# Three-Dimensional, Time-Dependent Analysis of High- and Low-\(Q\) Free-Electron Laser Oscillators
## 2 Numerical Formulation
### Performance of the MINERVA FEL Code
#### 2.3.2 The LCLS SASE FEL  
To illustrate the performance of MINERVA at much shorter wavelengths, we also briefly compare with experimental data from LCLS [198], which is a SASE FEL user facility that became operational in 2009, operating at a 1.5 A wavelength.  
To operate at 1.5 A, LCLS employs a 13.64 GeV/250 pC electron beam with a flat-top temporal pulse shape of 83 fs duration. The normalized emittance (\(x\) and \(y\)) is 0.4 mm-mrad and the rms energy spread is 0.01 percent. The undulator line consisted of 33 segments with a period of 3.0 cm and a length of 113 periods. In the simulation, each segment is modeled using Equation (17), with one period each in entry and exit tapers. A mild down-taper in field amplitude of \(-0.0016\) kG/segment starting with the first segment (with an amplitude of 12.4947 kG and \(K_{\text{rms}}=2.4748\)) and continuing from segment to segment was used. This is the so-called gain taper. The electron beam was matched into a FODO lattice consisting of 32 quadrupoles, each with a field gradient of 4.054 kG/cm and a length of 7.4 cm. Each quadrupole was placed a distance of 3.96 cm downstream from the end of the preceding undulator segment, and the gap lengths between the undulators followed a repetitive sequence of short (0.48 m)-short (0.48 m)-long (0.908 m).  
The LCLS produces pulses of about 1.89 mJ at the end of the undulator line [198], and saturation is found after about 65-75 m along the undulator line. A comparison between the measured pulse energies (red circles), as obtained by giving the electrons  
Figure 3: Simulated (blue line) and measured (red data points) relative linewidth as a function of the propagation distance \(z\) for the SPARC experiment. The error bar indicates the error in the measurement. The simulation result is, on average, over 20 noise realizations.  
Figure 2: Simulated (blue line) and measured (red crosses) pulse energy as a function of the propagation distance \(z\) for the SPARC experiment. The different points correspond to individual measurements and the error bar indicates the error in each of the measurements. The simulation result is an average over 20 noise realizations.  
a kick to disrupt the FEL process, and the simulation (blue) is shown in Figure 4. The experimental data are courtesy of [PERSON]. [PERSON] and [PERSON] at SLAC, and the simulation results represent an average over an ensemble of 25 runs performed with different noise seeds. As shown in Figure 4, the simulations are in good agreement with the measurements in the exponential growth region with close agreement for the gain length. The simulation exhibits saturation at the same distance as the experiment in the range of 65-75 m at a pulse energy of 1.5 mJ. After saturation, in view of the gain taper, the pulse energy grows more slowly to about 2.02 mJ at the end of the undulator line, which is approximately 8 percent higher than the observed pulse energy.  
The agreement between simulation and experiment for the pulse energy is poorer during the early stages of the interaction. This may be due to a variety of reasons. On the experimental side, as the pulse energy grows by 5 to 6 orders of magnitude from the initial shot noise to saturation, it is difficult to calibrate the detectors for the low pulse energies at the early stages of the interaction. Moreover, while kicking the electrons provides a fast measurement method, it is accompanied by a larger background signal and the possibility of restarting the FEL process downstream the undulator line when the kick is performed at the beginning of the undulator line [198]. On the simulation side, there may be some inaccuracies in the shot noise algorithm or the finite number of optical modes used that underestimates the initial noise level.