# Forward Transformation from Reactive Near-Field to Near and Far-Field at Millimeter-Wave Frequencies

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## Abstract

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## 1. Introduction

## 2. Equivalent Current Reconstruction and FT Approach

## 3. Evaluation Antennas

- Horn antennas loaded with a slot array, y linear polarization. One at 10 GHz, 30 GHz and 90 GHz. Short descriptions: SlottedHorn10G, SlottedHorn30G, and SlottedHorn90G.
- A cavity-fed dipole array, y linear polarization. One at 10 GHz, 30 GHz and 60 GHz. Short descriptions: DipoleArr10G, DipoleArr30G, and DipoleArr60G.
- A planar $3\times 3$ array of half-wavelength square patches at 60 GHz. The antennas were excited independently for each x/y polarization port. The following two configurations were produced: (i) circular polarization (short description: PatchArray60G_circular) and (ii) two simultaneous beams pointing at $\theta ={30}^{\xb0},\varphi ={0}^{\xb0}$, and $\theta ={10}^{\xb0},\varphi =-{90}^{\xb0}$, for x- and y-linear polarizations, respectively (short description: PatchArray60G_L30xL10y).

## 4. Error Metric

## 5. Plane Size Requirements

## 6. Algorithm Evaluation

## 7. Simulation Results

## 8. Validation with Measurements

## 9. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Antenna configurations used for finite-difference time-domain (FDTD) simulations. (

**a**) Slot array on a horn antenna designed for 10 GHz, 30 GHz and 90 GHz. A 20 dBi commercial horn (SAR-2013-28-S2) manufactured by SAGE Millimeter Inc. was used, while the array of rectangular slots was fabricated in a 0.15 mm stainless steel sheet (42.5 mm × 33.8 mm). (

**b**) Cavity-fed dipole array designed for 10 GHz, 30 GHz and 60 GHz. At 30 GHz, this antenna consists of an array of nine dipoles arranged in an irregular lattice. The dipoles (0.8 mm × 1.86 mm) are excited by non-resonant slots (1.9 mm × 0.4 mm) that share a 0.508 mm dielectric substrate of relative permittivity ${\u03f5}_{r}$ = 3.63. The excitation modes are generated by a resonant cavity of size 21.18 mm × 21.18 mm × 5.0 mm, which is fed by a 2.92 mm (K) connector situated underneath. (

**c**) Planar $3\times 3$ patch array at 60 GHz. Details for the slot array and cavity-fed dipole array can be found in [32]. Details for the planar 3 × 3 patch were as follows: grounded substrate ($t=0.15\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$, ${\epsilon}_{\mathrm{r}}=2.5$, $\sigma =0.0038\mathrm{S}/\mathrm{m}$), $S=2.5\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$, $P=1.44\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$, $\Delta x=\Delta y=0.33\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ (offset of the feed with respect to the center of the patch for each linear polarization).

**Figure 2.**Different sizes for the plane of equivalent sources (ranging from $4\times 4$ samples, or $1\phantom{\rule{0.166667em}{0ex}}{\lambda}^{2}$, to $80\times 80$ samples, or $400\phantom{\rule{0.166667em}{0ex}}{\lambda}^{2}$) used for forward transformation for the 30 GHz cavity-fed dipole array overlapped on the normalized electric field distribution. The plane is at a distance of 2 mm from the antenna under test (AUT) plane and ${E}_{0}=72.3\phantom{\rule{0.166667em}{0ex}}$ V/m.

**Figure 3.**Error in peak spatial-average power density (${psPD}_{tot}$) for different sizes of the plane of equivalent currents (or the closest measurement plane) for the different evaluation antennas. (

**a**) shows the dependence of the absolute error for the FT algorithm with respect to the simulated fields, and (

**b**) shows the error caused by plane truncation with respect to the reference case with the largest plane size.

**Figure 4.**Forward transformation (FT) to $z=50\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ based on simulated equivalent currents for the different plane sizes in Figure 2. The results from FT improve as the plane is enlarged, but results change very little for planes larger than $16\times 16$ samples.

**Figure 5.**Example of first (

**a**–

**d**) and third (

**e**–

**h**) planes used as inputs of the algorithm at different noise levels for the 30 GHz dipole array.

**Figure 6.**Example simulation and FT of the spatial-average power density (sPD, over a 1 cm${}^{2}$ circle) at $z=50\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ without noise for the dipole array at 30 GHz.

**Figure 7.**Example simulation and FT of the sPD (over a 1 cm${}^{2}$ circle) at $z=50\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ with a noise level of −24 dB for the dipole array at 30 GHz.

**Figure 8.**Mean absolute error in ${psPD}_{n}$ (

**a**) and ${psPD}_{tot}$ (

**b**) for different noise levels (y-axis of each figure) and for different distances. The plane of equivalent currents was placed at 2 mm for all cases. Mean values and standard deviations over 20 simulations per noise level are shown.

**Figure 9.**Mean absolute error in ${psPD}_{n}$ (

**a**) and ${psPD}_{tot}$ (

**b**) for different noise levels (see legend) depending on distance. The plane of equivalent currents was placed at 2 mm for all cases. Mean values over all antennas are shown.

**Figure 10.**Mean absolute error in ${psPD}_{n}$ (left column, (

**a**,

**c**,

**e**)) and ${psPD}_{tot}$ (right column, (

**b**,

**d**,

**f**)) for different distances ${z}_{1}$ of the plane of equivalent currents for the 10 GHz antennas. Different noise levels (y-axis of each figure) and FT to different distances are shown (title of each subplot). The plane of equivalent currents was placed at ${z}_{1}=2\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ (

**a**,

**b**), ${z}_{1}=4\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ (

**c**,

**d**), and ${z}_{1}=6\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ (

**e**,

**f**). Mean values and standard deviations over 20 simulations per noise level are shown. The uncertainty is bounded by the acceptable level for ${z}_{1}\ge 4\phantom{\rule{0.166667em}{0ex}}$ mm.

**Figure 12.**Errors in psPD (averaged over 1 cm${}^{2}$) for FT from measurements at ${z}_{1}=2\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ to evaluation planes at different distances (indicated in the subplot titles). The result from FT is compared to a reference value from simulation (vs. sim.) and to reference values obtained through a measurement (vs. meas.) at the respective distance. Measurements of four different antennas were used as indicated in the legend.

**Figure 13.**Radiated power density at $z=5$ mm obtained through simulation abd measurement and the proposed forward transformation algorithm for (

**a**) the cavity-fed dipole array antenna and (

**b**) the horn antenna loaded with slot arrays. Note that the measurements are performed over a $3\lambda \times 3\lambda $ area, whereas in the simulations, a larger area is considered for calculations.

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**MDPI and ACS Style**

Pfeifer, S.; Fallahi, A.; Xi, J.; Neufeld, E.; Kuster, N.
Forward Transformation from Reactive Near-Field to Near and Far-Field at Millimeter-Wave Frequencies. *Appl. Sci.* **2020**, *10*, 4780.
https://doi.org/10.3390/app10144780

**AMA Style**

Pfeifer S, Fallahi A, Xi J, Neufeld E, Kuster N.
Forward Transformation from Reactive Near-Field to Near and Far-Field at Millimeter-Wave Frequencies. *Applied Sciences*. 2020; 10(14):4780.
https://doi.org/10.3390/app10144780

**Chicago/Turabian Style**

Pfeifer, Serge, Arya Fallahi, Jingtian Xi, Esra Neufeld, and Niels Kuster.
2020. "Forward Transformation from Reactive Near-Field to Near and Far-Field at Millimeter-Wave Frequencies" *Applied Sciences* 10, no. 14: 4780.
https://doi.org/10.3390/app10144780