# A Collection of Formulae for the Design of Compton Back-Scattering X-ray Sources
## 3 Spectral Properties and Inhomogeneous Broadening
 \tag{39}\]  
where \(f(x,x^{\prime};y,y^{\prime};\varepsilon,t)\) is the e-beam six dimensional phase-space distribution. We assume that the distributions can be factorized as  
\[f(x,x^{\prime};y,y^{\prime};\varepsilon,t)=F(x,x^{\prime};y,y^{\prime})\phi( \varepsilon)\phi(t) \tag{40}\]  
Figure 7: Geometry of the Compton Backscattering including angular shifts. where \(F(x,x^{\prime};y,y^{\prime})=g(x,x^{\prime})g(y,y^{\prime})\), with  
\[g(\eta,\eta^{\prime})=\frac{1}{2\pi\epsilon_{\eta}}\exp\left(-\frac{\gamma_{\eta} ^{2}\eta^{2}-2\alpha_{\eta}\eta\eta^{\prime}+\beta_{\eta}\eta^{\prime 2}}{2 \epsilon_{\eta}}\right),\ \ \eta=x,y\ \ \,\]  
\[\phi(\epsilon)=\frac{1}{\sqrt{2\pi}\sigma_{\epsilon}}\exp\left(-\frac{\epsilon ^{2}}{2\sigma_{\epsilon}^{2}}\right),\ \ \ \epsilon=\frac{\gamma-\gamma_{0}}{\gamma_{0}}, \tag{41}\]  
\[\phi(t)=\frac{1}{\sqrt{2\pi}\sigma_{t}}\exp\left(-\frac{t^{2}}{2\sigma_{t}^{2 }}\right).\]  
The normalization of the phase space \(g(\eta,\eta^{\prime})\) distribution is ensured if the Twiss parameters satisfy the condition [43]  
\[\gamma_{\eta}\beta_{\eta}-\alpha_{\eta}^{2}=1. \tag{42}\]  
The level curves of \(g(\eta,\eta^{\prime})\) can be exploited to visualize the geometrical content of Courant-Snyder invariant (see Figure 8), the rms values of the radial and vertical beam sections and of the angular divergences placed on the curve line in correspondence with the tangent points of the lines parallel to the axis \(\eta\) and \(\eta^{\prime}\), respectively. The technicality associated with the evaluation of the convolution integral is just a matter of working out multi-dimensional Gaussian integrals. The use of the integral representation on the r.h.s. of Equation (38) simplifies such a task and leads to an analytically manageable form, as reported below [34]  
\[ \begin{array}{l}\langle S(\
u)\rangle=2 Re\int_{0}^{\tau}(1-t)\frac{e^{-i\
u t -\frac{1}{2}(\pi\mu_{\epsilon}t)^{2}}}{\sqrt{R_{x}(t)R_{y}(t)}}\,dt,\\ \\ R_{\eta}(t)=(1+\alpha_{\eta}^{2})(1-i\pi\mu_{\eta}t)(1-i\pi\mu_{\eta^{\prime}} t)-\alpha_{\eta}^{2},\end{array}  \tag{43}\]  
Figure 8: Courant-Snyder ellipse, \(\gamma_{\eta}^{2}\eta^{2}-2\alpha_{\eta}\eta\eta^{\prime}+\beta_{\eta}\eta^{ \prime 2}=\epsilon_{\eta}\). The abscissa of the point denoted by a dot yields the rms of the e-beam angular divergence, the ordinate of the point denoted by a star yields the associated rms transverse section. where we have included either the effects of the energy spread and emittances, which are specified through the inhomogeneous broadening parameters [34]  
\[ \begin{split}\mu_{\varepsilon}&=4\widehat{N}_{l}\sigma_ {\varepsilon},\\ \mu_{\eta^{\prime}}&=\frac{4\widehat{N}_{l}\gamma^{2 }\varepsilon_{\eta}}{\left(1+\frac{K_{l}^{2}}{2}\right)\beta_{\eta}};\ \ \ \mu_{\eta}=\frac{4\,\widehat{N}_{l}\gamma^{2}\varepsilon_{\eta}}{\left(1+\frac{ K_{l}^{2}}{2}\right)\gamma_{\eta}}k_{\beta}^{2},\\ \varepsilon_{\eta}&=\frac{\varepsilon_{\eta}}{\gamma} ;\ \ \eta=x,y\ \,\\ k_{\beta}&=\frac{\pi K_{l}}{\gamma\lambda_{l}} \equiv\text{betatron motion wave number}.\end{split}  \tag{44}\]  
Taking into account that  
\[\sigma_{\eta^{\prime}}^{2}=\gamma_{\eta}\varepsilon_{\eta},\ \ \sigma_{\eta}^{2}= \beta_{\eta}\varepsilon_{\eta}, \tag{45}\]  
we can also write  
\[ \begin{split}\mu_{\eta^{\prime}}&=\frac{4\, \widehat{N}_{l}\gamma^{2}\sigma_{\eta^{\prime}}^{2}}{\left(1+\frac{K_{l}^{2}}{ 2}\right)(1+\alpha_{\eta}^{2})},\\ \mu_{\eta}&=\frac{4\,\widehat{N}_{l}\gamma^{2}}{ \left(1+\frac{K_{l}^{2}}{2}\right)(1+\alpha_{\eta}^{2})}\left(\frac{\sigma_{ \eta}}{\beta_{T}^{*}}\right)^{2},\\ \beta_{T}^{*}&=k_{\beta}^{-1},\end{split}  \tag{46}\]  
where \(\beta_{T}^{*}\) denotes the value of the Twiss parameter associated with the betatron wave number. The \(\beta_{T}^{*}\) is usually referred to as the undulator \"natural\" focusing parameter and, in absence of further focusing elements is the reference quantity for the electron beam matching.