## 3 High Gain FEL Equations and Harmonic Generation
We have so far considered the harmonic linear dynamics, in which within certain limits (to be discussed in the final section), the evolutions of the associated intensities are independent. It should be noted that harmonics too can be triggered by a bunching coefficient, as implicitly contained in Equation (37). As it is well known, the bunching grows with the intensity itself; the dynamics of each harmonic are the result of intrigued feedback between induced bunching and higher order-bunching (namely, at higher values of \(n\)). We crudely describe the process by noting that each harmonic, while the intensity grows, induces further bunching, determining the onset of the sub-harmonics \(m\) of the \(n\)-th harmonic--namely, the first harmonic indices higher order harmonics and each of them determines a further set of harmonics. In terms of bunching coefficients, we have: The fundamental harmonics (\(n=1\)) contributes to the increase of the bunching of order 1 and of other higher order bunching coefficients. The same holds for the higher order harmonics; the third is, e.g., responsible for the bunching of its own harmonics \((3,6,9,\dots\) with respect to the fundamental). The bunching process is not \"free,\" and indeed an energy spread, associated with gain degradation and saturation mechanisms, is produced. This phenomenology is not considered within the present context. The inclusion of nonlinear terms in our analysis occurs by keeping intensity dependent terms [17; 18; 41] which allow the following redefinition of the \(FEL\) integral equation:  
\[ \begin{split}&\frac{d}{d\tau}a=-2\pi g_{0}e^{-i\
u_{0}\tau}\sum_{m=- \infty}^{\infty}(-i)^{m}b_{1-m}\frac{J_{m}(\mid\Pi\mid)}{\mid\Pi\mid^{m}} \left(\int_{0}^{\tau}(\tau-\tau^{\prime})a(\tau^{\prime})e^{i\
u_{0}\tau^{ \prime}}d\tau^{\prime}\right)^{m},\\ &\int_{0}^{\tau}(\tau-\tau^{\prime})a(\tau^{\prime})e^{i\
u_{0} \tau^{\prime}}d\tau^{\prime}=\mid\Pi\mid e^{i\psi}\end{split}  \tag{41}\]  
where  
\[J_{m}(\mid\Pi\mid)=\sum_{r=0}^{\infty}\frac{(-1)^{r}}{r^{t}(m+r)!}\left(\frac {\mid\Pi\mid}{2}\right)^{m+2r}. \tag{42}\]  
If we consider a perturbative solution of the above equation in terms of the fundamental intensity, we can write the first three terms as  
\[ \begin{split}&\frac{d}{d\tau}a_{l}=-2\pi g_{0}b_{1}e^{-i\
u_{0} \tau}+i\pi g_{0}\int_{0}^{\tau}(\tau-\tau^{\prime})a_{l}(\tau^{\prime})e^{-i \
u_{0}(\tau-\tau^{\prime})}d\tau^{\prime},\\ &\frac{d}{d\tau}a_{nl,2}=-2\pi g_{0}e^{-i\
u_{0}\tau}\frac{b_{-1} }{8}\left(\int_{0}^{\tau}(\tau-\tau^{\prime})a_{l}(\tau^{\prime})e^{i\
u_{0} \tau^{\prime}}d\tau^{\prime}\right)^{2},\\ &\frac{d}{d\tau}a_{nl,3}=-2\pi g_{0}e^{-i\
u_{0}\tau}\frac{b_{-2} }{2^{3}3!}\left(\int_{0}^{\tau}(\tau-\tau^{\prime})a_{0}(\tau^{\prime})e^{i \
u_{0}\tau^{\prime}}d\tau^{\prime}\right)^{3}\end{split}  \tag{43}\]  
which have been derived using the limit \(\lim\limits_{\mid\Pi\mid\to 0}\frac{J_{m}(\mid\Pi\mid)}{\mid\Pi\mid^{m}}= \frac{1}{m!2^{m}}\). The subscripts \(l\) and \(nl\) stand for linear and nonnlinear, respectively. The \(nl\)-contributions stand for the onset of the nonnlinear harmonic generation. Considering a linearly polarized undulator not providing coupling to even on axis harmonic generation, we obtain from the third Equation (43)\[a_{nl,3}=-2\pi g_{0}\frac{b_{-2}}{2^{3}3!}\int_{0}^{\tau}e^{-iv_{0} \tau^{\prime}}h_{3}(v_{0},\tau^{\prime})d\tau^{\prime}\, \tag{44}\] \[h_{3}(v_{0},\tau)=\left(\int_{0}^{\tau}(\tau-\tau^{\prime})a_{0}( \tau^{\prime})e^{iv_{0}\tau^{\prime}}d\tau^{\prime}\right)^{3}.\]  
Keeping the fast growing root only and assuming \(v_{0}=0\), we find a further contribution to the harmonic intensity depending on the power of the fundamental and characterized (as already anticipated) by the gain length  
\[L_{g}^{(nlhg)}=\frac{L_{g}}{n} \tag{45}\]  
where \(nlhg\) stands for nonnlinear-harmonic generation. The conclusion of this discussion is that there are two distinct contributions to the higher order harmonic emission.