# 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)
A tunable seed around 50 nm in the second modulator would allow to overcome the limitations of the wavelength separation of the harmonics and provide access to a continuous wavelength range and high bunching. For example, a final target wavelength of 4 nm with more than 13% bunching could be achieved either by a 47.4 nm or a 51.3 nm seed. The preferred setup with the second modulator enclosed in a cavity and thus being resonant to a shorter seed is shown in Figure 3. As a final remark, we note that one cavity could be employed for both modulators, which would be preferred in terms of cavity length requirements. However, in this case, the peak power of the radiation cannot be tuned independently at the two modulators which is an important aspect of the optimization of EEHG. To demonstrate the feasibility of the proposed EEHG configuration, a single-pass full simulation with the FEL code Genesis 1.3 [39] is carried out. The wavelengths of the first and the second seed laser are 300 nm and 50 nm, respectively. The electron beam parameters are the same as those used in the already presented analytical calculations above, and in addition, the normalized emittance is 0.6 mm mrad, the electron bunch length is 314 fs full width at half max (FWHM), and the current profile is Gaussian with a peak of 500 A. The duration of the Gaussian seed laser pulses is set to 150 fs and 50 fs FWHM for the first and second seed laser, respectively. The simulation is optimized for an output wavelength of 2.013 nm with longitudinal dispersions of \(R_{56,1}=2.649\) mm and \(R_{56,2}=17.50\) um. The radiator has a period length of \(\lambda_{u}=19\) mm and is tuned to the output wavelength. The bunching along the electron bunch upstream from the radiator, the evolution of the FEL peak power along the radiator, as well as the spectrum and power profile at the same position in the radiator are presented in Figure 4. The bunching amplitude is approximately 9.5% and thus slightly smaller than the 11.2% predicted by  
Figure 3: In an oscillator-based EEHG scheme, one or two optical cavities can be attached. In this figure, the first modulation occurs with a conventional external seed laser, while the second energy modulation is achieved by employing an optical cavity around the second modulator. The optical cavity is fed by a seed laser and maintains its properties in order to seed consecutive electron bunches. Figure 2: (a) Maximum bunching factor for different combinations of seed laser wavelengths. (b) Optimum setup of the chicanes to maximize the bunching factor. The working point resulting in a lower \(R_{56,1}\) is shown for each configuration. the simple analytical model (see Figure 2a), but still more than sufficient for an efficient amplification in the radiator. Employing an Oscillator-Based Seeding Scheme in an Accelerator in Continuous-Wave or Burst-Mode Operation  
A seeded oscillator-amplifier scheme is suitable for accelerators that can generate electron bunches at high repetition rates, as it requires a cavity length which matches the electron bunch repetition rate. The cavity roundtrip length should be \(L_{cap}=c/(m\cdot f_{rep})\), where \(f_{rep}\) refers to the electron bunch separation and \(m\) is an integer which represents the number of roundtrips of the radiation before it meets again an electron bunch. For instance, when the electron bunches arrive with a frequency of \(1\,\mathrm{MHz}\), the total roundtrip cavity length should be \(L_{cap}\approx 300\,\mathrm{m}\) for \(m=1\). Alternatively, the radiation pulse can perform more than one roundtrip in between two consecutive bunches. However, in this case the total resonator reflectivity decreases with the number of passes \(m\) as \(R^{m}\). A superconducting accelerator can run in continuous wave (CW) or burst-mode operation.