# Advanced Scheme to Generate MHz, Fully Coherent FEL Pulses at nm Wavelength
## 3 Resonator Considerations
### Cavity Design Considerations  
The numbers quoted so far are needed for the system to work, and should be complementary with a discussion on the technical feasibility of the resonator. The important questions here are if the downstream mirror, which will have the maximum power density, will be able to withstand it, and if mirrors with the required properties actually exist. We consider two operation regimes for the resonator: one at a wavelength between \(200\,\)nm and \(300\,\)nm, and one between \(50\,\)nm and \(100\,\)nm. Regarding the reflectivity requirements, we expect that for wavelengths around \(300\,\)nm, the mirror choice will not pose an issue as there are options to choose from. Optics in this wavelength regime are used for current laser systems, such as dielectric  
Figure 6: The color bar indicates the energy modulation achieved for combinations of seed laser peak powers (\(P_{in}\)) and modulator lengths (\(L_{mod}\)) and is calculated with Equation (3). The horizontal axis shows the normalized modulator length to the gain length (\(L_{mod}/L_{g}\)). The vertical dashed lines show the reflectivity \(R\) required for equilibrium between amplification and losses for different normalized modulator lengths, and is calculated with Equation (2). mirrors, with reflectivity and damage threshold that guarantee sustainable operation and have been studied for other storage ring FELs in the past as well [48]. The main challenge is faced for the working point in the XUV range between 100 nm and 50 nm, where no commonly used options are available. Here, we consider the upper limit in gain, where the roundtrip loss should not exceed a factor 1000 to avoid electron beam heating. Under normal incidence, this means that each mirror should reflect at least \(1/\sqrt{1000}\approx 1/33\) or 3%. In case of a ring resonator with mirrors at 45 degree incidence angle, each should reflect more than \(1/\sqrt{\sqrt{1000}}\approx 1/5.6\) or 18%. For example, we consider Molybdenum mirrors. At normal incidence, the reflectivity at 40 nm is \(\sim\)6%, at 45 degree around 40% [49]. Both values exceed the requirements. Note that a gain of 1000 is an upper limit that would require a relatively long modulator. However, it is preferred to operate at a lower gain if the reflectivity of mirrors allows it. Here, we consider simple estimations in order to calculate the power density for a Gaussian beam. Assuming a Gaussian beam with a waist at the end of the undulator, the size of the spot at the mirror is [47]:  
\[w^{2}(L)=w_{0}^{2}\Bigg{(}1+\left(\frac{L}{\ell}\right)^{2}\Bigg{)}\,, \tag{4}\]  
where \(L\) is the distance from the undulator to the mirror, \(w_{0}\) is the spotsize at the waist and \(\ell\) is the Rayleigh length. With the distance to the mirror much larger than the Rayleigh length and remembering that for a Gaussian beam \(\pi w_{0}^{2}=\lambda\ell\) with \(\lambda\) the radiation wavelength, the dependence of the beam radius on the distance becomes nearly linear and we can rewrite Equation (4) as  
\[w^{2}(L)\approx\left(\frac{L\lambda}{\pi w_{0}}\right)^{2}\approx\left(\frac{ L\lambda}{\pi\sigma_{b}}\right)^{2}, \tag{5}\]  
where we have approximated the spotsize of the radiation with the electron beam size \(\sigma_{b}\). Since the mirror has an angle with respect to the radiation in one plane only, the area of the radiation on the mirror for a transversely symmetric beam can be approximated as:  
\[S\approx\left(\frac{L\lambda}{\pi\sigma_{b}}\right)^{2}\frac{1}{\sin\alpha}\,, \tag{6}\]  
with \(\alpha\) the glancing angle. Assuming that the fraction of the pulse energy that is not reflected by the mirror is in fact absorbed, the power density \(P_{d}\) absorbed is  
\[P_{d}=\frac{E_{p}}{S}\cdot(1-R)=E_{p}(1-R)\sin\alpha\Big{(}\frac{\pi\sigma_{b} }{L\lambda}\Big{)}^{2}\,, \tag{7}\]  
with \(E_{p}\) the pulse energy. Here, we take the example of FLASH2 and the existing mirrors commonly used in FLASH operation to demonstrate a feasible working point.