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.