# 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)  
HGHG schemes are characterized by a limited up-frequency conversion efficiency due to the fact that the \(n\)th harmonic requires the energy modulation to be \(n\) times larger than the slice energy spread to maximize the bunching. This is typically limiting the conversion to \(n=15\) and critically depends on the energy spread [37]. The EEHG scheme [13; 14; 15] was proposed to overcome this limitation, achieve higher harmonics and, thus, shorter wavelengths. In this scheme, there are two seed lasers with two modulators, two dispersive sections, and one radiator. The first modulator and seed laser are used to induce an energy modulation, and then the first dispersive section, which has a large longitudinal dispersion, shreds the longitudinal phase space of the electron beam creating thin energy bands. Each of these bands has a lower energy spread than the initial one, and this way a lower energy modulation is required in the second modulator compared to HGHG. The second dispersive section is weaker and compresses the energy bands. Similarly to what happens in HGHG, it converts the energy modulation from the second modulator into a density modulation, which in this case can have higher harmonic content. In a regular single-pass EEHG, two modulators and two seed lasers are needed. In order to convert the classic scheme to a high repetition rate cavity-FEL, one possibility is to include two cavities, one for each modulator. In the case of two cavities, the wavelength can be chosen independently and the high repetition rate is secured. Another solution is to feed one modulator with an external seed laser and place the other modulator in a cavity. In this case, the repetition rate of the external seed laser source determines the overall repetition rate. This seed laser should have a longer wavelength which is at present already available at high repetition rate. Then, the other modulator which is enclosed in the optical cavity is resonant to a shorter wavelength. It is important to investigate if it is more advantageous to have the shortest wavelength at the first or the second modulator. We study the specific case of a combination of two seed laser wavelengths of \(300\,\mathrm{nm}\) and \(50\,\mathrm{nm}\) by using an electron beam with a nominal energy of \(1.35\,\mathrm{GeV}\), energy spread of \(120\,\mathrm{keV}\), and energy modulation amplitudes of \(A_{1}=3\) and \(A_{2}=5\) times the energy spread in the first and second modulator, respectively. These parameters fit the choices of the FLASH2020+ project [38]. The resulting maximum bunching factor \(b\)[13] for final wavelengths between \(2\,\mathrm{nm}\) and \(6\,\mathrm{nm}\) is shown in Figure 2a. Using a seed with a wavelength of \(50\,\mathrm{nm}\) in the first modulator and \(300\,\mathrm{nm}\) in the second modulator is not beneficial in terms of bunching compared to the classic scheme with two seed lasers with a wavelength of \(300\,\mathrm{nm}\), whereas much higher bunching can be achieved by utilizing the shorter seed in the second modulator. Both \(50\,\mathrm{nm}\) configurations drastically reduce the required longitudinal dispersion of the first chicane, as can be seen in Figure 2b. As the second chicane converts the energy modulation from the second modulator, a seed wavelength of \(50\,\mathrm{nm}\) in this modulator results in an approximately six times smaller optimum dispersive strength than the one needed for a \(300\,\mathrm{nm}\) seed. Figure 1: In an oscillator-based HGHG scheme, an optical cavity is added and encloses the modulator. The optical cavity acts as a feedback system which maintains the peak power of the stored radiation field and, under perfect synchronism, this field is used to seed consecutive electron bunches arriving from the linac upstream the cavity. Note that in reality, the optical cavity design will be more complex than this simplified sketch.