# Advanced Scheme to Generate MHz, Fully Coherent FEL Pulses at nm Wavelength
## 3 Resonator Considerations  
### A Simple Model for the Reflectivity Requirements and Estimated Power Level in the Cavity  
The transition between the build-up and the steady-state regime in the case of start-up from shot noise is discussed in more detail in Section 4, while here we focus on the steady-state operation of the modulator-amplifier. We maintain the generality of the discussion by using approximations to build a simple model that can be used to investigate the parameter space for the design requirements. In the steady-state regime, there is a number of conditions that need to be fulfilled:  
* The input seed power needs to exceed the shot noise power of the electron beam by several orders of magnitude; otherwise, the SASE is not suppressed and the seeding process is not successful. Only a part of the seed power contributes to the exponential growth. Using for estimation the 1D cold FEL model this fraction amounts to 1/9. Assuming an excess of 3 orders of magnitude, the minimum input seed laser pulse peak power needs to be at least several 10 kW to 100 kW, depending on the exact electron beam parameters [45]. In addition, for seeding techniques it is required to induce an energy modulation of several times the initial energy spread which depends on the target harmonic to be amplified, the exact seeding scheme and the modulator length for given electron beam parameters. Typically, this requires a peak power that is larger than 100 kW. * The saturation power downstream of the modulator needs to be well below the \"natural\" saturation to avoid large induced energy spread, which would suppress the amplification process at the amplifier. As a general rule, the energy spread downstream of the modulator \(\sigma_{E}\) relative to the electron beam energy \(E\), should be considerably less than the FEL parameter of the amplifier \(\rho_{amp}\)[46], thus \(\sigma_{E}/E\ll\rho_{amp}\)[45]. The maximum acceptable seed peak power after amplification in the modulator strongly depends on the length of the modulator with respect to the gain length, and thus on the power amplification and on the energy spread increase. For the sake of avoiding a specific parameter set, here we assume that saturation at the seed laser wavelength yields between 1 GW to several 10 GW. Assuming a margin of 3 orders of magnitude to avoid \"heating\" of the beam, the seed peak power after amplification needs to be limited to not more than several tens of MW. The gain from shot noise to saturation of an FEL is around 9 orders of magnitude, which corresponds to about 20 power gain lengths (\(L_{g}\)). This means that there are 3 orders of magnitude between the minimum input peak power (\(P_{in}\)) and the maximum output peak power which are allowed to be lost in the cavity. Otherwise, either the minimum power is too close to shot noise or the maximum power too close to saturation. It is clear that these boundaries are not very strict and should only be seen as an approximation. It is known that the power along z develops as [46]:  
\[P(z)=\frac{P_{in}}{9}\cdot e^{z/L_{g}}\,. \tag{1}\]  
With a roundtrip reflectivity \(R\), the power after a modulator length of \(L_{mod}\) should be \(P(L_{mod})=P_{in}/R\). This leads to  
\[\frac{P_{in}}{R}=\frac{P_{in}}{9}\cdot e^{L_{mod}/L_{g}}\to L_{mod}=L_{g} \cdot\ln(\frac{9}{R}) \tag{2}\]  
For the first approximately three power gain lengths we expect no FEL power amplification, and this is referred to as the letharge regime. Assuming three orders of magnitude for the maximum allowed power amplification, the maximum modulator length is \(9\cdot L_{g}\) to compensate losses. The same equation can be used for design considerations; for instance, for a total reflectivity of 6%, the modulator should be roughly \(5\cdot L_{g}\).