# Investigation of Spatial Chirp Induced by Misalignments in a Parallel Grating Pair Pulse Stretcher
## 1 Introduction  
A linear chirped pulse, of which the spectrum can be linearly mapped into the time domain, is often adopted as the probe pulse in the ultrafast dynamic measurements, such as the shock wave experiments [1; 2], the dynamic metrology of plasmas or ionization [3; 4], the single shot terahertz detection [5; 6], the ultrafast transient absorption experiments [7; 8], and the dynamic ellipsometry metrology of materials under extreme conditions [9]. In the above experiments, a pulse stretcher is needed to generate a linearly chirped pulse with a certain pulse width, in which chirp characteristics determine the practical measurement accuracy. Therefore, the accurate adjustment of the stretch system is of great importance.  
In a chirped pulse stretcher system, gratings, prisms, or optical fibers are often employed as the chromatic dispersion elements [10; 11; 12; 13]. Compared to the grating-based stretcher, a prism pair has smaller dispersion, and it is hard to obtain a long-duration chirped pulse. A long fiber possesses a high nonlinear dispersion and is limited only for relatively low pulse energy. Hence, the parallel grating pair pulse stretcher is more suitable to create a linear chirped pulse, in which pulse width can be easily adjusted by only changing the distance between the gratings. Since the misalignments of the gratings and mirrors in the stretcher (or compressor) can not only cause dispersion in the time domain (temporal chirp) [14; 15] but also cause dispersion in the space domain (spatial chirp) [16; 17; 18; 19; 20], accurate adjustment of the stretcher is necessary to achieve an ideal linearly chirped pulse. Spatial chirp induced by the specific misalignment of the parallel grating pair has been theoretically analyzed, and several adjustment methods have been proposed [21; 22; 23]. However, it is difficult to evaluate the overall effects when two or more misalignments exist simultaneously, especially when taking the mirror adjustment errors into consideration. By utilizing extremely accurately manufactured roof mirrors or corner cubes,single-grating based stretchers or compressors have been proposed, which are more compact and show significant simplification in their adjustment [24, 25]. In practice, limited by the element size and the great difficulty in manufacturing, the roof mirrors and corner cubes usually consist of two or three separate mirrors, leading the single grating structure to still possess the similar adjustment errors with the dual-grating structure. Therefore, it is highly desirable to investigate the criteria to describe the spatial chirp when several misalignments exist and summarize a feasible procedure for the misalignment calibration.  
In this paper, a single grating pulse stretcher based on the dispersion theory of parallel grating pair has been built, which is designed for the shock compression experiment. By constructing a virtual stretch system in the Zemax software, the spatial chirp of the stretched pulse induced by different possible misalignments has been analyzed. Since the simulation results show little coupling effects among different misalignments, the spatial chirp induced by specific misalignment can be calibrated individually, and the overall effect is possible to be evaluated using superposition. According to the sensitivity analysis of the spatial chirp to different misalignments, we summarized the adjustment procedure. By applying the proposed procedure on the adjustment of our home-made pulse stretcher, the spatial chirp has been reduced to an ignorable level, demonstrating the good adjustment of our stretcher. The temporal chirp of the stretched pulse is also measured with the spectral interferometer [26]. Results show the output pulses have good linearity between wavelength and time, and its pulse width is well consistent with the designed value.