Unlike differential-based methods, MoM simulators do not need a radiation box to calculate the far field data. The far-zone fields are calculated directly by integrating the currents on the traces and across the embedded objects using the asymptotic form of the background structureâs dyadic Green's functions:
:<math> \mathbf{E^{ff}(r)} = \iiint_V \mathbf{ \overline{\overline{G}}_{EJ,ff}(r|r') \cdot J(r') } \, d\nu ' + \iiint_V \mathbf{ \overline{\overline{G}}_{EM,ff}(r|r') \cdot M(r') } \, d\nu '</math>Â :<math> \mathbf{H^{ff}(r)} = \dfrac{1}{\eta_0} \mathbf{ \hat{r} \times E^{ff}(r) }</math><!--[[File:PMOM112.png]]-->
where η<sub>0</sub> = 120π is the characteristic impedance of the free space. As can be seen from the above equations, the far fields have the form of a TEM wave propagating in the radial direction away from the origin of coordinates. This means that the far-field magnetic field is always perpendicular to the electric field and the propagation vector, which in this case happens to be the radial unit vector in the spherical coordinate system. In other words, one only needs to know the far-zone electric field and can easily calculate the far-zone magnetic field from it. In EM.Cube's mixed potential integral equation formulation, the far-zone electric field can be expressed in terms of the asymptotic form of the vector electric and magnetic potentials '''A''' and '''F''':
:<math>\mathbf{E^{ff}}(x,y,z) = j k_0 \eta_0 \hat{r} \times [\hat{r} \times \mathbf{A}(r \to \infty)] +j k_0 \hat{r} \times \mathbf{F}(r \to \infty)</math><!--[[File:PMOM113.png]]-->
The asymptotic form of these vector potentials are calculated using the "'''Method of Stationary Phase'''" when k<sub>0</sub>r → ∞. In that case, one can use the approximation:
:<math> k_0 |\mathbf{r-r'}| \approx k_0 (r - \mathbf{\hat{r} \cdot r'}) </math><!--[[File:PMOM115.png]]-->
After applying the stationary phase method, one can extract the spherical wave factor exp(-jk<sub>0</sub>r)/r from the far-zone electric field, leaving the rest as functions of the spherical angles θ and φ. In other words, the far field is normalized to r, the distance from the field observation point to the origin. It is customary to express the far fields in spherical components E<sub>θ</sub> and E<sub>φ</sub>. Note that the outward propagating, TEM-type, far fields do not have radial components, i.e. E<sub>r</sub> = 0.
:<math> \mathbf{E_{\theta}}(\theta, \phi) = \cos\theta \cos\phi E_x + \cos\theta \sin\phi E_y - \sin\theta E_z </math>:<math> \mathbf{E_{\phi}}(\theta, \phi) = -\sin\phi E_x + \cos\phi E_y </math><!--[[File:PMOM114.png]]-->
=== Visualizing The Far Fields ===