INTRODUCTION
The model adopted has two open problems. The first one regards coherence (Fernandez, 1999): the vector transport equation behaves linearly only for an incoherent source (Pomraning, 1973), i.e. when there is not a prevailing phase at the source beam. Coherent radiation is not considered yet in the transport models used to study X-ray diffusion.
The second problem regards variance reduction. Actually, the variance reduction on the angular variables is performed using the average kernel, while the Stokes components Q, U, and V are computed using weights. As a result, we use a mixed method in which only the first component of the Stokes parameters, the intensity I, is optimized
Coherence effects
The influence of coherence is very important since it have been shown that the vector transport equation behaves linearly only for an incoherent source (Pomraning, 1973), i.e. when there not exist defined relationships connecting the phases of the waves in the source beam. By computing the time average of the momentary intensity for two beams along the y direction corresponding to a retardation e and having a phase difference d, it can be shown that an uniform phase produces the following non-linear transformation of the Stokes parameters:
(1)
where Il1, Il2,Ir1 and Ir2 denote, respectively, thel or r time average component of the square electric field amplitude of the beams 1 or 2 (in the Stokes’ (L) representation). The square roots in Eqn. (1) express the well known result in optics that phased waves interact overlapping the moduli of their electric fields (in place of overlapping their squared ones) which gives place to the wave interference effect. Such a result would suggest that the whole transport equation should have a non-linear term in case of a coherent source. It is worth to note, however, that since the non-linearity involves only the first two components of the Stokes vector (and not its components U and V), theinfluence of coherence would not modify the polarisation state but only the intensity of the resulting beam.[1]
The diffusion of coherent radiation is not considered yet in the transport models used to describe x-ray diffusion, even if, clearly, the only way to explain properly the exchange between states of coherence and incoherence should be that of describing both components with the same transport equation.
An alternative approach has been followed in the frame of radar wave diffusion (Mishchenko, 1996), to explain the presence of coherent states (in albedo reflection) which appear in the neighbourhood of the backscattering direction, even when the source beam is incoherent.
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Figure 1. Schematic explanation of the interference nature of coherent backscattering. When the viewing direction and the illuminating direction are exactly opposite (a=0) we get constructive interference. By adding all the contributions from the different number of collisions we obtain the coherent solution.
For the moment this effect has been studied using the Bethe-Salpeter equation (Barabanenkov et al., 1991). As it is shown in figure 1, the coherence term is built by adding the conjugate contributions of all the n-collision chains followed in the right and back sense. It has been claimed that this effect explains the light reflection by ice particles floating in the Saturn rings. No general solution is still available, except in the case of Rayleigh interactions where a complete solution has been found (Ozrin, 1992) which describes the angular dependence of the additive term.
The total solution for both components, incoherent and coherent, is calculated by adding the two solutions, I and Icoh, which denote respectively, the one obtained by solving the Boltzmann vector equation and the coherent contribution obtained by solving the Bethe-Salpeter equation:
(2)
The coherent term described above is narrowly defined near the backscattering direction, and decreases very rapidly when the scattering angle departs from 180 degrees. This approximation is considered sufficient in the radar waves regime because the two components have a low chance to exchange states, and therefore, can be considered as practically independent.
This kind of coherence is also of interest in the x-ray regime, since there is some evidence of the existence of coherent states produced by incoherent sources. Paradoxically, this states have been studied in correspondence with the so called ‘incoherent’ or Compton scattering (Schülke, 1989) but not in relation with ‘coherent’ or Rayleigh scattering. It is hoped that this mention may attract the attention of the readers on this point.
REFERENCES
Barabanenkov YuN, Kravtsov YuA, Ozrin VD and Saichev AI (1991) Enhanced backscattering in optics, in: Progress in Optics XXIX. E. Wolf (Ed.), pp. 65-197. Elsevier, Amsterdam.
Fernández, J.E. (1999) Polarization effects in multiple scattering photon calculations using the Boltzmann vector equation Radiation Physics and Chemistry 56, 27-59
Mishchenko MI (1996) Diffuse and coherent backscattering by discrete random media--I. Radar reflectivity, polarisation ratios, and enhancement factors for a half-space of polydisperse, nonabsorbing and absorbing spherical particles. Journal of Quantitative Spectroscopy & Radiative Transfer 56, 673-702
Ozrin VD (1992) Exact solution for coherent backscattering of polarised light from a random medium of Rayleigh scatterers. Waves in random Media 2, 141-164.
Pomraning, G.C. (1973) The Equation of Radiation Hydrodynamics. Pergamon Press, Oxford.
Schülke W (1989) Inelastic x-ray scattering. Nucl. Instrum. Methods A280, 338-348.
FOOTNOTES
[1] The only parameter that can change is the polarisation degree which measures the ratio between the unpolarised and polarised parts. See Eqn. (18)-(20).
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