## 3 OAM-Temporal Couplings in the Most Intense Ring of Transform-Limited Ultrafast Vortices
### Pulse Characterization  
As a preliminary, we introduce a characterization of transform-limited pulses by means of two measurable parameters that will be seen to be directly involved in the OAM-temporal couplings. This characterization appears to be particularly relevant to nonlinear optics applications with these pulses because it describes quite accurately the shape of the main temporal lobe of amplitude about its maximum, and ignores the temporal parts of the pulse of low amplitude. First, we consider the instantaneous frequency at the time \(\tau=0\) of maximum amplitude, defined as  
\[\bar{\omega}_{c}(\rho)=\left.\frac{d\arg E(\rho,\tau)}{d\tau}\right|_{\tau=0}\,, \tag{14}\]  
which is called the central frequency to distinguish it from \(\bar{\omega}(\rho)\), since their values are generally different. Writing \(\arg E=\tan^{-1}(\operatorname{Im}E/\operatorname{Re}E)\) with \(\operatorname{Re}E=(E+E^{\star})/2\) and \(\operatorname{Im}E=(E-E^{\star})/2i\), and using Equation (5) and its derivative with respect to time, one arrives at the alternate expression  
\[\bar{\omega}_{c}(\rho)=\frac{\int_{0}^{\infty}\hat{E}_{\omega}(\rho)\omega d \omega}{\int_{0}^{\infty}\hat{E}_{\omega}(\rho)d\omega}=\frac{\int_{0}^{\infty }|\hat{E}_{\omega}(\rho)|\omega d\omega}{\int_{0}^{\infty}|\hat{E}_{\omega}( \rho)|d\omega}\,. \tag{15}\]  
The introduction of the absolute values in the last step does not alter the result with the spectral phases in Equation (12) independent of frequency. In this way, the value of \(\bar{\omega}_{c}\) is easily obtainable from the square root of the spectral density \(|\hat{E}_{\omega}|^{2}\) measured by a spectrometer. Second, a measure of the width of the central lobe of the pulse based on its concavity is provided by the expression  
Figure 1: (**a**) Radial profile of fluence, \(F\), and peak intensity, \(|E(\rho,0)|^{2}\), of the ultrafast vortex with topological charge \(l=27\) and spectrum \(\hat{a}_{\omega}=\omega^{0.5}e^{-\omega/25}\) of mean frequency \(\bar{\omega}=2.5\) fs\({}^{-1}\), normalized to their respective maxima on the respective caustic surfaces \(\rho_{F}=2.32\) fs\({}^{1/2}\) and \(\rho_{E}=1.90\) fs\({}^{1/2}\). (**b**) Respective pulse shapes (real field and amplitudes) on these two caustic surfaces, oscillating at different carrier frequencies, \(\bar{\omega}(\rho_{F})=2.5\) fs\({}^{-1}\simeq\bar{\omega}\) and \(\bar{\omega}(\rho_{E})=3.616\) fs\({}^{-1}\), and with different durations. For a better comparison, the four curves are normalized to the peak amplitude of the pulse at \(\rho_{E}\). \[\Delta t_{c}^{2}(\rho)\equiv-2\left(\left.\frac{d^{2}|E(\rho,\tau)|/d\tau^{2}}{|E( \rho,\tau)|}\right|_{\tau=0}\right)^{-1}\,, \tag{16}\]  
yielding the \(1/e\)-decay half-duration for a Gaussian-shaped amplitude. For other pulse shapes, Equation (16) provides the duration of a Gaussian pulse with the same concavity of the maximum at \(\tau=0\), and hence is called the central duration. Using the equivalent expression \(\Delta t_{c}^{2}=-4|E|^{2}/(d^{2}|E|^{2}/d\tau^{2}|_{\tau=0})\), writing \(|E|^{2}=EE^{*}\), using Equation (5) and its derivatives with respect to time at \(\tau=0\), one easily arrives at the alternate expression  
\[\Delta t_{c}^{2}(\rho)=4/\Delta\omega_{c}^{2}(\rho)\,, \tag{17}\]  
where  
\[\Delta\omega_{c}^{2}(\rho)=2\frac{\int_{0}^{\infty}\hat{E}_{\omega}(\rho)[ \omega-\bar{\omega}_{c}(\rho)]^{2}d\omega}{\int_{0}^{\infty}\hat{E}_{\omega}( \rho)d\omega}=2\frac{\int_{0}^{\infty}|\hat{E}_{\omega}(\rho)|[\omega-\bar{ \omega}_{c}(\rho)]^{2}d\omega}{\int_{0}^{\infty}|\hat{E}_{\omega}(\rho)|d\omega} \tag{18}\]  
defines a spectral bandwidth based on the spectral amplitude instead of the spectral intensity that yields, again, the \(1/e\)-decay half-bandwidth for a Gaussian-like spectral amplitude \(|\hat{E}_{\omega}(\rho)|\), and therefore coincides with \(\Delta\omega(\rho)\) for a Gaussian spectrum.