# Two-Beam Free-Electron Lasers and Self-Injected Nonlinear Harmonic Generation
## 1 Introduction
Linear and NLHG contributions are indeed summed incoherently, namely through the sum of two intensities (\(\Lambda_{n}(z)\) and \(\Pi_{n}(z)\)), without including the respective phases determining at that point an insignificant oscillation. we have denoted with \(\Lambda_{n}(z)\) and \(\Pi_{n}(z)\) the terms accounting for the small-signal and nonlinear harmonic generation parts respectively  
\[P_{1}=P_{1}(z)=P_{0}\frac{A(z)}{1+\frac{P_{0}}{P_{F,1}}[A(z)-1]},\]  
\[A(z)=\frac{1}{9}\Bigg{[}3+2\cosh\Bigg{(}\frac{z}{L_{g,1}}\Bigg{)}+4\cos \Bigg{(}\frac{\sqrt{3}}{2}\frac{z}{L_{g,1}}\Bigg{)}\cosh\Bigg{(}\frac{z}{2L_{g,1}}\Bigg{)}\Bigg{]},\]  
\[P_{n}(z)=\Lambda_{n}(z)\,+\Pi_{n}(z)\,,\quad n=3,5...\quad\Lambda_{n}(z)\,=P_{ 0,n}\,A_{n}(z)\;.\]  
For the case with \(n>1\) the gain length should be replaced by  
\[L_{g,n}^{*}=\frac{\lambda_{u}}{4\pi\sqrt{3}\rho_{n}^{*}}=n^{-\frac{1}{3}}L_{g,n}\,,\qquad\rho_{n}^{*}=n^{\frac{1}{3}}\!\!\beta_{n},\,\,\,P_{0,n}\,\,\,\text{ input seed power}\;, \tag{4}\]  
\[\rho_{n}\equiv n-\text{th harmonic Pierce parameter},\]  
\[\rho_{n}=\bigg{(}\frac{f_{b,n}}{f_{b,1}}\bigg{)}^{2/3}\rho_{1},\]  
\[f_{b,n}=\frac{J_{n}-1}{2}\,(n\xi)-J_{\frac{n}{2}+1}\,(n\xi),\,\,\xi=\frac{1}{4 }\frac{K^{2}}{1+\frac{K^{2}}{2}},\]  
\[\rho_{1}\cong\frac{8.63\times 10^{-3}}{\gamma}\big{[}J\,(\lambda_{u}Kf_{b,1})^{ 2}\big{]}^{1/3}.\]  
The formula for the Pierce parameter of the harmonic (\(\rho_{1}\)) is expressed in a practical form, where \(J\) is given in \(A/m^{2}\) and \(\lambda_{u}\) in \(m\). The NLHG contributions are specified by the formulae  
\[\Pi_{n}(z)=\Pi_{0,n}\frac{\exp\Bigg{(}\frac{nz}{L_{g,1}}\Bigg{)}}{1+\frac{\Pi _{0,n}}{\Pi_{F,n}}\Big{[}\exp\Big{(}\frac{nz}{L_{g,1}}\Big{)}-1\Big{]}},\,\,\Pi _{0,n}=c_{n}\Big{(}\frac{P_{0}}{\rho_{1}P_{k}}\Big{)}^{n}\,\Pi_{F,n}\,,\quad c _{3}=8\,,\quad c_{5}=116, \tag{5}\]  
\[P_{1}=P_{1}(z)=P_{0}\frac{A(z)}{1+\frac{P_{0}}{P_{F,1}}[A(z)-1]},\]0  
where the saturated power \(P_{F,1}\) reads  
\[P_{1}=P_{1}(z)=P_{0}\frac{A(z)}{1+\frac{P_{0}}{P_{F,1}}[A(z)-1]},\]1  
and \(P_{E}\) is the e-beam power. We note therefore that  
1. The first harmonic power grows initially by exhibiting the lethargic phase followed by the exponential behavior, characterized by the gain length \(L_{g,1}\). 2. The same occurs for the higher-order harmonics (with gain length \(L_{g,n}^{*}\)), till the bunching effects trigger the mechanism of nonlinear harmonic generation. 3. This last phase is characterized by a sudden change in the growth rate followed by a kind of saturation. The characteristic gain length is a fraction of that of the fundamental (namely \(L_{n}^{(NH)}=\frac{L_{g,1}}{R}\)). The inflexion point (before the change in the intensity growth derivative) occurs at \(z=z*\), where the \(\Lambda_{n}(z)\) and \(\Pi_{n}(z)\) contributions balance. The dots refer to a numerical benchmark with the code Prometeo, for a set of specific parameters. After the saturation the figure exhibits a flat top, without displaying the characteristic power oscillations. This is an artefact of the analytical approximations. It should be noted that, although not significant for the present discussion, these oscillations can be included in the analytical scheme as shown in refs. [15; 16]. _Bi-Harmonic undulator_  
The use of bi-harmonic undulators [10; 11] has been suggested in the past as a device allowing the enhancement of the nonlinear harmonic generation in high-gain FEL devices. In the case of an undulator exhibiting a field with linear orthogonal polarization, specified by the vector  
\[P_{1}=P_{1}(z)=P_{0}\frac{A(z)}{1+\frac{P_{0}}{P_{F,1}}[A(z)-1]},\]2  
The corresponding FEL pendulum equation can be written by extending those reported in (2) and reads  
\[