# A Collection of Formulae for the Design of Compton Back-Scattering X-ray Sources
## 4 Non Linear Contributions to Thomson Scattering  
The increase of the laser intensity determines the increase of the associated \(K_{l}\). As in the case of the magnetic undulator, a new and richer phenomenology arises, such as the possibility of extending the range of tunability through the emission of higher harmonics. In this section we will add a few comments on how these aspects of the CBS phenomenology should be accounted for. As already noted, the wavelength shift due to the laser parameter \(K_{l}\) is one of these effects known since the early sixties of the last century [14]. The signature for the non-linear corrections is the emergence of a further broadening of the spectral line and the emission at higher order harmonics [7; 14; 28; 38; 39; 44; 45; 46; 47; 48; 49; 50]. As previously underscored, the laser intensity effects can be quantified through the \(K_{l}\) parameter, using the analogy between the Thomson scattering and the emission in magnetic undulators. According to the present discussion, the undulator can be viewed as an ensemble of coherent photons, with a volumetric density [8]  
\[\bar{n}_{u}\cong\frac{\alpha}{4}\frac{K_{u}^{2}}{\lambda_{u}r_{0}^{2}}, \tag{49}\]  
providing a number of (virtual) photons, per cubic meter, of the order of \(10^{29}\). The use of Equations (8) and (49) allows to derive the number of scattered photons (the photon yield) by a single electron as reported below  
\[N_{s}\cong c\Delta T\sigma_{th}\bar{n}_{u}=\frac{2}{3}\pi r_{0}^{2}N_{u}K_{u} ^{2}. \tag{50}\]If we transpose the last identity to the case of the Thomson scattering, we naively infer that an increase of the laser intensity and therefore a corresponding increase of \(K_{l}\), determines a significant enhancement of the scattered photons. This is not the only signature of the high intensity (non linear) regime. If we pursue the analogy with the emission in undulators, we understand that with the increase of the output photon flux a richer phenomenology, including the appearance of higher order harmonics, becomes significant. In order to address the problem of specifying, in less vague terms, the high field effects in Thomson backscattering, we use a more appropriate theoretical framework. We remind the reader that the Lienard-Wiechert potentials yield the spectrum per unit solid angle as [7]  
\[ \begin{split}\frac{d^{2}I_{x}}{d\omega d\Omega}&= \sum_{m}\frac{d^{2}I_{x,m}}{d\omega d\Omega}=\frac{16\,\alpha}{\pi}N_{e}\Big{(} \gamma\,\widetilde{N}_{l}\Big{)}^{2}\xi_{l}^{2}\sum_{m}m^{2}f_{b,m}^{2}( \xi_{l})\langle S_{m}(\
u_{m})\rangle,\\ &\frac{d^{2}I_{x,m}}{d\omega d\Omega}&=\frac{16\, \alpha}{\pi}N_{e}\Big{(}\gamma\,\widetilde{N}_{l}\Big{)}^{2}\xi_{l}^{2}m^{2}f _{b,m}^{2}(\xi_{l})\langle S_{m}(\
u_{m})\rangle,\\ & f_{b,m}(\xi_{l})=(-1)^{\frac{m-1}{2}}\Big{[}I_{\frac{m-1}{2}}(m \xi_{l})-J_{\frac{m+1}{2}}(m\xi_{l})\Big{]},\end{split}  \tag{51}\]  
where the sum is taken on the index of the harmonic m, \(\langle S_{m}(\
u_{m})\rangle\) is the same as in Equation (43), with the (significant) difference that the inhomogeneous broadening parameters are multiplied by the order of the harmonics and  
\[ \begin{split}\
u_{m}&=2\pi\Big{(}m\,\widetilde{N} _{l}\Big{)}\frac{\omega_{s,m}-\omega}{\omega_{s,m}},\\ &\omega_{s,m}=m\omega_{s,1},\ \ \omega_{s,1}=\omega_{s}.\end{split}  \tag{52}\]  
It is evident that the spectrum distortion effects are more significant for higher harmonics. This is easily understood, since the \(\mu\) parameters are the ratio between the induced homogeneous to the natural broadening and the latter is reduced by a factor \(m\) in the case of the higher order harmonics. We are now ready to start a more complete discussion. The harmonic spectrum in Equation (51) is relevant to the radiation emitted on axis (even though it contains the angular corrections associated with the electron beam divergences).