When a plane wave is incident upon a dielectric wedge of arbitrary wedge angle, the scattered field may be obtained by solving a dual integral equation in the spectral domain. This dual integral equation is the two dimensional Wiener-Hopf-Fock type integral equations. Factorization of this equation is not possible and an approximate solution is tried.
A geometric optics solution for the wedge may easily be obtained. The physical optics solution for the wedge is then obtained from the boundary fields of the geometric optical approximation. one may define a correction field that is added to the physical optics approximation to satisfy the original dual integral equation. It may be shown that the correction field is to correct the edge diffracted field of the physical optics solution. The fields reflected, refracted, and transmitted from the boundaries are retained as in the geometric optics solution for the corrected solution.
The correction of the edge diffracted field may be calculated asymptotically by assuming a multipole time source at the edge. Then from the dual integral equation for the correction field, One may-derive a dual series equation for the multipole expension coefficients, which is easily amenable to numerical calculation.
The other way to correct the edge diffraction is to assume sheet currents along the dielectric boundaries. One may require these sheet currents to satisfy the edge condition and the dual integral equation. Arbitrary sheet currents may be expanded in a series of Bessel functions (Neumann's expansion), where the order of Bessel function is the fractional order to satisfy the edge condition of the static limit. Dual integral equation then yields the dual series equation for the Neumann's expansion coefficients.
The calculation of these expansion coefficients is to correct the edge diffraction of the physical optics approximation where the far field pattern is well known. The Bessel series satisfying the edge condition, however, gives fields which converge rather slowly in the far field region. One may therefore convert this series expansion into another series that converges fast in the far field region and numerical calculation for the correction field is made.
The numerical calculation of the correction field shows that the sheet current method yields more accurate results than the multipole expansion method. Corrected edge diffracted field patterns and the total field patterns are calculated and are shown in figures.
