## 1. Introduction

[2] Although its geometry is simple, there is no rigorous solution to the diffraction by a dielectric wedge until now [*Lewin and Sreenivasiah*, 1979]. The ordinary ray-tracing in its physical region provides geometrical optics (GO) field. Here each ray denotes the corresponding plane wave with the same amplitude and propagation angle. The physical optics (PO) approximation renders its diffraction coefficients approximate but analytic as finite series of cotangent functions with angular period 2*π* [*Kim et al.*, 1991a]. It should be noted that there is the one-to-one correspondence between the ordinary rays of GO field and the cotangent functions of PO diffraction coefficients. However, the PO diffraction coefficients cannot satisfy the boundary condition at wedge interfaces [*Bowman et al.*, 1969] and the edge condition at wedge tip [*Meixner*, 1972]. Some heuristic modifications of the PO diffraction coefficients have been used to account finite dielectric constant [*Burnside and Burgener*, 1983], finite conductivity [*Luebbers*, 1984], and composite [*Booysen and Pistorius*, 1992]. Their heuristic solutions yielded acceptably accurate results only in some limited cases. Numerical calculations of the diffraction coefficients have also been performed using the method of moment [*Wu and Tsai*, 1977] and the FDTD method [*Stratis et al.*, 1997]. In spite of those valuable results, numerical techniques could not provide comparable achievements in the physical understanding of edge diffraction.

[3] The PO diffraction coefficients of a dielectric wedge have been corrected by employing the dual integral equations [*Kim et al.*, 1991b]. The error of the PO solution was interpreted by the nonzero of the fictitious field emanating from the PO currents in the complementary region, in which the media inside and outside a dielectric wedge are exchanged each other. The concept of the complementary region may be considered as an extended version of the extended boundary condition [*Bates*, 1980] or the null-field method [*Storm and Zheng*, 1987]. Then the nonuniform currents were approximated by the multipole expansion at the wedge tip or the Neumann's expansion along the wedge interfaces. Those expansion coefficients should be calculated numerically under the condition that those radiated field had to cancel out the fictitious PO field in the complementary region. Although it provided an improved solution to the diffraction by a dielectric wedge, its numerical calculation suffered from instability.

[4] Recently the method of hidden rays was suggested as an analytic procedure on the correction of the error posed in the PO diffraction coefficients of composite wedge [*Kim*, 2007]. Its basic idea was inspired from investigating the perfectly conducting wedge. Its exact diffraction coefficients are expressed by sum of four cotangent functions with the angular period 2*πν*_{∞}, where *ν*_{∞} can be derived from the edge condition at the tip of a perfectly conducting wedge [*Sommerfeld*, 1954]. In contrast, the corresponding PO diffraction coefficients consist of two cotangent functions with the 2*π* angular period. After replacing the angular period of the PO diffraction coefficients by 2*πν*_{∞}, one may multiply the PO cotangent functions by 1/*ν*_{∞} to keep the corresponding residues equal to the GO field. Then the PO diffraction coefficients may be changed into sum of two cotangent functions among the exact diffraction coefficients. Let us apply the one-to-one correspondence in a reverse sense. Then one may find two additional rays, which correspond to the remaining two cotangent functions among the exact diffraction coefficients. Two additional rays are geometrical rays which obey the usual principle of GO but do not exist in the physical region. These rays can be traced only in the complementary region. This new ray-tracing law provides an extended GO field consisting of two ordinary rays in the physical region and two hidden rays in the complementary region. Employing the one-to-one correspondence between geometrical rays and cotangent functions, one may construct the exact diffraction coefficients routinely.

[5] It is well known that PO approximation of ordinary rays provides not only GO term but also edge-diffracted field in the physical region. In contrast, PO approximation of hidden rays contributes only to edge-diffracted field in the physical region. Hence the presented method is called the hidden rays of diffraction (HRD). Its procedure may be generalized as shown in Figure 1. The usual principle of GO provides ordinary rays in the physical region. After termination of ordinary ray-tracing in the physical region, the usual principle of GO is also applied to trace hidden rays in the complementary region. Then, the diffraction coefficients are constructed by finite series of cotangent functions, which have one-to-one correspondence with not only ordinary rays in the physical region but also hidden rays in the complementary region. The angular period of the cotangent functions is adjusted to satisfy the edge condition at wedge tip. The accuracy of the diffraction coefficients in the physical region can be measured by checking how closely the diffraction coefficients satisfy the null-field condition in the complementary region.

[6] In this paper, the method of hidden rays is applied to the E-polarized diffraction by a dielectric wedge. The formulation of dual integral equations is summarized in section 2. The ordinary ray-tracing provides the complete expression on the GO field including multiple reflections inside the dielectric region. And the corresponding PO solution consists of the GO field and the edge-diffracted field, of which diffraction coefficients are given by finite series of cotangent functions in section 3. Section 4 shows the trajectory of hidden rays in the complementary air and dielectric regions. The last actually reflected ray in the physical dielectric region becomes the first hidden ray in the complementary dielectric region. The one-to-one correspondence between the geometrical rays and the cotangent functions provides the improved diffraction coefficients in analytic form. In section 5, the diffraction coefficients and field patterns are plotted in figures. It is shown that the presented diffraction coefficients converge to zero in the complementary regions more closely than the PO solution. The conclusions are summarized in section 6. The time convention exp(−*iωt*) is adopted and suppressed here.