## 4 Ultrafast Vortices with Prescribed Pulse Shape in the Most Intense Ring
### Ultrafast Vortices with Power-Exponential Laguerre-Gauss Spectrum  
In practice, one has a spectrum \(\hat{a}_{\omega}\) that is substantially determined by the laser source and is therefore independent of the topological charge \(l\), and with this spectrum one may wish to synthesize an ultrafast vortex of certain charge \(l\). The above OAM-temporal coupling effects imply that the temporal properties of the main lobe of the pulse in the most intense ring of the ultrafast vortices synthesized with the same spectrum \(\hat{a}_{\omega}\) change with \(l\). This is particularly true if the bandwidth of \(\hat{a}_{\omega}\) is very large and/or the topological charge high. As a sufficiently flexible model, we take the power-exponential spectrum  
\[\hat{a}_{\omega}=\frac{\pi}{\Gamma(\beta)}\left(\frac{\beta}{\bar{\omega}_{c} }\right)^{\beta-1}\bar{\omega}_{c}^{\beta}e^{-\frac{\omega}{2\bar{\omega}_{c} }\beta}e^{-i\Phi}\,, \tag{31}\]  
corresponding in time domain to the same pulse \(a(\tau)=[-i\beta/(\bar{\omega}_{c}\tau-i\beta)]^{\beta}e^{-i\Phi}\) as \(P(\tau)\) in Equation (28) of arbitrary central frequency \(\bar{\omega}_{c}\) and central duration \(\Delta t_{c}=\sqrt{2\beta}/\bar{\omega}_{c}\). Integral in Equation (1) with the LG beams in Equation (1) and with the power-exponential spectrum in Equation (31) can be carried out to yield the closed-form expression  
\[E=D\left(\sqrt{\frac{2}{|l|}}\frac{r}{s_{\omega_{c}}(z)}\right)^{|l|}\left[ \frac{-i\left(\beta+\frac{|l|}{2}\right)}{\bar{\omega}_{c}\left(t^{\prime}- \frac{r^{2}}{2 cq(z)}\right)-i\beta}\right]^{\beta+\frac{|l|}{2}}e^{-i\Phi} \tag{32}\]  
which is, this time, nonsingular and localized for any value of \(l\). A multiplicative factor has been introduced in Equation (32) to adjust to unity the peak amplitude at \(r_{E}(0)=\sqrt{|l|/2}\,s_{\bar{\omega}_{c}}\) at the waist \(z=0\). The peak amplitude at any point \((r,z)\) is given by  
\[|E(\tau=0)|=D\left(\sqrt{\frac{2}{|l|}}\frac{r}{s_{\omega_{c}}(z)}\right)^{|l| }\left(\frac{\beta+\frac{|l|}{2}}{\beta+\frac{r^{2}}{s_{\omega_{c}}^{2}(z)}} \right)^{\beta+\frac{|l|}{2}} \tag{33}\]  
which takes a maximum value on the caustic surface \(r_{E}(z)=\sqrt{|l|/2}\,s_{\bar{\omega}_{c}}(z)\). On this caustic surface, the pulse shape is  
\[E(\tau)=D\left[\frac{-i\left(\beta+\frac{|l|}{2}\right)}{\bar{\omega}_{c} \tau-i\left(\beta+\frac{|l|}{2}\right)}\right]^{\beta+\frac{|l|}{2}}e^{-i\Phi}\,, \tag{34}\]  
which is the same as \(a(\tau)\) but with \(\beta\) replaced with \(\beta+|l|/2\). Thus, for the fundamental Gaussian beam (\(l=0\)), i.e., the fundamental isodiffracting pulsed Gaussian beam [33; 34], the pulse shape at \(r=0\) is just \(a(\tau)\) of the central frequency \(\bar{\omega}_{c}\) and duration \(\Delta t_{c}=\sqrt{2\beta}/\bar{\omega}_{c}\), as determined by the full source spectrum \(\hat{a}_{\omega}\). For \(|l|\
eq 0\), the pulse temporal shape in the most intense ring maintains the frequency \(\bar{\omega}_{c}(\tau_{E})=\bar{\omega}_{c}\) of the source spectrum regardless of the value of \(l\), but the duration increases with the magnitude of the topological charge as  
\[\Delta t_{c}(r_{E})=\frac{\sqrt{2\beta+|l|}}{\bar{\omega}_{c}}\,; \tag{35}\]  
in particular, the pulse adapts itself so that its duration is always above the lower bound \(\sqrt{|l|}/\bar{\omega}_{c}(r_{E})=\sqrt{|l|}/\bar{\omega}_{c}\). Figure 4 illustrates the adaptation of pulse shape on the most intense caustic surface to the value of the topological charge. With the ultra-broadband spectrum of Figure 4a of central frequency in the near infrared, the half-cycle pulse represented by blue curves in Figure 4b-d could be built with \(l=0\) at \(r=0\). At the maxima of the radial distribution of peak intensity depicted in Figure 4e for several values of \(|l|\), the pulses represented in Figure 4b-d as black curves are seen to increase their duration with increasing \(|l|\).