ELECTROMAGNETIC FIELDS IN THE PRESENCE OF AN INFINITE DIELECTRIC WEDGE: THE PHASED LINE SOURCE EXCITATION CASE by M. A. SALEM † and A. H. KAMEL (New Jersey Institute of Technology, Newark, NJ 07102, USA and Advanced Industrial, Technological and Engineering Center, PO Box 433, Heliopolis Center, Cairo 11757, Egypt) [Received 6 August 2007. Revise 27 November 2007. Accepted 29 November 2007] Summary Electromagnetic fields, excited by an electric phased line source in the presence of an infinite dielectric wedge, are determined by application of the Kontorovich–Lebedev transform. The Maxwell’s equations together with the conditions of continuity of the tangential field components at the material interfaces are formulated as a vector boundary-value problem. By representing the field components as Kontorovich–Lebedev integrals, the problem is reduced to a system of singular integral equations for the unknown spectral functions. We construct numerical solutions to those equations that permit fields evaluation for values of the wedge refractive index, not necessarily close to unity, and for arbitrary positioned source and observer. Numerical results showing the influence of a wedge presence on the directivity of a phased line source are presented and verified through finite-difference frequency-domain simulations. 1. Introduction If the refraction index takes different values inside and outside a material wedge, the boundary-value problem is not separable and no analytical closed-form solution has been found so far despite the use of a broad variety of methods of mathematical analysis. The methods based on the Kontorovich– Lebedev transform (1) have been successfully applied in the past to a number of scalar problems of diffraction by wedges. In order to solve these problems by an application of the Kontorovich– Lebedev transform, the wave numbers are first converted into pure imaginary numbers and then analytic continuation is used to obtain the wave diffraction solution. This technique was originally used by Oberhettinger (2) and successfully used later on by a number of authors (3 to 7). In (7), the results of (6) were extended to a more general source, an unphased line source, and wedges of arbitrary electric and magnetic constants and with arbitrary source–observer configurations. This paper extends the results of (7) to a vector problem. It shows that the singular integral equation formulation is suitable for numerical analysis and develops numerical procedures that provide numerical results. In section 2, the problem is formulated, the fields are represented by Kontorovich–Lebedev spectral integrals with unknown spectral amplitudes and the system of singular integral equations satisfied by the Kontorovich–Lebedev spectra is derived. In section 3, the singular integral equa- tions are solved numerically by using Rawlins’ perturbation scheme when the transverse wave † 〈ms2.718@gmail.com〉 Q. Jl Mech. Appl. Math, Vol. 61. No. 2 c  The author 2008. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org Advance Access publication 24 January 2008. doi:10.1093/qjmam/hbm029 Downloaded from https://academic.oup.com/qjmam/article/61/2/219/1931170 by KIM Hohenheim user on 22 April 2022