## 5 Spectral Interferometry
To improve the SNR of the SPIDER, we develop a complete-TSPS SPIDER (CTP-SPIDER) [68], where both the test spectrum and reference spectrum can introduce TSPS, as shown in Figure 6. Here, complete TSPS acts as a balanced detection that can not only remove the effect of the DC term of the interferogram but also reduce the measurement noises. Available shear can be down to 1.5% of the spectral width in this device. To improve the compactness and the robustness of the SPIDER, people propose some simplifications for SPIDER. The MI pulse replicator has two orthogonal light paths, which is take up more space. Moreover, it is very sensitive to the environment, easily causing jitters or shift between two pulse replicas, which is required to be controlled at a sub-fs level (or even 10 as for few-cycle pulses) [63]. A glass plate (or etalon) [66; 69] causing multiple reflections is a substitute for MI. However, the output is a fade-out pulse train where the fade-in additional dispersions introducing to the sub-pulses. All-transmission pulse replicator [70; 71] based on birefringence is a better substitute, where the optical axis of the birefringence crystal is set at 45\({}^{\circ}\) relative to the polarization of the input light. The o-ray and e-ray pulses separate with a delay for their different group-velocities. In this replicator, the dispersion difference between the o-ray and e-ray inducing an additional dispersion, which is much lower than that by etalon or beamsplitter of MI, thus can be applied to characterize the few-cycle pulses [71]. Reference [71] also shows that such common-path, all-transmission pulse replicator can improve the SNR by a factor of 3.7. Table 2 shows a comparison of some SPIDER geometries on the operating mode, data-dimension, spectral resolution, shear, phase error, et al. An approach based on the asymmetric phase-matching function [8] to generate spectral shear without a pulse stretcher was proposed [70; 72; 73]. The pulse stretcher can be removed in this configuration. The highly asymmetric phase-matching function is due to group-velocity matching between FH o-ray with SF and group-velocity mismatching between FH e-ray with SF for a type-II \(\chi^{(2)}\)-medium. In such a medium, the acceptance bandwidth can be designed very large at the o-axis whereas very narrow at the e-axis. Thus, the e-polarized FH acts as quasi-monochromatic light to carry the o-polarized FH to high frequency without altering the spectral phase. The phase-matching  
Figure 6: Schematic of a compact SPIDER with a birefringent pulse replicator and a compact stretcher consisting of two oppositely placed rectangular prisms [68]. function is angularly dependent, two beams at slightly different propagation angles will therefore generate spectrally sheared upconverted replicas. In 2010, a new method called Self-Referenced Spectral Interferometry (SRSI) [13] for femtosecond pulses characterization was introduced. Different from the shearing interferometry, SRSI is a linear method like traditional interferometry with no need for neither shear nor SF generation. In SRSI, the reference pulse is generated from the unknown test pulse based on Crossed-Polarized Wave generation (XPW), and the reference field can be written as \(E_{\text{ref}}(t)\)-\(|E_{\text{test}}(t)|^{2}E_{\text{test}}(t)\), where \(E_{\text{test}}(t)\) is the test field. Here, the XPW acts as a time filter [75], and flatten the spectral phase. The phase extracted directly from the interferogram can get a rough estimation of the phase. To obtain an accurate result, several iterations of algorithms are needed. The use of third-order nonlinear optical processes, need not consider the phase-matching compared to SPIDER using \(\chi^{(2)}\)-nonlinear, however, has a higher energy requirement. Except XPW [13, 31], other \(\chi^{(3)}\)-nonlinear generation can also be considered in SRSI, such as SD [76, 77] or TG [78, 79].