# State of the Art of High-Flux Compton/Thomson X-rays Sources
## 2 Generalities on Thomson/Compton Scattering  
In boosted sources, the radiation frequency depends on the electron Lorentz factor \(\gamma\), on the observer position \(\theta\), on the input pump frequency \(\omega_{L}\) and on the interaction angle \(\alpha\) according to:  
\[\omega_{X}=\frac{\omega_{L}(1+\cos\alpha)}{(1-\beta\cos\theta+2\hbar\omega_{L }/(mc^{2}\gamma))}. \tag{1}\]  
This expression, where \(\hbar\) is the reduced Planck constant, \(\beta=v_{e}/c\), \(v_{e}\) being the electron velocity, \(c\) the speed of light and \(m\) the electron mass, shows the frequency-angle correlation due to the boost which concentrates the highest energy photons at the center of the pulse, while the less energetic photons occupy the outer zone. In the limit \(4h\ \omega_{L}\gamma/(mc^{2})<<1\) the last term in the denominator is negligible and the process is classical. The total achievable X-ray flux is approximately given for head-on collision (\(\alpha=0\)) by:  
\[N_{x}\approx\sigma_{Th}f_{RF}\frac{N_{e}N_{L}}{2\pi(\sigma_{e}^{ 2}+\sigma_{L}{}^{2})} \tag{2}\]  
where \(\sigma_{Th}=6.65\times 10^{-29}m^{2}\) is the Thomson cross section, \(N_{e}=Q/e=6.25\times 10^{9}Q(nC)\), \(N_{L}=E_{L}/(h\
u_{L})=E_{L}\lambda_{L}/(hc)\simeq 5\times 10^{24}\ E_{L}(J) \lambda_{L}(m)\) is the number of laser photons, \(f_{RF}\) the repetition rate of the system and \(\sigma_{L}\) and \(\sigma_{e}\) are the rms transverse dimension of the laser and electron distribution, respectively. The rms X-ray beam bandwidth \(bw\), in the linear approach, is approximately given by [61; 62]:  
\[bw\approx\sqrt{\left[\frac{\Psi^{2}/\sqrt{12}}{1+\Psi^{2}/2}+ \frac{2\epsilon_{n}^{2}}{\sigma_{e}^{2}}\right]^{2}+\left[2\frac{\Delta\gamma }{\langle\gamma\rangle}\right]^{2}+\left[\frac{\Delta\omega_{L}}{\omega_{L}} \right]^{2}+\left[\frac{M^{2}\lambda}{4\pi\sigma_{L}}\right]^{4}} \tag{3}\]  
where the acceptance \(\Psi=\gamma\theta_{acc}\) scales with the collimation angle \(\theta_{acc}\), \(\epsilon_{n}\) is the emittance of the electron beam, \(\frac{\Delta\omega_{L}}{\omega_{L}}\) the proper rms bandwidth of the laser and \(M\) its purity factor. The number of X-ray photons produced per second in a given rms bandwidth \(bw\) and collected within a collimation forward angle \(\theta_{acc}\), given in practical units, is:  
\[N_{x,bw}\approx 5.8\times 10^{11}f_{RF}\frac{E_{L}(J)Q(nC)}{\sigma_{x}{}^{2}( \mu m^{2})}\gamma^{2}\theta_{acc}^{2} \tag{4}\]  
where, for the sake of simplicity, both laser and electron beams are supposed to arrive at the interaction point (IP) with rms transverse sizes \(\sigma_{x}\). The polarization properties of the Compton radiation follow those of the laser, so linear polarized lasers generate mainly linear polarized X-rays, while circular polarized lasers induce circular polarization. At the outer border, a crown of radiation characterized by different polarization occurs, due to the formation of the relativistic cone. The effect is weak and usually involves few photons [63]. The longitudinal coherence is extremely scarce. On the transverse plane, however, due to the small radius of the source, coherence areas of linear dimension in the micron range and widening during propagation occur. This structure can be exploited in phase contrast imaging. In the mild non-linear regime, where the laser parameter \(a_{0}=eE_{0}/(mc\omega_{L})\) is order or larger than 1 (\(E_{0}\) and \(\omega_{L}\) being the peak electric field and the angular frequency of the laser pulse, respectively), the bandwidth presents a shift of the peak towards lower frequency values, the occurrence of spectral fluctuations and an ample broadening due to non-linear multi-photon effects in the back-scattering [4; 64; 65; 66], followed by the insurance of the harmonics, whose characteristics have been detected and studied [67]. The use of high intensity ultra short laser systems as interaction laser presents therefore serious limitations.