SBR Method

Ray Reflection & Transmission

The Incident, Reflected and Transmitted Rays at the Interface Between Two Dielectric Media

The incident, reflected and transmitted rays are each characterized by a triplet of unit vectors:

• [math]( \mathbf{ \hat{u}_{\|}, \hat{u}_{\perp}, \hat{k} } )[/math] representing the incident parallel polarization vector, incident perpendicular polarization vector and incident propagation vector, respectively.
• [math]( \mathbf{ \hat{u}_{\|}^{\prime}, \hat{u}_{\perp}', \hat{k}' } )[/math] representing the reflected parallel polarization vector, reflected perpendicular polarization vector and reflected propagation vector, respectively.
• [math]( \mathbf{ \hat{u}_{\|}^{\prime\prime}, \hat{u}_{\perp}^{\prime\prime}, \hat{k}^{\prime\prime} } )[/math] representing the transmitted parallel polarization vector, transmitted perpendicular polarization vector and transmitted propagation vector, respectively.

The reflected ray is assumed to originate from a virtual image source point. The three triplets constitute three orthonormal basis systems. Below, it is assumed that the two dielectric media have permittivities ε₁ and ε₂, and permeabilities μ₁ and μ₂, respectively. A lossy medium with a conductivity σ can be modeled by a complex permittivity ε_r = ε'_r –jσ/ε₀. Assuming n to be the unit normal to the interface plane between the two media, and Z₀ = 120Ω , the incident polarization vectors as well as all the reflected and transmitted vectors are found as:

[math] \mathbf{ \hat{u}_{\perp} = \frac{\hat{k} \times \hat{n}}{|\hat{k} \times \hat{n}|} } [/math]

[math] \mathbf{ \hat{u}_{\|} = \hat{u}_{\perp} \times \hat{k} } [/math]

The reflected unit vectors are found as:

[math] \mathbf{ \hat{k}' = \hat{k} - 2(\hat{k} \cdot \hat{n}) \hat{n} } [/math]

[math] \mathbf{ \hat{u}_{\perp}' = \hat{u}_{\perp} } [/math]

[math] \mathbf{ \hat{u}_{\|}' = \hat{u}_{\perp}' \times \hat{k}' } [/math]

The transmitted unit vectors are found as:

[math] \mathbf{ \hat{k}^{\prime\prime} = \hat{n} \times a - \sqrt{1-a \cdot a} \; \hat{n} } [/math]

[math] \mathbf{ \hat{u}_{\perp}^{\prime\prime} = \hat{u}_{\perp} } [/math]

[math] \mathbf{ \hat{u}_{\|}^{\prime\prime} = \hat{u}_{\perp}^{\prime\prime} \times \hat{k}^{\prime\prime} } [/math]

where

[math] \mathbf{a} = (k_1/k_2) \mathbf{\hat{k} \times \hat{n}}[/math]

[math] k_1 = k_0 \sqrt{\varepsilon_1 \mu_1} [/math]

[math] k_2 = k_0 \sqrt{\varepsilon_2 \mu_2} [/math]

[math] \eta_1 = Z_0 \sqrt{\mu_1 / \varepsilon_1} [/math]

[math] \eta_2 = Z_0 \sqrt{\mu_2 / \varepsilon_2} [/math]

[math] \sin\theta^{\prime\prime} = \frac{k_1}{k_2}\sin\theta \text{ if } \sin\theta \le k_2/k_1[/math]

The reflection coefficients at the interface are calculated for the two parallel and perpendicular polarizations as:

[math] R_{\|} = \frac { \eta_2(\mathbf{ \hat{k}^{\prime\prime} \cdot \hat{n} }) - \eta_1(\mathbf{ \hat{k} \cdot \hat{n} }) } { \eta_2(\mathbf{ \hat{k}^{\prime\prime} \cdot \hat{n} }) + \eta_1(\mathbf{ \hat{k} \cdot \hat{n} }) } = \frac{\eta_2 \cos\theta^{\prime\prime} - \eta_1 \cos\theta} {\eta_2 \cos\theta^{\prime\prime} + \eta_1 \cos\theta} = \frac{Z_{2\|} - Z_{1\|}} {Z_{2\|} + Z_{1\|}} [/math]

[math] R_{\perp} = \frac { \eta_2(\mathbf{ \hat{k} \cdot \hat{n} }) - \eta_1(\mathbf{ \hat{k}^{\prime\prime} \cdot \hat{n} }) } { \eta_2(\mathbf{ \hat{k} \cdot \hat{n} }) + \eta_1(\mathbf{ \hat{k}^{\prime\prime} \cdot \hat{n} }) } = \frac{\eta_2 / \cos\theta^{\prime\prime} - \eta_1 / \cos\theta} {\eta_2 / \cos\theta^{\prime\prime} + \eta_1 / \cos\theta} = \frac{Z_{2\perp} - Z_{1\perp}} {Z_{2\perp} + Z_{1\perp}} [/math]

Penetration Through Thin Walls Or Surfaces

The Incident and Transmitted Rays through a Thin Wall

In "Thin Wall Approximation", we assume that an incident ray gives rise to two rays, one is reflected at the specular point, and the other is transmitted almost in the same direction as the incident ray. The reflected ray is assumed to originate from a virtual image source point. Similar to the case of reflection and transmission at the interface between two dielectric media, here too we have three triplets of unit vectors, which all form orthonormal basis systems.

The transmission coefficients are calculated for the two parallel and perpendicular polarizations as:

[math] T_{\|} = \frac{(1-{\Gamma_{\|}}^2) \exp(-jk_2 d (\mathbf{ \hat{k}^{\prime\prime} \cdot \hat{n}}))} { 1-{\Gamma_{\|}}^2 \exp( -2jk_2 d (\mathbf{ \hat{k}^{\prime\prime} \cdot \hat{n} }) ) } [/math]

[math] T_{\perp} = \frac{(1-{\Gamma_{\perp}}^2) \exp(-jk_2 d (\mathbf{ \hat{k}^{\prime\prime} \cdot \hat{n}}))} { 1-{\Gamma_{\perp}}^2 \exp( -2jk_2 d (\mathbf{ \hat{k}^{\prime\prime} \cdot \hat{n} }) ) } [/math]

where

[math] \Gamma_{\|} = \frac{ \eta_2(\mathbf{ \hat{k}^{\prime\prime} \cdot \hat{n} }) - \eta_1(\mathbf{ \hat{k} \cdot \hat{n} }) } { \eta_2(\mathbf{ \hat{k}^{\prime\prime} \cdot \hat{n} }) + \eta_1(\mathbf{ \hat{k} \cdot \hat{n} }) } = \frac{\eta_2 \cos\theta^{\prime\prime} - \eta_1 \cos\theta} {\eta_2 \cos\theta^{\prime\prime} + \eta_1 \cos\theta} = \frac{Z_{2\|} - Z_{1\|}} {Z_{2\|} + Z_{1\|}} [/math]

[math] \Gamma_{\perp} = \frac{ \eta_2(\mathbf{ \hat{k} \cdot \hat{n} }) - \eta_1(\mathbf{ \hat{k}^{\prime\prime} \cdot \hat{n} }) } { \eta_2(\mathbf{ \hat{k} \cdot \hat{n} }) + \eta_1(\mathbf{ \hat{k}^{\prime\prime} \cdot \hat{n} }) } = \frac{\eta_2 / \cos\theta^{\prime\prime} - \eta_1 / \cos\theta} {\eta_2 / \cos\theta^{\prime\prime} + \eta_1 / \cos\theta} = \frac{Z_{2\perp} - Z_{1\perp}} {Z_{2\perp} + Z_{1\perp}} [/math]

Wedge Diffraction From Edges

The Incident Ray and Diffract Ray Cone at the Edge of a Building

For the purpose of calculation of diffraction from building edges, we define a "Wedge" as having two faces, the 0-face and the n-face. The wedge angle is a = (2-n)p, where the parameter n is required for the calculation of diffraction coefficients. All the diffracted rays lie on a cone with its vertex at the diffraction point and a wedge angle equal to the angle of incidence in the opposite direction. A diffracted ray is assumed to originate from a virtual image source point. Three triplets of unit vectors are defined as follows:

• [math]\mathbf{(\hat{u}_0, \hat{u}_l, \hat{t})}[/math] representing the unit vector normal to the edge and lying in the plane of the 0-face, the unit vector normal to the 0-face, and the unit vector along the edge, respectively.
• [math]\mathbf{(\hat{u}_f, \hat{u}_b, \hat{t})}[/math] representing the incident forward polarization vector, incident backward polarization vector and incident propagation vector, respectively.
• [math]\mathbf{(\hat{u}_f', \hat{u}_b', \hat{t}')}[/math] representing the diffracted forward polarization vector, diffracted backward polarization vector and diffracted propagation vector, respectively.

The three triplets constitute three orthonormal basis systems. The propagation vector k' of the diffracted ray has to be constructed based on the diffraction cone as follows:

[math] \mathbf{\hat{k}'} = \cos\phi_w \mathbf{\hat{u}_0} + \sin\phi_w \mathbf{\hat{u}_l} + \mathbf{(\hat{k} \cdot \hat{t}) \hat{t}}, \quad 0 \le \phi_w \le \alpha[/math]

where the resolution of the angle θ_w is chosen to be the same as the resolution of the incident ray.

The other unit vectors for the incident and diffracted rays are found as:

[math] \mathbf{ \hat{u}_f = \frac{\hat{k} \times \hat{t}}{|\hat{k} \times \hat{t}|} } [/math]

[math] \mathbf{ \hat{u}_b = \hat{k} \times \hat{u}_f } [/math]

[math] \mathbf{ \hat{u}_f' = \frac{\hat{k}' \times \hat{t}}{|\hat{k}' \times \hat{t}|} } [/math]

[math] \mathbf{ \hat{u}_b' = \hat{k}' \times \hat{u}_f' } [/math]

The diffraction coefficients are calculated in the following way:

[math] D_s = \frac{-e^{-j\pi/4}}{2n \sqrt{2\pi k} \sin\beta_0'} \left\lbrace \begin{align} & \cot \left(\frac{\pi + (\phi-\phi')}{2n}\right) F[kLa^+(\phi-\phi')] + \cot \left(\frac{\pi - (\phi-\phi')}{2n}\right) F[kLa^-(\phi-\phi')] + \\ & R_{0 \perp} \cot \left(\frac{\pi - (\phi+\phi')}{2n}\right) F[kLa^-(\phi+\phi')] + R_{n \perp} \cot \left(\frac{\pi + (\phi+\phi')}{2n}\right) F[kLa^+(\phi+\phi')] \end{align} \right\rbrace [/math]

[math] D_h = \frac{-e^{-j\pi/4}}{2n \sqrt{2\pi k} \sin\beta_0'} \left\lbrace \begin{align} & \cot \left(\frac{\pi + (\phi-\phi')}{2n}\right) F[kLa^+(\phi-\phi')] + \cot \left(\frac{\pi - (\phi-\phi')}{2n}\right) F[kLa^-(\phi-\phi')] + \\ & R_{0 \|} \cot \left(\frac{\pi - (\phi+\phi')}{2n}\right) F[kLa^-(\phi+\phi')] + R_{n \|} \cot \left(\frac{\pi + (\phi+\phi')}{2n}\right) F[kLa^+(\phi+\phi')] \end{align} \right\rbrace [/math]

where F(x) is the Fresnel Transition function:

[math] F(x) = 2j \sqrt{x} e^{jx} \int_{\sqrt{x}}^{\infty} e^{-j\tau^2} \, d\tau [/math]

In the above equations, we have

[math] \begin{align} s = |\rho_D - \rho_S| \\ s' = |\rho_D - \rho_r| \end{align} [/math]

[math]L = \frac{s s' \sin^2 \beta'}{s + s'} [/math]

[math]a^{\pm}(\nu) = 2\cos^2 \left( \frac{2n\pi N^{\pm} - \nu}{2} \right), \quad \nu = \phi \pm \phi' [/math]

where [math]N^{\pm}[/math] are the integers which most closely satisfy the equations [math] 2n\pi N^{\pm} - \nu = \pm \pi [/math].

 

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