# Light-to-Plasma Momentum Transfer
## 1 Introduction  
We consider the momentum transfer from light to plasma when light propagates through plasma. Many experiments involving high-power lasers depend directly or indirectly on the momentum transfer from light to plasma. For example, the electron density profile of plasma material expanding from laser-irradiated solid targets has been modified due to light pressure [1; 2; 3], and there have been proposals to use direct light momentum to compress plasma material for inertial fusion studies [4; 5; 6]. Light pressure can accelerate electrons in laser-plasmas to high energies that are useful for many applications [7; 8; 9; 10; 11]. Electrons accelerated by light pressure may be used as a heating source for fusion ignition when the electrons or co-moving ions are injected into laser-compressed deuterium/tritium [12; 13; 14]. The interaction of light with electrons in the conduction band of solids can be modeled as light-plasma interactions. For a first approximation, the refractive indices of a conductor or semiconductor can exhibit similar dependencies to plasmas with \"free\" electrons. Quantum plasmonics involves the study of the quantum properties of light and its interaction with such electrons, usually when there are solid nanoscale interfaces, and often involving applications where the light is confined to below-diffraction distances [15]. The momentum of light and light-momentum transfer to electrons in solids is important in solid-state plasmonics. Special relativity predicts the vacuum momentum of a photon of light to be \(\hbar\omega/c=|\hbar\mathbf{k_{0}}|\), where \(\mathbf{k_{0}}\) is the vacuum wave-vector of the photon, \(\omega\) is the photon angular frequency, and \(c\) is the vacuum speed of light. To many commentators, the appropriate value of the momentum of a photon in a dielectric medium has been less clear-cut. Over 100 years ago, [PERSON] [16] proposed a photon momentum of \(\eta\,\hbar\mathbf{k_{0}}\) and [PERSON] [17], a photon momentum of \((1/\eta)\,\hbar\mathbf{k_{0}}\), where \(\eta\) is the refractive index of the medium of propagation. The different and seemingly incompatible proposals for photon momentum have become known as the Abraham-Minkowski controversy with many experiments, \"thought experiments\", and much theoretical work undertaken in an effort to resolve the controversy (for reviews, see [18; 19; 20; 21; 22]). Quantum theories of light propagation have determined a momentum operator for a radiation field in order to evaluate the progression of electromagnetic waves [23; 24]. Quantum optics then associated the [PERSON] photon momentum with the momentum ofthe field, plus the momentum associated with the medium polarization, while the Abraham photon momentum considered the momentum of the electromagnetic field alone [25]. The momentum of light has become an increasingly important parameter in structured light studies where, for example, orbital angular momentum due to the field spatial distribution is produced [26]. In plasmas, an electromagnetic wave interacts with free electrons. Considering the classical (non-quantum) electron motion in an electromagnetic wave, we show in this paper that for a plasma, the Minkowski and Abraham expressions for photon momentum are equivalent. With focused laser light, the number of photons in a laser pulse can be large. For example, for a peak irradiance of \(10^{16}\) W cm\({}^{-2}\) and \(1\) micron wavelength, there are typically \(\approx\)\(10^{16}\) photons per picosecond per micron focused radius in a laser pulse. Nevertheless, the energy or momentum transferred from light to plasma can be usefully evaluated per unit of photon energy \(\hbar\omega\) or photon momentum \(\hbar\mathbf{k}\). We follow this \"per photon\" approach in our treatment in this paper, as it aids comment on the Abraham-Minkowski controversy. We evaluate the energy and momentum per photon transferred from light to plasma over a range of irradiances, including the extremely high irradiances now possible with focused high-power, short-duration laser pulses.