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Giuliano Manara, Sergio Terranova, and Ludger Klinkenbusch: "Comparison between Two Uniform Asymptotic Evaluations of a Typical Diffraction Integral"

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URSI EM Theory Symposium, EMTS 2019, San Diego, CA, 27 – 31 May 2019


A comparison between two different uniform asymptotic high-frequency procedures for the evaluation of a typical diffraction integral is presented in this paper. In particular, attention is focused on the specific case of evanescent wave diffraction from a straight wedge [1]. The two procedures are the Pauli-Clemmow (PC) method [2], [3]  and the Van der Waerden (VW) method [4], [5]. As well known, the usual leading term of the PC method is not able to provide the proper discontinuity compensation when the poles cross the Steepest Descent Path (SDP) away from the saddle point. However, by considering all higher order terms in the PC asymptotic expansion of the diffraction integral [6], it is shown that each higher order term provides a contribution of order K −1/2 . By suitably collecting all these terms of order K −1/2 , a modified leading term of the PC method is obtained, which results in a compact expression consisting of the standard UTD multiplicative form [1], plus a UTD slope-like correction term [7]. It can be easily demonstrated that the above modified leading term of the PC method exactly coincides with the usual leading term of the VW method. Consequently, in all those cases where the usual leading term of the PC method fails to be uniform, the UTD slope-like correction can be added to obtain a uniform solution. 

The modified leading term of the PC method shows an apparent advantage from a numerical point of view, consisting in the fact that its first term coincides with the standard UTD multiplicative form. Moreover, this form has shown to work surprising well in the case of diffraction of a Complex Source Beam by a straight wedge [8], in most case without any need for the UTD slope-like correction. 

[1]  T.B.A Senior and J.L. Volakis, Approximate Boundary Conditions in Electromagnetics, IEE Electromagnetic Waves Series, The Institution of Electrical Engineers, London, United Kingdom, 1995, pp. 332-336. 

[2] P. C. Clemmow, “Some Extensions of the Method of Integration by Steepest Descent,” Quart. J. Mech. Appl. Math., vol. 3, Jan. 1950, pp. 241-256. 

[3] P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields, Piscataway, NJ, USA: IEEE Press, pp. 56-58. 

[4] B. L. Van der Waerden, “On the Method of Saddle Points,” Appl. Sci. Res. B, vol. 2, no. 1, 1952, pp. 33-45. 

[5]. L.B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, Englewood Cliffs, NJ, USA: Prentice-Hall, 1973. 

[6] C. Gennarelli and L. Palumbo, “A Uniform Asymptotic Expansion of a Typical Diffraction Integral with Many Coalescing Single Pole Singularities and a First Order Saddle Point,” IEEE Trans. Antennas Propag., vol. AP-32, no. 10, Oct. 1984, pp. 1122-1124. 

[7]  R. G: Kouyoumjian, G. Manara, P. Nepa, and B. J. E. Taute, “The Diffraction of an Inhomogeneous Plane Wave by a Wedge,” Radio Science, vol. 31, no. 6, Nov./Dec. 1996, pp. 1387-1397. 

[8]  H.-T. Chou, P. H. Pathak, Y. Kim, and G. Manara, “On Two Alternative Uniformly Asymptotic Procedures for Analyzing the High-Frequency Diffraction of a Complex Source Beam by a Straight Wedge,” IEEE Trans. Antennas Propag., vol. AP-66, no. 7, July 2018, pp. 3631-3641.