# Analytical Solutions for the Propagation of UltraShort and UltraSharp Pulses in Dispersive Media
## 6 Smooth Pulses
### Smooth Rectangular Pulse  
Similarly, the smooth rectangular pulse  
\[A(z=0,t)=\frac{A_{0}}{2}\bigg{[}erfc\bigg{(}\frac{t-\theta/2}{\Delta}\bigg{)} -erfc\bigg{(}\frac{t+\theta/2}{\Delta}\bigg{)}\bigg{]} \tag{63}\]  
will appear at the end of the medium in the following form:  
\[\begin{array}{ll}A(z>0,t)&=A_{0}\text{srect}_{1}\Big{(}\frac{t}{\theta}, \sqrt{2\ln(2)}\frac{\theta}{\pi\Delta},\frac{\beta_{2}z}{\theta^{2}}\Big{)}= \\ &\frac{A_{0}}{2}\bigg{[}erfc\bigg{(}\frac{t-\theta/2}{\sqrt{-2i\beta_{2}z+ \Delta^{2}}}\bigg{)}-erfc\bigg{(}\frac{t+\theta/2}{\sqrt{-2i\beta_{2}z+\Delta ^{2}}}\bigg{)}\bigg{]}\end{array} \tag{64}\]  
where  
\[\begin{array}{l}\text{srect}_{\xi}(\tau,\delta,\zeta)\equiv\Big{\{}erfc \Big{[}(\tau-\xi/2)/\sqrt{i2\zeta+2\ln(2)/\pi^{2}\delta^{2}}\Big{]}-\\ \qquad\qquad erfc\Big{[}(\tau+\xi/2)/\sqrt{i2\zeta+2\ln(2)/\pi^{2}\delta^{2}} \Big{]}\Big{\}}/2\end{array} \tag{65}\]  
is the smooth rectangular function (see Ref. [23; 24; 25]), which describes the dispersion dynamics of a smooth rectangular pulse.  
In Figure 17, this smooth rectangular pulse's profile is presented for the same values as in Figure 6, i.e., the perfect rectangular pulse. However, the difference is that in Figure 17, the pulse's boundaries are smooth instead of discontinuous. As can be seen, due to the smooth transition, the oscillations, which appear in Figure 6, disappear in Figure 17.  
This pulse has numerous applications in optical communications. For example, if the transmitted data is encoded in the amplitudes of these smooth rectangular pulses, then the transmitted signal profile can be written as  
\[A(t,z>0)=\sum_{n}a_{n}srect_{\xi}\bigg{(}t/\theta-n,\sqrt{2\ln(2)}\frac{ \theta}{\pi\Delta},\beta_{2}z/\theta^{2}\bigg{)} \tag{66}\]  
where \(a_{n}\) are coefficients which carry the data.  
These pulses have two properties that make them especially suitable for stream of pulses. The \(\xi\) parameter can determine the signal's protocol. The Return-to-Zero (RZ) protocol occurs for \(\xi<1\), when \(\xi\) determines the duty cycle. However, when \(\xi=1\), then this is a perfect Non-Return-to-Zero (NRZ) protocol. The second property, which is more important, emerges due to symmetry of the erfc function. Since  
\[erfc\bigg{(}\frac{t}{\Delta}\bigg{)}+erfc\bigg{(}\frac{-t}{\Delta}\bigg{)}=2 \tag{67}\]  
then two adjacent smooth rectangular pulses look exactly like a single but twice as wide a pulse:  
\[\text{srect}_{1}(\tau-0.5,\delta,\zeta)+\text{srect}_{1}(\tau+0.5,\delta, \zeta)=\text{srect}_{2}(\tau,\delta,\zeta) \tag{68}\]  
Sample Application: In 40 Gb/s optical communication channels (over smf28 fiber) with a carrier wavelength of \(\lambda=1.55\) um, the pulses can be simulated by rectangular ones with \(\theta=25\) ps, \(\beta_{2}\cong 20\) ps\({}^{2}\)/Km, in which case \(\zeta=0.3\) and \(\zeta=0.01\) in Figure 17 correspond to \(z=\zeta\theta^{2}/\beta_{2}\cong 9.4\) Km and \(z\cong 0.31\) Km, respectively. Since \(\Delta=0.2\theta\), then the 10-90% rise time corresponds to \(\Delta t_{10-90}\cong\Delta/0.67=0.2\theta/0.67\cong 7.5\) ps.