## 2 Principle of SRSI  
Nonlinear processes are required for most ultrashort-pulse measurements to perform time-gating in the sampling of short temporal segments of the pulses to be characterized [29; 30; 36; 49]. This requirement increases the complexity of an ultrashort-pulse measurement system because of the need for suitable nonlinear media and relatively high peak power [49]. The use of a reference pulse with a known amplitude and phase in SI drastically simplifies the measurement with respect to the equipment needed and algorithm used [36; 37]. The principle of SI is shown in Figure 1a. An unknown pulse and a reference pulse, separated by a suitable time delay \(\tau\), are collinearly guided into a spectrometer. A spectral interferogram that includes the phase information of both the reference and the unknown pulse is obtained and expressed as \(2\sqrt{S_{\rm r}(\omega)}\sqrt{S_{\rm u}(\omega)}\cos[\varphi_{\rm u}(\omega)- \varphi_{\rm r}(\omega)+\omega\tau]\), where \(S_{\rm r}(\omega)\) and \(S_{\rm u}(\omega)\) are the spectra of the reference pulse and the unknown pulse and \(\varphi_{\rm r}(\omega)\) and \(\varphi_{\rm u}(\omega)\) are the spectral phase of the reference pulse and the unknown pulse. Fourier transform spectral interferometry (FTSI) is introduced into the SI process to determine the spectral phase of the pulse to be measured [36]. If \(E_{\rm r}(\omega)\) and \(E_{\rm u}(\omega)\) are the complex spectral amplitudes at angular frequency \(\omega\) of the reference and the unknown pulse, respectively, then the interferogram obtained by the spectrometer is expressed as  
\[ \begin{array}{ll}D(\omega,\tau)&=\left|E_{\rm r}(\omega)+E_{\rm u}(\omega)e ^{i\omega\tau}\right|^{2}=\left|E_{\rm r}(\omega)\right|^{2}+\left|E_{\rm u}( \omega)\right|^{2}+E_{\rm r}^{*}(\omega)E_{\rm u}(\omega)e^{i\omega\tau}+ \text{c.c.}\\ &=S_{0}(\omega)+f(\omega)e^{i\omega\tau}+c.c.\end{array}  \tag{1}\]  
where \(S_{0}(\omega)\) is the sum of the spectral intensities of the reference and the unknown pulse and \(f(\omega)\) is the interference of the two spectra. The spectrum and spectral phase of the unknown pulse can be obtained using a Fourier transform-based procedure and an algebraic calculation, as outlined in the following steps and in Figure 2:  
1. Fourier transform the interference signal \(D(\omega,\tau)\) from the frequency domain to the time domain and obtain the temporal signal containing \(S(\tau)\) and \(f(\tau)\). 2. Separate \(S(\tau)\) and \(f(\tau)\) from the temporal signals obtained in Step 1 by using a suitable window function, e.g., a super Gaussian window. The width of the window function should be half of the gap between \(S(\tau)\) and \(f(\tau)\) (or half of the delay time between the reference pulse and the test pulse to be measured). 3. After inverse Fourier transforming \(S(\tau)\) and \(f(\tau)\) obtained in Step 2, we obtain the signals of \(S_{0}(\omega)\) and \(f(\omega)\) in the frequency domain, respectively. 4. Using \(S_{0}(\omega)\) and \(f(\omega)\) obtained in Step 3, the spectral amplitudes of the unknown pulse and the reference signal are analytically calculated using the following expressions [37]: \[\left|E_{\rm u}(\omega)\right|=\frac{1}{2}\bigg{(}\sqrt{S_{0}(\omega)+2|f( \omega)|}-\sqrt{S_{0}(\omega)-2|f(\omega)|}\bigg{)}\] (2) \[\left|E_{\rm r}(\omega)\right|=\frac{1}{2}\bigg{(}\sqrt{S_{0}(\omega)+2|f( \omega)|}+\sqrt{S_{0}(\omega)-2|f(\omega)|}\bigg{)}.\] (3)  
Figure 1: Principles of (**a**) spectral interferometry (SI) and (**b**) self-referenced spectral interferometry (SRSI), in which a nonlinear process is used to generate a reference pulse. After calculating the phase using \(f(\tau)[\text{arg}f(\omega)]\), the preliminary spectral phase of the pulse to be measured is calculated using Equation (4): \[\phi_{\text{u}}(\omega)=\phi_{\text{r}}(\omega)+\text{arg}[f(\omega)]\] (4) where \(\phi_{\text{u}}(\omega)\) and \(\phi_{\text{r}}(\omega)\) are the spectral phases of the test pulse and the reference pulse, respectively. 6.