# Advanced Scheme to Generate MHz, Fully Coherent FEL Pulses at nm Wavelength
## 3 Resonator Considerations
### Optical Klystron  
Another well-established method of gain control in FELs is the use of an Optical Klystron (OK), first introduced in [57]. It was originally introduced for gain control in oscillator FELs [58], but its application has been expanded. It has been used as a method to speed up the FEL process in SASE operation, when the total amplifier length is not sufficient for a given wavelength [59; 60; 61]. In addition, it is used in a seeding scheme when the seed laser peak power is not sufficient to increase the energy modulation required in seeding [17]. The simplest configuration of an optical klystron consists of two undulators tuned at the same resonant wavelength and a dispersive section in between them. The electron beam travels in the first undulator starting from some initial conditions (noise, or external seed) and a relatively weak energy modulation is induced. Then, the dispersive element modifies the electron beam phase space. This way, the bunching at this fundamental wavelength is increased, and the bunched electron beam generates coherent emission in the second undulator with increased gain. The dependence of the power gain on the longitudinal dispersion is a useful knob for our setup. In an oscillator, the two modulator sections separated by the dispersive section are in the resonator as shown in Figure 9. A 1D theory of optical klystron is discussed in [61] and a recent revision can be found in [62]. The optimum longitudinal dispersion depends on the energy spread and in our case can be estimated as  
\[R_{56,1}=\frac{\lambda_{res}}{2\pi\delta}\,, \tag{8}\]  
where \(\delta\) is the relative energy spread. With the studied parameter space, the optimum longitudinal dispersion is predicted as \(R_{56,1}=89\) um. Note that the sum of the length of modulator 1 and modulator 2 in Figure 9 is equal to the length of the modulator in Figure 1, so the power gain increase is introduced by chicane 1 only, and not by increasing the length of the modulator. In order to transition to the zero net gain regime, the \(R_{56,1}\) should initially be set to a value close to the optimal, and later on tuned to another value which would reduce the gain in the second modulator. In Figure 10a, we show the net gain achieved for different reflectivities and \(R_{56,1}\). The \(R_{56,1}\) at the steady state is determined by the intersection of the curves and the horizontal dashed line, which shows the zero net gain. We are interested in the range \(R_{56,1}<75\) um, because a too large \(R_{56,1}\) would cause an over-rotation of the longitudinal phase space which is not useful, as we still need to increase the bunching at a harmonic of the seed wavelength with the \(R_{56,2}\). The optimum longitudinal dispersion appears at around \(R_{56,1}=73\) um, which is approximately in agreement with Equation (8). Note, here, that the reflectivities required with the optical klystron are dramatically reduced, by more than an order of magnitude, when we compare to Figure 8a. As an example, with a reflectivity \(R_{set}=0.38\%\), we can build-up the peak power needed for seeding with \(R_{56,1}=42.5\) um, and after 19 passes change the longitudinal dispersion of the first chicane to \(R_{56,1}=30\) um to achieve zero net gain, and stable peak power of the radiation field per  
Figure 9: In an oscillator-based HGHG scheme, an optical klystron can be employed. To do so, the cavity contains two modulators separated by a chicane. This way this chicane can be tuned to control the gain per pass. pass as shown in Figure 10b. Note, here, that the input peak power is considerably lower in the order of 120 kW compared to the roughly 3.5 MW needed in all other gain-control methods presented already, to achieve the same energy modulation \(A=7\). In addition, the reflectivity required, \(R_{set}=0.38\%\), which considerably relaxes the requirements on the mirror specifications. The optical klystron has many advantages. As already explained, the first one is that it makes the transition from positive to zero net gain possible.