# Advanced Scheme to Generate MHz, Fully Coherent FEL Pulses at nm Wavelength
## 5 Comparison of Simulation Results  
Until now, we have only discussed about the process in the modulator and resonator. In this section, we compare simulation results at a final wavelength of 4.167 nm, reached with different schemes and this time we show the final FEL pulses generated at the amplifier. For the HGHG simulations, this wavelength is the 12 th harmonic of a 50 nm resonant modulator. We consider the following four cases:  
* A SASE setup, starting from shot noise and without changing any electron beam parameters. The FEL pulse is extracted at the same position as the seeding simulations.
* A single-pass standard HGHG setup, starting with an ideal Gaussian seed laser pulse instead.
* An HGHG seeded oscillator-amplifier starting with a low repetition rate seed laser. This scheme was discussed in detail in [22]. For the first electron bunch an external seed laser pulse is injected, and then the seed pulse is stored in the cavity.
* An HGHG seeded oscillator-amplifier system starting from shot noise. This was described in detail in Section 4. A reflectivity change from \(R_{build-up}=14\%\) to \(R_{equil}=10.6\%\) was used to transition from positive to zero net gain.  
In Table 2, we have summarized the main simulation results for the four different cases, and in Figure 11 we show the final spectra for the four different cases with the same final wavelength of 4.167 nm. In addition, for completeness, we have added the pulse properties of the output FEL at 2 nm with the EEHG simulations discussed in Section 2.1.2. The output FEL is shown in Figure 4. Note that the peak power is comparable for all HGHG seeded pulses as expected; however, as the resulting pulse duration differs, the bandwidth cannot be directly compared. It is important to emphasize that a single-spike spectrum was generated in all seeded schemes. The power spectral density in the multi-pass HGHG starting with a seed laser, and in the standard single-pass HGHG are almost identical, while the multi-pass HGHG starting from shot noise seems to have almost an order of magnitude higher spectral density as shown in Figure 11d. In this case, we have used a monochromator with an rms bandwidth of \(\Delta\lambda/\lambda=2.5\times 10^{-4}\) in the resonator, which stretches the radiation pulses and filters the radiation in the frequency domain. Because of this, the result in Figure 11d deviates compared to the other two HGHG cases.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Peak Power** & \(\Delta\lambda_{FWHM}/\lambda\) & **rms Pulse Duration** \\ \hline
**SASE** & 3 MW & \(2\times 10^{-3}\) & 75 fs \\
**Standard HGHG** & 1.2 GW & \(1.6\times 10^{-4}\) & 20 fs \\
**multi-pass HGHG (seed)** & 1.2 GW & \(2\times 10^{-4}\) & 27 fs \\
**multi-pass HGHG (shot-noise)** & 1.1 GW & \(5\times 10^{-5}\) & 60.6 fs \\
**single-pass EEHG** & 0.18 GW & \(1.6\times 10^{-4}\) & 11.35 fs \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation results for final FEL pulse at the same position along amplifier. For the multi-pass simulations, we examine the FEL pulse after 100 passes. For the SASE, we calculate based on the average over 50 simulations with different shot noise. For EEHG, we consider the simulation results of a 2 nm output FEL shown in Figure 4.