# Two-Beam Free-Electron Lasers and Self-Injected Nonlinear Harmonic Generation
## 1 Introduction
\begin{array}{rcl}\frac{\partial^{2}\xi_{x,1}}{\partial\tau^{2}}&=&|a_{x,1}| \cos(\zeta_{x,1}+\varphi_{x,1}),\\ \\ \frac{\partial a_{x,3}}{\partial\tau}&=&-j_{x,3}\langle e^{-i\xi_{x,3}}\rangle,\\ \\ \frac{\partial a_{x,1}}{\partial\tau}&=&-j_{x,1}\langle e^{-i\xi_{x,1}}\rangle,\\ \\ \frac{\partial a_{y}}{\partial\tau}&=&-j_{y}\langle e^{-i\xi_{y}}\rangle;\ \ \xi_{x,3}=\xi_{y}=3\zeta_{x,1}.\end{array}  \tag{8}\]  
The relevant physical meaning is not dissimilar from that discussed as a comment to Equation (2). In this case, we have considered 1 st and 3 rd harmonics only, with the significant difference that two \"third\" harmonics, with orthogonal polarization, are driven by the fundamental. The third harmonics bears both vertical and linear polarization components and the relevant dimensionless amplitude are defined as  
\[|a_{\sigma}|^{2}=0.8\pi^{4}\frac{I_{\sigma}}{I_{s,\sigma}}, \tag{9}\]  
where \(\sigma=x,y\) and \(I_{s}\) is the associated saturation intensity  
\[I_{s,\sigma}\bigg{[}\frac{MW}{\text{cm}^{2}}\bigg{]}=\frac{6.9\times 10^{2} \bigg{(}\frac{\gamma}{N_{\sigma}}\bigg{)}^{4}}{[\lambda_{u,\sigma}[\text{cm}] Kf_{b,\sigma}(\xi)]^{2}}. \tag{10}\]  
Please note that \(N_{x}=3N_{y},\ \ \lambda_{u,x}=\lambda_{u,y}/3\). The corresponding Bessel factors are specified below (the wavelength of the first harmonic is \(\lambda_{1}=(\lambda_{u}/2\gamma^{2})(1+K^{2})\). Further comments are provided in Section 3)  
\[ \begin{array}{l}f_{b,1x}={}^{(3)}J_{0}\Big{(}\xi,\frac{\zeta}{3}\Big{)}\,- \,{}^{(3)}I_{1}\Big{(}\xi,\frac{\zeta}{3}\Big{)},\\ \\ f_{b,3x}=-{}^{(3)}I_{1}(3\,\xi,\,\xi)\,+\,{}^{(3)}I_{2}(3\,\xi,\,\xi),\\ \\ f_{b,3y}={}^{(3)}J_{0}(3\,\xi,\,\xi)\,-\,{}^{(3)}I_{2}(3\,\xi,\,\xi),\\ \\ \xi=\frac{1}{4}\frac{K^{2}}{1+K^{2}},\end{array}  \tag{11}\]and \({}^{(m)}J_{n}(x,\,y)\) are generalized Bessel functions defined through the generating function  
\[\sum_{n=-\infty}^{+\infty}t^{n}\Big{[}^{(m)}J_{n}(x,\,y)\Big{]}=e^{\frac{x}{2} \left(t-\frac{1}{2}\right)+\frac{y}{2}\left(t^{m}-\frac{1}{m}\right)}, \tag{12}\]  
and the series expansions (for further comments see ref. [17])  
\[{}^{(m)}J_{n}(x,\,y)=\sum_{l=-\infty}^{+\infty}J_{l}(x)\,J_{n-m\,l}(y). \tag{13}\]  
In Figure 2 an example is given of bi-harmonic power growth. The 1-D simulation displays the already anticipated behavior: along with the fundamental \"two\" third harmonics, with radial and vertical components, are generated through the nonlinear mechanism. The physical content of the figure is not different from what we have already discussed when we commented Figure 1 and is worded as reported below. The first harmonic grows and induces, via NLHG, two third harmonics with distinct orthogonal polarizations. The signature for the nonlinear growth is specified by the abrupt change of the growth rate. The two phases of the growth are characterized by the respective gain lengths, namely  
\[ \begin{array}{c}L_{g,3x}^{*}=\frac{1}{3^{1/3}}\frac{\lambda_{g}}{4\,\pi \sqrt{3}\rho_{3x}},\\ L_{g,y}^{*}=\frac{\lambda_{g}}{4\,\pi\sqrt{3}3^{1/3}\rho_{y}},\end{array}  \tag{14}\]  
and  
\[L_{g}^{(NH)}=\frac{L_{g}!x}{3}, \tag{15}\]  
Figure 2: Power growth of main and third harmonics for a bi-harmonic undulator with, \(E=1078\) Mev, \(\lambda_{u}=6\) cm, \(K=0.99\), \(\rho=1.258\times\,10^{-3}\) (numerical simulation PROMETEO). where  
\[\rho_{1x}\cong\frac{8.63\times 10^{-3}}{\gamma}\big{[}J\left(\lambda_{u}Kf_{b,1x} \right)^{2}\big{]}^{1/3},\]  
\[\rho_{3x}=\rho_{1x}\left(\frac{f_{b,3x}}{f_{b,1x}}\right)^{\frac{3}{2}}, \tag{16}\]  
\[\rho_{y}=\rho_{1x}\left(\frac{1}{3}\frac{f_{b,3x}}{f_{b,1x}}\right)^{\frac{3}{2 }}.\]  
The y-polarized component is significantly larger than its x-counterpart and grows almost at the level of the first. This is an interesting result since, for the single harmonic undulator, the third exhibits a power of two orders of magnitude below that of the fundamental. The proposal, the design and construction of this type of undulator traces back to more than three decades ago [18; 19].