The fast algorithm to calculate the physical optics scattered fields with concave phase variations

In this work, we develop the numerical steepest descent path (NSDP) algorithm to calculate the physical optics (PO) integrals on quadratic surface with concave phase variations. Numerical results demonstrate that the NSDP method could achieve the frequency independent computational workload and error controllable accuracy. Furthermore, to illustrate the transition effect of the critical points, we study the high frequency critical points and extend the NSDP method to calculate the PO integrals with the coalescence of the critical points.