# Light-to-Plasma Momentum Transfer
## 2 Photon Momentum at Low Irradiance
We find the following:  
\[W_{P}=\frac{n_{e}<U>}{U_{L}/(\hbar\omega)}=\frac{n_{e}e^{2}}{2m_{0}\epsilon_{ 0}\omega^{2}}\hbar\omega=(1/2)\frac{\omega_{p}^{2}}{\omega^{2}}\hbar\omega \tag{4}\]  
where the natural oscillation frequency \(\omega_{p}\) of the electrons in a plasma (known as the plasma frequency, see, e.g., Section 8.1 of Rybicki and Lightman [30]) is given in SI units by the following:  
\[\omega_{p}^{2}=\frac{n_{e}e^{2}}{m_{0}\epsilon_{0}}. \tag{5}\]  
The relationship between the wave-vector \(\mathbf{k}\) in plasma and the wave-vector \(\mathbf{k}_{0}\) in a vacuum is readily obtained by the dispersion relationship of light in a plasma (see, e.g., Section 4.4 of [PERSON] [28]). A plasma medium has a dispersion relationship according to the following:  
\[k^{2}=\frac{\omega^{2}-\omega_{p}^{2}}{c^{2}}. \tag{6}\]  
Re-arranging Equation (6), we find the following:  
\[k=\left(1-\frac{\omega_{p}^{2}}{\omega^{2}}\right)^{1/2}\frac{\omega}{c}= \left(1-\frac{\omega_{p}^{2}}{\omega^{2}}\right)^{1/2}k_{0} \tag{7}\]  
Refractive indices vary depending on whether the index is related to the phase or the group velocity of a propagating electromagnetic wave. The phase velocity \(v_{p}\) is defined by \(\omega/k\), where \(k\) is the amplitude of the wave-vector of the wave, while the group velocity \(v_{g}=d\omega/dk\). In a vacuum, both the phase and group velocities are at the same speed-of-light \(c\), but in a dielectric medium, \(v_{p}=c/\eta_{p}\), and \(v_{g}=c/\eta_{g}\), where \(\eta_{p}\) is the phase refractive index and \(\eta_{g}\) is the group refractive index. For a plasma, the phase refractive index is given by the following:  
\[\eta_{p}=\left(1-\frac{\omega_{p}^{2}}{\omega^{2}}\right)^{1/2}. \tag{8}\]  
Similarly, the group refractive index is given by the following:  
\[\eta_{g}=\left(1-\frac{\omega_{p}^{2}}{\omega^{2}}\right)^{-1/2}. \tag{9}\]  
The standard derivation of the phase and group refractive index expressions has been given, for example, in Section 8.1 of Rybicki and Lightman [30] and in Section 2.1.2 of [PERSON] [29]. Using Equations (7)-(9), we have two expressions for the photon momentum:  
\[p_{phot}=\hbar k=\eta_{p}\,\hbar k_{0}, \tag{10}\]and  
\[<\)\(U\)\(>=\frac{e^{2}E_{0}^{2}}{4m_{0}\omega^{2}}. \tag{2}\]0  
Equation (10) corresponds to the [PERSON] description of the photon momentum [16], while Equation (11) corresponds to the [PERSON] description of the photon momentum [17]. The [PERSON] controversy has been generally resolved if an appropriate phase or group refractive index is employed, respectively, in the [PERSON] and [PERSON] expressions [31; 32], though this conclusion has been obscured by associating [PERSON]'s interpretation with the words \"canonical or wave momentum\" and [PERSON]'s interpretation with the \"kinetic momentum\", rather than referring directly to the phase ([PERSON]) or group ([PERSON]) velocities of light (see [19; 20; 33; 34]). For example, the diffraction of light was determined by the phase of the light, so in diffraction calculations, the [PERSON] photon momentum should be used (see, for example, the diffraction \"thought experiment\" discussed by [PERSON] [35]). The propagation of information and the energy of light in a medium depends on the group velocity and, thus, requires the [PERSON] photon momentum.Our short discussion here has demonstrated that the distinction between the Minkowski and the Abraham evaluations of photon momentum vanishes for a plasma medium.