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>
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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>
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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>
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