# Two-Beam Free-Electron Lasers and Self-Injected Nonlinear Harmonic Generation
## 3 FEL Operating with HCB and Bi-Harmonic Undulators  
In the present section, we show how HCB and bi-harmonic undulators can be combined to obtain a laser beam with different harmonics and polarizations.  
The idea we would like to present is that of two beams with different energies injected in an undulator displaying the following on-axis magnetic field  
\[\vec{B}_{u}\equiv[d\,B_{0}\sin(h\,k_{u}z),\,B_{0}\sin(k_{u}z),0], \tag{23}\]  
where \(h\) is an integer and \(d\) is a numerical factor, smaller or larger than unit and not necessarily an integer.  
The spectral content of the on-axis emitted radiation is fairly rich [25] and the order of the harmonics depend on the polarization of the emitted harmonics, we have indeed  
\[\begin{array}{l}\lambda_{n,x}=\frac{\lambda_{n}}{2\,n\gamma^{2}}\Big{[}1+ \frac{K^{2}}{2}\Big{(}1+\frac{d^{2}}{h^{2}}\Big{)}\Big{]},\\ \\ \lambda_{n,y}=\frac{\lambda_{u}}{2\,(hn)\gamma^{2}}\Big{[}1+\frac{K^{2}}{2} \Big{(}1+\frac{d^{2}}{h^{2}}\Big{)}\Big{]}.\end{array} \tag{24}\]  
The derivation of the FEL evolution equations involves a rather awkward algebra, partially alleviated by the use of multi-variable Bessel functions, which become an efficient tool to evaluate the associated Pierce parameter.  
The advantage of the bi-harmonic undulator is that we are not obliged to consider odd harmonics of the fundamental only. If we keep \(h=2\) in Equation (23) we expect higher harmonics of even order. According to Equation (24) \(\lambda_{1,y}\) is the second harmonic of the fundamental, with \(y\) polarization.  
The general expression of the Bessel factor terms reads  
\[\begin{array}{l}f_{b,\,n_{x}}=(-1)^{\frac{n-1}{2}}\Big{[}(h)\,I_{\frac{n-1} {2}}\Big{(}n\,\xi,\ (-1)^{h+1}\frac{d^{2}}{h^{2}}\Big{(}\frac{n\,\xi}{h}\Big{)}\Big{)}\,-\, (h)\,I_{\frac{n+1}{2}}\Big{(}n\,\xi,\ (-1)^{h+1}\frac{d^{2}}{h^{2}}\Big{(}\frac{n\,\xi}{h} \Big{)}\Big{)}\,\Big{]},\\ \\ f_{b,\,(lm)_{y}}=\frac{d}{h}(-1)^{\frac{(n-1)}{2}}\Bigg{[}(h)\,I_{\frac{n(n-1 )}{2}}\Big{(}hn\xi,(-1)^{h+1}\frac{d^{2}}{h^{2}}n\xi\Big{)}+(-1)^{h(h)}\,I_{ \frac{n(n+1)}{2}}\Big{(}hn\xi,(-1)^{h+1}\frac{d^{2}}{h^{2}}n\xi\Big{)}\Bigg{]}, \\ \\ \xi=\frac{1}{4}\frac{K^{2}}{1+\frac{K^{2}}{2}\left(1+\left(\frac{n}{2}\right) ^{2}\right)}.\end{array} \tag{25}\]  
Regarding therefore \(h=2\), \(d=2\) we obtain for the Bessel factors  
\[\begin{array}{l}f_{b,\,1_{x}}=\,\Big{[}(^{2})\,I_{0}\Big{(}\,\xi,\ -\frac{\xi}{2}\Big{)}\,-\,\ (^{2})\,I_{1}\Big{(}\,\xi,\ -\frac{\xi}{2}\Big{)}\,\Big{]},\\ f_{b,2_{y}}=\,\Big{[}(^{2})\,I_{0}(2\,\xi,\ -\xi)\,+\,(^{2})\,I_{2}(2\,\xi,\ -\xi) \Big{]},\\ f_{b,3_{x}}=-\,\Big{[}(^{2})\,I_{1}\Big{(}3\,\xi,\ -\left(\frac{3\,\xi}{2} \right)\Big{)}\,-\,(^{2})\,I_{2}\Big{(}3\,\xi,\ -\left(\frac{3\,\xi}{2} \right)\Big{)}\,\Big{]}\end{array} \tag{26}\]and for the Pierce parameters  
\[\begin{split}&\rho_{3x}=\left(\frac{f_{b,3x}}{f_{b,1x}}\right)^{ \frac{2}{3}}\rho_{1x},\\ &\rho_{y}=\left(\frac{f_{b,2y}}{2f_{b,1x}}\right)^{\frac{2}{3}} \rho_{1x},\\ &\rho_{1x}=U(\lambda_{u},K)\sqrt[3]{J}\left(f_{b,1x}(K)\right)^{ \frac{2}{3}},\\ & U(\lambda_{u},K)=\frac{8.36\times 10^{-3}}{\gamma}[\lambda_{u}K]^{ \frac{2}{3}}.\end{split} \tag{27}\]  
We consider the same parameter in Figure 2 which for \(h=2\) yields three harmonics at saturation, with wavelengths \(13.36\) nm (x-pol), \(6.68\) nm (y-pol) and \(4.43\) (x-pol).  
In Figure 5 we have reported the behavior of the Pierce parameters vs. K and in Figure 6 the growth of the three harmonics vs. the undulator length for \(E_{2}/E_{1}=\sqrt{3}\). The second harmonic (with \(y\)-polarization) grows at the same level of the first and the third, seeds (with \(x\)-polarization) the first associated with the higher energy beam.  
We have so far shown that the mechanism of two harmonic undulators and of harmonically coupled beams, works, at least in principle. In the forthcoming section we will add further elements, stressing the prospective usefulness of these concepts.