# FEL Pulse Duration Evolution along Undulators at FLASH
## 2 SASE FEL Amplification Process
During the experiments discussed in this paper, the beam formation system  
Figure 1: (**a**) Simulated average radiation pulse energy and fluctuations in the radiation pulse energy along the undulator length normalized to the saturation length. The fluctuations are scaled with \(\sqrt{\tau_{el}}\) to compensate for the pulse duration dependence as described below in the text. (**b**) Simulated evolution of the radiation pulse duration along the undulator length. Thin curves show the result of the 1D model for three different electron pulse lengths \(M\)\(\sim\)\(\tau_{el}=4,8\) and 16. The black bold curve shows the 3D model for the pulse length of \(M\)\(\sim\)\(\tau_{el}=5\) without energy chirp, and the bold red curve shows the 3D model including energy chirp. was tuned such that the LSC produced a significant chirp of about 5 MeV peak-to-peak, resulting in an increase in the radiation spectrum bandwidth by about a factor of two with respect to the natural FEL bandwidth. The results of the simulation including the LSC chirped electron beam are shown in Figure 1 as a red bold line. While there is no difference between the chirped and monoenergetic (unchirped) cases in the linear regime, we see visible differences in the postsaturation regime. The pulse duration becomes shorter and the radiation pulse energy grows faster in the chirped case. The explanation of this phenomenon can be found in the positive energy chirp along the lasing fraction of the bunch (see Figure 2). It is well known that a linear energy chirp \(\gamma/t\) along the bunch is equivalent to a linear undulator tapering such that [1]  
\[\frac{1}{H_{\rm w}}\frac{dH_{w}}{dz}=-\frac{(1+K^{2})^{2}}{2K^{2}}\frac{1}{ \gamma^{3}}\frac{d\gamma}{cdt}\, \tag{5}\]  
where \(H_{\rm w}\) is the peak magnetic field and \(K\) is the rms value of the undulator parameter. We see that the positive energy chirp is equivalent to an undulator tapering with decreasing field strength along the undulator. It is well known that the application of the undulator tapering with the decreasing field allows one to preserve the synchronism between electrons and the electromagnetic wave, thus increasing the FEL efficiency [13]. With an appropriate optimization of the energy chirp, one can realize conditions which are equivalent to an optimum undulator tapering, when a significant fraction of particles is trapped in the effective ponderomotive potential. The trapped electrons interact stronger with the radiation, and the radiation power grows along the undulator length, while the radiation pulse duration remains nearly constant and the slippage effect is suppressed as well. Of course, an increase in the energy chirp results also in an increase in the radiation bandwidth, as we see from the experimental data. In the following sections, we compare experimental results with 3D simulations performed for a chirped electron beam. The energy chirp along the lasing fraction of the bunch was set to 150 keV/fs (Figure 2), leading to a spectral width of about 1% for the XUV pulse, in agreement with several spectral measurements performed at FLASH [5; 22; 23; 24]. The chirp in the electron bunch leads to a chirp of 30 meV/fs in the 13.5 nm XUV pulse, or, expressed as second-order dispersion, it yields a value of 25 fs\({}^{2}\), which is comparable to the measurements described in Ref. [25], with a second-order dispersion of 50 fs\({}^{2}\), where an exceptionally large bandwidth (i.e., chirp) was requested. Figure 2: Energy chirp along electron bunch induced by the longitudinal space charge field (blue curve) as it was used for the chirped 3D simulation. The grey dashed curve shows the longitudinal profile of the electron bunch.