# Directly Measuring the Pulse Front Distortion of High-Peak-Power Femtosecond Lasers
## 2 Measurement Method  
In the previous measurement scheme [24], the pulse under investigation was equally divided into two beams; then, injected into the two arms of a Michelson interferometer; and finally, spatially overlapped after a beam splitter. After adjusting the time delay between two arms, the time delay distribution between the two pulse fronts of the space overlapped pulse is measured by scanning with a second-order autocorrelator. It is notable that this method is based on the quadratic curve assumption of PFD, and thus the relative delay between the two pulse fronts of overlapped pulse is a linear function of radial position \(r\). Through fitting the measured results, we can achieve a linear coefficient \(k\). Then, according to the linear coefficient (\(k\)) and the radius (\(r_{0}\)) of the investigated pulse, the quadratic coefficient (\(a=k/2r_{0}\)) and corresponding curve (\(T=ar^{2}\)) of PFD can be obtained. The measured PFD of a 80 mm-diameter pulse is \(-54.4\) fs, which should be a little different from the actual value through the consideration of beam divergence. Moreover, the PFD at any position of the investigated pulse cannot be directly measured by this method. The setup of the improved self-reference cross correlator is described in Figure 1. Compared to that in [24], an extra iris diaphragm is installed at the entrance and \(M4\) is mounted on a high-precision linear translation stage. The iris diaphragm is concentric with the pulse under investigation, and is utilized to sample pulses with different diameters. The radius of the iris diaphragm corresponds to the radial position of the investigated pulse. Therefore, by changing the aperture of the iris diaphragm and measuring the maximal PFD of each sampled pulse, we can achieve the PDF at any position of the investigated pulse. The maximal PFD of each sampled pulse is the time delay between the pulse fronts at the edge and the center of pulse. However, along with the aperture of the iris diaphragm changing, the two pulse beams after \(BS\) will spatially separate. By linearly translating \(M4\) until the two pulse beams are overlapped at the near field hole and far field CCD, in which case the spatial overlap of the two pulse beams is realized, the time delay between the pulse fronts can be measured. In order to realize a precise alignment, a high-resolution CCD (3.7 \(\upmu\)m, Beamgage-SP928) and a long-focal-length positive lens (f = 4 m) are used in the far field monitoring system. The misalignment of these two pulse beams can be controlled below 5 \(\upmu\)rad, which induces a time delay across the 40 mm overlapped laser beam of about 0.6 fs. The influence is so little it can be ignored. As Figure 1 shows, the pulse under investigation is size limited by the iris diaphragm firstly, and then equally divided into two pulse beams by \(M1\). These two equal-sized pulse beams are injected into the two arms of the [PERSON] interferometer respectively, and finally spatially overlapped after the \(BS\) by linearly translating \(M4\). In addition, the two pulse fronts of the overlapped pulse are completely separated by adjusting the delay line. Based on the above description, the pulse front of the overlapped pulse can be described as in Figure 2. As we know, the lens's induced PFD is symmetrical, even though the pulse has divergence. Therefore, the delay times between the pulse front at the center (O) and the two edges (A/B) are the same; i.e., \(\Delta T_{1}\) = \(\Delta T_{2}\). Then the maximal PFD of each sampled pulse can be illustrated as Equation (1). \[\Delta\mathrm{T}=\Delta T_{1}=\Delta T_{2}=\frac{1}{2}(T_{2}-T_{1}), \tag{1}\]  
It means that the maximal PFD of each sampled pulse can be directly obtained just by measuring the time delay between the pulse fronts at two edges of the overlapped pulse, i.e., \(T_{1}\) and \(T_{2}\). In order to measure the time delay, the two edge parts (OA and BO) of the overlapped pulse are delivered into a commercial second-order autocorrelator, with a beam diameter of 2 mm.