Non-linear effects in polarized photon transport

Studies on radiative transfer (Chandrasekhar, 1950; Fano et al., 1959; Pomraning, 1973; Fernandez and Molinari, 1993) lead to a clean formulation of polarized photon transport in terms of the vector Boltzmann equation whose solution gives the four Stokes components of the flux, from which the full polarization state of the photons can be determined at any given position, wavelength (energy) and solid angle. One of the relevant results observed during the formulation of the vector transport equation is the non-linearity of the equations for the single components of the flux, despite the linearity of the complete vector equation. This phenomenon has strong implications in photon transport because it makes meaningless the use of a scalar model for describing the intensity component of the flux in the presence of polarization. The non-linearity of the single equations can be illustrated by studying the spectral intensity of X-rays. In fact, exact results for multiple scattering involving the photoelectric effect and Rayleigh and Compton scattering are obtained with recourse to an orders of interactions solution of the vector transport equation in a plane geometry (Fernandez et al., 1993; Fernandez, 1995). The intensity so obtained is compared with that corresponding to a scalar model of transport and average polarized kernels for the same interactions. The extent of the difference between the solutions (for an unpolarized photon source) is then used to illustrate the phenomenon mentioned above. It is worth noting that the non-linearity of the equation prevents Monte Carlo designers from obtaining purely scalar description of the intensity, when polarization is taken into account. It is shown that for a correct Monte Carlo simulation of the intensity a vector description is required as inspired by the vector transport equation (Fernandez and Sumini, 1995; Bastiano and Fernandez, 1996).