In this example, FDTD is used to simulate a X-band (WR-90) rectangular open-ended-waveguide-probe (OEWG) antenna mounted on a finite and infinite PEC ground plane. The radiation pattern and radiation efficiency are investigated using the Directivity Analysis group. This example demonstrates how the source window feature of the Directivity Analysis group can be used to accurately capture the total radiated power in instances the antenna's source (feed) passes through the directivity group's monitors.

The rectangular probe antenna is an aperture antenna used in RF applications such as in terrestrial communications and antenna measurements. Its widespread use is attributed to the antenna’s moderate bandwidth and gain, high-power capability, ease of integration with waveguide components, simple construction, and well-known near- and far-field response. Traditionally, a coaxial pin is used to excite the waveguide’s fundamental TE_{10} mode whose dominant field polarization and magnitude shown in the above figure. As with any aperture-type antenna, the generated fields on the open aperture are treated as the source of the far fields which are related to equivalent electric and magnetic current densities through field equivalence (Balanis [1]). In the case the probe antenna is mounted on an infinite ground plane, the radiation equation can be directly applied to find the theoretical far-fields. However, when the ground plane is finite, the far-fields must be approximated using diffraction theory or found numerically, such as in FDTD.

This example utilizes the directivity analysis group to find the radiation features of the rectangular probe antenna. First, the antenna is considered mounted on an infinite ground plane. Image theory is used to obtain the far-field FDTD and theoretical results which are then compared. Next, the antenna is considered mounted on a finite ground plane and FDTD results are presented.

## Simulation Setup

The simulation file rect_oewg.fsp contains the simulation model for this example. It consists of a vacuum filled rectangular waveguide of width b=22.86mm and a=10.16mm surrounded by a PEC wall of thickness 2.5mm. These are constructed using the Rectangle structure. The ground plane is a 2D PEC sheet which extends outside the simulation domain in the infinite ground plane simulation but has a finite dimensions of g_{x}=40 mm,g_{y}=20 mm in the finite ground plane simulation.

X-Y View: Infinite Ground Plane | Perspective View: Infinite Ground Plane |

X-Y View: Finite Ground Plane | Perspective View: Finite Ground Plane |

The Directivity analysis group is used to setup the monitors and find the far fields of the OEWG antenna. In the analysis group’s Setup Variable tab, the x, y, and z span are chosen such that the monitors are at least λ/4 away from the edge of the open aperture (infinite GP) or the edge of the ground plane (finite GP). The variable inf gp is set to either 1 (infinite ground plane) or 0 (finite ground plane). The down sample variable is set to 1, so that no down sampling occurs on the monitors. In the infinite GP simulation, we set source window to 0 (since the antenna’s feed does not pass through a monitor used in calculating directivity).

In the finite GP simulation, the fields inside the waveguide pass through the z1 boundary and would contribute to the calculated farfields and radiated power in the directivity analysis group. To avoid this, we allow the user to specify a window to cancel out this contribution. This window’s size and position should coincide with the region the waveguide’s source fields intersects the monitor, as shown in the above figure. To achieve this in this example, the directivity analysis group’s parameters are set as: source window=1, (window width, window height)=(22.86mm,10.16mm), window center (i,j)=(0,0), and window surface=z1.

To excite and measure the rectangular OEWG probe’s fundamental TE_{10} mode, a port is placed inside the waveguide’s PEC walls near the bottom edge of the simulation region. The TE_{10} mode is injected at 10GHz, which is above the mode’s cutoff frequency at 6.56GHz.

A mesh override region is placed over the entire simulation domain to enforce that the mesh size is at least λ/10 on the directivity analysis group’s monitors. Further mesh over regions are used to resolve the cross-section of the waveguide and the ground plane/vacuum interface

PML boundaries are specified on all boundaries to be at least λ/4 thick such that no reflections will occur outside of the Directivity analysis group’s monitors. To speed up the simulation, symmetry and antisymmetry boundaries are specified on the x min and y min boundaries, respectively.

Note : Advanced Option - Snap PEC to yee cell boundary When the "snap pec to yee cell boundary" option is enabled in the FDTD object's advanced option tab, the interface of any PEC is forced to align with the Yee cell boundary (more details can be found in Simulation - FDTD). It is recommended that this option be always turned |

## Results and Analysis

After opening up the FDTD file *rect_oewg.fsp*, the *rect_oewg.lsf* script is used to run the simulation and generate the results. Using the original file, the script initializes and runs two simulations of the OEWG: one with an infinite ground plane and one with a finite ground plane. The script modifies the directivity analysis group and FDTD region to each simulation’s respective settings discussed in the Simulation setup section. With the current mesh settings, both simulations take a few minutes to run. It should be noted that these results are obtained with a finer mesh over the probe's aperture.

### Infinite Ground Plane

The directivity patterns and radiation performance shown above are automatically generated by the script. In this example, the E-plane corresponds to the Y-Z plane and the H-plane corresponds to the X-Z plane. As expected, the infinite ground plane doesn’t redirects all energy forward and the pattern is only plotted from [-90,90] degs. The return loss is about 13 dB.

Aperture Fields: Infinite Ground Plane

=====Radiation Performance: Infinite Ground Plane===== Design Frequency: 10 GHz Input Power: 113.57 nW Accepted Power: 106.65 nW Radiated Power: 104.76 nW Radiation Efficiency from Input Power: 92.2 % Radiation Efficiency from Accepted Power: 98.2 % Maximum Directivity: 6.48 dB Total Realized Gain: 6.13 dB =====Radiation Performance: Finite Ground Plane===== Design Frequency: 10 GHz Input Power: 113.17 nW Accepted Power: 107.51 nW Radiated Power: 108.18 nW Radiation Efficiency from Input Power: 95.6 % Radiation Efficiency from Accepted Power: 100.6 % Maximum Directivity: 8.13 dB Total Realized Gain: 7.94 dB

Whereas theory assumes that the aperture fields are the ideal TE_{10} mode profile, the simulated aperture fields (shown in Fig. 5) differ due to the weak excitation of other modes at the transition from the waveguide to free space. This causes an observable deviation of the theoretical and simulated directivity patterns in the H-plane and a nearly 1.8dB difference in the total directivity. The radiation efficiency, which is taken as the ratio of accepted power over radiated power is at its expected value of 100% (no metallic or dielectric losses are present). Please refer the script file for details for those quantities, which is at the end of the script. Definitions and explanations of those quantities can be found in the Methodology page.

### Finite Ground Plane

For the OEWG with a finite ground plane, the size of the ground plane significantly alters the radiated power. Although no theory is presented for this case ( see the geometrical theory of diffraction in Balanis [1]), the directivity exhibits the expected pattern with a large main lobe, a H-plane beamwidth smaller than the E-plane beamwidth, a small backlobe with a front-to-back ratio of 12dB, and a reasonable directivity.

We can be assured that the source window is functioning correctly be observing that the radiation efficiency is nearly 100%. The directivity analysis group is subtracting out the source’s fields power contribution to the total radiated power calculation, which allows us to accurately obtain the radiation efficiency.

### Related references

[1] C. A. Balanis, Antenna Theory and Design, 4th Edition. John Wiley & Sons (2016).