Asymptotic solutions for backscattering by smooth 2D surfaces

High-frequency asymptotic expansions of electric and magnetic fields are obtained at a perfectly conducting smooth 2D surface illuminated by a plane incident wave in two cases of TE and TM linear polarization. Diffraction corrections up to the second order of the inverse large parameter p = ak ( where a is a curvature radius at the specularly reflected point, and k is a field wavenumber) to the geometrical optics fields, and specifically to their phases, backscattering cross-sections (HH and VV for TE and TM polarizations, correspondingly), as well as the polarization ratio HH/VV, are derived for the specular points of a general form. These general results are applied to backscattering from cylinders with conical section directrixes ( circle, parabola, ellipse and hyperbola), and a number of new compact explicit equations are derived, especially for elliptic and hyperbolic cylinders illuminated at an arbitrary incidence angle relative to their axes of symmetry.