Click here to learn more about the theory of [[SBR Method]].
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=== Ray Reflection & Transmission ===
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[[File:reflect.png|thumb|350px|The Incident, Reflected and Transmitted Rays at the Interface Between Two Dielectric Media]]
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The incident, reflected and transmitted rays are each characterized by a triplet of unit vectors:
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* <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.
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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 ε<sub>1</sub> and ε<sub>2</sub>, and permeabilities μ<sub>1</sub> and μ<sub>2</sub>, respectively. A lossy medium with a conductivity σ can be modeled by a complex permittivity ε<sub>r</sub> = ε'<sub>r</sub> âjσ/ε<sub>0</sub>. Assuming '''n''' to be the unit normal to the interface plane between the two media, and Z<sub>0</sub> = 120Ω , the incident polarization vectors as well as all the reflected and transmitted vectors are found as:
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:<math> \mathbf{ \hat{u}_{\perp} = \frac{\hat{k} \times \hat{n}}{|\hat{k} \times \hat{n}|} } </math>
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:<math> \mathbf{ \hat{u}_{\|} = \hat{u}_{\perp} \times \hat{k} } </math>
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The reflected unit vectors are found as:
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:<math> \mathbf{ \hat{k}' = \hat{k} - 2(\hat{k} \cdot \hat{n}) \hat{n} } </math>
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:<math> \mathbf{ \hat{u}_{\perp}' = \hat{u}_{\perp} } </math>
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:<math> \mathbf{ \hat{u}_{\|}' = \hat{u}_{\perp}' \times \hat{k}' } </math>
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The transmitted unit vectors are found as:
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:<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>
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:<math> \mathbf{ \hat{u}_{\|}^{\prime\prime} = \hat{u}_{\perp}^{\prime\prime} \times \hat{k}^{\prime\prime} } </math>
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where
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:<math> \mathbf{a} = (k_1/k_2) \mathbf{\hat{k} \times \hat{n}}</math>
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:<math> k_1 = k_0 \sqrt{\varepsilon_1 \mu_1} </math>
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:<math> k_2 = k_0 \sqrt{\varepsilon_2 \mu_2} </math>
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:<math> \eta_1 = Z_0 \sqrt{\mu_1 / \varepsilon_1} </math>
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:<math> \eta_2 = Z_0 \sqrt{\mu_2 / \varepsilon_2} </math>
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:<math> \sin\theta^{\prime\prime} = \frac{k_1}{k_2}\sin\theta \text{ if } \sin\theta \le k_2/k_1</math>
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The reflection coefficients at the interface are calculated for the two parallel and perpendicular polarizations as:
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:<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>
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:<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>
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=== Penetration Through Thin Walls Or Surfaces ===
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[[File:thinwalltrans.png|thumb|350px|The Incident and Transmitted Rays through a Thin Wall]]
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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.
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The transmission coefficients are calculated for the two parallel and perpendicular polarizations as:
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:<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>
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:<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>
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where
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:<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>
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:<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>
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=== Wedge Diffraction From Edges ===
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[[File:diffract.png|thumb|350px|The Incident Ray and Diffract Ray Cone at the Edge of a Building]]
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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:
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* <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.
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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:
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:<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>
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where the resolution of the angle θ<sub>w</sub> is chosen to be the same as the resolution of the incident ray.
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The other unit vectors for the incident and diffracted rays are found as:
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:<math> \mathbf{ \hat{u}_f = \frac{\hat{k} \times \hat{t}}{|\hat{k} \times \hat{t}|} } </math>
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:<math> \mathbf{ \hat{u}_b = \hat{k} \times \hat{u}_f } </math>
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:<math> \mathbf{ \hat{u}_f' = \frac{\hat{k}' \times \hat{t}}{|\hat{k}' \times \hat{t}|} } </math>
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:<math> \mathbf{ \hat{u}_b' = \hat{k}' \times \hat{u}_f' } </math>
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The diffraction coefficients are calculated in the following way:
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:<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>
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:<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>
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where ''F(x)'' is the Fresnel Transition function:
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:<math> F(x) = 2j \sqrt{x} e^{jx} \int_{\sqrt{x}}^{\infty} e^{-j\tau^2} \, d\tau </math>
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In the above equations, we have
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:<math> \begin{align} s = |\rho_D - \rho_S| \\ s' = |\rho_D - \rho_r| \end{align} </math>
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:<math>L = \frac{s s' \sin^2 \beta'}{s + s'} </math>
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:<math>a^{\pm}(\nu) = 2\cos^2 \left( \frac{2n\pi N^{\pm} - \nu}{2} \right), \quad \nu = \phi \pm \phi' </math>
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where <math>N^{\pm}</math> are the integers which most closely satisfy the equations <math> 2n\pi N^{\pm} - \nu = \pm \pi </math>.
=== SBR As An Asymptotic EM Solver ===
[[EM.Cube]]'s SBR simulation engine can be used as a versatile and powerful asymptotic electromagnetic (EM) solver. If you compare [[EM.Cube]]'s [[Propagation Module]] with its other computational modules, you will notice a lot of similarities. While other modules group objects primarily by their material properties, [[Propagation Module]] categorizes the types of obstructing surfaces. Besides sharing the same ray-surface interaction mechanisms, all the objects belonging to a surface group also share the same material properties. [[Propagation Module]] offers similar source types and similar observable types as the other computational modules. For instance, the Hertzian dipole sources used in a SBR simulation are identical to those offered in PO, MoM3D and Planar modules. The plane wave sources are identical across all computational modules. [[Propagation Module]]'s sensor field planes, far field observables (either radiation patterns or RCS) and Huygens surfaces are all fully compatible with [[EM.Cube]]'s other computational modules.
As an asymptotic EM solver, the SBR engine can be used to model large-scale electromagnetic radiation and scattering problems. An example of this kind is radiation of simple or complex antennas in the presence of large scattering platforms. You have to keep in mind that by using an asymptotic technique in place of a full-wave method, you trade computational speed and lower memory requirements for modeling accuracy. In particular, the [[SBR Method|SBR method ]] cannot take into account the electromagnetic coupling effects among nearby radiators or scatterers. However, when your scene spans thousands of wavelengths, an SBR simulation might often prove to be your sole practical solution.
=== Novelties Of EM.Cube's SBR Solver ===
=== The Need For Discretization Of Propagation Scene ===
In a typical SBR simulation, a ray is traced from the location of the source until it hits a scatterer. The [[SBR Method|SBR method ]] assumes that the ray hits either a flat facet of the scatterer or one of its edges. In the case of hitting a flat facet, the specular point is used to launch new reflected and transmitted rays. The surface of the facet is treated as an infinite dielectric medium interface, at which the reflection and transmission coefficients are calculated. In the case of hitting an edge, new diffracted rays are generated in the scene. However, only those who reach a nearby receiver in their line of sight are ever taken into account. In other words, diffractions are treated locally.
[[EM.Cube]]'s [[Propagation Module]] allows you to draw any type of surface or solid CAD objects under impenetrable and penetrable surface groups. Some of these objects have flat faces such as boxes, pyramids, rectangle or triangle strips, etc. Some others contain curved surfaces or curved boundaries such as cylinders, cones, etc. All the non-flat surfaces have to be discretized in the form of a collection of smaller flat facets. [[EM.Cube]] uses a triangular surface mesh generator to discretize the penetrable and impenetrable [[Surface Objects|surface objects]] of your propagation scene. This mesh generator is very similar to the ones used in [[EM.Cube]]'s two other modules: MoM3D and Physical Optics (PO).