# Light-to-Plasma Momentum Transfer
## 3 Light Momentum Transfer to Plasma at High Irradiance  
At higher electromagnetic wave irradiances, the motion of the accelerated electrons parallel to the direction of the wave propagation becomes important. At the highest irradiances, we also need to use an appropriate relativistic expression for the electron kinetic energies, and during the electron-acceleration calculations, we need to make an allowance for a relativistic \"mass increase\" from the electron rest mass of \(m_{0}\) to \(\gamma m_{0}\), where \(\gamma\) is the Lorentz parameter. We also need to consider the dielectric plasma effect on the electric and magnetic fields of the electromagnetic wave. When the electron density is below the critical electron density, the plasma refractive index \(\eta_{P}\) modifies the vacuum electric field \(E\) to \(E/\sqrt{\eta_{P}}\) and the vacuum magnetic field \(B\) to \(\sqrt{\eta_{P}}\,B\). Electrons accelerated by an electromagnetic wave in the direction of the electric field are accelerated by the magnetic field \(\mathbf{B}\) of the wave due to the Lorentz \(-e\mathbf{v}\times\mathbf{B}\) force. The magnetic field of a linearly polarized electromagnetic wave produces acceleration in the direction of the beam propagation (the wave-vector \(\mathbf{k}\)-direction) with an acceleration magnitude of \((-e/\gamma m_{0})\,v\,B\,\sqrt{\eta_{P}}\), where \(B\) is the vacuum magnetic field. Given a velocity oscillation in the form of \(v=-eE_{0}/(\sqrt{\eta_{P}}\,\gamma m_{0}\omega)\sin(\omega t)\), where \(E_{0}\) is the vacuum electric field and the magnetic field oscillation in the form of \(B=B_{0}\cos(\omega t)\), the acceleration in the \(\mathbf{k}\) direction has a magnitude \(e^{2}E_{0}^{2}/(\gamma^{2}m_{0}^{2}\omega c)\sin(\omega t)\cos(\omega t)\), provided that the electron motion in the \(\mathbf{k}\)-direction is small. Integrating in time, we obtain the velocity \(v_{k}\) of an electron in the direction of the \(\mathbf{k}\)-vector when irradiated by an electromagnetic wave:  
\[v_{k}=\frac{e^{2}E_{0}^{2}}{2\gamma^{2}m_{0}^{2}\omega^{2}c}\sin^{2}(\omega t). \tag{12}\]  
Integrating again, we can obtain the electron position \(z_{k}\) parallel to the \(\mathbf{k}\)-vector:  
\[z_{k}=\frac{e^{2}E_{0}^{2}}{4\gamma^{2}m_{0}^{2}\omega^{3}c}(\omega t-(1/2) \sin(2\omega t)) \tag{13}\]  
For a small relativistic mass increase (i.e., \(\gamma\) approaching unity) and linearly polarized light, an electron oscillates in the \(\mathbf{k}\)-direction at a frequency of \(2\omega\) with a time-averaged drift velocity in the \(\mathbf{k}\)-direction. The electron oscillates at a frequency of \(\omega\) perpendicular to \(\mathbf{k}\) and \(2\omega\) parallel to \(\mathbf{k}\), so the electron trajectory follows a figure-eight motion with a superimposed time-averaged drift velocity in the direction of the \(\mathbf{k}\)-vector (see Figure 1). After averaging a period of oscillation of the electromagnetic wave, the time-averaged drift velocity in the \(\mathbf{k}\)-direction becomes the following:  
\[<v_{k}>=\frac{e^{2}E_{0}^{2}}{4<\gamma>^{2}m_{0}^{2}\omega^{2}c} \tag{14}\]  
where \(<\gamma>\) is the cycle-averaged Lorentz factor for the oscillating electron. After averaging over the time of a cycle of electromagnetic oscillation, the net velocity of an electron in an electromagnetic field is given by Equation (14). The quantities in Equation (14) can be conveniently grouped into a single parameter, known as the reduced vector potential \(a_{o}\), and defined by the following:  
\[a_{o}^{2}=\frac{e^{2}E_{0}^{2}}{m_{0}^{2}\omega^{2}c^{2}}. \tag{15}\]  
For small values of \(a_{o}\), the drift velocity of an electron in an electromagnetic field is \(<v_{k}>=a_{o}^{2}/4\). A useful quantitative indication of the value of \(a_{o}\) in laser-plasma experiments has been found by relating the electric-field amplitude \(E_{0}\) to the irradiance \(I\) of an electromagnetic wave measured in units of W cm\({}^{-2}\).