) and \(b=1/\theta\) on the right one (final distance corresponds to \(\zeta=\beta_{2}z/\theta^{2}=0.2\)).  
Moreover, and this is the reason that Equation (54) is very useful in replacing step functions, the smooth step (Equation (54)) can be written as a convolution between the step function and a Gaussian:  
\[\begin{array}{l}\frac{1}{2}erfc\big{(}\frac{t}{\Delta}\big{)}=\frac{1}{\sqrt{ \pi}}\int\limits_{t/\Delta}^{\infty}\exp\big{(}-x^{2}\big{)}dx=\frac{1}{\Delta \sqrt{\pi}}\int\limits_{-\infty}^{\infty}u(\tau-t)\exp\big{(}-\tau^{2}/\Delta^ {2}\big{)}d\tau=\\ u(-t)*\frac{1}{\Delta\sqrt{\pi}}\exp\big{(}-t^{2}/\Delta^{2}\big{)}\end{array} \tag{58}\]  
where the asterisk represents temporal convolution. However, since the effect of dispersion can also be represented as a convolution with the kernel (Equation (4)), the propagation of the initial pulse\({}^{\prime}\) profile  
\[A(t,0)=\frac{A_{0}}{2}erfc\bigg{(}\frac{t}{\Delta}\bigg{)} \tag{59}\]  
can be written simply as  
\[A(t,0)=A_{0}u(-t)*\frac{1}{\Delta\sqrt{\pi}}\exp\Big{(}-t^{2}/\Delta^{2}\Big{)} *K(t,z)=\frac{1}{2}A_{0}erfc\bigg{(}\frac{t}{\sqrt{\Delta^{2}-2i\beta_{2}z}} \bigg{)} \tag{60}\]  
Therefore, every step function from the previous section can be replaced with the smooth step function. In particular, the boosted smooth step function  
\[A(z=0,t)=\frac{A}{2}\exp(-i\omega_{0}t)erfc\bigg{(}\frac{t}{\Delta}\bigg{)} \tag{61}\]  
will have the following profile at the end of the medium:  
\[A(z>0,t)=\frac{A}{2}\exp\bigg{(}i\frac{\beta_{2}}{2}\omega_{0}^{2}z-i\omega_{ 0}t\bigg{)}erfc\bigg{(}\frac{t-\beta_{2}\omega_{0}z}{\sqrt{\Delta^{2}-2i\beta _{2}z}}\bigg{)}. \tag{62}\]  
Sample Applications: The rise time of pulses in optical communications are usually defined as the time period it takes to rise from 10% to 90% maximum intensity value. Therefore, provided this rise-time (\(\Delta t_{10-90}\)) is given, \(\Delta\) can be derived by \(\Delta\cong 0.67\Delta t_{10-90}\). If the rise time is defined as the period in which the signal rises from 25% to 75% of its maximum intensity value (\(\Delta t_{25-75}\)), then \(\Delta\cong 1.28\Delta t_{25-75}\).