The numerical steepest descent path method to calculate physical optics radiations with concave phase variations

In this work, we develop the numerical steepest descent path (NSDP) algorithm to calculate the physical optics (PO) integrals with the quadratic concave phase variations. All the high frequency critical points contributions, including the stationary phase points (SPP), the boundary resonance points (RP) and the vertex points (VP) are clearly expressed. In order to illustrate the transition effect of the critical points, we extend the NSDP method to calculate the PO integrals with the coalescence of the critical points. Numerical results of triangular domains are given to verify the efficiency. The NSDP method could achieve the frequency independent computational workload and obtain accurate results. Together with the PO integrals with the quadratic parabolic and hyperbolic phase terms, this work makes the NSDP theory be complete for treating the PO integrals with quadratic phase variations.