Extracting Fields in the Radiation Zone

When run in the Driven or Eigenmode modes, Palace can extrapolate fields from the near-field region (being simulated) to the far-field zone. This capability can be used to study the radiative properties of a system.

The mathematical details on how this is accomplished are available in the reference. The key points are as follows:

• Computing fields requires evaluation of surface integrals, so users must specify a surface of integration (more on this later).
• For the result to be physically accurate, it is important to properly model the propagation of waves to infinity. This can be accomplished by enclosing the system inside a sphere or box and applying config["Boundaries"]["Absorbing"] boundary conditions.
• The result is provided as complex vectors $r \mathbf{E}(\theta, \phi)$, where $(\theta, \phi)$ identify a point on sphere at infinite distance and the result is defined up to a global phase.

Palace outputs $r\mathbf{E}$ because this is a well-defined, finite quantity ($\mathbf{E}$ itself goes to zero at infinity as 1/r). You can also think of this as the electric field measured at a distance of one unit of length. With this output, you can immediately compute various quantities of interest. For instance, $|r\mathbf{E}|^2$ gives the relative radiative power.

Setup

A typical setup consists of starting from the system under consideration and enclosing the system inside an outer boundary (typically a sphere or a box), if it is not already. Then, we set "Absorbing" boundary conditions on this surface and choose it as the surface for the integration. For best accuracy, it is a good idea to make sure that this outer boundary is meshed finely enough to resolve the expected wavelength.

Turning on far-field extraction requires activating the feature in the configuration JSON file. To do so, we look at the "FarField" section under "Postprocessing" in "Boundaries". Here, we need to specify the identifier of the integration surface in "Attributes" and specify how we want to sample the sphere at infinity. As in many other parts of Palace, "Attributes" expects a vector, as it can happen the boundary is split in multiple pieces.

Once we define the surface of integration, we need to specify where we want to evaluate target far-field points. The simplest way to do this is by setting "NSample", so that the far-field sphere is uniformly sampled with NSample points. (Note that uniform on a sphere means that the polar angle $\theta$ of the sampled points is not uniformly distributed, to avoid bunching of points on the poles.)

Often, uniform sampling on the far-field sphere might be a good first step, but not the best way to accurately capture the radiative properties (e.g., when the radiation is highly directional). In this case, you can specify at what angles $\theta, \phi$ you want to evaluate the fields. This is done by passing 2-vectors to the "ThetaPhis" array with the angular coordinates of your choosing (in degrees).

This can be combined with scripts to target specific regions. For instance, the following Python code produces the required "ThetaPhis" section to sample over the xy plane:

import json
dphi = 1  # degree
print(json.dumps({"ThetaPhis": [[90.0, p] for p in range(0, 361, dphi)]}))

Both "NSample" and "ThetaPhis" can be specified simultaneously and the results will be combined and duplicates removed.

Output

Once a simulation is run, Palace generates a CSV file named farfield-rE.csv in the folder specified by the config["Problem"]["Output"] configuration.

The CSV contains a header line that describes the columns, and one row for each $(\theta, \phi)$ pair. The columns are:

• f (GHz) : Frequency
• theta (deg.) : Polar angle in degrees
• phi (deg.) : Azimuthal angle in degrees
• r*Re{E_x} (V), r*Im{E_x} (V) : Real and imaginary parts of $r E_x$ component
• r*Re{E_y} (V), r*Im{E_y} (V) : Real and imaginary parts of $r E_y$ component
• r*Re{E_z} (V), r*Im{E_z} (V) : Real and imaginary parts of $r E_z$ component

All field values are in SI units.

To obtain the magnetic fields, you can assume that propagation occurs in free space, so that

\[r \mathbf{H}_p(\mathbf{r}_0) = \frac{\mathbf{r}_0 \times r \mathbf{E}_p(\mathbf{r}_0)}{Z_0}\]

with $Z_0$ impedance of free space.

This type of output can be processed by several different packages and languages. For instance, see the antenna example for an example of a Julia script that plots antenna patterns.