## 2 Ultrafast Vortices and Previous Results  
We express an ultrafast vortex of topological charge \(l\), or \(l\) units of orbital angular momentum, as the superposition  
\[E(r,\phi,z,t^{\prime})=\frac{1}{\pi}\int_{0}^{\infty}\hat{E}_{\omega}(r,\phi,z)e^{-i\omega t^{\prime}}d\omega\,, \tag{1}\]of monochromatic LG beams  
\[\hat{E}_{\omega}(r,\phi,z)=\hat{a}_{\omega}D(z,\phi)\left[\frac{\sqrt{2}r}{s_{ \omega}(z)}\right]^{|l|}e^{\frac{i\omega r^{2}}{2\omega(z)}}\,, \tag{2}\]  
of the same topological charge \(l\), zero radial order, weights \(\hat{a}_{\omega}\), and different angular frequencies \(\omega\). In the above expressions \((r,\phi,z)\) are cylindrical coordinates, \(c\) is the velocity of light in vacuum, and \(t^{\prime}=t-z/c\) is the local time for a plane pulse. The factor \(D(z,\phi)=e^{-i(|l|+1)\tan^{-1}(z/z_{R})}e^{-i\phi t\phi}/[1+(z/z_{R})^{2}]^{ 1/2}\), where \(z_{R}\) is the Rayleigh distance, accounts for the azimuthal phase variation, and the \(z\)-dependent amplitude attenuation and [PERSON]'s phase shift associated with diffraction. The complex beam parameter is \(q(z)=z-iz_{R}\), which is usually written in the form  
\[\frac{1}{q(z)}=\frac{1}{R(z)}+i\frac{2c}{\omega s_{\omega}^{2}(z)}\,, \tag{3}\]  
where  
\[s_{\omega}(z)=s_{\omega}\sqrt{1+\left(\frac{z}{z_{R}}\right)^{2}}\,,\quad s_{ \omega}=\sqrt{\frac{2z_{R}c}{\omega}}\,, \tag{4}\]  
are the width and waist width, respectively, of the fundamental (\(l=0\)) Gaussian beam, and \(1/R(z)=z/(z^{2}+z_{R}^{2})\) is the curvature of the wave fronts of Gaussian and LG beams. Being limited to positive frequencies, the optical field \(E\) in Equation (1) is the analytical signal complex representation of the real field \(\mathrm{Re}\,E\)[38]. As in preceding works [3; 5; 36], the Rayleigh distance \(z_{R}\) is assumed to be independent of frequency, i.e., we adopt the so-called isodiffracting model [33; 34; 39; 40] because it is the only situation in which the pulse temporal shape does not experience any change during propagation (except for the global complex amplitude \(D\)) no matter how high \(l\) is, as shown recently [3] and as desired for applications. This choice is also a way to isolate the pure effects of OAM on temporal shape, i.e., the OAM-temporal coupling, and thus distinguish it from propagation effects on pulse shape in models other than the isodiffracting model, and whose study is left to further research. For a simpler analysis, we introduce the normalized radial coordinate \(\rho=r/\sqrt{2z_{R}c[1+(z/z_{R})^{2}]}\). A constant value of \(\rho\) represents a revolution hyperboloid, or caustic surface, expanding as the monochromatic LG beams do. Equation (1) with Equation (2) now reads  
\[E(\rho,\tau)=\frac{1}{\pi}\int_{0}^{\infty}\hat{E}_{\omega}(\rho)e^{-i\omega \tau}d\omega\,, \tag{5}\]  
where  
\[\hat{E}_{\omega}(\rho)=\hat{a}_{\omega}D\left(\sqrt{2}\rho\right)^{|l|}\omega ^{|l|/2}e^{-\rho^{2}\omega}\,, \tag{6}\]  
and where \(\tau=t^{\prime}-r^{2}/2 cR(z)\) is the local time for the ultrafast vortex. The pulse front \(\tau=0\) determines the time of arrival of the pulse at each point of space, and is the same family of spherical surfaces (in the paraxial approximation) as the phase fronts of the superposed isodiffracting LG beams. According to Equations (5) and (6), the pulse temporal shape depends on the particular caustic surface \(\rho\), but not on propagation distance \(z\). In [3], the caustic surface \(\rho_{F}\) where the time-integrated intensity, energy density, or fluence, \(F(\rho)=\int_{-\infty}^{\infty}(\mathrm{Re}E)^{2}d\tau=(1/2)\int_{-\infty}^{ \infty}|E|^{2}d\tau=(1/\pi)\int_{0}^{\infty}|\hat{E}_{\omega}|^{2}d\omega\) is maximum, i.e., the bright ring as recorded by a time-integrating detector, is considered the most relevant caustic surface, and the effects of OAM on pulse temporal shape on that caustic surface are described.