Introduction
In this article we present a workflow to calculate the phase center (PC) of an antenna. The PC of an antenna is the position of the corresponding isotropic dipole emitter which in the far field emits the same spherical wave as a section of the antenna radiation pattern.
For the PC calculation we are going to use the Least Mean Squares Solution [1]. For determining the PC we are using far field data from a box of monitors.
The yellow solid line below represents the far field while the gray dashed line represents the phase front, the center of which is the PC.
By considering a set of discrete far field phase samples \(ψ_i\) within the main lobe, a sum of squares difference (SSD) model can be defined as in [1]:
$$SSD = \sum_i[\psi_i(\psi_0+k(d_zcos(\theta_i)+d_ysin(\theta_i)))]^2$$
where \(ψ_0\) is the unwrapped phase at θ=0° and k is the wavenumber.
Setting the derivatives of the equation above with respect to the unknowns (dz, dy, and \(ψ_0\)) to zero, a system can be assembled in matrix form:
\[ \left[\begin{array}{ccc} k\sum\limits_{i}w_isin^2\theta_i & k\sum\limits_iw_isin(\theta_i)cos(\theta_i) & \sum\limits_iw_isin(\theta_i) \\ k\sum\limits_iw_isin(\theta_i)cos(\theta_i) & k\sum\limits_iw_icos^2\theta_i & \sum\limits_iw_icos(\theta_i) \\ k\sum\limits_iw_isin(\theta_i) & k\sum\limits_iw_icos(\theta_i) & \sum\limits_iw_i \\ \end{array}\right] \left[\begin{array}{ccc} d_y \\ d_z \\ \psi_0 \\ \end{array}\right] = \left[\begin{array}{ccc} \sum\limits_iw_i\psi_isin(\theta_i) \\ \sum\limits_iw_i\psi_icos(\theta_i) \\ \sum\limits_iw_i\psi_i \\ \end{array}\right] \]
where \(w_i\) terms correspond to a weighting function. For a planar Efield pattern, the amplitude weighting function is defined as:
$$w_i = \frac{E_i^2}{ \sum\limits_{i=1}^n E_i^2}$$
where the \(E_i\) are the far field electric field samples corresponding to \(θ_i\) and \(ψ_i\).
Run and results
In this example, we perform a 3D simulation using an electric dipole source at the center of the simulation region (0,0,0) and a nanoparticle close to the dipole on the yz plane. The axis of the dipole is oriented along the xdirection. We use a box of monitors surrounding the dipole to calculate the far field. After running the simulation, the far field from the box monitor can be visualized as below. We can identify the main radiation lobe in the yz plane. For an Ag nanoparticle with 100nm radius, placed at (0,0.15um,0.15um), the main lobe is positioned at θ between 20° to 110°.
In the script phase_center.lsf used for the PC calculation, we select the “yz” plane for the calculation of the PC which is the plane of the main lobe. Furthermore, we set the range of angle between 20° to 110°, which are the angles of the main lobe as identified from the far field visualization (scat_ff analysis group). The script will also automatically select the dominant field component for the calculation of the PC (\(E_θ\) or \(Ε_φ\)). By running the script, dy and dz are calculated. For this example, dy=dz=0.039um, hence, the phase center is located at (0,0.039um,0.039um).
The “make_plots” option can be enabled, i.e. set to 1, to plot the phase for far field projection from the center of the box of monitors (0,0,0) and from the PC as well as to calculate the maximum phase difference.
Notes:

[1] https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/kw52j844n