## 2 Laser-Driven Very-High-Energy Electron/Photon Beam Radiation Therapy System
### Design of Laser Plasma Accelerators for VHE Electron Beams  
The laser plasma accelerator comprises an injector cell filled with a mixed gas, e.g., 98% helium gas mixed with 2% nitrogen, and an accelerator cell filled with a pure gas, e.g., hydrogen or helium gas. Figure 4 illustrates schematically a physical process for the wakefield excitation and electron trapping and acceleration in wakefields [26], which are generated when an intense laser pulse propagates a neutral mixed gas in the injector cell. The evolution of plasma electron density is shown in Figure 4a and the excited longitudinal wakefield is shown in Figure 4c. As shown in Figure 4b, helium and the outer shell electrons of nitrogen up to N\({}^{5+}\) are fully ionized in the leading front of the laser pulse with an intensity of 1.5 \(\times\) 10\({}^{16}\) W/cm\({}^{2}\) to produce plasma electrons in the outer region of the laser pulse, of which the boundary is indicated by a thin dotted line. Since two inner shell (\(K\)-shell) electrons of N\({}^{6+}\) and N\({}^{7+}\) are ionized at a laser intensity higher than 1 \(\times\) 10\({}^{19}\) W/cm\({}^{2}\), the inner shell electrons are produced only near the peak intensity of the laser pulse, of which the intensity profile is indicated by a thick dotted line for the normalized vector potential \(a_{0}=0.855\times 10^{-9}\)\(L^{1/2}(\mathrm{W/cm^{2}})\)\(\lambda_{\mathrm{L}}(\mathrm{\mu m})=2\), where \(L\) is the intensity and \(\lambda_{\mathrm{L}}\) is the laser wavelength. In Figure 4a, a solid curve indicates the evolution of the ionization level of nitrogen (the electron number of ionized nitrogen atom) along the propagation axis. A boundary of the plasma region containing the inner shell electrons from ionized N\({}^{6+}\) and N\({}^{7+}\) is indicated by a thin dashed line. Plasma electrons contained in the boundary are blown out by radiation pressure (ponderomotive force) of the laser pulse with the relativistic intensity \(a_{0}>>1\) and form a narrow, dense electron sheath surrounding a spherical ion column behind the laser pulse, often referred to as a bubble, as shown in Figure 4b. Such charge separation generates a strong longitudinal electric field of the order of 100 GV/m at a plasma electron density of 10\({}^{18}\) cm\({}^{-3}\), which is three orders of magnitude higher than an accelerating field of conventional RF accelerators. Inside the bubble, an electron undergoes a strong focusing force simultaneously. Hence, once electrons are trapped into a bubble, they are efficiently accelerated up to high energy of the order of 1 GeV over a dephasing length of the order of 1 cm, where accelerated electrons outrun a proper accelerating phase. In the bubble (blowout) regime for \(a_{0}\geq 2\), since an electron-evacuated cavity shape is determined by balancing the Lorentz force of the ion sphere exerted on the electron sheath with the ponderomotive force of the laser pulse, the bubble radius \(R_{\mathrm{B}}\) is approximately given as \(k_{\mathrm{\rho}}R_{\mathrm{B}}=2\sqrt{a_{0}}\), where \(k_{\mathrm{\rho}}=(4\pi\sigma n_{\mathrm{e}})^{1/2}\) is the plasma wave number evaluated with the unperturbed on-axis density \(n_{\mathrm{e}}\), and the classical electron radius \(r_{\mathrm{e}}=e^{2}/mc^{2}=2.818\times 10^{-13}\) cm with electron charge \(e\), mass \(m_{\mathrm{e}}\), and vacuum light velocity \(c\). The accelerating field \(E_{Z}\) is given by \(E_{Z}/E_{0}=(1/2)\omega k_{\mathrm{\rho}}R_{\mathrm{B}}\), where \(E_{0}=mc\omega_{\mathrm{\rho}}/e=96\) (GV/\(m\)) [\(n_{\mathrm{e}}/10^{18}(\mathrm{cm}^{-3})\)]\({}^{1/2}\) is the non-relativistic wave-breaking field and \(\alpha\) represents a factor taking into account the beam loading and the difference between the simulation and theoretical estimation.