## 3 FEL Performances  
### Longitudinal Coherence and Pulse Duration  
FEL-1 and FEL-2 may be operated even without the presence of an external seed, i.e., in SASE mode. The emission may be indeed enhanced by tuning the dispersive section of FEL-1 [20] or the multiple dispersion sections of FEL-2 [21], to increase the electron bunching at entrance of the final amplifier. In this configuration, the light pulse resembles the longitudinal structure of the beam current, with a correlation length much shorter than the electron bunch length. The consequence is a longer pulse, of several hundred of fs, with broad spectrum and spiky spectral features, characteristic  
Figure 2: Schematic layout of FERMI (acronym of Free Electron laser Radiation for Multidisciplinary Investigations) FEL-2, implementing the HGHG double stage cascade in the fresh-bunch high-gain harmonic generation configuration. The first stage (mod1-disp1-rad1) is analogous to the high-gain harmonic generation scheme of FEL-1 shown in Figure 1. The second stage (mod2-disp2-rad2) is based on the same concept, but the seed is the radiation produced in the first stage. The two stages are separated by a delay line (delay) which lengthens the electron path with respect to the radiation path allowing to shift the seed over a “fresh” portion of the electron beam. of SASE FELs. This mode of operation can be used in those experiments requiring longer pulses and broadband emission, for alignments, or in the eventuality the seed laser system is not available. In Figure 3 we show the seeded FEL spectrum measured at harmonic order 8 compared to an optimized Optical-Klystron-SASE spectrum (OK-SASE spectrum). In in the example shown in the figure, in seeded mode the relative linewidth drops by about an order of magnitude. Typical relative linewidths of 2 to 5 \(\times\) 10\({}^{-4}\) (rms) are routinely achieved on both FEL-1 and FEL-2. Less evident from the images in Figure 3, the presence of the seed also affects the pulse duration. In seeded mode the pulse duration depends primarily on the duration of the seed, scaled according to the harmonic order \(n\) by a coefficient \(7/6n^{1/3}\)[22]. Typical pulse duration with a seed duration of 120-150 fs are 50-70 fs (fwhm) on FEL-1. FEL-2 seeded with a short seed laser (70 fs) is expected to deliver FEL pulses of about 20 fs, depending on the laser setup and harmonic conversion order. We may therefore compare the longitudinal coherence of the source in the two modes of operation, in terms of distance from the Fourier transform limit for a Gaussian pulse (FTL),  
\[\Delta t\Delta\lambda=0.44\lambda^{2}/c\]  
where \(\Delta t\) and \(\Delta\lambda\) correspond to the full width half maximum of the temporal and spectral distributions of the optical pulse. If we consider the FERMI SASE conditions of operation in Figure 3b where the spectral width is about 0.3 nm, and where the optical pulse duration should be comparable to the electron bunch length, of the order of 0.5-0.7 ps, a rough estimate would indicate the pulse about 120-200 \(\times\) FTL. This value, with the input seed and with the proper tuning of the FEL parameters, drops to \(\approx\)1.2-1.5 FTL [23]. Specific tuning of the e-beam and the seed laser parameters may be set to bring the FEL even closer to the ideal Fourier limit [24]. The example in Figure 3 corresponds to harmonic order 8 of FEL-1. A narrow, single mode spectral line, such as the one shown in Figure 3 corresponding to harmonic 8 of FEL-1 seeded at 260 nm, is available in the entire spectral window of photon energies emitted by the FERMI FELs (20-60 eV on FEL-1 and 60-330 eV on FEL-2). As a second example in Figure 4, the spectrum of the second stage of FEL-2 is shown. The high end of the nominal photon energy range is reached upshifting the seed frequency by harmonic order 13 in the first stage and harmonic 5 in the second stage, for a total frequency upshift of 65.