The authors have studied the diffraction pattern produced by a particle field illuminated by an elliptic and astigmatic Gaussian beam. They demonstrate that the bidimensional fractional Fourier transformation is a mathematically suitable tool to analyse the diffraction pattern generated not only by a collimated plane wave [J. Opt. Soc. Am A 19, 1537 (2002)], but also by an elliptic and astigmatic Gaussian beam when two different fractional orders are considered. Simulations and experimental results are presented.
We delimit a regime, valid for most ground-based lidar probings of cirrus clouds, in which the field-of-view dependence of multiple scattering reaches a plateau. In this regime and assuming the phase function to be constant around pi, we formally demonstrate Platt's modification of the single-scattering lidar equation, with a parameter eta(P) accounting for the reduction of the effective scattering coefficient defined so that (1 - eta(P)) is the amount of energy scattered in the forward peak. Then, to cope with nonconstant backscattering functions, we discuss the introduction of an effective backscattering coefficient that is an average of the scattering probabilities around pi.
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