# Two-Beam Free-Electron Lasers and Self-Injected Nonlinear Harmonic Generation
## 1 Introduction  
Nonlinear harmonic generation (NLHG) is one of the key mechanisms adopted to extend the spectral tunability range of Free-Electron Laser (FEL) [1, 2, 3, 4, 5]. Harmonic seeding [6, 7] and NLHG have been widely exploited in the past to push the FEL and its application towards shorter wavelengths. Other suggestions, although promising, have been carefully studied, but have not found specific applications so far. In this article, we discuss the possibility of combining two different proposals, developed in the past, concerning the use of devices exploiting NLHG in high-gain FEL devices [4, 8]. We deal with the harmonically coupled two-beam self-injection [9] and bi-harmonic undulators [10, 11], conceived to enhance the FEL coherent harmonic content emission and extend the capabilities of the device itself. Although the mechanism of nonlinear harmonic generation occurs in a FEL device operating with linearly and helically polarized undulators, here we treat the linear case only and consider undulators with on-axis field vector  
\[\vec{B}_{u}\equiv B_{0}[0,\sin(k_{u}z),0]. \tag{1}\]  
The pattern to the higher harmonic emission is an intrigued interplay between all the processes underlying the FEL lasing and involving energy modulation, bunching, coherent emission, higher-order bunching, and saturation [7]. The relevant study is afforded by exploiting a generalization of the FEL pendulum equations, including the evolution of the fields of the individual harmonics, namely [1]  
\[ \begin{split}&\frac{d^{2}}{d\tau^{2}}\zeta=\sum_{n=0}^{\infty} \lvert a_{n}\rvert\cos(\psi_{n}),\ \ \ \psi_{n}=n\zeta+\phi_{n},\\ & a_{n}=\lvert a_{n}\rvert e^{i\phi_{n}},\\ &\frac{d}{d\tau}a_{n}=-j_{n}\big{\langle}e^{-in\zeta}\big{\rangle}.\end{split}  \tag{2}\]where \(n\) refers to the order of the harmonic, \((a,\,j)\) are the Colson's dimensionless amplitude and current respectively, \(\zeta\) is the dimensionless electron phase, finally the brackets \(\left\langle\ldots\right\rangle\) denote averages on the initial phase distribution. The modulus of \(a\) is linked to the field intensity (power density ) \(I\) by the identities  
\[ \begin{split}&\left|a\right|^{2}=8\,\pi^{2}\frac{I}{I_{s}},\\ & I_{s}=\frac{c}{8\pi}\left(\frac{m_{e}c^{2}}{e}\right)^{2}\left( \frac{\gamma}{N}\right)^{4}\left(\lambda_{u}\frac{K}{\sqrt{2}}f_{b}\right)^{- 2}\quad.\end{split}  \tag{3}\]  
The quantity \(I_{s}\) denotes the FEL saturation intensity, a key quantity either in FEL and conventional lasers. It controls the onset of saturation, which starts playing a role when the FEL intensity is close to \(I_{s}\)[12]. The physical content of the previous set of equations can be summarized as follows. The fundamental harmonic (\(n=1\)) drives the bunching, which allows the growth of the different harmonics if the corresponding dimensionless currents \(j_{n}\) are nonvanishing. In the case of FEL devices operating with linear undulators, this is automatically ensured by the on-axis coupling to odd harmonics. An example of nonlinear harmonic generation intensity growth is given in Figure 1, drawn using the analytical formulae summarized in Equation (4). The relevant agreement with numerical computation has been benchmarked with the Prometeo code [13], as discussed below. The process consists of two distinct parts. In the first, the fundamental (\(n=1\)) and higher-order harmonics (3,5) grows independently, till the bunching at higher harmonics, induced by the fundamental, triggers the nonlinear harmonic generation mechanisms. The analytical procedure developed in the past [14] yields a fairly straightforward description of the different evolution steps. The model equations are reported below, and  
Figure 1: Harmonic power growth vs. the undulator length in m; (**a**) First Harmonic (green) (**b**) Third harmonic (blue) (**c**) Fifth harmonic (red). The encircled region around the inflection points, where the transition to the NLHG regime occurs, is not well reproduced by the analytical formulae.