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/* General Huygens Sources */
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== Physical Optics as an Asymptotic Technique ==
:<math> \mathbf{J(r)} = \mathbf{\hat{n}\times} \begin{bmatrix} 1-+R_{||} & 0 \\ 0 & 1-R_{\perp} \end{bmatrix} \cdot \begin{bmatrix} \mathbf{H_{||}^{inc}(r)} \\ \mathbf{H_{\perp}^{inc}(r)} \end{bmatrix} </math>
:<math> \mathbf{M(r)} = -\mathbf{\hat{n}\times} \begin{bmatrix} 1+-R_{||} & 0 \\ 0 & 1+R_{\perp} \end{bmatrix} \cdot \begin{bmatrix} \mathbf{E_{||}^{inc}(r)} \\ \mathbf{E_{\perp}^{inc}(r)} \end{bmatrix} </math>
where it is assumed that both the incident electric field and incident magnetic field have been decomposed into two parallel and perpendicular polarizations and R<sub>||</sub> and R<sub>⊥</sub> denote the reflection coefficients at the interface between air and the impedance surface for the cases of parallel and perpendicular polarizations, respectively. These reflection coefficients are given by:
:<math> R_{\perp} = \frac{Z_s\eta_s \cos\theta - \eta_01} {Z_s\eta_s \cos\theta + \eta_01} </math>
:<math> R_{\|} = \frac{Z_s - \eta_0\cos\theta- \eta_s } {Z_s + \eta_0\cos\theta+ \eta_s } </math>
where θ is the incident angle between the propagation vector of the incident field and the normal to the surface , <math>\eta_s = Z_s/\eta_0</math> and <math>\eta_0 = 120\pi \; \Omega</math> is the intrinsic impedance of the free space.
From the surface impedance boundary condition, it can easily be shown that
:<math> \mathbf{M(r)} = -Z_s \mathbf{\hat{n}\times} \mathbf{J(r)} </math>
In the case of an impedance-matched surface (, Z<sub>s</sub> = η<sub>0</sub>), η<sub>s</sub> = 1, and one can write:
:<math> R_{\|} = -R_{\perp} = \frac{1-\cos\theta- 1} {1+\cos\theta+ 1} </math>
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Two special limiting cases of an impedance surface are perfect electric conductor (PEC) and perfect magnetic conductor (PMC) surface. For a PEC surface, Z<sub>s</sub> = 0, η<sub>s</sub> = 0, R<sub>||</sub> = 1, R<sub>⊥</sub> = -1, and one can write:
:<math> \mathbf{J(r)} = 2 \mathbf{\hat{n} \times H(r)} </math>
:<math> \mathbf{M(r)} = 0 </math>
while for a PMC surface, Z<sub>s</sub> = ∞, η<sub>s</sub> = ∞, R<sub>||</sub> = -1, R<sub>⊥</sub> = 1, and one can write:
:<math> \mathbf{J(r)} = 0 </math>
:<math> \mathbf{E^{ff}(r)} = \frac{jk_0 e^{-jk_0 r}}{4\pi r} \left\{ Z_0 \mathbf{ \hat{r} \times \hat{r} } \times \iint_{S_J} \mathbf{J(r')} e^{-jk_0 \mathbf{\hat{r}\cdot r'}} ds' + \mathbf{\hat{r}} \times \iint_{S_M} \mathbf{M(r')} e^{-jk_0 \mathbf{ \hat{r} \cdot r' } } ds' \right\} </math>
:<math> \mathbf{H^{ff}(r)} = \frac{1}{Z_0} \mathbf{\hat{r} \times E^{ff}(r)} </math>
where the summation over index ''j'' is carried out for all the elementary cells Δ<sub>j</sub> that make up the Huygens box. In [[EM.Cube]] Huygens surfaces are cubic and are discretized using a rectangular mesh. Therefore, Δ<sub>j</sub> represents any rectangular cell located on one of the six faces of Huygens box. Note that the calculated near-zone electric and magnetic fields act as incident fields for the scatterers in your [[EM.Illumina]] project. The Huygens source data are normally generated in one of [[EM.Cube]]'s full-wave computational modules like [[EM.Tempo]] (FDTD), [[EM.Picasso]] (Planar MOM) or MoM3D[[EM.Libera]] (3D MOM). Keep in mind that the fields scattered (or reradiated) by your physical structure do not affect the fields inside the Huygens source.
The far fields of the Huygens surface currents are calculated from the following relations:
:<math> \mathbf{E^{ff}(r)} = \frac{jk_0}{4\pi} \frac{e^{-jk_0 r}}{r} \sum_j \iint_{\Delta_j} \left[ Z_0 \, \mathbf{ \hat{r} \times \hat{r} \times J_j(r') } + \mathbf{ \hat{r} \times M_j(r') } \right] e^{ -jk_0 \mathbf{\hat{r} \cdot r'} } \, ds' </math>
:<math>\mathbf{H^{ff}(r)} = \frac{1}{Z_0} \mathbf{\hat{r} \times E^{ff}(r)} </math>
