Extinction paradox and actual power scattered in light beam scattering: a two-dimensional study

The extinction paradox is examined by applying partial-wave analysis to a two-dimensional light beam interacting with a long transverse cylinder without absorption, assuming always short wavelengths. We show that the (conventional) power scattered, Psca, except for a very narrow beam hitting a transparent cylinder on axis, is always double the power directly intercepted by the scatterer, Pitc, including a zero result for Psca when the incident beam is basically off the material surface. This contradicts the interpretation that attributes one half of Psca to edge diffraction by the scatterer. Furthermore, we identify the shadow-forming wave (SFW) from the partial-wave sum in the forward direction and show that the actual power scattered or, equivalently, the power depleted from the incident beam is equal to one unit of Pitc for a narrow beam, gets larger for a broader beam, and approaches 2Pitc for a very broad beam. The larger value in the latter cases is due to the extent of divergence of the SFW beam out of the incident beam at distances well beyond the Rayleigh range.