# Multistage Smoothing Compressor for Multistep Pulse Compressors
## 2 Beam Smoothing Owing to the Spatial Chirp  
### Beam Smoothing in the Near Field  
Beam smoothing by inducing spatial chirp is the key method to reduce the spatial intensity modulation of a laser beam, and then improve the input/output laser pulse energy in the compressor of a PW laser system. To date, there are two typical optical designs, prism pairs and grating pairs, which can induce spatial chirp to smooth the laser beam [22; 23]. In both setups, the first prism or grating induces angular dispersion to the laser beam, and the second parallel prism or grating is used to collimate the output laser beam. The induced spatial chirp width \(D\) is related to the laser spectral bandwidth and the distance between the prism pair or the grating pair. \(D\) can be expressed as \(D=L(\tan(\theta_{s})-\tan(\theta_{i}))\text{cos}\alpha\), where \(L\) is the perpendicular distance of the prism pair or the grating pair; \(\theta_{s}\) and \(\theta_{l}\) are the diffraction angles of the shortest wavelength \(\lambda_{s}\) and the longest wavelength \(\lambda_{l}\), respectively, of the input laser pulse; and \(\alpha\) is the incident angle of the grating or the apex angle of the prism. For a laser pulse with a narrow-band spectrum, the difference in \(\tan(\theta_{s})-\tan(\theta_{l})\) is small; therefore, a much longer distance \(L\) is required to achieve the same width \(D\) compared with that required for a broadband spectrum. In comparison to a prism pair, a grating pair can induce a relatively large spatial chirp width with a relatively short distance \(L\). The basic optical schemes of a prism pair and a grating pair for beam smoothing are shown in Figure 1a,b, respectively. The beam-smoothing effect in one direction is directly related to the induced spatial chirp width \(D\). For both the prism pair and the grating pair, the same beam-smoothing effect will be achieved if they induce the same \(D\). Owing to the relatively weak angular-dispersion ability of prism, a distance \(L\) of nearly 70 m is required to smooth the beam effectively [22]. Although a prism pair with large apex angle can shorten the distance, the prism thickness will be increased, which is more expensive. Two perpendicular prism pairs can also shorten the distance [24]. Transmitting gratings may efficiently reduce the distance in the future. In this study, an asymmetric four-grating compressor was used, which has two grating pairs with reflective gratings. This design is able to induce a suitable spatial chirp width with a relatively short distance difference between the two grating pairs. Sincean asymmetric four-grating compressor is able to smooth the output laser beam directly, the pre-compressor can induce a relatively small spatial chirp width to the input laser beam using a prism pair with a small apex angle, making it more practical and feasible. To clearly explain the beam smoothing capability in the near field based on the induced spatial chirp, laser spatial intensity modulation (LSIM, the ratio of the maximum and average laser spatial intensities) with a random high spatial frequency of 1.0 mm\({}^{-1}\) is intentionally induced into the input laser beam. Obvious beam smoothing is obtained for the output beam after either a prism pair or a grating pair, as shown in Figure 2. A pulse centered at 925 nm with a full spectral bandwidth of 200 nm and a 370 \(\times\) 370 mm\({}^{2}\) 10 th-order super-Gaussian beam is used as the simulation laser source. As shown in Figure 2a, the LSIM experiences a rapid decline from approximately 2.0 to 1.3, when the spatial chirp width increases from zero to approximately 10 mm, and a further slight decline to approximately 1.1 when the spatial chirp width is continually increased to approximately 60 mm. Figure 2b shows the relationship between the induced spatial chirp width and the perpendicular distance \(L\) of the prism pair with a 15\({}^{\circ}\) apex  
Figure 1: Optical scheme of beam smoothing based on (**a**) a prism pair, and (**b**) a grating pair.