## 2 Experimental Setup and Methods
The auto- correlator was used with a pulse width measurement range of (10 fs-12 ps) and it was connected to the PC through USB and by using the APE's Standardized Software Interface, allowing either remote control or integration into automated setups. Therefore, the deflected beam is tuned until the intensity is stabilized and maximized, and then, the pulse is acquired and the data analysis is completed by using the Gaussian fitting curve function _cftool_ of MATLAB 2020. To evaluate the spectral resolution, the exact estimation of the pulses' duration is required. Normally, the full width at half-maximum (FWHM) of the unknown pulse \(\tau_{p}\) is proportional to the FWHM of the measured fringe-resolved intensity autocorrelation function \(\tau_{ac}\):  
\[\tau_{ac}=k\cdot\tau_{p} \tag{1}\]  
where \(k\) is the proportionality factor, also known as the deconvolution factor. \(k\) differs significantly for different pulse shapes; therefore, to evaluate the pulse width from the intensity autocorrelation requires some previous knowledge of the pulse shape. Typically, the deconvolution factor can be calculated for analytical pulse shapes or computed numerically for complicated pulses; however, for some common pulse shapes, the deconvolution factor is known (\(k=1.414\) for Gaussian shape, \(k=1.543\) for each shape, \(k=1\) for square shape) [28]. It is well known that the Fourier transformation correlates \(\beta\), which defines the linear slope of the central frequency, to the group delay dispersion (GDD) applied to the laser pulse as reported in the following [19]:  
\[\beta=\frac{2 GDD}{\tau_{0}^{4}+4 GDD^{2}}\approx\frac{1}{2 GDD}\ for\ \tau\gg\tau_{0} \tag{2}\]  
The GDD leads to a stretch of the pulse width, which passes from \(\tau_{0}\) in the case of the FTL pulse to the chirped width \(\tau\):  
\[\tau=\tau_{0}\sqrt{1+\left(\frac{4\ln 2\cdot GDD}{\tau_{0}^{2}}\right)^{2}} \approx 2.77\frac{|GDD|}{\tau_{0}}\ for\ \tau\gg\tau_{0} \tag{3}\]  
From Equations (2) and (3), we found that \(\tau\cdot\tau_{0}\approx 4\ln 2|GDD|\approx 2\ln 2/|\beta|\); this product is considered a direct measure of the chirp parameter \(\beta\) for \(\tau\gg\tau_{0}\)[19]. We note that two chirped pulses have the same slope only if the resultant products \(\tau\cdot\tau_{0}\) are equal. Information obtained by autocorrelation measurements allows us to monitor the pulse duration and chirp of the laser beam, which are very important parameters to optimize the non-linear interaction in microscopy. Moreover, considering that SRS is a two-pulse technique, its spectral resolution is not defined by the spectrum of the individual exciting pulses but by the spectrum of their temporal interference. Consequently, cross-correlation characterization is equally important for providing information about the entire system; in particular, the FWHM of pump and probe beams' cross-correlation allows us to evaluate the experimental spectral bandwidth [29]. Typically, SF is obtained by equally chirping the pump and the Stokes pulses by using glass elements of known group-velocity dispersion without significant intensity losses [30]. As a general rule, the spectral resolution is restricted both by the level to which the pulses are chirped and by the similarity of these chirps. A better spectral resolution can be obtained when the pump and Stokes pulses with larger bandwidths are chirp-matched. In the case of transform limited laser pulses, the spectral resolution can be calculated by the formula [19]:  
\[\Delta\widetilde{\
u}=\frac{2\ln 2}{\pi c}\sqrt{2\Big{(}\tau_{p}^{-2}+\tau_{S}^{-2 }\Big{)}}=20.8\;\text{ps-cm}^{-1}\sqrt{\tau_{p}^{-2}+\tau_{S}^{-2}} \tag{4}\]  
and in the case of our fs laser sources its evaluated value is of 181 cm\({}^{-1}\).