# Characteristics and Applications of Spatiotemporally Focused Femtosecond Laser Pulses
## 2 The Basic Principles of the SSTF Technique and Characteristics of SSTF Pulses  
Figure 1 schematically shows a typical SSTF system [27]. In the following discussions, in order to avoid any confusion, we refer to SSTF as this scheme. As it can be seen in Figure 1, the pre-chirped femtosecond laser pulses propagating along the \(z\) direction are firstly dispersed laterally along the \(x\) direction by a pair of gratings (G1, G2). The spatially-separated frequencies are then recombined by a focusing lens (L). Only at the geometric focus of the focusing lens, the different spectral components spatially overlap, leading to the shortest pulse duration as well as the highest peak intensity. However, when moving away from the geometrical focal spot, the peak intensity will decrease rapidly due to the broadening of the pulse duration, as shown in Figure 2[18; 21]. In such a manner, the peak intensity is strongly localized near the geometrical focus, and the focal depth of the SSTF pulses are greatly reduced in comparison with conventional focused pulses (see Figure 2b-d). The characteristics of SSTF pulses mentioned above can be simulated accurately based on Fresnel diffraction integral theory, which can be found in the Appendix. Figure 1: Schematic illustration of the typical SSTF (simultaneous spatial and temporal focusing) system. G1, G2: grating pair; L: lens. Reproduced with permission from [27], Copyright The Optical Society, 2015. Interestingly, it was discovered that the SSTF beam naturally creates a focal spot with a tilted pulse front [28; 29], which can induce a nonreciprocal writing effect in isotropic transparent materials [23]. As it can be seen in Figure 3, it is assumed that the input laser pulses carries the spatial chirp along the \(x\) axis before the focusing lens, and the laser pulses will arrive at the focal plane at different times, which is known as the pulse front tilt (PFT) effect. With experimental and theoretical investigations, it is verified that the PFT at the focus of SSTF pulses has several prominent features. The origin of the PFT effect is proved to be mainly related to the lateral separation of the different spectral components of the pulses introduced by a pair of gratings, which cannot be eliminated by introducing the initial PFT in the opposite direction [28]. Additionally, the degree of PFT, which is several orders of magnitude higher than that of the conventional femtosecond laser pulses, can be controlled by changing the spatial chirp before the focusing lens. In order to quantitatively investigate the degree of PFT of a SSTF pulse at the focus, an interferometric method was proposed by [PERSON] et al. in 2015. The basic principle of the measuring method is illustrated in Figure 4[30]. A conventional focused pulse is introduced as a reference pulse, and interference occurs in the focal plane between the reference pulse and SSTF pulse. By changing the time delay between the reference pulse and the SSTF pulse, the interference pattern of two pulses forms in the focal plane will shift linearly. Thus, by plotting the shift of the interference fringe as a function of the time delay between the two pulses, the tilting angle of the SSTF pulse can be obtained [30]. Figure 3: The simulation result of PFT (pulse front tilt) in the focal plane of an SSTF pulse. \(X\) refers to the direction of the spatial chirp. Reproduced with permission from [30], Copyright The Optical Society, 2014. Figure 2: **(a)** Pulse duration in the vicinity of the geometric focus for SSTF pulses. In the simulations, the spectral bandwidth is 30 nm. The focal length of the focusing lens is 100 mm. Reproduced with permission from [21], Copyright American Physical Society, 2011. Intensity distributions at the focus produced by an objective lens (**b**) without and (**c,d**) with the SSTF technique in the \(X\)Z and \(Y\)Z planes, respectively. The NA (numerical aperture) of the objective lens is 0.46. Reproduced with permission from [18], Copyright The Optical Society, 2010.