## 2 Laser-Driven Very-High-Energy Electron/Photon Beam Radiation Therapy System
### Design of Laser Plasma Accelerators for VHE Electron Beams
The maximum energy gain \(W_{\rm max}\), limited due to dephasing, is given by \(\Delta\gamma_{\rm max}=W_{\rm max}/m_{\rm e}c^{2}=(2/3)\alpha\kappa_{\rm self} \alpha_{0}(n\epsilon n\epsilon)\), where \(n_{\epsilon}=m_{\rm e}\omega_{\epsilon}^{2}/4\pi e^{2}=\pi/(r,\lambda_{\epsilon }^{2})\) is the critical plasma density, \(\kappa_{\rm self}=(a_{0}^{2}\,/\,8)\{(1+a_{0}^{2}\,/\,2)^{1/2}-1-\ln([(1+a_{0} ^{2}\,/\,2)^{1/2}+1]/2)\}^{-1}\) is a correction factor of the group velocity for a self-guided relativistic laser pulse, of which the relativistic factor related to the group velocity \(\beta_{\rm g}=v_{g}/c\) is given by \(\gamma_{g}^{2}=1/(1-\beta_{g}^{2})=\kappa_{\rm self}(\omega_{\epsilon}^{2}/ \,\omega_{\epsilon}^{2})=\kappa_{\rm self}(n_{\epsilon}\,/\,n_{\epsilon})= \kappa_{\rm self}\gamma_{g0}^{2}\), where \(\gamma_{g0}=\omega L/\omega_{p}\) is the relativistic factor for the linear group velocity for \(a_{0}^{2}<<1\). The dephasing length \(L_{\rm dp}\) for the self-guided bubble regime is given by \(k_{p}L_{\rm dp}=(2\,/\,3)k_{p}R_{g}\gamma_{g}^{2}=(4\,/\,3)\sqrt{a_{0}}\,\kappa _{\rm self}(n_{\epsilon}\,/\,n_{\epsilon})\). The important parameters of laser plasma accelerators (a list of symbols for designing parameters see in Table 1) for reaching a given energy \(E_{b}\) are summarized as [31]:  
The operating electron plasma density is determined by  
\[n_{\epsilon}=\frac{2}{3}\alpha\kappa_{\rm self}a_{0}\frac{n_{\epsilon}}{ \Delta\gamma_{\rm max}}=3\times 10^{18}[\rm cm^{-3}]\kappa_{\rm self}a_{0} \frac{0.8\mu m}{\lambda_{\epsilon}})^{2}(\frac{200\rm MeV}{E_{b}\,/\,\alpha}) \tag{1}\]  
The accelerator length is set to be equal to the dephasing length as  
\[L_{\rm exc}=L_{\rm dp}=\sqrt{\frac{3}{2}}\frac{(\Delta\gamma_{\rm max}\,/\, \alpha)^{3/2}}{\pi\kappa_{\rm self}^{\rm(1/2}a_{0}})\lambda_{\epsilon}=\frac {2.5\,[\rm mm]}{\kappa_{\rm self}^{\rm(1/2)}a_{0}}(\frac{\lambda_{\epsilon}}{ 0.8\mu m})(\frac{E_{b}\,/\,\alpha}{200\rm MeV})^{v_{g}2} \tag{2}\]  
The pump depletion length due to pulse front erosion becomes  
\[L_{\rm pd}=c\tau_{{}_{L}}\frac{n_{\epsilon}}{n_{\epsilon}}=\frac{3}{2}\frac{ c\tau_{{}_{L}}\Delta\gamma_{\rm max}\,/\,\alpha}{\kappa_{\rm self}a_{0}}= \frac{5[\rm mm]}{\kappa_{\rm self}a_{0}}(\frac{\tau_{{}_{L}}}{30\rm fs})( \frac{E_{b}\,/\,\alpha}{200\rm MeV}) \tag{3}\]  
The pulse duration required for satisfying a dephasing length longer than a pump depletion length is  
\[\tau_{{}_{L}}\geq 14\,[\rm fs]\kappa_{\rm self}^{\rm(1/2)}(\frac{\lambda_{ \epsilon}}{0.8\mu m})(\frac{E_{b}\,/\,\alpha}{200\rm MeV})^{v_{g}2} \tag{4}\]  
The matched spot radius is given by  
\[r_{{}_{L}}=3.1[\mu m]\frac{R_{{}_{n}}}{\sqrt{\kappa_{\rm self}a_{0}}}(\frac{ \lambda_{\epsilon}}{0.8\mu m})(\frac{E_{b}\,/\,\alpha}{200\rm MeV})^{v_{g}2} \tag{5}\]  
where  
\[R_{{}_{n}}=k_{p}r_{{}_{L}}=\{\frac{\ln(1+a_{0}^{2}\,/\,2)}{\sqrt{1+a_{0}^{2} \,/\,2-1-2\ln[  
A laser-plasma driven VHE electron beam radiation therapy system is presented for the electron beam energies--50 MeV (Case A), 100 MeV (Case B), 150 MeV (Case C), 200 MeV (Case D), and 250 MeV (Case E). A drive laser pulse from the laser system is focused on the entrance of gas cells at the normalized laser field \(a_{0}=2\) corresponding to the laser intensity \(I=5.5\times 10^{18}\) W cm\({}^{-2}\). Self-guided propagation of such laser pulses in the gas cell requires the group velocity correction factor \(\kappa_{\mathrm{self}}=1.19\) and the matched spot radius \(R_{m}=3.2\). The wakefield reduction factor \(\alpha\) due to loaded charge \(Q_{b}\) is calculated from \(\alpha^{2}+C\alpha^{3/2}-1=0\), assuming the electron beam radius \(k_{b}\sigma_{b}=1\), where the coefficients are \(C=0.50\), 0.35, 0.29, 0.25, and 0.22 as \(\alpha=0.80\), 0.85, 0.88, 0.89, and 0.90, respectively, for Case A to E. The main parameters for the laser plasma accelerator are provided as shown in Table 2.