# Analytical Solutions for the Propagation of UltraShort and UltraSharp Pulses in Dispersive Media
## 5 Singular Pulses
### Generalization and Applicable Examples  
The mathematical tools that were presented in the previous sections can be implemented in numerous practical examples. In what follows we will present some examples. First example:  
Generalization to any power of the bounded cosine pulse, i.e.,  
\[A(z=0,t)= \begin{cases}A_{0}\cos^{n}(\pi t/T)&|t|<T/2\\ 0&|t|\geq T/2\end{cases}  \tag{43}\]  
which can be written as  
\[A(z=0,t)=\frac{1}{2^{n}}A_{0}\Big{[}\exp\Bigl{(}i\frac{\pi}{T}t\Bigr{)}+\exp \Bigl{(}-i\frac{\pi}{T}t\Bigr{)}\Big{]}^{n}[u(-t+T/2)-u(-t-T/2)] \tag{44}\]  
and since the brackets can be expanded  
\[A(z=0,t)=\frac{1}{2^{n}}A_{0}\sum_{k=0}^{n}\left( \begin{array}{c}n\\ k\end{array} \right)\exp\Bigl{[}i(2k-n)\frac{\pi}{T}t\Bigr{]}[u(-t+T/2)-u(-t-T/2)] \tag{45}\]  
then, using Equation (24), the pulse's profile at the end of the medium is  
\[ \begin{split} A(z=0,t)=\frac{1}{2^{n}}A_{0}\times\sum\limits_{k= 0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right)\exp\Bigl{(}i\frac{\beta z^{2}}{2^{n}}\bigl{(}\frac{2\pi}{ T}\bigr{)}^{2}\Bigr{)}\times\\ \left[\exp\bigl{(}i(2k-n)\pi\frac{t}{T}\bigr{)}erfc\Bigl{(}\frac{t-T/2+(2k-n) \pi\beta z/T}{\sqrt{-2i\beta z^{2}}}\Bigr{)}-\right.\\ \left.\exp\bigl{(}-i(2k-n)\pi\frac{t}{T}\bigr{)}erfc\Bigl{(}\frac{t-T/2-(2k-n )\pi\beta z/T}{\sqrt{-2i\beta z^{2}}}\Bigr{)}\right]\end{split}  \tag{46}\]  
Figure 15: Comparison between the intensities of the three singular pulses represented by Equation (27)—dashed curve, Equation (39)—solid curve, and Equation (42)—dotted curve in a logarithmic scale. Second example:  
The initial singular pulse  
\[A(z=0,t)=A_{0}\exp(-a|t|) \tag{47}\]  
consists of two exponential pulses, and therefore its propagation can be expressed in a relatively simple form, namely:  
\[A(z>0,t)=\frac{A_{0}}{2}\exp(-i\frac{\beta_{2}}{2}a^{2}z)\left\{\exp(at)erfc \left(\frac{t-i\beta_{2}az}{\sqrt{-2i\beta_{2}z}}\right)+\exp(-at)erfc\left( \frac{-t-i\beta_{2}az}{\sqrt{-2i\beta_{2}z}}\right)\right\} \tag{48}\]  
Third example:  
In the previous example, we sewed two singular pulses; in general, however, one can sew several pulses together to create a combined pulse, which is much smoother than its singular components. For example, take the following pulse profile:  
\[A(z=0,t)= \begin{cases}A_{0}\cos(kt)&|t|<T/2\\ A_{0}\cos(kT/2)\exp[-k(|t|-T/2)\tan(kT/2)]&|t|\geq T/2\end{cases}  \tag{49}\]  
which consist of three singular and discontinuous pulses; however, the combined pulse and its derivative are both continuous. The discontinuity occurs only in the second derivative. In this case the pulse's amplitude profile obeys  
\[ \begin{array}{ll}A(z,t)=\frac{A_{0}}{4}&\exp(i\frac{\beta_{2}z}{2}k^{2}) \left[\exp(ikt)erfc\left(\frac{t-T/2+k\beta_{2}z}{\sqrt{-2i\beta_{2}z}}\right) +\exp(-ikt)erfc\left(\frac{t-T/2-k\beta_{2}z}{\sqrt{-2i\beta_{2}z}}\right)- \\ &\exp(ikt)erfc\left(\frac{t+T/2+k\beta_{2}z}{\sqrt{-2i\beta_{2}z}}\right)-\exp (-ikt)erfc\left(\frac{t+T/2-k\beta_{2}z}{\sqrt{-2i\beta_{2}z}}\right)\right]+ \\ &\frac{1}{2}A_{0}\cos(kT/2)\exp\left(-i\frac{\beta_{2}}{2}(k\tan(kT/2))^{2}z \right)\times\\ &\left[\exp(k\tan(kT/2)t)erfc\left(\frac{t-T/2-i\beta_{2}k\tan(kT/2)z}{\sqrt{ -2i\beta_{2}z}}\right)\right.\\ &\exp(-k\tan(kT/2)t)erfc\left(\frac{-t-T/2-i\beta_{2}k\tan(kT/2)z}{\sqrt{-2i \beta_{2}z}}\right)\right]\end{array}  \tag{50}\]  
Fourth example:  
The discontinuity of the exponential pulse can be reduced by multiplying the pulse by a sine function:  
\[A(z=0,t)=-A_{0}\exp(at)\sin(bt)u(-t) \tag{51}\]  
Since the pulse can be written as  
\[A(z=0,t)=\frac{iA_{0}}{2}[\exp((a+ib)t)-\exp((a-ib)t)]u(-t) \tag{52}\]  
then the pulse's amplitude after a distance \(z\) reads  
\[ \begin{array}{ll}A(t,z)&=\frac{iA_{0}}{4}\exp\left(-i\frac{\beta_{2}}{2}(a^ {2}-b^{2})z+at\right)\times\\ &\left[\exp(\beta_{2}abz+ibt)erfc\left(\frac{t-i\beta_{2}z(a+ib)}{\sqrt{-2i \beta_{2}z}}\right)-\exp(-\beta_{2}abz-ibt)erfc\left(\frac{t-i\beta_{2}z(a-ib )}{\sqrt{-2i\beta_{2}z}}\right)\right]\end{array}  \tag{53}\]  
In Figure 16, this pulse is presented for two cases with different parameters.