## 5 Spectral Interferometry  
As mentioned in Part II, the spectral phase is the key quantity for the complete measurement of an ultrashort pulse. Compared to the Spectrogram method based on a 2D combining measurement, the spectral interferometry method is a 1D frequency-domain measurement, which is implemented simply by using a reference pulse with a well-known phase to interfere with the test pulse. However, the determination of the reference phase relies on another measurement. Accordingly, what we need firstly is a self-reference spectral interferometry. This kind of interferometry was firstly proposed by [PERSON] et al. [7; 8], which is called SPIDER, based on shearing interferometry. Now, it already becomes an alternative measuring technique. In the original version of SPIDER, the laser divides into three, one is stretched sufficiently into a chirped pulse, the other two are combined with a delay \(\tau\), as shown in Figure 4. Then the pulse replicas are frequency-up-conversed in a \(\chi^{(2)}\)-crystal with the different quasi continuous wave (CW) frequency part of the chirped pulse, generating two SF-pulses with a frequency difference (or a frequency shear) \(\Omega\). The interferogram of the SF-pulses is given by  
\[S(\omega)=|E(\omega)|^{2}+|E(\omega-\Omega)|^{2}+2|E(\omega)||E(\omega-\Omega) |\text{cos}[\omega\tau+\varphi(\omega)-\varphi(\omega-\Omega)]. \tag{11}\]  
From \(S(\omega)\), the differential phase \(\theta_{\tau}(\omega)=\varphi(\omega)-\varphi(\omega-\Omega)+\omega\tau\) can be easily extracted by Fourier domain filtering which was first suggested by [PERSON] et al. for applications in holography [57]:  
\[\theta_{\tau}(\omega) = \text{arg}\int_{\tau_{0}-t_{\text{win}}/2}^{\tau_{0}+t_{\text{win }}/2}\int_{-\infty}^{\infty}S(\omega^{\prime})\exp(-i\omega^{\prime}t^{ \prime})d\omega^{\prime}\exp(-i\omega t^{\prime})dt^{\prime} \tag{12}\]  
where the integral region [\(\tau-t_{\text{win}}\)/2, \(\tau+t_{\text{win}}\)/2] is to filter the positive (or negative) alternating current (AC) term of the spectrum, \(t_{\text{win}}\) is the width of the filter windows. The linear phase \(\omega\tau\) should be calibrated experimentally, to determine the differential phase \(\theta(\omega)=\varphi(\omega)-\varphi(\omega-\Omega)\), by recording a reference interferogram without spectral shear additionally. The spectral phase \(\varphi(\omega)\) can be obtained by concatenating \(\theta(\omega)\)[8]. If \(\Omega\) is small sufficiently within that the spectral phase does not vary obviously, \(\varphi(\omega)\approx\int\!\theta(\omega)\mathrm{d}\omega/\Omega\). The algorithm of SPIDER is simple, robust, and direct compared to FROG, more easily supporting real-time display at a much higher frame rate. There are two important development directions for SPIDER recently. One is to improve the capability for retrieving pulses those with complex temporal/spectral structure, e.g., few-cycle pulse, white-light supercontinuum. The other is to improve the usability or practicability of SPIDER, e.g., compactness, stability, robust, and cost. Figure 4: Schematic of SPIDER. \(\chi^{(2)}\) is a nonlinear crystal. The fundamental (FH) pulse replicas are frequency-conversed in the nonlinear crystal with a shear \(\Omega\). In the original SPIDER, the pulse replicator based on MI brings unbalanced dispersions to the replicas then limits its application for ultra-broadband pulses. Zero-additional phase (ZAP) [58] SPIDER uses one FH test pulse and two chirped-pulse replicas, avoiding dispersive distortion by the unbalanced pulse replicator, and broaden the measuring spectral range of SPIDER to hundreds of nanometers. Promoting the spectral resolution is much more challenging for shearing spectral interferometry. One of the causes is that spectral interferometry encodes the differential spectral phase into the interval of interferential fringes, so SPIDER should operate at a high spectral sampling rate typically 5-10 times of the Nyquist limit [59].