# FEL Pulse Duration Evolution along Undulators at FLASH
## 2 SASE FEL Amplification Process  
The amplification process in SASE FELs develops from the shot noise in the electron beam, passes an exponential stage of amplification, and finally evolves in the nonlinear regime. Figure 1 shows the evolution of the radiation pulse energy and its fluctuations, as well as the evolution of the radiation pulse duration along the undulator. The results are obtained by the time-dependent FEL simulation code FAST using 1D and 3D FEL models [12]. To be specific, we consider the case of an electron beam with a longitudinalGaussian profile with an rms duration \(\tau_{el}\). Fluctuations in the radiation pulse energy reach a maximum value at the end of the exponential gain regime and subsequently decrease in the nonlinear regime, while the radiation pulse energy continues to grow. In the high-gain exponential regime, the number of modes \(M\) in the radiation pulse is defined as the inverse-squared deviation of the radiation pulse energy, \(M=1/\sigma_{E}^{2}\), where \(\sigma_{E}^{2}=\langle(E_{r}-\langle E_{r}\rangle)^{2}\rangle/\langle E_{r}\rangle^ {2}\)[2; 13]. The saturation point corresponds to the maximum brilliance of the radiation [14; 15]. At the same time, the fluctuations in radiation pulse energy decrease by a factor of three with respect to the maximum value. In the framework of the one-dimensional model, the saturation length and coherence time at saturation given by  
\[z_{sat}\simeq\frac{\lambda_{W}}{4\pi\rho}\Big{(}3+\frac{\ln N_{c}}{\sqrt{3}} \Big{)},\quad(\tau_{c})_{max}\simeq\frac{1}{\rho\omega}\sqrt{\frac{\tau\mathrm{ ln}N_{c}}{18}}, \tag{1}\]  
are expressed in terms of the FEL parameter \(\rho\)[16] and the number of cooperating electrons \(N_{c}=I/(e\rho\omega)\)[2; 13; 17]. Here, \(\omega\) is the radiation frequency, \(I\) is the beam current, \(-e\) is the charge of the electron, and \(\lambda_{W}\) is the undulator period. A practical estimate for the parameter \(\rho\) comes from the observation that in the parameter range of SASE FELs operating in the VUV and X-ray wavelength ranges, the number of field gain lengths to reach saturation is about 10 [14]. Thus, the parameter \(\rho\) and the coherence time \(\tau_{c}\) relate to the saturation length by:  
\[\rho\simeq\lambda_{W}/z_{sat},\quad\tau_{c}\simeq\lambda z_{sat}/(2\sqrt{\tau }c\lambda_{W}). \tag{2}\]  
For the number of modes \(M\gtrsim 2\), the rms electron pulse length \(\tau_{el}\) and the minimum radiation pulse length \(\tau_{ph}^{min}\) given in full-width half-maximum (FWHM) at the end of the exponential gain regime are given by [3; 18; 19]:  
\[\tau_{ph}^{min}(FWHM)\simeq\tau_{el}\simeq\frac{M\lambda}{5\rho}\simeq\frac{ M\lambda z_{sat}}{5c\lambda_{W}}. \tag{3}\]  
The minimum radiation pulse duration expressed in terms of coherence time (given in Equation (2)) is  
\[\tau_{ph}^{min}(FWHM)\simeq 0.7\times M\times\tau_{c}. \tag{4}\]  
The radiation pulse duration is mainly defined by the length of the lasing fraction of the electron bunch with some corrections related to the slippage effect. In the beginning of the amplification process, the radiation pulse shape just repeats the longitudinal shape of the electron bunch. In the exponential high-gain regime, the power amplification (and beam bunching) is stronger for higher currents; thus, the radiation pulse duration is reduced as the electron bunch travels along the undulator. When the amplification approaches saturation (full bunching) in the central part of the bunch, the tails of the electron bunch begin to contribute more to the radiation power. Beam bunching continues to grow there and the radiation pulse duration starts to grow as well. The effect of the lasing tails gives the same relative radiation pulse lengthening, as is illustrated in Figure 1. The pulse length at the saturation point is about 1.4 times higher than the minimum pulse for the linear regime given by Equation (2), and it is increased further up to about a factor of two in the deep nonlinear regime.