## 2 Theoretical Calculations  
The electric field and intensity characteristics of an optical pulse with respect to time (\(t\)) can be assumed to have Gaussian profiles [2]. \[\widetilde{E}(t)=E_{0}e^{-aT}e^{i\left(\omega t+bt^{2}\right)} \tag{1}\]  
\[I(t)=\left|E_{0}\right|^{2}e^{-2 at^{2}} \tag{2}\]  
where \"\(a\)\" is related to the pulse duration, \(\Delta t\), as shown below, and \"\(b\)\" is a chirp parameter. \[\sqrt{\frac{2\text{ln}2}{a}}\approx\Delta t \tag{3}\]  
The second-order FRAC signal can be expressed as  
\[I_{\text{FRAC}}^{2}(\tau)=\int\limits_{-\infty}^{\infty}\left|\left( \widetilde{E}(t-\tau)+\widetilde{E}(t)\right)^{2}\right|^{2}\text{d}t \tag{4}\]where \(\tau\) is the time delay between the two pulses in the interferometer [21]. This parameter can be set to zero because the beam is not separated in this study (_i.e._, no interferometer is used), and this leads to Equation (5). \[I_{\text{FRAC}}^{2}(\tau=0)=\int\limits_{-\infty}^{\infty}\left|(2\widetilde{E}( t))\right|^{2}\mathrm{d}t=16\int\limits_{-\infty}^{\infty}\left|(\widetilde{E}( t))\right|^{2}\mathrm{d}t=16\left|E_{0}\right|^{4}\int\limits_{-\infty}^{ \infty}e^{-4\mathrm{d}t^{2}}\mathrm{d}t=16A^{2}\sqrt{\frac{a}{\pi}} \tag{5}\]  
where the parameter, \(A\), is defined as a pulse energy by the following equation. \[\int\limits_{-\infty}^{\infty}I(t)\mathrm{d}t=\left|E_{0}\right|^{2}\int \limits_{-\infty}^{\infty}e^{-2\mathrm{d}t^{2}}\mathrm{d}t=A \tag{6}\]  
When the signal intensity becomes one half of this value via a chirp of the pulse,  
\[I_{\text{FTL}}^{2}(\tau=0)=2I_{\text{CP}}^{2}(\tau=0) \tag{7}\]  
where \(I^{2}_{\text{FTL}}\) and \(I^{2}_{\text{CP}}\) are the signal intensities of the transform-limited and chirped pulses, respectively. The following equation is then obtained. \[\sqrt{a_{\text{FTL}}}=2\sqrt{a_{\text{CP}}} \tag{8}\]  
where \(a_{\text{FTL}}\) and \(a_{\text{CP}}\) are the parameters that were calculated using Equation (3). Then,  
\[\Delta t_{\text{CP}}=2\Delta t_{\text{FTL}} \tag{9}\]  
where \(\Delta t_{\text{FTL}}\) and \(\Delta t_{\text{CP}}\) are the pulse durations of the transform-limited and chirped pulses, respectively. The relationship between \(\Delta t_{\text{CP}}\) and \(\Delta t_{\text{FTL}}\) can be expressed as Equation (10) [21]. \[\Delta t_{CP}=\sqrt{(\Delta t_{FTL})^{2}+\left(\frac{\varphi_{2}\times 4\text{ ln}2}{\Delta t_{FTL}}\right)^{2}} \tag{10}\]  
where \(\varphi_{2}\) is the group delay dispersion (GDD), which can be expressed as a product of the group velocity dispersion (GVD) determined using the Sellmeier equation [30; 31] and the effective length of the optical material in the beam path, \(\ell-\ell_{0}\) (see Figure 1a). Then,  
\[I(t)=\left|E_{0}\right|^{2}e^{-2 at^{2}} \tag{2}\]0  
where \(\ell\) = (2 tan 34\({}^{\circ}\)) \(\times\)\(L\) for a Brewster prism used at approximately 250 nm. The factor of 2 in Equation (11) is multiplied due to a double pass of the beam in the prism. When the dispersion is canceled by a prism pair, \(GDD\) = 0 and \(\ell_{0}\) = (2 tan 34\({}^{\circ}\)) \(\times\)\(L_{0}\) where \(\ell_{0}\) and \(L_{0}\) are the parameters of \(\ell\) and \(L\) at which the dispersion is canceled (the two-photon ionization signal is maximal). A parameter of \(X\) can be defined as a displacement of the prism position from the optimum location that cancels the dispersion (_i.e._, \(GDD\) = 0), then \(X\) = \(L-L_{0}\). When the intensity of the two-photon ionization signal decreases to one half of the maximum value, \(L\) = \(L_{1/2}\) and then \(X_{1/2}\) = \(L_{1/2}-L_{0}\). From Equations (9) and (10), \(\Delta t_{\text{FTL}}\) can then be rewritten as  
\[I(t)=\left|E_{0}\right|^{2}e^{-2 at^{2}} \tag{2}\]1  
where \(\varphi_{2}\) (\(X_{1/2}\)) is the value of \(GDD\) at \(X\) = \(X_{1/2}\). This equation suggests that the transform-limited pulse width can be calculated by measuring the parameter of \(X_{1/2}\) in the experiment.