## 2 Principle of SRSI
Fourier transforming the laser spectrum obtained in Step 4 with the spectral phase obtained in Step 5 yields the temporal profile and duration of the pulse to be measured. SI is a linear, analytical, sensitive, and accurate method for measuring ultrashort pulses [37]. By combining SI with FROG, a method named temporal analysis by dispersing a pair of light \(e\) fields (TADPOLE) was developed and demonstrated in 1996 [50]. Owing to the simplicity and linearity; hence, the high sensitivity of SI, TADPOLE can characterize the temporal profile of a pulse train with an average energy of 42 zJ (42 \(\times\) 10\({}^{-21}\) J) per pulse. Thus, SI can be used to measure the temporal contrast of a femtosecond laser pulse, as discussed in the following section. However, a separate FROG measurement makes this method very complicated. Figure 2: Fourier transform spectral interferometry (FTSI) procedure for calculating the temporal profile of an unknown pulse in the SI method. FT Transform = Fourier transform and FT\({}^{-1}\) Transform = inverse Fourier transform [43]. Development of SRSI was driven by the recent research performed to improve the temporal contrast of the ultraintense femtosecond laser using the XPW method [51; 52; 53]. The generated XPW wave has a broadened and smoothed spectrum and a smoother and flatter spectral phase than that of the incident pulse because of the temporal filtering effect of the XPW process [35]. The principle of the SRSI method is presented in Figure 1b and described as follows: A reference pulse is self-created from a fraction of the input unknown pulse using a frequency-conserved nonlinear optical process. The self-created reference pulse and the input unknown pulse are then collinearly guided into a spectrometer with a time delay \(\tau\), which results in spectral interference in the spectrometer. The spectral phase of the generated reference pulse is almost flat in the entire spectral range, usually owing to the pulse-cleaning process if the pulse duration of the input unknown pulse is smaller than twice that of its Fourier-transform-limited pulse duration [37]. Then the spectral amplitude and phase of the input unknown pulse can be reconstructed from the spectral interference using the classical FTSI process presented in Figure 2. SRSI originated from SI; the main difference between the two is that, in the former, the reference pulse is generated from the incident pulse using frequency-conserved third-order nonlinear processes. Previously, we assumed that the spectral phase of the generated XPW wave is absolutely flat or equal to zero. This assumption limits the application range of the SRSI method in measuring a pulse with low dispersion. The spectral bandwidth of the generated reference pulse is narrower than that of the incident pulse if the chirp of the incident pulse makes the duration of the unknown pulse broader than twice that of its Fourier-transform-limited pulse [33]. Furthermore, although the spectral phase of XPW is very flat, it usually is not equal to zero in the entire spectral range. To obtain a precise measurement of the spectral phase, an iterative calculation is performed as follows:  
1. Using the expression \(E_{\mathrm{r}}(t)\propto|E_{\mathrm{u}}(t)|^{2}(t)E_{\mathrm{u}}(t-\tau)\), we obtain the temporal profile of the reference pulse using the temporal profile of the unknown pulse obtained in Step 6 above. After taking the inverse Fourier transform of \(|E_{\mathrm{r}}(t)|^{2}\), the spectrum \(|E_{\mathrm{r}}(\omega)|^{2}\) and spectral phase \(\phi_{\mathrm{r}}(\omega)\) of the reference pulse are obtained. 2. The new spectral phase of the reference pulse, \(\phi_{\mathrm{r}}(\omega)\), is then used to find the new spectral phase of the test pulse, \(\phi_{\mathrm{u}}(\omega)=\phi_{\mathrm{r}}(\omega)+\arg[f(\omega)]+C\), where \(C\) is the constant spectral phase with respect to the wavelength induced by dispersive optics, such as that in a beamsplitter, and can be obtained by a simple calculation.