A New Aspect on Wedge Diffraction: Hidden Rays on the Shadow Boundary

The phenomenon of diffraction may be defined by any deviation of geometrical rays from rectilinear paths which cannot be interpreted as reflection and refraction. Although the ascending series solutions to some canonical structures as cylinders, spheres, conducting wedges, etc. are available, they are not directly useful in high frequency region. Those ascending series solutions suffer from very slow convergence in short wavelength region and no way out for explaining the diffraction phenomenon. For the perfectly conducting half-plane illuminated by a plane wave, Sommerfeld [1] provided an asymptotic solution, of which the leading and second terms agreed with the geometrical optics and edge-diffracted field, respectively. The Sommerfeld-type diffraction coefficients of perfectly conducting wedges led Keller [2] to formulate the geometrical theory of diffraction (GTD), and later many researchers to develop its improved versions. However, the application of the GTD and its improved versions to penetrable scatterers has been hampered by the lack of exact diffraction coefficients of penetrable wedges. A number of analytical techniques have been investigated to obtain diffraction coefficients of penetrable wedges. But there is no rigorous solution to those diffraction problems until now. One of severe difficulties posed in these problems is that two propagating waves inside and outside a penetrable wedge make it impossible to satisfy the boundary condition on the wedge interfaces. While accurate values of the diffraction coefficients have been obtained using the method of moment and FDTD method, those numerical results could not provide comparable achievements in the physical understanding of the wedge diffraction. As a detour, the simplest way to obtain an approximate but analytic expression on the diffraction coefficients of penetrable wedges may be the physical optics (PO) approximation. But the PO diffraction coefficients cannot include the reflection and refraction on the shadow boundary of wedges. Hence one of the historically tantalizing problems in electromagnetic and optical area may be how to account the field behavior on the shadow boundary of penetrable wedges. Recently the concept of hidden rays was suggested to account the effect of the shadow boundary of penetrable wedges using the same analytic process as the ordinary rays on the lit boundary. This approach was called the hidden rays of diffraction (HRD), which have already been used to construct analytic expressions of the diffraction coefficients of two penetrable wedges. One is a composite wedge composed of a perfect conductor and a lossless dielectric [3], and the other is a lossless dielectric wedge [4]. The accuracy of the HRD diffraction coefficients was assured in comparison with the conventional PO diffraction coefficients. In this paper, E-polarized diffraction by a perfectly conducting wedge is reconsidered to show the relationship between the geometrical rays and the diffraction coefficients. Hidden rays are also geometrical rays satisfying the usual principle of geometrical optics (GO). While ordinary rays exist in the physical region, hidden rays occur only in the complementary region, in which the original media inside and outside wedges are exchanged each other [5]. And the rule of hidden-ray tracing in the complementary region can be found easily. Then the HRD diffraction coefficients are constructed by sum of cotangent functions, which correspond to the geometrical rays on by one. This HRD approach is applied to the diffraction by a composite wedge composed of a perfect conductor and lossy dielectric, of which HRD diffraction coefficients are plotted here.