Radiating Systems, Multipole Fields and Radiation — John David Jackson
Physics — E&M · shadow-scattering-optical-theorem
Problem
(a) Show from (10.125) that the integral of the shadow scattering differential cross section, summed over outgoing polarizations, can be written in the short-wavelength limit as
σsh=∫d2x⊥∫d2x⊥′⋅4π21∫ei(x⊥−x⊥′)⋅k⊥d2k⊥
and therefore is equal to the projected area of the scatterer, independent of its detailed shape.
(b) Apply the optical theorem to the ``shadow'' amplitude (10.125) to obtain the total cross section under the assumption that in the forward direction the contribution from the illuminated side of the scatterer is negligible in comparison.