# Two-Beam Free-Electron Lasers and Self-Injected Nonlinear Harmonic Generation
## 4 Final Comments  
In the previous section we have discussed the possibility of merging two concepts, regarding FEL devices, which can operate with \"harmonic\" beams and undulators. The concepts we have developed can be framed within analogous efforts discussed in the past involving the use of undulators with orthogonal polarizations [20; 26; 27] and segmented undulators [28], as well. Regarding this last point, we note that FELs operating with segmented undulators in the SASE mode employ two (or more) undulator sections, tuned at a sub-harmonic of the preceding one. \[\begin{array}{l}P_{n}(z)=P_{b,n}^{L}F_{n}^{L}(z)+P  
The saturated power can however be increased by injecting a second beam according to the prescription reported below. The power emitted in the last section can be enhanced using a second beam not necessarily harmonically coupled to the first or with larger energy. We consider indeed a FEL operating in the SASE regime with a first undulator section with \(\lambda_{u}^{(1)}\cong 2.8\) cm, \(K^{(1)}\cong 2.133\), a first beam energy of \(E^{(1)}\cong 346\) MeV, an associated resonant wavelength \(\lambda^{(1)}\cong 100\) nm and the remaining parameters arranged to have a Pierce parameter \(\rho^{(1)}\cong 2.64\times 10^{-3}\). A second undulator section is foreseen to have a period \(\lambda_{u}^{(2)}\cong 1.4\) cm, \(K^{(2)}\cong 1\), the energy of the second beam is supposed to be \(E^{(2)}\cong 288\) MeV and therefore the wavelength emitted in the second part is tuned at the third harmonic of the first. If the first undulator is cut at the onset of the saturation induced at the third harmonic of the first section (\(Z\cong 10\) m) and if the parameters are arranged in such a way that \(\rho^{(1)}\cong\rho^{(2)}\) the FEL power at 33 nm, seeded by the maximum field in the first section, reaches the same power level of the field at 100 nm (as already noted we have managed to have equal Pierce parameters in the two sections this means current of the order of hundreds of Amperes and beam power ranging around \(10^{4}\) MW). It is evident that such a configuration is not too much realistic for an actual FEL operation; however, it shows that sophisticated FEL architectures using multi-beam configuration can be exploited, which may also involve oscillator FEL devices. The use of two-beam oscillator FELs has not been explored so far, in a forthcoming investigation we will discuss the possible advantages offered by a solution of this type. Figure 7: First and third harmonic evolution, in a segmented undulator. At each undulator segment (4 and 8 m) the field growth is dominated by the bunching acquired during the interaction inside the undulator segment. The red and green lines represent the growth with the inclusion of the intensity (considered to be a kind of seeding). The final section, where nonlinear harmonic generation occurs, is dominated by the bunching, when the second beam is superimposed the power intensity grows according to the usual pattern. As a matter of facts, FEL is a device of pivotal importance in applied sciences. The extreme versatility in terms of tunability, pulse duration, peak and average power... of fourth generation synchrotron radiation X-ray sources is the reason of their spectacular success in materials science, chemistry and biology [29]. Following the successful operation of Flash [30] and LCLS [31] and justified by their unique achievements, further XFEL facilities have been operated or are under construction around the world with different peculiarities of operation modes [32; 33; 34; 35; 36; 37; 38]. The elements of discussion outlined in this article suggest the use of pulses at different colors, with different polarizations. These additional elements foresee the assembling of hybrid FEL architectures, with characteristics useful for users, interested in applications demanding for polarization control of the laser light and/or in spectrally resolved pump and probe experiments.