An Alternative Treatment of Saddle Stationary Phase Points in Physical Optics for Smooth Surfaces

In a recent paper on computing the physical optics (PO) integral with a saddle stationary phase point (SPP) by numerical steepest descend method (NSDM) (F. Vico-Bondia, “A new fast physical optics for smooth surfaces by means of a numerical theory of diffraction,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 773-789, Mar. 2010), a one dimensional integral was obtained, on the path of which there existed higher-order (up to third) poles. To avoid tackling the singular integral, the Abel's summation technique was used to solve the problem ingeniously. However, it has been shown in some literatures that the integral would be divergent if there were even-order poles on the path of integral. Superficially, the integral in F. Vico-Bondia, 's paper, does not obey this law and the main aim of the communication is to clear this contradiction. We show rigorously that higher-order poles have no contributions to the final result. To acquire a general law, we further consider arbitrary polynomials of higher degrees for the PO integral, in which the orders of poles can be arbitrary number. In such a general case, we still show that all higher-order poles have no contributions and thus solve the contradiction satisfactorily. Numerical examples are presented to validate the new derivations and illustrate the accuracy and efficiency of NSDM.