Near- to Far-Field Transform

The finite-difference time-domain (FDTD) method calculates electromagnetic fields within the near-field of a simulated structure. Enlarging the simulation space to include the structure's far-field region is computationally demanding and not practical for most analyses, such as those performed with radar scattering or antenna radiation. The most efficient method to compute far-zone results is a mathematical transform that produces far-field values based on near-field data.

In XF, the process is referred to as the near- to far-field transform (NTFF). Here, near-field electric and magnetic fields that are tangential to a closed surface containing the simulated structure are integrated to arrive at the far-field response. It utilizes the surface equivalence theorem, which states that fields outside a closed fictitious surface are calculated by placing nonphysical current densities on the surface such that the fields inside the closed surface are zero and those outside are equal to the fields radiated by the structure inside the surface [1].

XF computes far-field results using a three-step process:

Compute near-field values during FDTD simulation.
Compute electric and magnetic current densities on the far-zone box.
Perform near- to far-field transform.

References

Balanis, Constantine A., Advanced Engineering Electromagnetics, John Wiley & Sons, 1999.

Near-Field Computation

Yee cell with field components. — Yee cell with electric and magnetic field components.

An XF simulation solves Maxwell's equations in the time-domain using the FDTD method, where the electromagnetic field calculations progress at discrete steps in space and time.

\begin{equation} \epsilon\frac{\partial E}{\partial t} = \nabla\times H \end{equation} \begin{equation} -\mu\frac{\partial\ H}{\partial t} = \nabla\times E \end{equation}

Space is segmented into a rectilinear grid of relatively small cells with an electric field, $E$, centered on each edge and a magnetic field, $H$, centered on each face. This field orientation is known as the Yee cell, on which the FDTD method is based.

Time is quantized into a small increment, referred to as a timestep, that represents the time required for an electromagnetic wave to travel from one field location to the next. Updates occur as the electric fields and then the magnetic fields are computed at each timestep using Maxwell's curl equations.

This full wave solution captures all near-field effects within the FDTD simulation space, including surface currents on conductors and dissipated power in dielectrics.

Far-Zone Box Representation

Dipole antenna with currents. — Location of far zone box in blue.

Using the surface equivalence theorem, the 3-D FDTD simulation volume is represented as a closed surface, referred to as the far-zone box, with a current density on each of its six sides. In XF, the far-zone box is located four cells from each outer edge of the grid and is independent of the structure being simulated.

Consider a radiating source, such as a dipole antenna placed at the center of the FDTD simulation space. During the FDTD simulation, the electric, $J_1$, and magnetic, $M_1$, currents on the surface of the dipole antenna and the $E$ and $H$ fields in the near-field of the antenna are computed.

Electric fields on far-zone box. — Electric fields on the far-zone box.

The tangential $E$ and $H$ fields on the far-zone box are visualized by placing planar sensors at the location of the far-zone box. Crossing the far-zone box, the fields radiated from the dipole reach the end of the simulation space where they are absorbed by the outer boundary conditions.

Applying the surface equivalence principle, the sources $J_1$ and $M_1$ at the dipole are replaced with nonphysical electric and magnetic currents flowing tangentially along the far-zone box given by the following equations:

\begin{equation} J_s = \hat n \times (H_1 - H_0) \end{equation} \begin{equation} M_s = - \hat n \times (E_1 - E_0) \end{equation}

where,

$E_0 $ = electric field one cell inside the far-zone box

$H_0 $ = magnetic field one cell inside the far-zone box

$E_1 $ = electric field one cell outside the far-zone box

$H_1 $ = magnetic field one cell outside the far-zone box

$\hat n $ = unit outward normal vector to the far-zone box

The virtual electric and magnetic currents $J_s$ and $M_s$, respectively, radiate both inside and outside the far-zone box, and serve as the sources for the far-field values. In order to compute the far-field values, the $E_0$ and $H_0$ fields are set to zero.

The resulting equations compute the current densities on each cell of the far-zone box:

\begin{equation} J_s = \hat n \times H_1 \end{equation} \begin{equation} M_s = - \hat n \times E_1 \end{equation}

The surface currents $J_s$ and $M_s$ at every cell edge on the far-zone box are integrated to arrive at the far-field results for a single observation angle, defined by a $\theta$ and $\phi$ direction.

First, vector potentials are defined:

\begin{equation} N = \iint_S J_s e^{j \beta r^\prime cos(\psi)} ds^\prime \end{equation} \begin{equation} L = \iint_S M_s e^{j \beta r^\prime cos(\psi)} ds^\prime \end{equation}

where,

$J_s $ = electric current density on the far-zone box

$M_s $ = magnetic current density on the far-zone box

$\beta $ = phase constant

$r^\prime\ cos(\psi) $ = difference in paths between the source and the far-field observation point

$ds^\prime $ = differential area selected on the far-zone box which is equivalent to a cell edge in XF

$S $ = surface of the far-zone box

In the far-field region, the $E$ and $H$ fields are orthogonal to each other with the $\hat\theta$ and $\hat\phi$ components as the dominant contributors. The radial components are not necessarily zero but are negligible compared to the $\hat\theta$ and $\hat\phi$ components.

Therefore, for far-field observations, $N$ and $L$ reduce to the following equations in the spherical coordinate system:

\begin{equation} N_\theta = \iint_S \big(J_x \cos (\theta) \cos (\phi) + J_y \cos (\theta) \sin(\phi) - J_z \cos (\theta)\big) e^{j\beta r^\prime \cos(\psi)} ds^\prime \end{equation} \begin{equation} N_\phi = \iint_S \big(- J_x \sin (\phi) + J_y \cos (\phi)\big) e^{j\beta r^\prime \cos(\psi)} ds^\prime \end{equation} \begin{equation} L_\theta = \iint_S \big(M_x \cos(\theta) \cos(\phi) + M_y \cos(\theta) \sin(\phi)- M_z \cos(\theta)\big) e^{j\beta r^\prime \cos(\psi)} ds^\prime \end{equation} \begin{equation} L_\phi = \iint_S \big(- M_x \sin(\phi) + M_y \cos(\phi)\big) e^{j\beta r^\prime \cos(\psi)} ds^\prime \end{equation}

Far-field observation point. — One cell's contribution to the far-field data.

These equations are numerically implemented by accounting for the location and orientation of each rectangular face of the far-zone box. The far-zone box's symmetrical placement around the center of the FDTD simulation space results in six faces involved in the integration process.

The figure provided shows one far-zone box surface with the far-zone reference point, $o$, placed at the center of the dipole antenna. In this image, the rectangular far-zone box surface is discretized using the incremental area $ds^\prime$ that contains the current densities acting as the far-field source points. The line $r^\prime$ denotes the distance between the far-zone reference point and the far-field source point. Line $r$ denotes the distance between the far-zone reference point and the far-field observation point, and has a default value of 1 m for far-field computations in XF. The angle between $r$ and $r^\prime$ is represented by $\psi$.

The last step in the NTFF transform computes the electric fields using the vector potentials:

\begin{equation} E_r \simeq 0 \end{equation} \begin{equation} E_\theta \simeq - \frac {j \beta e^{-j \beta r}} {4 \pi r} (L_\phi + \eta N_\theta) \end{equation} \begin{equation} E_\phi \simeq - \frac {j \beta e^{-j \beta r}} {4 \pi r} (L_\theta - \eta N_\phi) \end{equation}

Far-field antenna of dipole antenna. — 3-D radiation pattern of the dipole antenna.

Performing the NTFF transform for a set of observation angles allows the dipole antenna's 3-D radiation pattern to be displayed.

Implementation

The following considerations apply to XF's near- to far-field transform:

Users can set observation points through a far zone sensor.
Users can move the default reference point, $o$, from the center of the simulation space by editing the far zone reference settings.
The far-zone box is located four cells inside the simulation space.
The incremental area on the far-zone surface, $ds'$, is set by the grid and corresponds to every cell on the far-zone box.
The distance to the observation point, $r$, is one meter.