Radio Science

Hidden rays of diffraction for dielectric wedge

Authors


Abstract

[1] Electromagnetic diffraction by a dielectric wedge was analyzed using the method of hidden rays. After ordinary ray-tracing is terminated in the physical region the usual principle of geometrical optics is also applied to trace the hidden rays in the complementary region, in which original media of background and dielectric are exchanged for each other. 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 constructed diffraction coefficients was checked by showing how closely the null-field condition is satisfied in the complementary region.

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 2π [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 2πν, 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 2π angular period. After replacing the angular period of the PO diffraction coefficients by 2πν, 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.

Figure 1.

Schematic diagram on the method of hidden rays.

[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.

2. Dual Integral Equations

[7] Figure 2 shows the geometry of a dielectric wedge with relative dielectric constant equation image in Sd. Two boundaries of the air region S0, equation image = 0 and θd, satisfy 0 < θd < π. When an E-polarized unit plane wave ui(ρ, θ) with an arbitrary angle θi is incident on the dielectric wedge, the z – component of the total electric field u(ρ, θ ) may be written into the dual integral equations [Kim et al., 1991a] as

equation image
equation image

where k0 = wequation image and kd = k0equation image denote the wave numbers in S0 and Sd, respectively. The operator F−1 denotes two-dimensional inverse Fourier transform, which is defined here by F−1 [G(α, β)] = g(x, y) = g(ρ cos θ, ρ sin θ ) = g(ρ, θ). And J0, M0, Jd and Md are given by

equation image
equation image
equation image
equation image
Figure 2.

A dielectric wedge illuminated by an E-polarized plane wave.

[8] One of interesting features of the above dual integral equations (1) and (2) is that the total field becomes zero in the complementary regions, as shown in (1b) and (2b). The region S0 denotes the physical region filled with air. However, its complementary region Sd(0) is a fictitious region of Sd, of which medium is replaced by air. Then (1) means that Jd + Md on θ = θd and J0 + M0 on θ = 2π may generate the scattered field in S0, but cancel out the incident field in Sd(0). In the same manner, S0(d) denotes the fictitious dielectric region corresponding to S0. Then (2) describes such a radiation phenomenon that Jd + Md on θ = θd and J0 + M0 on θ = 0 provide not only the total field in Sd but also the null-field in its complementary dielectric region S0(d).

3. Physical Optics Approximation

[9] Figures 3a and 3b show the trajectories of ordinary rays in cases that an E-polarized plane wave ui is incident on the dielectric boundaries of θ = 2π and θd, respectively. The propagation angle of each ray is easily determined by the Snell's law as

equation image
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equation image
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The above internal reflections continue until the propagation angle of the last ordinary ray uN+1 (uN+1+), θN+1N+1+) is located between 0 and θd. Then the total number of internal reflections, N(N+) is sum of N0 (N0+) and Nd (Nd+), which denote the total numbers of internal reflections on the dielectric boundaries of θ = 0 and θd, respectively.

Figure 3a.

Ordinary rays traced in the physical region for the incidence on the θ = 2π boundary.

Figure 3b.

Ordinary rays traced in the physical region for the incidence on the θ = θd boundary.

[10] The amplitudes of the geometrical rays un(un+) and un.t(un.t+) are denoted by Kn(Kn+) and Kn.t(Kn.t+), respectively, which are given by multiplication of the Fresnel's reflection coefficients as

equation image
equation image

where

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[11] Then the GO field ug may be expressed by

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where

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In (7) and (8), D(a, b) is 1 for a ≤ θ ≤ b and 0 elsewhere. And in (8c) and (8d), δN (δN+) is 1 for N(N+) = odd and 0 elsewhere.

[12] Replacing the exact field u(ρ, θ ) in (3) by the GO field ug(ρ, θ) in (7), and inserting the PO approximation terms of J0, M0, Jd, and Md in the dual integral equations (1a) and (2a) provide the PO field up(ρ, θ) consisting of the GO field ug and integral of diffraction coefficients along the steepest descent (SDP) path [Felsen and Marcuvitz, 1973] as

equation image

The PO diffraction coefficients outside the dielectric wedge f1(w) may be expressed by

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where

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In the same manner, the PO diffraction coefficient inside the dielectric region, f2(w) can be expressed by

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4. Hidden Rays of Diffraction

[13] As described in the previous section, the GO field was given by sum of ordinary rays in the physical region, and the PO field was given by the integral of the PO diffraction coefficients f1,2(w) along the Sommerfeld integral path P. As shown in (10)–(12), f1,2(w) was given by finite series of cotangent functions. Deforming the path P into the SDP, one may express the PO field by sum of the GO field and the edge-diffracted field. It implies that the GO field is the residues of f1,2(w) in the closed P-SDP and the edge-diffracted field is the SDP integral of f1,2(w). Hence one may find the one-to-one correspondence between the ordinary rays of the GO field and the cotangent functions of the PO diffraction coefficients. However, it is well recognized that the PO field cannot satisfy the boundary condition at the wedge interfaces and the edge condition at the wedge tip. To remove the error posed in the PO field, f1,2(w) should be corrected under the condition that the residues of the corrected f1,2(w) in the closed P-SDP provides the same terms as the GO field. Then one may consider two strategies on the modification of f1,2(w). At first, the angular period 2π of f1,2(w) is changed into 2πνɛ, where νɛ is taken to satisfy the edge condition at the dielectric wedge tip. To keep the residues of the modified diffraction coefficients f1,2(a)(w) in the closed path P-SDP equal to the GO field, the factor 1/νɛ is multiplied to the modified diffraction coefficients. The second modification is implemented by adding another term f1,2(ν)(w) to f1,2(a)(w). It is assumed that f1,2(ν)(w) is also expressed by finite series of cotangent functions with angular period 2πνɛ. Applying the one-to-one correspondence to f1,2(ν)(w) in a reverse sense, the cotangent functions of f1,2(ν)(w) generate the same number of the geometrical rays. However, the residues of f1,2(ν)(w) in the closed P-SDP should be zero in the physical region because the residues of f1,2(a)(w) in the closed path P-SDP has already provided the exact GO field. It implies that the geometrical rays corresponding to f1,2(ν)(w) cannot exist in the physical region. According to the formulation of dual integral equations, the exact solution to wedge diffractions should become zero in the complementary region. Then the geometrical rays corresponding to f1,2(ν)(w) may be interpreted by the hidden rays in the complementary region. The principle of the ordinary ray-tracing was also applied until all of the hidden rays should be located only in the complementary region. Then the one-to-one correspondence provides the additional diffraction coefficients f1,2(ν)(w) from the hidden rays in the complementary region. The final diffraction coefficients f(e)(w) may be expressed by sum of f1,2(a)(w) and f1,2(ν)(w).

[14] The hidden rays can be obtained easily by applying the usual principle of GO in the complementary region. As shown in Figure 4a, the first hidden ray uN+1 is generated inside the complementary dielectric region after the last ordinary ray uN+1 is captured inside the physical dielectric region. Then the first hidden ray uN+1 is incident on the supplementary dielectric boundary. Let u2m−1 denote the hidden ray undergoing the (2m − 2 − N) - th internal reflection from the first hidden ray uN+1. Then as shown in Figure 4b, the ray u2m−1 impinges upon the dielectric boundary, and then generates the refracted ray u2m − 1.t and the reflected ray u2m. The hidden ray u2m is incident on the dielectric boundary, and then generates the refracted ray u2m.t and the reflected ray u2m+1. The propagation angle of each hidden ray is easily determined by employing the conventional Snell's law. It should be noted that (4e) and (4g) are also satisfied in the hidden ray-tracing. Unlike (4d) and (4f), the propagation angle of the refracted ray um.t, θm.t is specified differently by

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Figure 4a.

Analytic continuation of the last ordinary ray into the complementary region.

Figure 4b.

Hidden rays traced in the complementary region.

[15] The amplitudes of the hidden rays um and um.t, denoted by Km and Km.t are given by multiplication of the Fresnel's reflection coefficients as the same forms as (5a) and (5b), respectively. Furthermore, the mth Fresnel's reflection coefficient Rm is also expressed by (6a). In the same manner, the hidden rays corresponding to Figure 2b can be easily traced.

[16] Any ray inside the dielectric region is reflected on one of its two boundaries and then incident on the other boundary. Hence the propagation direction of any ordinary or hidden ray in the dielectric region should be counted differently as

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The angles θ2m.0+, θ2m.d+, θm+1.d+, and θ2m+1.0+ are also defined in the same manner as (14).

[17] The internal reflection continues before any hidden ray penetrates into the next periodic physical dielectric regions as [2πνɛ, θd + 2πνɛ] or [−2πνɛ, −2πνɛ + θd]. If a hidden ray falls in those physical dielectric regions, the hidden ray may be changed into an ordinary ray. The total number of internal reflections, M is sum of Md and M0, which denote the total numbers of internal reflections on the dielectric boundaries of θ = θd and 0, respectively. Then θ2m and θ2m+1 should satisfy the following inequalities

equation image
equation image

where the constant νɛ is taken by the minimum positive value satisfying the edge condition at the tip of the dielectric wedge [Meixner, 1972] as

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In the same manner, Md+ and M0+ are also defined.

[18] The one-to-one correspondence between geometrical rays and cotangent functions provides the extended diffraction coefficients in the physical and complementary air regions, f1(e)(w) as

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where

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[19] In the same manner, the HRD diffraction coefficients in the physical and complementary dielectric regions, f2(e)(w) can be expressed by

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where

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[20] Then the total field u(ρ, θ) may be expressed by

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where ug (ρ, θ) is identical to the exact GO field.

5. Diffraction Coefficients and Field Patterns

[21] Applying (20) into (19b), one may easily prove that f2(e)(w) becomes zero in S0(d). It means that f2(e)(w) does always satisfy the null-field condition in the complementary dielectric region automatically. Then the accuracy of the HRD diffraction coefficients is dependent on f1(e)(w). Consider the first limiting case, in which the relative dielectric constant ɛ increases to infinite. In this case, R0 = R0+ = −1, Km.t = Km.t+ = 0, and νɛ in (16) becomes ν = (2π − θd)/π. One may easily prove that f1(e)(w) becomes identical to the exact diffraction coefficients of the perfectly conducting wedge. Next consider the second limiting case as ɛ decreases to 1. Since R0 = R0+ = 0, K1.t = K1.t+ = 1 K2.t = K2.t+ = ⋯ = 0, θ1.t = θ0.i, θ0.t+ = θ0.i+, and νɛ = 1, f1(e)(w) becomes zero. In fact, the corresponding problem includes no diffraction term.

[22] In general, it is not clear mathematically whether f1(e)(w) satisfies the null-field condition in the corresponding complementary region or not. As a typical example, consider the dielectric wedge with θd = 90° illuminated by an E-polarized unit plane wave with θi = 225°. Figure 5a shows the real parts of the PO diffraction coefficients f1(w) in the physical and complementary air regions for ɛ = 1.4, 5, 10, 100, and 1000. It should be noted that the imaginary parts of f1(w) are zero. The dotted line marked byɛ = ∞ denotes the exact diffraction coefficients of the corresponding perfectly conducting wedge. As expected, the PO diffraction coefficients intersect the exact solution near the dielectric boundaries. In contrast, Figure 5b shows the real parts of the HRD diffraction coefficients f1(e)(w) in the physical and complementary air regions. As ɛ increases 1000, f1(e)(w) approaches to the exact diffraction coefficients of ɛ = ∞ monotonically. According to our formulation of dual integral equations, the exact diffraction coefficients must become zero in the complementary region. In comparison with the PO diffraction coefficients in Figure 5a, one may easily find that the HRD diffraction coefficients in Figure 5b approach to zero more closely in the complementary region in S0(d). It implies the accuracy of the HRD diffraction coefficients. Since the imaginary parts of f1(e)(w) are negligible, they are not plotted here.

Figure 5a.

Real parts of physical optics (PO) diffraction coefficients in air and its complementary regions.

Figure 5b.

Real parts of hidden rays of diffraction (HRD) diffraction coefficients in air and its complementary regions.

[23] Figures 6a–6c illustrate the PO and HRD diffraction coefficients in the physical and complementary dielectric regions for ɛ = 2, 5, 10, 100, and 1000. As shown in Figure 6a, the PO diffraction coefficients f2(w) cannot become zero in the complementary region S0(d). In contrast, Figure 6b illustrates that the HRD diffraction coefficients f2(e)(w) satisfy the null-field condition exactly in S0(d). The total reflection inside the complementary dielectric region may generate the imaginary parts of f2(e)(w) for ɛ = 10, 100, and 1000, as shown in Figure 6c.

Figure 6a.

Real parts of PO diffraction coefficients in dielectric and its complementary regions.

Figure 6b.

Real parts of HRD diffraction coefficients in dielectric and its complementary regions.

Figure 6c.

Imaginary parts of HRD diffraction coefficients in dielectric and its complementary regions.

[24] The uniform asymptotic integration [Felsen and Marcuvitz, 1973] is applied to (9) and (21). Then the PO and HRD fields are calculated at 5λ (wavelength) away from the wedge tip for θd = 90°, θi = 225°, and ɛ = 1.4, 10, and 1000. Figures 7a–7b show the amplitude patterns of the edge-diffracted and total fields forɛ = 1.4, where the bold and dotted lines denote the HRD and PO patterns, respectively. Figures 7a and 7b illustrate the close convergence of the PO field to the HRD field. It agrees with the fact that the PO approximation works well in low-dielectric scatterer as ɛ = 1.4. However, for ɛ = 10, Figures 8a and 8b show some difference between PO and HRD patterns. While the PO patterns suffer from abrupt jump on the dielectric boundaries, the HRD patterns reveal nearly continuous. Figures 9a–9b and 10a–10b illustrate the amplitude patterns for ɛ = 1000, in which dotted lines denote the exact patterns of the corresponding perfectly conducting wedge. As expected, the PO fields deviate from the dotted lines significantly and reveal abrupt discontinuities on the dielectric boundaries, as shown in Figures 9a and 9b. In contrast, Figures 10a and 10b illustrate that the HRD fields are nearly overlapped to the exact pattern of ɛ = ∞. It assures the accuracy of the HRD diffraction coefficients.

Figure 7a.

PO (dashed curve) and HRD (bold curve) edge-diffracted fields.

Figure 7b.

PO (dashed curve) and HRD (bold curve) total fields.

Figure 8a.

PO (dashed curve) and HRD (bold curve) edge-diffracted fields.

Figure 8b.

PO (dashed curve) and HRD (bold curve) total fields.

Figure 9a.

PO (bold curve) and exact (dashed curve, ɛ = ∞) edge-diffracted fields.

Figure 9b.

PO (bold curve) and exact (dashed curve, ɛ = ∞) total fields.

Figure 10a.

HRD (bold curve) and exact (dashed curve, ɛ = ∞) edge-diffracted fields.

Figure 10b.

HRD (bold curve) and exact (dashed curve, ɛ = ∞) total fields.

6. Conclusion

[25] The method of hidden rays provided the E-polarized diffraction coefficients of a dielectric wedge in such an analytical form as finite series of cotangent functions. As the relative dielectric constant of the dielectric part increases to infinite, the presented diffraction coefficients approach to the exact pattern of the corresponding perfectly conducting wedge. The accuracy of the suggested method was also checked by showing that the presented diffraction coefficients in the complementary regions approached to zero more closely than the PO solution. The presented diffraction coefficients, expressed in GTD format [Keller, 1962], may be suitable to treat more complicated dielectric scatterers.

Acknowledgments

[26] This work was supported in part by the Korean Department of Commerce, Industry and Energy under grant 10023090 and in part by the Korea Institute of Science and Technology under contract 2E20040.

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