## 2 Algorithmic and Analytical Solutions of the \(Fel\) Integral Equation
Analytic solutions can be obtained too; they require some algebra, and in particular the use of [PERSON]'s method to get the solution of Equation (13). Putting everything together we find:  
(a) The Fang-Torre formula (this Formula appeared in an unpublished manuscript by [PERSON], and was later derived with minor refinements by [PERSON], and reported in [27]) valid for \(a_{0}\
eq 0\), \(b_{1}=0\):  
\[ \begin{split} a(\tau,\
u_{0})&=\frac{a_{0}}{3(\
u_{ 0}\!+\!p\!+\!q)}e^{-\frac{2j}{3}\
u_{0}\tau}\Big{\{}(-\
u_{0}\!+\!p\!+\!q)e^{- \frac{j}{3}(p+q)\tau}\!+\!2(2\
u_{0}+p+q)e^{\frac{j}{6}(p+q)\tau}\!.\\ &\cdot\left[\cosh\left(\frac{\sqrt{3}}{6}(p-q)\tau\right)+i\frac{ \sqrt{3}\
u_{0}}{p-q}\sinh\!\left(\frac{\sqrt{3}}{6}(p-q)\tau\right)\right] \right\},\\ p&=\left[\frac{1}{2}(r+\sqrt{d})\right]^{\frac{1}{ 3}},\hskip 56.905512 ptq=\left[\frac{1}{2}(r-\sqrt{d})\right]^{\frac{1}{3}},\\ r&=27\pi g_{0}-2\
u_{0}^{3},\hskip 56.905512 ptd=27 \pi g_{0}(27\pi g_{0}-4\
u_{0}^{3}).\end{split}  \tag{19}\]  
The square modulus of \(a(\tau,\
u_{0})\) yields the intensity growth as a function of the dimensionless time and of the detuning parameter. Regarding Equation (19) we find  
(20)  
We have very loosely described in introductory remarks the \(SAE\) regime or seedless operation. In this case the field grows from a kind of prebunching characterizing the beam itself [28; 29]. (b) The field growing from a bunching coefficient associated with the electron distribution and the solution of the evolution problem reads:  
\[ \begin{split} a(\tau,\
u_{0})&=\frac{2\pi g_{0}b_{ 1}}{\Delta}e^{-i\frac{2}{3}\
u_{0}\tau}\Big{\{}\mu e^{-\frac{j}{6}(p+q)\tau}+ \\ &-\left(\chi_{-}\cosh\left[\frac{\sqrt{3}}{6}(p-q)\tau\right]+ \chi_{+}\sinh\left[\frac{\sqrt{3}}{6}(p-q)\tau\right]\right)e^{\frac{j}{6}(p+q )\tau}\Big{\}},\\ \Delta&=\frac{\sqrt{3}}{9}(p^{3}-q^{3}),\hskip 56.905512 pt \mu=i\frac{\sqrt{3}}{9}[\
u_{0}-(p+q)](p-q),\\ \chi&=\frac{1}{6}\Bigg{[}\
u_{0}+\frac{1}{2}(p+q)-i \frac{\sqrt{3}}{2}(p-q)\Bigg{]}\Bigg{[}p+q+i\frac{\sqrt{3}}{3}(p-q)\Bigg{]}, \hskip 14.226378 pt\chi_{\pm}=\chi\pm\chi^{*}\.\end{split}  \tag{21}\]  
Before proceeding further, we write Equation (21) in a form more appropriate for high gain/\(SASE\)\(FEL\) operation. To that end, we remind the reader that the small signal gain coefficient and Pierce parameter \(\rho\) (see references [8; 9; 10; 11; 12; 13; 14; 15; 16]) are linked by the identity  
\[\rho=\frac{\sqrt{\pi g_{0}}}{4\pi N}, \tag{22}\]where \(N\) is the number of undulator periods. The intensity field growth is ruled by the so-called gain length  
\[L_{g}=\frac{\lambda_{u}}{4\pi\sqrt{3}p}, \tag{23}\]  
which is the pivotal quantity controlling the small signal high gain dynamics; its inverse defines the growth per unit length. With these remarks in mind, we note that:  
1. Quantities such as \(v_{0}\) and \(\tau\) should be replaced by \[v_{0}\ \rightarrow\ \tilde{v_{0}}=\frac{v_{0}}{4\pi N\sqrt{3}\rho}=\frac{1}{2 \sqrt{3}\rho}\frac{\omega_{0}-\omega}{\omega_{0}},\qquad\qquad\tau\ \rightarrow\ \tilde{z}=\frac{z}{L_{g}}\] (24) and it is worth noting that the number of undulator periods does not appear anymore in the definition of the new dimensionless quantities (see below for further comments);
2. Quantities involving the product of \(p,q\) and \(\tau\), written in the new variables, read \[p\tau=\sqrt{3}\left(\sqrt[3]{1+\frac{\sqrt{1-4\tilde{v}_{0}^{3}}}{1-2\tilde{v }_{0}^{3}}}\right)\tilde{z},\qquad\qquad q\tau=\sqrt{3}\left(\sqrt[3]{1-\frac {\sqrt{1-4\tilde{v}_{0}^{3}}}{1-2\tilde{v}_{0}^{3}}}\right)\tilde{z}.\] (25)  
It is finally important to stress that the Colson's dimensionless amplitude in terms of dimensional quantities reads  
\[a=2\sqrt{2}\pi\sqrt{\frac{I}{I_{s}}} \tag{26}\]  
where \(I\) is the field intensity (power/surface) and \(I_{s}\) is the saturation intensity [30; 31; 32; 33], a quantity which, in the theory of lasers, is exploited as a reference quantity to fix the onset of saturation.