## 3 OAM-Temporal Couplings in the Most Intense Ring of Transform-Limited Ultrafast Vortices
### Lower Bound to the Ultrafast Vortex Duration in Its Most Intense Ring  
With the above pulse properties in mind, let us locate the caustic surface \(\rho_{E}\) of the ultrafast vortex where the peak pulse amplitude is maximum. After some calculations, the derivative of the peak amplitude in Equation (13) with respect to \(\rho\) is found to be  
\[\frac{d|E(\rho,0)|}{d\rho}=\frac{|E(\rho,0)|}{\rho}\left[|l|-2\rho^{2}\bar{ \omega}_{c}(\rho)\right]\,. \tag{19}\]  
The fluence will then be maximum or minimum at the caustic surface determined by  
\[\rho_{E}^{2}=\frac{|l|}{2\bar{\omega}_{c}(\rho_{E})}\,, \tag{20}\]  
which generally differs from the caustic surface \(\rho_{F}^{2}=|l|/2\bar{\omega}(\rho_{F})\) of maximum fluence [3]. Evaluation of the second derivative of \(|E(\rho,0)|\) with respect to \(\rho\) leads to an involved expression, which however at \(\rho_{E}\) simplifies to \[\left.\frac{d^{2}|E(\rho,0)|}{d\rho^{2}}\right|_{\rho_{E}}=-2\bar{\omega}_{E}(\rho_ {E})|E(\rho_{E},0)|\left[2-\frac{|l|}{2}\frac{\Delta\omega_{c}^{2}(\rho_{E})}{ \bar{\omega}_{c}^{2}(\rho_{E})}\right]\,. \tag{21}\]  
The condition of maximum in Equation (21) allows us to conclude that the bandwidth \(\Delta\omega_{c}\) on the caustic surface of maximum intensity and the topological charge of ultrafast vortices always satisfy the inequality  
\[\frac{\Delta\omega_{c}(\rho_{E})}{\bar{\omega}_{c}(\rho_{E})}<\frac{2}{\sqrt{| l|}}\,, \tag{22}\]  
i.e., the relative bandwidth on the most intense ring cannot exceed a maximum value determined by the topological charge of the vortex. The inequality in Equation (22) is formally equal as \(\Delta\omega(\rho_{F})/\bar{\omega}(\rho_{F})<2/\sqrt{|l|}\), but these two inequalities refer to different caustic surfaces and the quantities involved are different. The above inequality implies a lower bound to the duration of the central lobe of an ultrafast vortex at its most intense ring. From Equation (17), the inequality in Equation (22) transforms into  
\[\Delta t_{c}(\rho_{E})>\frac{\sqrt{|l|}}{\bar{\omega}_{c}(\rho_{E})}\,. \tag{23}\]  
This inequality imposes a lower bound to the central duration relative to the central period \(2\pi/\bar{\omega}_{c}(\rho_{E})\), and therefore a lower bound to the number of oscillations of the pulse in the most intense ring of the ultrafast vortex. If the number of oscillations \(N\) is measured as the full width at half maximum of intensity of the central lobe, \(\sqrt{2\ln 2}\Delta t_{c}(\rho_{E})\), over the central period, the inequality in Equation (23) leads to inequality \(N>\sqrt{\ln 2/2}\sqrt{|l|}/\pi\) for the number of oscillations. Since the lower bound in the right hand side increases monotonically with \(|l|\), the number of oscillations in the central lobe of the pulse in the most intense ring of the ultrafast vortices synthesized with the same source spectrum \(\hat{a}_{\omega}\) increases if the imprinted topological charge is increased. In practice, as seen below, the central frequency \(\bar{\omega}_{c}(\rho_{E})\) of the pulse in the most intense ring is substantially determined by the available frequencies in the superposition, i.e., by \(\hat{a}_{\omega}\), indeed \(\bar{\omega}_{c}(\rho_{E})\approx\bar{\omega}_{c}\), with  
\[\bar{\omega}_{c}=\frac{\int_{0}^{\infty}|\hat{a}_{\omega}|\omega d\omega}{\int _{0}^{\infty}|\hat{a}_{\omega}|d\omega}\,, \tag{24}\]  
With \(\bar{\omega}_{c}(\rho_{E})\approx\bar{\omega}_{c}\) independent of the topological charge, the inequality in Equation (23) imposes directly a lower bound \(\sqrt{|l|}/\bar{\omega}_{c}\) to the duration of the main lobe, and implies an increase of this duration with the magnitude of the topological charge.