## 5 Spectral Interferometry
When we extract the phase from the interferogram, the spectral resolution is determined by the width of the temporal filtering window: \(\Delta\omega_{\text{res}}\)\(\sim\)\(t_{\text{win}}\)\({}^{-1}\). Wavelet transform has been suggested [60; 61] as a substitute of Fourier transform for phase retrieval, and proven to increase the filtering window thus the minimum of the required sampling rate by 20%. However, the wavelet transform has lower computational efficiency compared to the Fourier transform. To relax the spectral resolution requirement of the spectrograph, a technique called spatially encoded arrangement for SPIDER (SEA-SPIDER) [59; 62] is proposed, where two SF-beams cross into the spectrograph, and the interferential fringes are resolved in space-dimension. This method can operate for single-shot measurement. However, the measurement based on spatial interference is easily affected by the beam quality or imaging aberrations. Comparatively, two-dimensional spectral shearing interferometry (2 DSI) [42; 63; 64], as shown in Figure 5, introduces a time-dimension instead of the space-dimension to resolve the interferential fringes, which can avoid the affection of the beam quality. Its \"time interference\", causing by the minor delay between two pulse replicas, is similar to that in an interferential autocorrelator. The 2 DSI-SPIDER need not determine the normal fringes-spacing for calibration. A disadvantage of 2 DSI-SPIDER is that it cannot operate for single-shot measurement. Both SEA-and 2 DSI-SPIDER are based on 2D measurement, needs inevitably more complex structures and expensive devices, e.g., imaging spectrograph for SEA-SPIDER, or motor-controlled element for 2 DSI-SPIDER. Moreover, the recorded 2D data means a longer time required to process than the usual 1D data. Recently, we have successfully combined two-step phase-shifting (TSPS) into SPIDER [65], which can expend the filtering window to the positive (or negative) semi-axis in the Fourier domain by using 1D measurement. A quarter waveplate is inserted into one arm of the MI pulse replicator to control the zero-order additional phase at \(\theta_{0}\) or \(\theta_{0}+\pi\), respectively, to generate two spectral interferograms with \(\pi\) phase-shifting. By combining these two interferograms, we can get the difference of two interferograms: \(\Delta S(\omega)=S(\omega)-S^{\pi}(\omega)=4|E(\omega)||E(\omega-\Omega)| \text{cos}[\omega\tau+\varphi(\omega)-\varphi(\omega-\Omega)]\), where the strong direct current (DC) term is canceled, helping to expend the width of the filtering window. Figure 5: Schematic of 2 DSI optics. The SPIDER is based on a collinear ZAP geometry, in the vertical arm of MI, a minor delay is introduced to generate time interferential fringes. There is another parameter, spectral shear \(\Omega\), that determines the resolution of the spectral phase. We obtain the spectral phase \(\varphi(\omega)\) by concatenating the differential phase \(\theta(\omega)\) with an interval \(\Omega\). Smaller \(\Omega\) means better spectral resolution. However, the choice of \(\Omega\) is limited by the measured SNR. If \(\Omega\) is chosen too small, the differential phase becomes unmeasurably small due to the detection noise [66]. Moreover, this noise is also accumulated in the course of phase concatenations. Small \(\Omega\) leads to a large concatenation number, at the wavelengths far away from the central wavelength, the recovered phase noise becomes very large. For finite SNRs, SPIDERs usually work at relative shear \(\Omega/\Delta\omega_{\text{BW}}=4.6\%\)-\(20\%\)[7; 8; 60; 62; 63; 66], where \(\Delta\omega_{\text{BW}}\) is the spectral bandwidth. [PERSON] et al. [67] proposed a method called multiple-shearing spectral interferometry, where interferential data with different shears are combined to suppress the accumulating phase noise and improve the spectral resolution. This method is relatively complex, involving shear changes in the measurements. It also asks for a relatively high consistency of the multi-shear data.