# Multi-GeV Laser Wakefield Electron Acceleration with PW Lasers
## 2 Basic Physics of LWFA  
In this section, we will describe the basic physical process and energy scalings of LWFA. LWFA can be realized by focusing a high-power laser pulse at a relativistic intensity above \(10^{18}\) W/cm\({}^{2}\) onto a gaseous medium, as shown in Figure 1. When such an intense laser pulse interacts with a gaseous medium, the atoms in the medium are ionized at the rising edge of the pulse to turn into an underdense plasma. Thus, the main peak of the laser pulse interacts with an underdense plasma. At the laser intensity in the relativistic regime, where the normalized vector potential a\({}_{0}>1\), electrons in the plasma acquire a relativistic quiver velocity in the laser field. The normalized vector potential a\({}_{0}\) is defined as eE\({}_{0}\)/(m\({}_{\rm e}\)c\(\omega_{0}\)), where E\({}_{0}\), \(\omega_{0}\), e, m\({}_{\rm e}\), and c is the laser electric field amplitude, the laser angular frequency, the electron charge, the electron rest mass, and the speed of light, respectively. When a\({}_{0}\) is comparable to or larger than unity, the maximum quiver velocity of a classically oscillating electron in the laser field is close to the speed of light. As the intense laser pulse interacts with the plasma, the electrons are pushed away from the laser propagation axis by ponderomotive force originated from the laser-intensity gradient of a tightly focused laser pulse. Because the electrons are expelled from the laser axis, leaving much heavier ions behind, an extremely high electrostatic field is induced by the charge separation, which acts as a restoring force for the displaced electrons. As the electrons are expelled and return to the laser axis repeatedly, periodic modulations of electron density following the laser pulse, known as a plasma wave or Langmuir wave, are created [10], as shown in Figure 1. The shape of the plasma wave can be a spherical shell [26; 27], called a plasma bubble, when the \(a_{0}\) is sufficiently higher than 1, and the pulse duration is shorter than half the plasma period. If an electron bunch happens to roll into the bubble by self-injection, the bunch can be rapidly accelerated in the laser propagation direction by the enormous electric field in the bubble: this field is usually stronger by three orders of magnitude than that of the conventional RF linear accelerators. As a comparison, while the current state-of-the-art linac can be driven by S-band RF having 0.1 GV/m [28], the electric field gradients from LWFA can reach as high as 200 GeV/m with a centimeter-scale plasma medium having an electron density of about \(10^{18}\) electrons/cm\({}^{3}\) driven by PW laser pulses. One of the most critical issues in LWFA is to enhance the electron energy for given laser parameters. The electron energy is limited by the effective acceleration length and the average acceleration field strength. The effective acceleration length is determined either by the laser etching (depletion) length \(\mathrm{L_{etch}}\approx\left(\omega_{0}/\omega_{\mathrm{p}}\right)^{2}c_{ \mathrm{TL}}\) or by the dephasing length \(\mathrm{L_{d}}=\left(4/3\right)\left(\omega_{0}^{2}/\omega_{\mathrm{p}}^{2} \right)\sqrt{a_{0}}c/\omega_{\mathrm{p}}\), where \(\tau_{\mathrm{L}}\) is the laser pulse duration, and \(\omega_{\mathrm{p}}=\sqrt{4\pi n_{0}e^{2}/m_{e}}\) is the plasma frequency with \(n_{0}\) being the plasma density. The etching length (or depletion length) denotes the distance limit of laser propagation by the loss of laser energy in plasma media. The dephasing length is the maximum acceleration length for electrons to overtake the accelerating phase of the wakefield. For a non-evolving plasma-bubble in the blowout regime, the average acceleration field strength approximately corresponds to \(\sqrt{a_{0}}/2\) when the dephasing length is larger than the etching length, and \(a_{0}\) is sufficiently larger than 1.