Bicomplex Waves in Electromagnetic Scattering and Diffraction Problems

Manuscript received April 8, 1999. Manuscript revised September 8, 1999. † The author is with the Dept. of Lightwave Sciences, Osaka Electro-Communication University, Neyagawa-shi, 572-8530 Japan. a) E-mail: [email protected] SUMMARY The mathematical theory of bicomplex electromagnetic waves in two-dimensional scattering and diffraction problems is developed. The Vekua’s integral expression for the two-dimensional fields valid only in the closed source-free region is generalized into the radiating field. The boundary-value problems for scattering and diffraction are formulated in the bicomplex space. The complex function of a single variable, which obeys the Cauchy-Riemann relations and thus expresses low-frequency aspects of the near field at a wedge of the scatterer, is connected with the radiating field by an integral operator having a suitable kernel. The behaviors of this complex function in the whole space are discussed together with those of the far-zone field or the amplitude of angular spectrum. The Hilbert’s factorization scheme is used to find out a linear transformation from the far-zone field to the bicomplex-valued function of a single variable. This transformation is shown to be unique. The new integral expression for the field scattered by a thin metallic strip is also obtained. key words: bicomplex, scattering, diffraction, Hilbert’s factorization technique, Vekua’s formulas