# Light-to-Plasma Momentum Transfer
## 3 Light Momentum Transfer to Plasma at High Irradiance
Converting the angular frequency \(\omega\) to wavelength \(\lambda_{\mu m}\) in microns, we can write the dimensionless reduced vector potential as the following:  
\[a_{o}^{2}=\frac{I\lambda_{\mu m}^{2}}{1.37\times 10^{18}}. \tag{16}\]  
Inertial fusion experiments have typically utilized laser pulses with \(a_{o}<<1\), but some short-pulse (\(<\)100 fs) laser-plasma experiments have achieved \(a_{o}>100\). It has been shown (see, for example, [PERSON] [36], [PERSON] [37] and Section 4.3 Tallents [28]) that the cycle-averaged relativistic Lorentz factor for the electrons accelerated in an electromagnetic field can be found by the following:  
\[<\gamma>=\left(1+\frac{a_{o}^{2}}{2}\right)^{1/2}. \tag{17}\]  
Allowing for the \"de-phasing\" of the electron in the electromagnetic wave oscillation due to the \(\mathbf{k}\)-directed motion, it was also readily shown (see Section 4.1 Tallents [28]) that, when  
Figure 1: Schematic showing the oscillating electric field \(\mathbf{E}\) (solid red curve) and magnetic field \(\mathbf{B}\) (broken curve) of an electromagnetic wave with a drawing of a figure-eight electron motion superimposed on the drift velocity \(\mathbf{v_{k}}\) which is directed parallel to the wave-vector \(\mathbf{k}\). All electrons (circles) in a uniform field exhibited the same drift and oscillation. motion in the \(\mathbf{k}\)-direction was large, the time-averaged velocity in the \(\mathbf{k}\)-direction at large values of \(a_{o}\) can be expressed as the following:  
\[\frac{<v_{k}>}{c}=\frac{a_{o}^{2}}{4+a_{o}^{2}}. \tag{18}\]  
In the laboratory frame, the kinetic energy of an electron in a frame of velocity \(v\) with Lorentz factor \(\gamma\) can be written as \((\gamma^{2}/(1+\gamma))m_{0}v^{2}\). The energy of electrons moving parallel to \(\mathbf{k}\) with a drift velocity \(<v_{k}>\) in a plasma of electron density \(n_{e}\) is expressed by the following: \(<\gamma>^{2}/(1+<\gamma>)\,n_{e}m_{0}<v_{k}>^{2}\). Using Equations (3) and (18), the energy per incident photon is given by the following:  
\[n_{e}<U_{k}>=n_{e}\bigg{(}\frac{<\gamma>^{2}}{1+<\gamma>}\bigg{)}\frac{m_{0}< v_{k}>^{2}}{U_{L}/(\hbar\omega)}=\bigg{(}\frac{<\gamma>^{2}}{1+<\gamma>} \bigg{)}\frac{n_{e}m_{0}c^{2}}{(1/2)e_{0}E_{0}^{2}/(\hbar\omega)}\bigg{(}\frac {a_{o}^{2}}{4+a_{o}^{2}}\bigg{)}^{2}.\]  
Using Equation (17) for the Lorentz parameter \(<\gamma>\), Equation (15) to convert the maximum electric field \(E_{0}\) to the reduced vector potential \(a_{o}\), and Equation (5) for the plasma frequency \(\omega_{p}\), we can write the following for the directed electron energy per incident photon:  
\[n_{e}<U_{k}>=\frac{\omega_{p}^{2}}{\omega^{2}}\bigg{(}\frac{1+a_{o}^{2}/2}{1+ (1+a_{o}^{2}/2)^{1/2}}\bigg{)}\frac{2a_{o}^{2}}{(4+a_{o}^{2})^{2}}\hbar\omega \tag{19}\]  
At low irradiances (small \(a_{o}\)), \(n_{e}<U_{k}>\) varies as \((1/16)(\omega_{p}^{2}/\omega^{2})a_{o}^{2}\,\hbar\omega\). At high irradiances (large \(a_{o}\)), the directed electron energy per photon \(n_{e}<U_{k}>\) varies as \((1/(9\sqrt{2}))(\omega_{p}^{2}/\omega^{2})\)\((a_{o})^{-1}\,\hbar\omega\), suggesting a drop in the \"efficiency\" of the energy transfer from the photons to the \(\mathbf{k}\)-directed electrons. The \(\mathbf{k}\)-directed electron energy \(n_{e}<U_{k}>\) decreases as the irradiance of the beam increases to extreme values (high \(a_{o}\)). The \"optimum\" irradiance for the maximum transfer of kinetic energy in the direction of the light propagation per photon is the reduced vector potential \(a_{o}\approx 3.7\) (see Figure 2).