It can be seen that the discrete Fourier transform multiplies the samples of time-domain field quantities by the time step Δt. This means that the resulting Fourier transforms of electric and magnetic field components now have units of V/m/Hz or A/m/Hz, respectively. Moreover, the Fourier transforms of the three waveform types have different spectral values at the observation frequency f<sub>0</sub>. This makes it difficult to compare the FDTD simulation results with the results from [[EM.Cube]]'s other computational modules. For example, in the Planar, MoM3D and Physical Optics Modules, a plane wave source typically has a complex-valued functional form of exp(-jk<sub>0</sub>'''k.r'''), which has a unit magnitude. In [[FDTD Module]], the time domain plane wave source has a functional dependence of the following form:
:<math>\mathbf{E^{inc}}(r,t) = (E_{\theta}^{inc} \hat{\theta} + E_{\phi}^{inc} \hat{\phi})
f \left[ (t-t_0) - \frac{\mathbf{\hat{k} \cdot r} - l_0}{c} \right] </math>
<!--[[Image:FDTD69.png]]-->
where f(t) is the temporal waveform, t<sub>0</sub> is the time delay, l<sub>0</sub> is a spatial shift, and c is the speed of light in the free space. The Fourier transform of the above temporal function evaluated at f = f<sub>0</sub> has an amplitude different than 1. For this purpose, EM.Cube's [[FDTD Module]] normalizes the temporal waveform by the magnitude of its Fourier transform at the observation frequency f<sub>0</sub>. As a result, a temporal plane source with any of the three waveform types will always create spectral incident source with |'''E<sup>inc</sup>'''('''r''', f<sub>0</sub>)| = 1. The temporal waveform normalization factors for the three waveform types are given below:
:<math>\begin{align}&(N.F.)_{Sinusoidal} = \frac{2}{T} = \frac{2}{N \Delta t} \\&(N.F.)_{Gaussian} = \frac{ e^{(\pi f_0 \tau)^2} }{ \sqrt{\pi} \tau } = \frac{1}{\sqrt{\pi} \tau \delta^{1/4}} \\&(N.F.)_{Modulated} = \frac{2}{\sqrt{\pi} \tau}\end{align}</math>
<!--[[Image:FDTD70.png]]-->
If your physical structure is excited by a Lumped Source or a Waveguide Source or a Distributed Source, and one or more ports have been defined, the FDTD engine calculates the scattering (S) [[parameters]], impedance (Z) [[parameters]] and admittance (Y) [[parameters]] of the selected ports. The S [[parameters]] are calculated based on the port impedances specified in the project's "Port Definition". If more than one port has been defined in the project, the FDTD engine runs an internal port sweep. Each port is excited separately with all the other ports turned off. When the ''j''th port is excited, all the S<sub>ij</sub> [[parameters]] are calculated together based on the following definition:
:<math>S_{ij} = \sqrt{\frac{Re(Z_i)}{Re(Z_j)}} \cdot \frac{V_j - Z_j^*I_j}{V_i+Z_i I_i}</math>
<!--[[Image:FDTD82(1).png]]-->
:<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}\quadk_0 r >> 1</math>
<!--[[Image:FDTD104(1).png]]-->
where '''J '''and '''M''' are the equivalent electric and magnetic surface currents on the surface of the enclosing box. '''G<sub>A,ff</sub>''' is the asymptotic form of the dyadic Green's function associated with the magnetic vector potential '''A''' and '''G<sub>EM,ff</sub>''' is the asymptotic form of the dyadic Green's function of the electric field due to a magnetic current. In most FDTD problems, the background medium of your physical structure is the free space and these functions reduce to the much simpler and familiar free-space Green's function: exp(-jk<sub>0</sub>r)/(4πr). In that case, one can define a pair of electric and magnetic radiation integrals:
:<math>\begin{align}& \mathbf{N(r)} = \iint_S \mathbf{J(r')} e^{ -jk_0 \mathbf{\hat{r} \cdot r'} } \, ds' \\& \mathbf{L(r)} = \iint_S \mathbf{M(r')} e^{ -jk_0 \mathbf{\hat{r} \cdot r'} } \, ds' \\\end{align}</math>
<!--[[Image:FDTD107.png]]-->
where
:<math>\mathbf{\hat{r}} = \sin\theta \cos\phi \mathbf{\hat{x}} +\sin\theta \sin\phi \mathbf{\hat{y}} + \cos\theta \mathbf{\hat{z}}</math>
<!--[[Image:FDTD108.png]]-->
In that case, the θ and φ components of the far fields can be computed from the following relationships:
:<math> \begin{align}& E_{\theta}^{ff}(\theta, \phi) = -\frac{jk_0}{4\pi} (L_{\phi} + \eta_0 N_{\theta}) \\& E_{\phi}^{ff}(\theta, \phi) = \frac{jk_0}{4\pi} (L_{\theta} + \eta_0 N_{\phi})\end{align} </math>
<!--[[Image:FDTD106.png]]-->
where the θ and φ components of the radiation integrals are given by:
:<math> \begin{align}& N_{\theta}(\theta,\phi) = \iint_S [J_x\cos\theta\cos\phi + J_y\cos\theta\sin\phi - J_z\sin\theta] e^{ jk_0 \mathbf{\hat{r} \cdot r'} } \, ds' \\
& N_{\phi}(\theta,\phi) = \iint_S [-J_x \sin\phi + J_y\cos\phi] e^{ jk_0 \mathbf{\hat{r} \cdot r'} } \, ds'
\end{align} </math>
:<math> \begin{align}& L_{\theta}(\theta,\phi) = \iint_S [M_x\cos\theta\cos\phi + M_y\cos\theta\sin\phi - M_z\sin\theta] e^{ jk_0 \mathbf{\hat{r} \cdot r'} } \, ds' \\
& L_{\phi}(\theta,\phi) = \iint_S [-M_x \sin\phi + M_y\cos\phi] e^{ jk_0 \mathbf{\hat{r} \cdot r'} } \, ds'
\end{align} </math>
Normally, the radiation box should enclose the entire FDTD structure. In this case, the calculated radiation pattern corresponds to the entire radiating structure. The radiation box may contain only parts of a structure, which results in partial radiation patterns. In calculating the far field quantities, using Poynting's theorem, one can define the radiated power density as:
:<math>\mathbf{W} = \frac{1}{2} \text{Re}(\mathbf{E \times H^*}) = \frac{\mathbf{\hat{k}}}{2\eta_0} |\mathbf{E}(r,\theta,\phi)|^2 \, |_{r \to \infty}</math>
<!--[[Image:FDTD110.png]]-->
To eliminate the dependency on r, a normalized quantity called "Radiation Intensity" 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>
<!--[[Image:FDTD111.png]]-->
The total radiated power can now be calculated as:
:<math>P_{rad} = \int\limits_0^{2\pi} d\phi \int\limits_0^{\pi} d\theta \, S(\theta,\phi) \sin\theta =\frac{1}{2\eta_0} \int\limits_0^{2\pi} \int\limits_0^{\pi} |\mathbf{E^{ff}}(\theta,\phi)|^2 \sin\theta \, d\theta \, d\phi</math>
<!--[[Image:FDTD112.png]]-->
The 3D plots can be viewed in the project workspace by clicking on each item. The view of the 3D far field plot can be changed with the available view operations such as rotate, pan and zoom. A legend box appears in the upper right corner of the 3D radiation pattern plot, which can be dragged around with the left mouse button. The (maximum) '''Directivity''' of the radiating structure is displayed at the bottom of the legend box and is calculated using the definition:
:<math>D_0 = \frac{4\pi [S(\theta,\phi)]_{max}}{P_{rad}} = \frac{ 4\pi \big| \mathbf{E^{ff}}(\theta,\phi) \big|^2 |_{max} }{ \int\limits_0^{2\pi} \int\limits_0^{\pi} \big| \mathbf{E^{ff}}(\theta,\phi) \big|^2 \sin\theta \,d\theta \,d\phi }</math>
<!--[[Image:FDTD113.png]]-->
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. [[EM.Cube]] can calculate the radar cross section (RCS) of a target defined as:
:<math>\sigma_{\theta} = 4\pi r^2 \dfrac{ \big| \mathbf{E}_{\theta}^{scat} \big| ^2} {\big| \mathbf{E}^{inc} \big|^2}, \quad\sigma_{\phi} = 4\pi r^2 \dfrac{ \big| \mathbf{E}_{\phi}^{scat} \big| ^2} {\big| \mathbf{E}^{inc} \big|^2}, \quad\sigma = \sigma_{\theta} + \sigma_{\phi} = 4\pi r^2 \dfrac{ \big| \mathbf{E}_{tot}^{scat} \big| ^2} {\big| \mathbf{E}^{inc} \big|^2}</math>
<!--[[Image:FDTD130.png]]-->