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<td>[[image:Cube-icon.png | link=Getting_Started_with_EM.Cube]] [[image:cad-ico.png | link=Building Geometrical Constructions in CubeCAD]] [[image:fdtd-ico.png | link=EM.Tempo]] [[image:prop-ico.png | link=EM.Terrano]] [[image:static-ico.png | link=EM.Ferma]] [[image:planar-ico.png | link=EM.Picasso]] [[image:metal-ico.png | link=EM.Libera]] [[image:po-ico.png | link=EM.Illumina]] </td>
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== Transverse Electromagnetic Waves ==
Electromagnetic waves are synchronized oscillations of electric and magnetic fields that propagate at the speed of light through a vacuum. The oscillations of the two fields are perpendicular to each other and perpendicular to the direction of energy propagation. The wavefront of electromagnetic waves emitted from a point source is spherical. As the wave propagates away from its source, its field amplitudes decay as 1/R, where R is the distance between the source and observation points. At far enough distances from the source, the wave is reduced to a transverse electromagnetic (TEM) wave with a planar wavefront. The electric and magnetic field vectors of a TEM wave satisfy the following equation:
<math> \mathbf{H(r)} = \frac{1}{\eta_0} \mathbf{\hat{k}} \times \mathbf{E(r)} </math>
where <math>\mathbf{\hat{k}}</math> is the propagation vector, and η<sub>0</sub> = 120π Ω is the intrinsic impedance of the free space.
In a spherical coordinate system where the source is located at the origin of coordinates, the propagation vector can be written as:
<math> \mathbf{\hat{k}} = \mathbf{\hat{r}} </math>
where <math>\mathbf{\hat{r}}</math> is the unit radial vector. Then the transverse field components of a TEM wave satisfy the following relations:
<math> H_\theta = -\frac{E_\phi}{\eta_0}, \quad H_\phi = \frac{E_\theta}{\eta_0} </math>
== Definition of the Far Radiation Zone ==
<math>r << \frac{2D^2}{\lambda_0}</math>
where r is the distance between the observation and source points, λ<sub>0</sub> is the free space wavelength and D is the largest dimension of the radiating structure. In [[EM.Cube]], the far-zone electric fields '''E<sup>ff</sup>'''(θ, φ) are functions of the spherical observation angles only and are defined as
<math> \begin{align} & \mathbf{E(r) = E} (r,\theta,\phi) = \frac{e^{-jk_0 r}}{r} \mathbf{E^{ff}}(\theta,\phi) \\ & \mathbf{H(r) = H} (r,\theta,\phi) = \frac{1}{\eta_0} \mathbf{ \hat{r} \times E^{ff}(r) } \end{align} \quad k_0 r >> 1 </math>
where k<sub>0</sub> = 2π/λ<sub>0</sub> and η<sub>0</sub> = 120π Ω is the intrinsic impedance of the free space.
== Computing the Far-Zone Fields from Near-Zone Field Data==
[[EM.Tempo]]'s FDTD engine calculates the far fields using a near-field-to-far-field transformation of the field quantities on a given closed surface, which is denoted the radiation box. You can use [[EM.Cube]]'s default radiation box or define your own. [[EM.Picasso]], [[EM.Libera]] and [[EM.Illumina]] compute the far fields directly by integrating the electric and magnetic surface current solutions to the numerical problem.
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To eliminate the dependency on r, a normalized quantity called "Radiation Intensity" is defined in the following way:
<math> S(\theta,\phi) = \lim_{r \to \infty} r^2 |\mathbf{W}| = \frac{1}{2\eta_0} | \mathbf{E^{ff}}(\theta,\phi)|^2 </math>
The right hand side of the above equation shows the product of the "Element Pattern" by the "Array Factor".
In a phased array antenna system, a uniform phase progression is applied between successive elements along the principal axes. If the phase progression is denoted by ΔΦ<sub>x</sub>, ΔΦ<sub>y</sub> and ΔΦ<sub>z</sub> along the X, Y and Z-axes, respectively, then the far-field expression reduces to: <math> \mathbf{E_{tot}^{ff}}(\theta,\phi) =\mathbf{E_0^{ff}}(\theta,\phi) \sum_{m= 1}^{N_x} \sum_{n=1}^{N_y} \sum_{l=1}^{N_z} w^x_i w^y_j w^z_k e^{j \left[ (m-1)(k_0 S_x sin \theta cos \phi + \Delta\Psi_x ) + (n-1)(k_0 S_y sin \theta sin \phi + \Delta\Psi_y ) + (l-1)(k_0 S_z cos \theta + \Delta\Psi_z ) \right] } </math> where w<sup>x</sup><sub>i</sub>, w<sup>y</sup><sub>j</sub> and w<sup>z</sup><sub>k</sub> are real-valued one-dimensional weight distribution sequences along the three principal axes, respectively. == Electromagnetic Scattering and Radar Cross Section ==
When the physical structure is illuminated by a plane wave source, the calculated far field data indeed represent the scattered fields. In that case, the incident and scattered fields can be separated. The radar cross section (RCS) of a target is defined as follows:
:<math>\sigma_{\theta} = 4\pi r^2 \dfrac{ \big| \mathbf{E}_{\theta}^{scat} \big| ^2} {\big| \mathbf{E}^{inc} \big|^2}, \quad </math> :<math>\sigma_{\phi} = 4\pi r^2 \dfrac{ \big| \mathbf{E}_{\phi}^{scat} \big| ^2} {\big| \mathbf{E}^{inc} \big|^2}, \quad </math> :<math>\sigma = \sigma_{\theta} + \sigma_{\phi} = 4\pi r^2 \dfrac{ \big| \mathbf{E}_{tot}^{scat} \big| ^2} {\big| \mathbf{E}^{inc} \big|^2}</math>
In a bistatic radar system configuration, the transmitting and receiving antennas are different and located at different locations. The radar equations, which related the received power to the transmitted power, can be expressed as:
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