# A Collection of Formulae for the Design of Compton Back-Scattering X-ray Sources
## 3 Spectral Properties and Inhomogeneous Broadening
It should be underlined that the optimization of the luminosity requires, as already noticed, extremely peculiar values of the Twiss coefficients at the IP. See Figure 9 where we have reported the bunch interaction at the overlapping point, where the \(\beta_{T}\) parameter of the electron beam is of the order of \(cm\) or even less. If the e-beam is at waist in the crossing point, the \(\gamma_{T}\) parameters (inversely proportional to \(\beta_{T}\)) determine a significant increase of the e-beam divergence. It is therefore evident that the inhomogeneous broadening associated with the angular spread (\(\mu_{\eta^{\prime}}\)) is dominant. It should furthermore be noted that \(\mu_{\eta}\propto K_{l}^{2}\) and therefore, to the laser intensity, for low \(I_{l}\), the associated effect on the spectral broadening can be neglected. Figure 9: Geometry of the electron (red) and photon (green) bunches interaction at the IP. To be less qualitative, we reconsider the definition of the laser equivalent strength parameter and note that  
\[I_{l}\left[\frac{W}{m^{2}}\right]\cong 1.38\times 10^{10}\frac{K_{l}^{2}}{ \lambda[m]^{2}}. \tag{47}\]  
Accordingly, for \(K_{l}\cong 10^{-1}\) and using a laser wavelength 1 \(\upmu\)m (a Ti: sapphire e.g.) the associated power density is \(1.38\times 10^{20}\ W/m^{2}\). For these values, the contribution of \(\mu_{y}\) to the spectrum broadening can be neglected. In Figures 10 and 11, we provide examples of spectrum distortion due to energy spread/\"emittances\" and Twiss parameters, respectively. The comments to the figures are summarized below  
1. Figure 10 yields an idea of the dependence of the spectral line on the energy spread. In the case of a symmetric energy distribution no spectral peak shift is exhibited and the most significant effects are the broadening and the reduction of the peak intensity. 2. Figure 11a displays the comparison between correlated and non correlated phase space e-beam distributions at the IP (see Figure 11b). A non-zero value of the \(\alpha_{T(x,y)}\) correlation parameters determines an increase of the angular divergence along with a further broadening and an increase of the asymmetry and decreasing of the peak spectrum. 3. Figure 11c shows the comparison of the effect on the spectral line of the beam phase space distributions exhibiting different Twiss parameters (see Figure 11d). It is evident that either the spectral asymmetry and broadening are dominated by the effect of the larger angular divergence. The central peak frequency shift displayed by the cases with large \(\gamma_{T(x,y)}\) is simply due to the fact that the e-beam divergences increase \(\left(\left\langle\theta_{x,y}\right\rangle=\sqrt{\gamma_{T(x,y)}\epsilon_{n,x,y} /\gamma}\right)\) and therefore the central peak frequency is given by  
\[\omega^{*}\cong\frac{4\gamma^{2}\omega_{l}}{1+\frac{K_{l}^{2}}{2}+\left( \gamma\langle\theta_{x,y}\rangle\right)^{2}}. \tag{48}\]  
In this section we have given a fairly simple picture of the distortion induced in the Thomson backscattering spectrum, by the electron beam qualities and by the relevant transverse phase-space distribution. The previous results compare fairly well with analogous (numerical) computations reported in ref. [42]. It is worth underlining that the use of a procedure yielding the full contribution to the spectral broadening, with the inclusion of the e-beam distribution in phase space is of noticeable importance. The necessity of squeezing the beam to get a larger photon flux is counteracted indeed by the insurgence of a large angular divergence, which hampers the spectral brightness; and therefore, an optimization procedure involving the beam transport design at the IP should carefully be accomplished. As a final remark we note that a small transverse section beam, like those reported here, demands an accurate analysis of the effects possibly induced by the Coulomb intra-beam scattering, which may eventually lead to an increase of the beam emittance and therefore to the deterioration of the spectral brightness.