# Advanced Scheme to Generate MHz, Fully Coherent FEL Pulses at nm Wavelength
## 2 Overview of Methods
### Employing an Oscillator in Standard Seeding Schemes
#### 2.1.2 Echo-Enabled Harmonic Generation (EEHG)
At FLASH [40, 41], which operates in burst-mode, the bunch trains arrive with a repetition rate of \(10\,\mathrm{Hz}\) with a flattop of \(800\,\mathrm{\SIUnitSymbolMicro s}\) and a bunch spacing of \(1\,\mathrm{\SIUnitSymbolMicro s}\) (\(1\,\mathrm{MHz}\) repetition rate). With a pulsed operation at \(10\,\mathrm{Hz}\) as well, the flattop of the European XFEL is \(600\,\mathrm{\SIUnitSymbolMicro s}\) with a \(0.22\,\mathrm{\SIUnitSymbolMicro s}\) bunch separation (\(4.5\,\mathrm{MHz}\)) [42]. The exact number of bunches available depends on the operation mode and the sharing of those bunches among different  
Figure 4: (**a**) Bunching along the electron bunch. (**b**) FEL peak power along the amplifier modules (gray). (**c**) Spectrum after the 5 modules of the amplifier. (**d**) Power profile after 5 modules of the amplifier. undulator beamlines. In the case of burst-mode operation, there is a specific number of bunches available to build-up the peak power and stability needed to deliver seeded FEL pulses. This is not an issue when the process starts with a low repetition rate seed laser source because the steady-state regime is reached within a few passes ([PERSON] et al., 2018) as shown in Figure 5a, but it is critical when starting from shot noise, as we show in Figure 5b. The build-up regime is marked with a green background color. During this process, there must be positive net gain, and the peak power in each pass increases. The steady-state regime is marked with blue color in the same figure, and refers to the passes in the oscillator where the net gain is zero and the peak power per pass is constant. Comparing Figure 5a,b, there are more power fluctuations in the case where we start with a seed laser. This might be due to the fact that in this case we do not use a monochromator. In burst-mode operation, the more bunches are used during the build-up process, the less bunches will be part of the steady-state regime when seeded radiation is generated. The steady-state can be maintained for a maximum number of passes defined by the difference between the available bunches in one bunch train and the number of bunches used during the build-up process. Taking as an example FLASH and the build-up regime shown in Figure 5b, we would need 18 bunches to take part in the build-up of the power, and the remaining 782 bunches would be part of the steady-state regime where the seeded radiation is generated. A machine operated in CW mode offers a continuous number of bunches with a constant separation between them. For instance, SHINE in Shanghai will be operated in a CW mode and is expected to provide bunches with a continuous 1 MHz repetition rate ([PERSON] et al., 2017). The same repetition rate is planned for LCLS-II ([PERSON] et al., 2017) as well. In this case, the build-up time needed becomes less important. It is possible to increase the number of passes in the build-up regime and ensure a smooth transition to the steady state. However, it becomes more important to verify how long the steady-state regime can be maintained before the process needs to be initiated again. Figure 5: (**a**) Example of the peak power per pass in an oscillator starting with a low repetition rate seed laser. From the first pass already the net gain should be zero. In practice, it takes a few passes for the system to self-stabilize. (**b**) Example of the peak power per pass in an oscillator starting from shot-noise. For 19 passes the build-up regime where the net gain is positive is highlighted with a green color. At pass 19, the desired peak power level is reached and the steady state regime is entered, marked with a blue color. From this pass and onward, the net gain is reduced to zero and the peak power level is maintained in each pass.