The scattering matrix of random media consisting of large opaque spheres calculated using ray tracing and accounting for coherent backscattering enhancement

Abstract

Phase curves of the scattering Mueller matrix elements are calculated for light scattering in media randomly packed with large, non-transparent spheres ( X = 125 ) , the surfaces of which reflect light according to the Fresnel equations. We here consider three values of the refractive index: m = 0.73 + 5.93 i (metal Al), 1.6 + 1.72 i (metal Fe), and 1.5 + 0.1 i (black glass). We use a Monte Carlo ray-tracing approach, taking into account the coherent backscattering effects. The ray-tracing procedure is based on calculation of the Jones matrix for each ray originating from a random starting point with a given incident direction. When the ray leaves the medium after n reflections, we try to found the so-called reciprocal ray, which is reflected by the same subset of particles, but in the reverse order. If such a ray exists, we calculate the corresponding Jones matrix and sum this with the Jones matrix of the initial ray coherently. Then the Mueller matrix is calculated. We find that the main contributions to the coherent opposition spike are the second and third orders of reflection. Higher reflection orders make only a minor contribution to the total intensity even for highly reflective materials like Al. For these media we detect weak negative polarization branches at small phase angles. Their depth and width are about 0.5% and 7 ∘ , respectively. The iron shows a deeper negative polarization branch, and aluminum (bright metal) demonstrates almost zero polarization; whereas, dark glass takes an intermediate position. We find that for the second scattering order, the normalized elements of the scattering matrix, M 22 / M 11 , M 33 / M 11 , and M 44 / M 11 , depend noticeably on density at small phase angles. All elements on the main diagonal of the scattering matrix are sensitive to the angle of incidence; whereas, those on the secondary diagonals are sensitive to the phase angle. Phase dependencies of the elements - M 21 / M 11 , - M 12 / M 11 , - M 43 / M 11 , and - M 34 / M 11 for oblique illumination shift relative to the case of normal illumination. In the case of - M 21 / M 11 this shift has been observed experimentally.