descriptions can be framed within the same context [2]. Many \(FEL\)s are presently operating and have operated in the past [3]. They are covering the electromagnetic spectrum from \(THz\) region to X-rays. The last generation of \(FEL\) sources (sometimes referred as fourth generation synchrotron radiation sources) has opened up unprecedent scenarios in different research fields because of the radiation characteristics, in terms of brightness (ten orders of magnitude above the ordinary sources) and the shortness of the pulses [4; 5; 6; 7]. The growth of the \(FEL\) field is ruled, in the small signal regime (namely, the dynamical conditions, when the field intensity is not large enough to induce nonlinear effects), by an equation known as the \(FEL\) integral equation; it is derived from the linearization of the pendulum equation, reported below [8; 9; 10; 11; 12; 13; 14; 15; 16]. \[ \begin{split}&\frac{d^{2}}{d\tau^{2}}\ \zeta=\ |\ a\ |\ \cos\ (\zeta+\phi)\, \frac{d}{d\tau}a=-j\langle e^{-i\zeta}\rangle,\\ &\tau=\frac{z}{L},\qquad\qquad j=2\pi g_{0},\qquad\qquad a=|\ a\ |\ e^{i\phi}, \qquad\qquad\
u=\frac{d}{d\tau}\zeta\end{split}  \tag{1}\]  
where \(z\) is the longitudinal coordinate; \(L\) is the undulator length; \(g_{0}\) is the small signal gain coefficient; and \(a\) and \(j\) are Colson's dimensionless amplitude and current respectively, \(\zeta,\
u\) the \(FEL\) longitudinal phase space variables and the brackets, denote the average of the phase space distribution. The use of dimensionless variables has several advantages. They constitute a set of comprehensive quantities such as \(a\) and \(g_{0}\), allowing one to merge all the significant \(FEL\) parameters into a single variable, providing quantities of interest (such as the gain and saturation intensity; see below) crucial in the descriptions and designs of \(FEL\) devices. The relevant small signal limit is achieved through the following approximations [17; 18]. 1. From the first of Equation (1), we obtain the lowest order (in the field strength) expansion of \(\zeta\), namely \[\zeta\simeq\zeta_{0}+\
u_{0}\tau+\delta\zeta,\qquad\qquad\delta\zeta=\int_{0} ^{\tau}(\tau-\tau^{\prime})Re\Big{(}a(\tau^{\prime})e^{i(\zeta_{0}+\
u_{0} \tau^{\prime})}\Big{)}d\tau^{\prime}\] (2) where \(\
u_{0}=2\pi N\frac{\omega_{0}-\omega}{\omega_{0}}\) is the detuning parameter. 2. Inserting \(\delta\zeta\) in the second of Equation (1) and averaging on the electron-field phase \(\zeta_{0}\), we get \[ \begin{split}\frac{d}{d\tau}a=&-2\pi g_{0}b_{1}e^{- i\
u_{0}\tau}+i\pi g_{0}\int_{0}^{\tau}(\tau-\tau^{\prime})a(\tau^{\prime})e^{- i\
u_{0}(\tau-\tau^{\prime})}d\tau^{\prime}+\\ &+i\pi g_{0}b_{2}\int_{0}^{\tau}(\tau-\tau^{\prime})a^{*}(\tau^{ \prime})e^{-i\
u_{0}(\tau+\tau^{\prime})}d\tau^{\prime}\end{split} \] (3) where the averages have been taken by considering the e-beam mono-energetic and without spatial and angular dispersion, thereby getting \[ \begin{split}\langle\kappa(\zeta_{0})\rangle_{\zeta_{0}}=\frac{1 }{2\pi}\int_{0}^{2\pi}f(\zeta_{0})\kappa(\zeta_{0})d\zeta_{0},\\ f(\zeta_{0})=\sum_{n=-\infty}^{\infty}b_{n}e^{in\zeta_{0}}, \qquad\qquad b_{n}=\langle e^{-in\zeta_{0}}\rangle_{\zeta_{0}}.\end{split} \] (4)  
The physical meaning of Equation (3) is transparent; the three terms on the right-hand side account for three different regimes:  
1. Absence of an initial bunching (\(f(\zeta_{0})\) constant); in this case Equation (3) reduces to \[\frac{d}{d\tau}a=i\pi g_{0}\int_{0}^{\tau}(\tau-\tau^{\prime})a(\tau^{\prime} )e^{-i\
u_{0}(\tau-\tau^{\prime})}d\tau^{\prime}.\] (5)The underlying physics is that energy modulation and consequent bunching are due to the input coherent seed \(a_{0}\). 2. Nonconstant \(f(\zeta_{0})\) and emergence of non zero \(b_{n}\) coefficients; the field may grow in a seedless mode, induced by the initial bunching coefficient, as illustrated below.