Far-Field Radiation Sensor¶

The far-field radiation sensor is designed to recover fields in the far-field zone and is instrumental in investigating radiation and scattering patterns. It utilizes a near-to-far field transformation, which facilitate analytical calculations of the electromagnetic field in the far-field region based on its near-field values.

The far-field electric field is calculated using an approximation of the Stratton-Chu formula, which is valid in the limit where \(kr \gg 1\) (with \(k\) representing the wave number and \(r\) denoting the observation distance). According to this approximation, the far-field electric field \(\vec{E}_p(\vec{r}_0)\) at the observation direction \(\vec{r}_0\), parameterized by the spherical angles \((\theta,\phi)\), is expressed as

\[ r \vec{E}_p(\vec{r}_0) = \frac{-j k}{4 \pi} \vec{r}_0 \times \int_S \left[ \hat{n} \times \vec{E} - Z\, \vec{r}_0 \times \left(\hat{n} \times \vec{H}\right) \right] e^{-jk\vec{r}_0 \cdot \vec{r}} dS. \]

Here \(S\) is an arbitrary surface enclosing the simulated object with an outward normal \(\hat{n}\). The vectors \(\vec{E}\) and \(\vec{H}\) represent the electric and magnetic fields at the surface \(S\), while \(Z\) is the impedance. Variable \(r\) indicates a point on the surface \(S\), and the integration encompasses all such points. The observation direction is expressed as \(\vec{r}_0=(\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)\) with the polar angle \(\theta\in[0,\pi]\) and the azimuthal angle \(\phi\in[0,2\pi]\).

Note that we obtain \(r \vec{E}_p\) rather than \(\vec{E}_p\) because the electric field decays as \(\exp(-jkr)/r\). Multiplying by \(r\) ensures that the field quantity remains finite. Additionally, the solution is defined up to a global phase factor.

From \(r \vec{E}_p\), the magnetic field can be obtained under the assumption that the waves are propagating in free space:

\[ r \vec{H}_p = \frac{\vec{r}_0 \times r\vec{E}_p}{Z_0}, \]

where \(Z_0\) is the impedance of free space.

Having determined the far-field electric field \(\vec{E}_\text{far}\), the E-field radiation pattern \(\vec{E}_\text{pattern}(\theta,\phi)\) is defined as

\[ \vec{E}_\text{far}(\theta,\phi,r) = \vec{E}_\text{pattern}(\theta,\phi) \frac{e^{-j\vec{k}\cdot\vec{r}}}{r}. \]

From this equation, we have \(\vec{E}_\text{pattern}(\theta,\phi)=|r\vec{E}_p|\). E-field radiation pattern has the unit of voltage.

The power \(U\) radiated per unit solid angle, also known as radiation intensity, is given by

\[ U(\theta,\phi) = \vec{E}_\text{pattern}(\theta,\phi) \times \vec{H}_\text{pattern}(\theta,\phi) = \frac{1}{2Z_0} \left|\vec{E}_\text{pattern}(\theta,\phi)\right|^2. \]

To create a FarFieldRadiationSensor object, the user must provide a custom sensor name along with range parameters, including the start angle, stop angle, and step size. These parameters define the values for the the polar angle \(\theta\) and the azimuthal angle \(\phi\). A section of the outer surface of the computational domain, where the AbsorbingBoundaryCondition is applied, serves as the integration surface \(S\). An example of how to create and use a FarFieldRadiationSensor is provided below:

sensor = FarFieldRadiationSensor(
      "My Far-Field Radiation Sensor",
      polar_start=0.0,
      polar_stop=180.0,
      polar_step=6.0,
      azimuthal_start=0.0,
      azimuthal_stop=360.0,
      azimuthal_step=6.0,
)

thetas = sensor.get_polar_angles()
phis = sensor.get_azimuthal_angles()
e_pattern = sensor.get_radiation_pattern()