Exact Evaluation of Time-Domain Physical Optics Integral on Quadratic Triangular Surfaces

This article presents the exact evaluation of the physical optics (PO) integral in time-domain. The surface of a perfect electrically conducting (PEC) scatterer is modeled by quadratic triangles. Using the Radon transform interpretation, the PO integral is reduced to a line integral, which is formed by the intersection of the quadratic surface and the plane defined by the Dirac delta function. It is shown that the resulting line is a quadratic curve, e.g., an ellipse, in barycentric coordinates of the quadratic triangle, since the incident wave is a plane wave and the scattered fields are observed at the far-field. To evaluate the PO integral: 1) the PO integral is represented in barycentric coordinates and the type of the intersecting curve is determined; 2) an appropriate coordinate transformation is applied, e.g., elliptic coordinates for the ellipse; and then 3) the Gauss-Legendre quadrature rule (GLQR) is applied. It is shown that a suitable GLQR order yields the PO integral exactly. The validity and accuracy of the proposed method are demonstrated via numerical examples.