In order to understand the role of the bunching coefficients during the early stages of the \(FEL\) interaction, we must treat Equation (5) in terms of a naive expansion in the small signal gain coefficient, limiting ourselves to the second-order expansion in the small and neglecting \(b_{2}\); then we find  
\[ \begin{split} a(\tau)\simeq a_{0}+g_{0}a_{1}+g_{0}^{2}a_{2},& a_{0}(\tau)=0,\\ \frac{d}{d\tau}a_{1}=-2\pi b_{1}e^{-i\
u_{0}\tau},& \frac{d}{d\tau}a_{2}=i\pi\int_{0}^{\tau}(\tau-\tau^{\prime})a_{1}(\tau^{\prime })e^{-i\
u_{0}(\tau-\tau^{\prime})}d\tau^{\prime}\end{split}  \tag{6}\]  
and the relevant solution is  
\[a(\tau)\!\simeq\!-2\pi b_{1}g_{0}\bigg{(}\frac{\sin(\frac{\
u_{0}\tau}{\
u_{0}} )}{\frac{\
u_{0}}{2}}e^{-i\
u_{0}\frac{\tau}{2}}-\!\frac{1}{2}\pi g_{0}\frac{ (i\
u_{0}^{2}\tau^{2}\!+\!4\
u_{0}\tau\!-\!6i)e^{-i\
u_{0}\tau}+6i+2\
u_{0} \tau}{\
u_{0}^{4}}\bigg{)}. \tag{7}\]  
According to the previous equation, the field grows initially because of being triggered by the bunching coefficient. It provides a kind of coherent spontaneous emission, which in the second phase is responsible for the onset of the exponential growth regime (see Figure 1), where we have reported the combined effects of bunching and initial seed. The previous results have been obtained using a perturbative expansion, which is not strictly necessary, since the \(FEL\) integral equation can be reduced to the ordinary differential equation reported below. \[\Big{[}\hat{D}_{\tau}^{3}+2i\
u_{0}\hat{D}_{\tau}^{2}-\
u_{0}^{2}\hat{D}_{ \tau}\Big{]}a(\tau)=i\pi g_{0}\Big{(}a(\tau)+b_{2}\;a^{*}(\tau)e^{-2i\
u_{0} \tau}\Big{)},\qquad\hat{D}_{\tau}=\frac{d}{d\tau}, \tag{8}\]  
obtained after noting that integrals of type  
\[i\pi g_{0}\int_{0}^{\tau}(\tau-\tau^{\prime})a(\tau^{\prime})e^{-i\
u_{0}( \tau-\tau^{\prime})}d\tau^{\prime}=i\pi g_{0}e^{-i\
u_{0}\tau}\int_{0}^{\tau} \!\bigg{(}\int_{0}^{\tau^{\prime}}a(\tau^{*})e^{i\
u_{0}\tau^{\prime\prime}}d \tau^{\prime\prime}\bigg{)}d\tau^{\prime} \tag{9}\]  
appearing in Equation (6) are double integrals which can be eliminated by keeping two successive derivatives [17; 18]. It is evident that, for negligible \(b_{2}\) (we have checked that  
Figure 1: Square modulus of the dimensionless amplitude vs. the detuning parameter for \(g_{0}=0.5\) and different values of \(\tau\): \((0.5,0.6,0.7,0.8,0.9,1)\). The intensity grows with incresing dimensionless time. the second-order bunching coefficient does not produce any appreciable contribution and can be safely neglected), Equation (8) reduces to a naive third-order ordinary differential equation, leading to the exponential growth also referred as \(FEL\) instability, the characteristic of which occurs in any Free Electron device [19; 20; 21; 22; 23; 24; 25]. (This type of instability is common to any free electron device (including gyrotrons and cyclotron auto-resonance masers (\(CARM\))). This was widely established before the studies developed in the previously quoted references. Some interesting papers within this respect are listed in the bibliography.) It is worth noting that, even though not explicitly contributing to Equation (8), the bunching coefficient \(b_{1}\) appears in the initial conditions provided by  
\[a(0)=a_{0},\qquad\qquad\hat{D}_{\tau}a\mid_{\tau=0}=-2\ \pi\ g_{0}\ b_{1},\qquad \qquad\hat{D}_{\tau}^{2}a\mid_{\tau=0}=2\ \pi\ i\ \
u_{0}\ g_{0}\ b_{1}. \tag{10}\]  
We have so far fixed the mathematical formalism to treat the problems associated with the effect of the bunching on the high gain (and not only) \(FEL\) evolution. The paper outline is reported below. In Section 2 we discuss solution methods to deal with either the integral and third-order differential equation. In Section 3 we deal with the extension of the integral equation to the harmonic generation and include effects associated with nonlinear regime. Section 4 is finally devoted to application of the formalism for the design of \(FEL\), exploiting segmented undulator devices.