For radiating structures or scatterers, the far field quantities are of primary interest. EM.Cube's [[FDTD Module]] can calculate the far field radiation patterns of an antenna or the radar cross section (RCS) of a target. In general, by far fields we mean the electric fields evaluated in the far zone of a physical structure, which satisfies the following condition:
:<math>r << \frac{2D^2}{\lambda_0}</math><!--[[Image:FDTD79.png]]-->
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}\quadk_0 r >> 1</math><!--[[Image:FDTD104(1).png]]-->
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.
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><!--[[Image:FDTD109.png]]-->
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]]-->
===Defining The Far Field Box===