# FEL Pulse Duration Evolution along Undulators at FLASH
## 2 SASE FEL Amplification Process
The second effect leading to pulse lengthening is the slippage of the radiation by one radiation wavelength per one undulator period. Evidently, the slippage effect is more pronounced for shorter pulses and longer wavelengths. The case of \"cold\" (zero energy spread) and monoenergetic electron bunches has been analyzed in earlier papers in the framework of a 1D FEL model [2; 3; 19; 20]. Using the normalized electron pulse duration \(\bar{\tau}_{el}=\rho\omega\tau_{el}\) allows us to describe the simulation results in a universal way for \(\tau_{el}\gtrsim 2\). The simulation results normalized this way show almost identical behavior for different \(\bar{\tau}_{el}\), as illustrated in Figure 1. Note that for \(\bar{\tau}_{el}\gtrsim 2\), \(\bar{\tau}\) is essentially equal to the number of radiation modes (\(\bar{\tau}_{el}\)\(\sim\)\(M\))[3; 19]. The one-dimensional model allows us to describe the physics of pulse length effects and temporal properties of the radiation in an elegant way involving a minimum number of parameters. However, to describe a real experiment more quantitatively, a three-dimensional model which describes diffraction effects and the effects of betatron motion is needed. In addition, the details of the electron bunch structure (e.g., energy chirp) may be very important for the interpretation of the experimental results. Thus, in this paper we extend the analysis of short pulse effects using the results of three-dimensional, time-dependent simulations carried out with the FAST code. The bold curves in Figure 1 refer to the 3D simulations applying typical FLASH2 experimental conditions: electron energy of 1 GeV, rms energy spread of 0.2 MeV, and rms normalized emittance of 1.4 mm mrad. The lasing fraction of the electron bunch is approximated with a Gaussian distribution of 16 fs rms pulse duration and 1.5 kA peak current, resulting in a radiation wavelength of 13.5 nm. These parameters correspond to the value of \(M\)\(\sim\)\(\tau_{el}=5.1\). The results of the simulations are presented with the same normalization procedure as for the 1D case. The first set of simulations refers to the case of a monoenergetic electron beam (bold black curves). We see that the 1D and 3D results for the monoenergetic case are rather similar, starting from the end of the high-gain exponential regime. The difference in the linear regime reflects the spatial mode competition effect which is absent in the 1D model. However, at the end of the high-gain linear regime, the fundamental TEM\({}_{00}\) FEL mode is significantly larger as compared to other higher spatial modes, and we see a good agreement of the results of the 1D and 3D model. In the real accelerator, the electron beam is not monoenergetic. The long-pulse, low-current electron beam produced in the electron gun at FLASH is compressed in several stages by a large factor (up to about one hundred), and the peak current is increased correspondingly [21]. To achieve such a large compression, an energy chirp along the electron bunch is applied in the accelerating sections. This energy chirp leads to a bunch compression while the electron bunch moves through dedicated magnetic bunch compressors. An RF-induced energy chirp can be minimized such that it only slightly changes FEL properties with respect to a monoenergetic beam. Still, different kinds of wakefields and collective effects in the electron beam generate an energy chirp along the electron bunch. An example of such an energy chirp induced by the longitudinal space charge field (LSC) is shown in Figure 2[21]. An important feature of an LSC wake is that the energy of electrons in the lasing fraction of the electron bunch is increased from the tail to the head of the electron bunch. Such a feature leads to a visible increase in the FEL efficiency in the nonlinear regime, which on the other hand leads to an increase in the FEL radiation bandwidth.