Exact and asymptotic reductions of surface radiation integrals with complex phases to contour integrals

We develop here an efficient method for the reduction of surface radiation integrals to non-singular contour integrals, when we suppose known, as in Physical Optics (PO), the analytic expression of surface currents whose electromagnetic radiation is calculated. This method applies to a large class of oscillatory surface integrals with complex phase, allowing to save computer time as the surface dimensions increase. Although many authors have studied this problem, we present here a novel practical solution, leading to simple non-singular contour expressions (without any special function), (a) in monostatic case, for the radiation of PO currents carried by a perfectly or imperfectly reflective surface, excited by a point source whose radiation can be of arbitrary pattern, without any approximation in the phase of the integrand, (b) in bistatic case, when the integrand has a complex phase which is a quadratic function of the coordinates, as in the case of gaussian beam propagation or in second order phase approximation for the free space Green function at large distance, which apply from infinity to a short distance to the source (near-field case) for plates of arbitrary contours. The complex scaling used in case (b) is used to specifically extend the results of case (a) for (c) the radiation of PO currents on curved plates whose principal curvatures have arbitrary signs, for purely convex, purely concave or saddle surfaces. Comparisons with classic surface integration results show excellent agreements for all cases.