Geometric particle-in-cell methods for the Vlasov–Maxwell equations with spin effects

We propose a numerical scheme to solve the semiclassical Vlasov–Maxwell equations for electrons with spin. The electron gas is described by a distribution function $f(t,{\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ that evolves in an extended 9-dimensional phase space $({\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ , where $\boldsymbol s$ represents the spin vector. Using suitable approximations and symmetries, the extended phase space can be reduced to five dimensions: $(x,{{p_x}}, {\boldsymbol s})$ . It can be shown that the spin Vlasov–Maxwell equations enjoy a Hamiltonian structure that motivates the use of the recently developed geometric particle-in-cell (PIC) methods. Here, the geometric PIC approach is generalized to the case of electrons with spin. Total energy conservation is very well satisfied, with a relative error below $0.05\,\%$ . As a relevant example, we study the stimulated Raman scattering of an electromagnetic wave interacting with an underdense plasma, where the electrons are partially or fully spin polarized. It is shown that the Raman instability is very effective in destroying the electron polarization.

Conservation of angular momentum in second harmonic generation from under-dense plasmas

Spin and orbital angular momentum of an optical beam are two independent parameters that exhibit distinct effects on mechanical objects. However, when laser beams with angular momentum interact with plasmas, one can observe the interplay between the spin and the orbital angular momentum. Here, by measuring the helical phase of the second harmonic 2ω radiation generated in an underdense plasma using a known spin and orbital angular momentum pump beam, we verify that the total angular momentum of photons is conserved and observe the conversion of spin to orbital angular momentum. We further determine the source of the 2ω photons by analyzing near field intensity distributions of the 2ω light. The 2ω images are consistent with these photons being generated near the largest intensity gradients of the pump beam in the plasma as predicted by the combined effect of spin and orbital angular momentum when Laguerre-Gaussian beams are used.