# Two-Beam Free-Electron Lasers and Self-Injected Nonlinear Harmonic Generation
## 2 Harmonically Coupled Two Beams and High-Gain FELs  
As already underlined the HBC scheme foresees two beams with energies \(\gamma_{1}\) and \(\gamma_{2}=\sqrt{n}\gamma_{1}\), both injected in a linearly polarized undulator. Instabilities of two-stream nature, spoiling the FEL dynamics do not play any role because of the (large) beam energies. The condition on the energies of beams (1) and (2), ensures that the first harmonic, radiated by the beam (2), coincides with one of the higher-order harmonics of the first (1). As already established, the two beams radiate independently and give rise to two distinct FEL process, until the n-th harmonic, of the beam with lower energy, seeds that emitted by the beam with larger energy. The Pierce parameter associated with the two beams are  
\[\rho^{(1)}\cong\sqrt[3]{J^{(1)}}\,U(\lambda_{u},K),\]  
\[\rho^{(2)}\cong\frac{1}{\sqrt{n}}\sqrt[3]{J^{(2)}}\,U(\lambda_{u},K), \tag{17}\]  
\[U(\lambda_{u},K)=\frac{8.36\times 10^{-3}}{\gamma_{1}}[\lambda_{u}Kf_{b}(K)]^{ \frac{3}{3}},\]  
The Pierce parameters are therefore linked by  
\[\frac{\rho^{(2)}}{\rho^{(1)}}=\frac{1}{\sqrt{n}}\sqrt[3]{\frac{J^{(2)}}{J^{(1) }}}. \tag{18}\]If we assume that the two beams have the same current density, we obtain that the Pierce parameter associated with the beam at higher energy is lower by a factor \(n^{-1/2}\). The two fields grow inside the undulator independently. If \(E_{2}/E_{1}=\sqrt{3}\), the wavelength radiated by the beam (2) is the same of the third harmonic associated with the emission process of the beam (1). The intensities of the fields (1,2) grow independently, until NLHG occurs, when the third harmonics of the beam (1) seeds that from (2) (see Figure 3). The nonlinear harmonic growth is superimposed to the emission of the beam 2 and after the plateau of the nonlinear harmonic generation, eventually reaches the saturation which follows an exponential growth with the same gain length of the initial phase (lethargic and exponential) \(L_{g,1}^{(2)}=\sqrt{3}L_{g,1}^{(1)}\). The last step, including saturation, can be figured out as a straightforward exponential growth, namely  
\[P^{(2)}(z)=\Pi_{n,F}\frac{\exp\left[\frac{\left(z-z_{1}\right)}{L_{g}^{(2)}} \right]}{1+\frac{\Pi_{n,F}}{r_{F}^{(2)}}\left[\exp\left[\frac{\left(z-z_{1} \right)}{L_{g}^{(2)}}\right]-1\right]},\]  
\[P_{F}^{(2)}\cong\sqrt{2}\rho^{(2)}P_{E}^{(2)}, \tag{19}\]  
\[P_{E}^{(2)}\equiv\text{power of the higher energy beam},\]  
\[L_{g}^{(2)}\equiv\frac{\lambda_{u}}{4\pi\sqrt{3}\rho^{(2)}}=\sqrt{n}L_{g}^{(1 )}.\]  
If the two beams bear the same peak current (this is our assumption), the saturated power of the two laser beams is, approximately, the same. Figure 3: Intensity associated with the FEL action of the low energy beam (continuous red), and of the higher energy beam (dashed blue). The green circle stresses the same comment regarding Figure 1. The small blue circles refer to numerical benchmark with PROMETEO. Regarding the high energy part, the length of the undulator section necessary to reach the saturation, from the first plateau, is simply given by  
\[\rho^{(2)}\cong\frac{1}{\sqrt{n}}\sqrt[3]{J^{(2)}}\,U(\lambda_{u},K), \tag{17}\]0  
which yields  
\[\rho^{(2)}\cong\frac{1}{\sqrt{n}}\sqrt[3]{J^{(2)}}\,U(\lambda_{u},K), \tag{17}\]1  
It is evident that we have considered a fairly idealized case. We have also assumed that the two beams have identical current densities. This means that if the beams bear the same current and are supposed to be round the following condition should be satisfied  
\[\rho^{(2)}\cong\frac{1}{\sqrt{n}}\sqrt[3]{J^{(2)}}\,U(\lambda_{u},K), \tag{17}\]2  
where \(\beta^{(1,2)}\) are the associated Twiss parameters and \(\epsilon^{(1,2)}\) the e-beam emittances. Furthermore, including the usual scaling of the emittance with energy, we may also conjecture that \(\beta^{(2)}\cong\frac{\beta^{(1)}}{\sqrt{3}}\). These are qualitative statements, to be better framed within a specific design context of the electron beams transport channel, which will be discussed elsewhere.