# Light-to-Plasma Momentum Transfer
## 2 Photon Momentum at Low Irradiance  
When an electromagnetic wave of low irradiance is incident into a plasma, the motion of the free electrons in the plasma are, to a good approximation, parallel to the electric field of the wave in a direction transverse to the wave-vector \(\mathbf{k}\). We consider a plasma with an electron number density \(n_{e}\) much less than the critical electron density, so that wave reflection, light scattering, and electron-collision processes are negligible. A plasma profile could reflect light (and to a lesser extent, scatter light), but in a \"gedanken\" experiment, we can assume sufficiently small refractive index deviations from unity that these processes are negligible or, alternatively, only consider the light impinging on a uniform plasma that had not been reflected or scattered. For most laser-plasma experiments, ignoring collisions in the light-plasma interaction has resulted in very accurate approximations. Even at electron densities approaching critical density, the mean free path for collisions is much greater than the electron oscillation range in an electromagnetic wave. For example, the electron-electron mean free path is \(\approx\)2 \(\upmu\)m at a temperature of 30 eV and an electron density of \(10^{20}\) cm\({}^{-3}\) (see Section 5.3 [27]), while the electromagnetic oscillation distance is \(\approx\)0.02 \(\upmu\)m for a wavelength of 1 micron and an irradiance of \(10^{16}\) W cm\({}^{-2}\) (see Section 4.1 [28]). Due to the mass difference between electrons and ions, the electron-ion interaction difference is much greater than the electron-electron mean free path. The assumption that electron oscillation in an electromagnetic field is collision-less and local over distances much smaller than typical focal point sizes has been an excellent approximation used widely to evaluate local refractive indices and other parameters in laser-produced plasmas. A linearly polarized electromagnetic wave varying in time as \(E=E_{0}\cos(\omega t)\) with a frequency of \(\omega\) accelerates an initially stationary free electron at a rate of \(-Ee/m_{0}\), where \(e\) is the charge on the electron and \(m_{0}\) is the rest mass of the electron. Integrating the electron acceleration with respect to time gives the electron velocity as \(v=-eE_{0}/(m_{0}\omega)\sin(\omega t)\) with an electron energy \(U\), given by:  
\[U=(1/2)m_{0}v^{2}=\frac{e^{2}E_{0}^{2}}{2m_{0}\omega^{2}}\sin^{2}(\omega t). \tag{1}\]  
The average of a period of oscillation yields the time-averaged ponderomotive, or quiver energy, \(<\)\(U\)\(>\) of an electron in a propagating electromagnetic wave with an electric-field amplitude of \(E_{0}\):  
\[<\)\(U\)\(>=\frac{e^{2}E_{0}^{2}}{4m_{0}\omega^{2}}. \tag{2}\]The ponderomotive energy of a single electron in an electromagnetic wave has been discussed in many texts (see, for example, Section 2.4.1 of Tallents [29]). The ponderomotive energy per unit volume in a plasma of electron number density \(n_{e}\) is given by \(n_{e}<U>\). The energy density \(U_{L}\) of an electromagnetic wave in a vacuum is related to the electric-field amplitude \(E_{0}\) by the following:  
\[U_{L}=(1/2)\epsilon_{0}E_{0}^{2} \tag{3}\]  
where \(\epsilon_{0}\) is the vacuum dielectric constant. The vacuum density of photons per unit of volume is then simply \(U_{L}/(\hbar\omega)\). The ponderomotive energy per incident photon \(W_{P}\) in a plasma is given by the ponderomotive energy per unit of volume \(n_{e}<U>\) divided by the vacuum density of photons per unit of volume \(U_{L}/(\hbar\omega)\).