## 4 Ultrafast Vortices with Prescribed Pulse Shape in the Most Intense Ring  
As a verification, suppose we need the expression for an ultrafast vortex of certain transform-limited temporal shape,  
\[P(\tau)=\frac{1}{\pi}\int_{0}^{\infty}\hat{P}_{\omega}e^{-i\omega\tau}d\omega\,, \tag{25}\]  
in the most intense ring. The pulse \(P(\tau)\) is characterized by certain central frequency \(\bar{\omega}_{c}\) and duration \(\Delta t_{c}\). Equating Equations (5) and (6) evaluated at \(\rho_{E}^{2}=|l|/2\bar{\omega}_{c}(\rho_{E})=|l|/2\bar{\omega}_{c}\) to \(DP(\tau)\), the needed laser source spectrum must be \(\hat{a}_{\omega}=\hat{P}_{\omega}\omega^{-|l|/2}e^{\rho_{E}^{2}\omega}(\sqrt{ 2}\rho_{E})^{-|l|}\). Introducing this expression of \(\hat{a}_{\omega}\) in Equation (6) for generic \(\rho\), the result in Equation (5), and performing the integral, we obtain the expression  
\[E(\rho,\tau)=D\left(\frac{\rho}{\rho_{E}}\right)^{|l|}P\left(\tau-i\rho^{2}+i \rho_{E}^{2}\right)\,, \tag{26}\]of the ultrafast vortex of pulse shape \(P(\tau)\) at \(\rho_{E}\). Coming back to the physical radius \(r\), the caustic of maximum peak intensity reads \(r_{E}(z)=\sqrt{|l|/2}\,s_{\omega_{c}}(z)\), where \(s_{\omega_{c}}(z)\) is given by Equation (4) evaluated at \(\bar{\omega}_{c}\), and the ultrafast vortex reads  
\[E=D\Bigg{(}\sqrt{\frac{2}{|l|}}\frac{r}{s_{\omega_{c}}(z)}\Bigg{)}^{|l|}P \left[t^{\prime}-\frac{r^{2}}{2 cq(z)}+i\frac{|l|}{2\bar{\omega}_{c}}\right]\,. \tag{27}\]  
With given \(P(\tau)\), and hence given \(\bar{\omega}_{c}\) and \(\Delta t_{c}\), Equation (27) represents indeed an ultrafast vortex with pulse shape \(P(\tau)\) in is most intense ring at \(r_{E}(z)=\sqrt{|l|/2}\,s_{\omega_{c}}(z)\) only if \(|l|<\bar{\omega}_{c}^{2}\Delta t_{c}^{2}\); otherwise, the intensity takes a relative minimum at \(r_{E}(z)=\sqrt{|l|/2}\,s_{\omega_{c}}(z)\), with a maximum at another radius with different pulse shape such that the inequality in Equation (23) is satisfied, or possibly a singular expression, as in the following example. We consider the transform-limited pulses  
\[P(\tau)=\left(\frac{-i\beta}{\bar{\omega}_{c}\tau-i\beta}\right)^{\beta}e^{-i \Phi}\,, \tag{28}\]  
where \(\bar{\omega}_{c}\) is just the central frequency, and the parameter \(\beta\geq 1\) determines the central bandwidth \(\Delta\omega_{c}=\sqrt{2/\beta}\,\bar{\omega}_{c}\) and the duration \(\Delta t_{c}=\sqrt{2\beta}/\bar{\omega}_{c}\). For \(\beta\gg 1\), \(P(\tau)\) approaches the Gaussian pulse \(\exp(-\tau^{2}/\Delta t_{c}^{2})e^{-i(\bar{\omega}_{c}\tau+\Phi)}\) based on its central parameters not only in its central portion but at any time because \(P(\tau)\) approaches a Gaussian-enveloped pulse, as in the example of Figure 3a showing a near infrared, single-cycle pulse (\(\bar{\omega}_{c}=2.417\) rad/fs, \(\beta=14.24\)). Condition \(|l|<\bar{\omega}_{c}^{2}\Delta t_{c}^{2}\) for the existence of the ultrafast vortex with this pulse shape in its most intense ring reads \(|l|<2\beta\). Indeed, Equation (27) with Equation (28) yields  
\[E=D\left(\sqrt{\frac{2}{|l|}}\frac{r}{s_{\omega_{c}}(z)}\right)^{|l|}\left[ \frac{-i\beta}{\bar{\omega}_{c}\left(t^{\prime}-\frac{r^{2}}{2 cq(z)}\right)+i \left(\frac{|l|}{2}-\beta\right)}\right]^{\beta}e^{-i\Phi}. \tag{29}\]  
Figure 3: (**a**) The real part (black solid) and amplitude (black dashed) of the pulse \(P\) in Equation (28) with \(\bar{\omega}_{c}=2.417\) rad/fs (780 nm wave length), \(\beta=14.24\), and \(\Phi=\pi/2\), containing a single oscillation within its full width at half maximum in intensity, and its approximation \(P_{c}=e^{-\tau^{2}/\Delta t_{c}^{2}}e^{-i(\bar{\omega}_{c}\tau+\Phi)}\), with \(\Delta t_{c}=2.208\) fs (gray curves).