Open Access
Add to BibSonomy
Abstract
Current near-field antenna measurement methods are commonly based on metal probes, with the accuracy limited and hard to be optimized due to the drawbacks they suffered, such as large volume, severe metal reflection/interference and complex circuit signal processing in parameter extracting. In this work, a novel method is proposed based on Rydberg atom in the near-field antenna measurement, which can offer a higher accuracy due to its intrinsic character of traceability to electric field. Replacing the metal probe in near-field measurement system by Rydberg atoms contained in a vapor cell (probe), amplitude- and phase- measurements on a 2.389 GHz signal launched out from a standard gain horn antenna are conducted on a near-field plane. They are transformed to far-field pattern and agree well with simulated results and measured results by using a traditional metal probe method. A high precision in longitudinal phase testing with an error below 1.7% can be achieved.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Antenna measurement, especially for the far-field pattern of antenna, is one of the main steps for antenna design in the field of wireless communication, meterage and detection, etc. Traditional methods are conducted directly in far-field region by metal probes, namely dipole antennas, horns and metal waveguides [1]. The test distance (d) between antenna and probe follows the rule defined by an equation d ≥ 2D2/λ with D as the aperture size of the antenna under test and λ as the tested microwave wavelength [2], which means a distance of several meters and more is required. At the same time, the size of microwave anechoic chambers needs to match the test distance d. This brings expensive construction costs. Near-field measurement methods can shorten the test distance by one order of magnitude and even more. Meanwhile, the far-field pattern of antenna can be decoded from the near-field results of amplitude and phase distribution by near-far field transformation (one of Fourier transformations) [3–6] which can compensate for the shortcomings of far-field testing. But there are still unavoidable problems faced. At the present, the probes of near-field are also metal probes. The measurements are taken at the sampling points of near-field plane and the amplitude and phase of the signal should be information at the ideal sampling point. However, the size of the metal probe is positively correlated with the wavelength of the electromagnetic wave. When the size of metal probe is too large to be treated as point sensor at low frequencies, and the measurement results of amplitude and phase will be just average values in the area near the sampling point. When the size of metal probe is too small to manufacture accurately at high frequency especially in the terahertz band, the probe will introduce a lot of testing errors. Furthermore, there is a reflection between metal probe and antenna, which brings errors to measurement results and can limit the accuracy of this method [3]. The metal probe has directional sensitivity, but it is difficult to compensate the angular calibration errors accurately based on traditional metal probe [3,6]. Besides, extracting the amplitude and phase information of microwave by the metal probe is based on the current signal obtained from electromagnetic induction, so the systems based on metal probe require complex circuit design.
Recently, Rydberg atom using electromagnetically induced transparency (EIT) has been demonstrated as a new type of sensor in wireless communications [7–9], subwavelength imaging [10], and radar systems [11]. Rydberg atoms can be viewed as a quantum oscillator that is perfectly frequency-matched to electromagnetic field [12–15], and its sensitivity has been realized as low as 55 nVcm−1Hz−1/2 [16]. Since there is a wide variety of different Rydberg states that are accessible by tuning the Rydberg-atom excitation lasers, Rydberg atom offers ultra-wide wave coverage from ∼MHz to ∼THz [17–19]. Due to the isotropy of atoms, Rydberg-atom probe measures electromagnetic waves without favor of direction, and it can be self-calibrated with precise calculation [20,21]. Besides, Rydberg atom can be encapsulated in a small vapor cell with a size of several centimeters, which achieves miniaturization of probes for detecting low-frequency electromagnetic waves [22–25].
2. Experiment setup
The experimental schematic and relevant four energy levels of Cs atom are illustrated in Fig. 1(a) and (b). A probe laser beam overlaps with a coupling beam and counter-propagate through a cylindrical Cesium (Cs) vapor cell along the x axis at the opposite direction. The probe laser with a wavelength of 852 nm (λp) is used to excite the ground state |6S1/2〉to the first excite state |6P3/2〉with its frequency locked by the saturation absorption spectroscopy (SAS) technique [27]. The coupling laser of 509 nm (λc) is resonant with the excite state |6P3/2〉and the Rydberg state |66D5/2〉, and its frequency is maintained by the modulation-demodulation method [19]. By applying a microwave of 2.389 GHz radiated from a tested antenna, it causes the transition between two Rydberg states |66D5/2〉and |67P3/2〉.
Fig. 1. Experiment setup of the Rydberg-atom probe. (a) The probe laser and the coupling laser counter-propagate through the Cs atom vapor cell along the x axis, and the outputs from both the tested and auxiliary antennas propagate along the z-axis. DM1 and DM2 are dichroic mirrors (DM1 is with high 509-nm transmittance and high 852-nm reflectivity, and DM2 is with high 852-nm transmittance and high 509-nm reflectivity). A photodetector is used to record the intensity of the probe laser. (b) Relevant four energy levels of Cs atom. (c) Cs atom vapor cell.
In order to acquire the phase distribution of the signal, an auxiliary antenna is introduced into the setup as a local oscillator field for superheterodyne detection [16]. The frequency difference between the tested field and the auxiliary field is maintained at 1 kHz. The tested antenna is a standard horn (HD-22SGAH10N) with a pyramidal shape, and the auxiliary antenna is a smaller horn that minimizes the reflection effect. Both antennas are positioned over 55 cm (around 4.4 λm, lying in the near field range of 3 λm to 6 λm [1] with the parameter λm being the wavelength of the tested microwave) away from the vapor cell, and the two beams of microwave propagate along the z axis. Both antennas are linearly polarized and perpendicular to the propagation direction of both laser beams. In our measurement, the Cs-atom vapor cell is fixed at the origin position in the near-field plane, and the tested antenna is moved relatively with Cs-atom vapor cell in the transverse plane (xoy) with a scanning step of 5 cm (0.4 λm). Figure 1(c) shows the Cs-atom vapor cell used in our experiment. To eliminate the influence of electromagnetic waves from the environment, the vapor cell is surrounded by the RF-absorbing material.
3. Results and discussion
In the superheterodyne detection of our near-field measurement system, the frequencies of the auxiliary field and the tested field are very close to each other, while the amplitude of the auxiliary field is much larger than the tested field. The total field can be simplified as equation ${{E}_{{total}}} \cong {{E}_{a}}{ + }{{E}_{t}}{\cos(}\Delta \mathrm{\omega +\ }\Delta \mathrm{\varphi )\;\ }$[26,27], where Ea and Et are the amplitudes of the auxiliary field and the tested field, respectively. Parameters $\Delta \mathrm{\omega}$ and $\Delta \mathrm{\varphi }$ are their frequency and phase differences, respectively. As demonstrated in Ref. [7], the transmission of the probe laser ${{T}_{{probe}}}$ is proportional to the tested microwave amplitude as ${{T}_{{probe}}} \propto |{{{E}_{{total}}}} |\cong {{E}_{a}}{ + }{{E}_{t}}{\cos(}\Delta \mathrm{\omega +\ }\Delta \mathrm{\varphi )}$. As a result, the amplitude and phase of the tested field can be measured using superheterodyne detection.
We first study the one-dimensional distribution of the microwave in near-field plane along the x-axis with fixed y = 0 and z = 55. It should be noted that, based on the symmetrical feature of the horn antenna structure, the field distribution on the horn antenna, the electric field distribution in the near-field plane and limitation of our experimental condition, we can only sample data from half of the central axis area and ultimately obtain the complete data of the entire near-field sampling plane based on symmetry. The two-dimensional image data with amplitude- and phase- information are also obtained by this method through measuring a quarter of the xoy plane.
Figure 2(a) shows the superheterodyne curves of the probe laser at different sampling points recorded by the photodetector, which exhibits a cosine waveform and are consistent with above demonstrations. The center of the Rydberg-atom vapor cell is set at the coordinate origin (0, 0, 0). The coordinates of the four sampling points are chosen at x = 5, 15, 25, 35, respectively, with identical values of y = 0, z = 55. Through curve fitting, the amplitudes at above four positions are 1.81 V, 1.63 V, 1.14 V, and 0.572 V, respectively, and the corresponding phases are 141.06°, 178.46°, 98.97° and 24.00°, respectively. The amplitudes of microwave decrease as the antenna moves away from the cell center.
Fig. 2. (a) The transmission curves of the probe laser at various sampling points. The coordinates (x, y, z) of the sampling points are y = 0, z = 55, and x = 5, 15, 25, 35, respectively. The unit of coordinates is centimeters, unless otherwise specified in the rest of paper. (b) Amplitude distribution. (c) Phase distribution at horizontal sampling line in near-field sampling plane (y = 0, z = 55).
The normalized amplitude distribution along the x axis is plotted in Fig. 2(b), where the amplitude at x = 0 is chosen as 1. The dynamic range of microwave amplitude using Rydberg-atom probe in our experiment is wide enough to meet the requirements of the antenna near field measurement [29,30], up to 42 dB. The microwave radiated from the antenna is mainly distributed in the center of near field plane. Its amplitudes decrease to 3 dB when the moving amount of x-coordinate exceeds the scope of [-22, 22]. The profile of its amplitude matches the feature of the microwave radiated from the standard horn as a pencil beam.
Figure 2(c) shows the phase distribution varying with x coordinate. The phase oscillates fast with the antenna moving to the edge of the test area, which implies that the chirp occurs in microwave phase when the amplitude becomes weak. It should be noted that the sampling test curves are obtained through an oscilloscope with an average of 8 times in Fig. 2. The errors bars of amplitudes and phases are obtained by a three-times testing. It is shown that the errors for amplitudes and phases are larger at the edge than the ones in center, and the total errors are less than 1%.
To get further insight into the near-field pattern of the tested antenna, we measure the two-dimensional distributions of amplitude and phase in the transverse plane by moving the tested antenna along the x- and y- axes with a fixed distance of z = 55 cm. According to Nyquist's law [2], the sampling interval should be less than half wavelength Δ$\mathrm{l\ < \lambda m/2\ =\ 6}{.3\; cm}$, and the scanning steps both along the x and y axis are chosen as Δ$\textrm{l}$=5 cm ($2\mathrm{\lambda m/5}$) in our experiments. The near-field patterns in the xoy plane with a tested area of 260× 200 cm2 are shown in Fig. 3. The two-dimensional distributions of both amplitude and phase present an elliptical shape with a bias along the x axis, which are the same as the long-side direction of the radiation port of the horn in our experiment. Moreover, their aspect ratios are about 1.5 and 1.6, respectively, for the amplitude and phase distributions. As the aspect ratio of the radiation port of our horn is 1.43, the near-field patterns especially for microwave amplitude are similar to the shape of the radiation port of the horn. Therefore, by comparing the relationship between the near-field image and the horn antenna port, it is found that the shape of the horn has a great impact on the near-field distribution of electromagnetic field.
Fig. 3. The near field has a size of and is 260 × 200 cm2 at 55-cm away from the port of the tested horn (a) Amplitude distribution in the two-dimensional plane. (b) Phase distribution in the two-dimensional near-field plane.
In order to acquire the far-field pattern of the tested antenna, we conduct near-far field transformation on the measured results in near-field [29] and show them in Fig. 4(a). For comparison, the numerical simulation calculated by a finite element difference method [11] and the experiment results by a waveguide method are also added. In Fig. 4(a), the curves overlap over a certain angle range. The main lobes of the antenna are close for both simulation and measurement by metal probe and Rydberg atoms. Besides, small sidelobes of three far-field patterns can be seen at the corresponding angles. As one of the most concerned technical parameters in the far-field antenna measurement, the 3-dB beam width parameters obtained by numerical simulation, waveguide probe and Rydberg-atom probe are 24°, 23° and 22°, respectively. The deviations between them are small and below 2°, which indicates that it is feasible to obtain the far-field pattern of antenna by near-field antenna measurement based on the Rydberg-atom probe.
Fig. 4. Far-field patterns of the tested antenna obtained by numerical simulation (blue line), waveguide probe (red line) and Rydberg-atom probe (black line). (b)Near-field measurement diagram based on metal waveguide. The waveguide is placed on the scanning frame and move in the sampling plane whose relative movement between antenna and probe is just opposite to the system on Rydberg atom probe.
The traditional near-field testing schematic is shown in Fig. 4(b). The commonly used probe is a metal waveguide whose probe pattern (detection capability in different directions) has analytical solutions, and we can make the relatively accuracy compensation in data analysis according to different detection capability in different directions. The size of waveguide is 10.92 × 5.46 × 45.12 cm3 at 1.72 GHz-2.61 GHz, but it is much larger than the Cs atoms vapor cell. During the measuring, the waveguide probe moves in the two-dimensional near-field scanning plane in front of the tested antenna and collects the amplitude and phase information at sampling points as shown in Fig. 4(b), which is similar to the measurement system based on Rydberg atom probe. For the amplitude of the electric field value, the difference of maximum and minimum must be more than 40 dB [30]. Therefore, it is necessary to determine the size of the sampling plane based on such requirement that the near-field measurement based on Rydberg atom probe has same requirement. The main purpose for such requirement is that the sampling plane can obtain most electric field energy radiated from the tested antenna and the final far-field pattern of the antenna can be accurately which is obtained and transformed from the results of near-field patterns. After the sampling is completed, the amplitude and phase data of the near field sampling points are analyzed by the near-far field transformation formulas to obtain the required far field radiation pattern of the antenna under test. The specific derivation can be found in the Ref. [29].
In the experiment, there are two main aspects of errors such as systematic errors and statistical errors. The system errors are mainly attributed from lasers, microwave and structure of vapor cell. Both the frequencies of the coupling and probe lasers are locked with saturation absorption spectrum, and the drifts of the frequency are less than 100 kHz. The linewidth of EIT is nearly 10 MHz. The fluctuation of output power of the probe laser is less than 1%. Therefore, the errors of laser stability bring an error within 1% into phase of the signal. The stability of microwave energy output varies less than 0.1 dBm and the achievable calibration accuracy frequency of microwave is less than 4 × 10−8. Therefore, the error of the microwave output is less than 0.5%. The vapor cell is incomplete symmetric structure and it can bring errors into directional sensitivity test as the antenna moving. And these systematic errors can be significantly improved in future. The statistical errors are mainly caused by sampling point coordinates and the calculation errors. The scale error of the sampling frame guide rail we use is less than 1 mm, so the probe position error is less than 1 mm. The calculation errors of amplitude and phase is less than 1% as the shown in Fig. 2.
In the near-field measurement, the amplitudes and phases are two main measured parameters. The accuracy of amplitude has been described by several publications [16,20,28]. Phase information is one of the key factors determining the accuracy and reliability of the near-field measurements and the antenna far-field radiation pattern as shown in Fig. 4, which is typical not considered in far field measurements. Therefore, we exploit accuracy and error of the phase to evaluate the measurement accuracy based on the Rydberg-atom probe. The longitudinal phase distribution of the microwave along the antenna’s propagation direction is measured by moving the tested antenna along the z axis. In this test, the frequency of the antenna under test is 5.047 GHz and the corresponding wavelength is λm = 5.94 cm, and the frequency difference between the tested and auxiliary antenna is set at 1 kHz. The amplitudes decrease from 0.0239 V to 0.0187 V as the antenna moves from z = 32 to z = 36 as in Fig. 5(a). The phase differences $\Delta \mathrm{\varphi }$ in various transverse planes perpendicular to propagation direction are proportional to the longitudinal distance $\mathrm{\Delta l}$, namely $\Delta \mathrm{\varphi =\ \beta \times \Delta l}$, where $\mathrm{\beta }$ is the phase propagation constant β=2π/λm. Presented in Fig. 5(b), the phase change behaves in a linear relationship with the longitudinal distance, and six periods (2$\mathrm{\pi }$) of phase distribution are included in the distance ranged from z = 35 to z = 70 and parallel to each other. By linear fitting, the average slope is 61.76 °/cm. Compared with the theoretical value of β as 60.56 °/cm, the longitudinal phase error based on the Rydberg-atom probe is about l.7%. According to the previous essay [31] when the phase error is within 5 °, its influence on the measurement of the sidelobe is less than 0.1 dB. As a result, the new near field measurement method based on Rydberg-atom probe is suitable in antenna near-field measurement.
Fig. 5. Transmission curves of the probe laser at different distance away from the test antenna along z axis. (a) Transmission curves at different positions in the z axis direction. (b) Fitted curves of phase vs. distance obtained at different position points.
4. Conclusion
In this work, we propose a new probe based on Rydberg-atom for antenna near-field measurement. It could overcome the limitations, such as probe pattern error, large size, metal reflection and interference, and complex circuit signal processing in parameter extracting, etc, faced by traditional metal probe-based methods. Measurements are taken in an anechoic chamber with a Rydberg atom probe fixed at receiver side and a horn antenna under test is positioned on a scanning frame for antenna measurement. Amplitude- and phase- measurements are conducted in a 260 × 200 cm2 near-field plane over 55 cm away from the horn antenna, and the patterns are similar to a typical horn. In far-field, it could obtain the antenna pattern with a deviation below 2°. We think this method could offer a higher measurement accuracy by increasing the stability of output power and linewidth of our laser sources.
Funding
National Natural Science Foundation of China (62071040).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. A. D. Yaghjw, “An Overview of Near-Field Antenna Measurements,” IEEE Trans. Antennas Propag. 34(1), 30–45 (1986). [CrossRef]
2. J. Borowick, A. E. Holley, W. L. Lange, R. W. Howard, and R. L. Cummings, “A near-field antenna measurement system,” IEEE Trans. Instrum. Meas. 32(1), 91–102 (1983). [CrossRef]
3. A. C. Newell, “Error analysis techniques for planar near-field measurements,” IEEE Trans. Antennas Propag. 36(6), 754–768 (1988). [CrossRef]
4. N. Wang, Z. H. Xue, S. M. Yang, R. X. Liu, and X. W. Xu, “Fast sample technique in antenna time domain near field measurement,” In 2007 International Symposium on Electromagnetic Compatibility(ISEC) (2007), pp. 507–510.
5. M. Cao, Z. H. Xue, W. Ren, W. M. Li, R. Q. Zhu, and H. W. Cai, “Multiple reflection error analysis for planar near field measurement in time domain,” In 2016 IEEE Conference on Antenna Measurements & Applications (CAMA) (2017), pp. 1–4.
6. M. Cao, Z. H. Xue, R. Q. Zhu, H. W. Cai, and Z. Pei, “The influence of the measurement probe in nearfield measure,” In 2016 IEEE International Conference on Microwave and Millimeter Wave Technology(ICMMT) (2016), pp. 257–260.
7. D. H. Meyer, K. C. Cox, F. K. Fatemi, and P. D. Kunz, “Digital communication with Rydberg atoms and amplitude-modulated microwave fields,” Appl. Phys. Lett. 112(21), 211108 (2018). [CrossRef]
8. C. L. Holloway, M. T. Simons, J. A. Gordon, and D. Novotny, “Detecting and receiving phase modulated signals with a rydberg atom-based mixer,” Antennas Wirel. Propag. Lett. 18(9), 1853–1857 (2019). [CrossRef]
9. D. A. Anderson, R. E. Sapiro, and G. Raithel, “An atomic receiver for AM and FM radio communication,” IEEE Trans. Antennas Propag. 69(5), 2455–2462 (2021). [CrossRef]
10. C. L. Holloway, J. A. Gordon, A. Schwarzkopf, D. A. Anderson, S. A. Miller, N. Thaichaeoen, and G. Raithel, “Sub-wavelength imaging and field mapping via electromagnetically induced transparency and Autler-Townes splitting in Rydberg atoms,” Appl. Phys. Lett. 104(24), 244102 (2014). [CrossRef]
11. A. K. Robinson, N. Prajapati, D. Senic, M. T. Simons, and C. L. Holloway, “Determining the angle-of-arrival of an radio-frequency source with a Rydberg atom-based sensor,” Appl. Phys. Lett. 118(11), 114001 (2021). [CrossRef]
12. J. A. Sedlacek, A. Schwettmann, H. Kübler, R. Löw, T. Pfau, and J. P. Shaffer, “Microwave electrometry with Rydberg atoms in a vapour cell using bright atomic resonances,” Nat. Phys. 8(11), 819–824 (2012). [CrossRef]
13. S. Sevincli and T. Pohl, “Microwave control of Rydberg atom interactions,” New J. Phys. 16(12), 123036 (2014). [CrossRef]
14. D. A. Anderson, S. A. Miller, G. Raithel, J. A. Gordon, M. L. Butler, and C. L Holloway, “Optical measurements of strong microwave fields with Rydberg atoms in a vapor cell,” Phys. Rev. Appl. 5(3), 034003 (2016). [CrossRef]
15. C. L. Holloway, M. T. Simons, A. H. Haddab, C. J. Williams, and M.W. Holloway, “A “real-time” guitar recording using Rydberg atoms and electromagnetically induced transparency: Quantum physics meets music,” AIP Adv. 9(6), 065110 (2019). [CrossRef]
16. M. Y. Jing, Y. Hu, J. Ma, H. Zhang, L. Xiao, and S. T. Jia, “Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy,” Nat. Phys. 16(9), 911–915 (2020). [CrossRef]
17. J. A. Gordon, C. L. Holloway, C. L. Holloway, A. Schwarzkopf, D. A. Anderson, S. Miller, N. Thaicharoen, and G. Raithel, “Millimeter wave detection via Autler-Townes splitting in rubidium Rydberg atoms,” Appl. Phys. Lett. 105(2), 024104 (2014). [CrossRef]
18. C. G. Wade, N. Šibalić, N. R. de Melo, J. M. Kondo, C. S. Adam, and K. J. Weatherill, “Real-time near-field terahertz imaging with atomic optical fluorescence,” Nat. Photonics. 11(1), 40–43 (2017). [CrossRef]
19. C. L. Holloway, N. Prajapati, J. Kitching, J. A. Sherman, C. Teale, A. Rufenacht, A. B. Artusio-Glimpse, M. T. Simons, A. K. Robinson, and E. B. Norrgard, “Electromagnetically induced transparency based Rydberg-atom sensor for traceable voltage measurements featured,” AVS Quantum Sci. 4(3), 034401 (2022). [CrossRef]
20. C. L. Holloway, M. T. Simons, J. A. Gordon, P. F. Wilson, C. M. Cooke, D. A. Anderson, and G. Raithel, “Atom-based RF electric field metrology: from self-calibrated measurements to subwavelength and near-field imaging,” IEEE Trans. Electromagn. Compat. 59(2), 717–728 (2017). [CrossRef]
21. C. T. Fancher, D. R. Scherer, M. C. S. John, and B. L. S. Marlow, “Rydberg atom electric field sensors for communications and sensing,” IEEE Trans. Quantum Eng. 2, 1–13 (2021). [CrossRef]
22. K. C. Cox, D. H. Meyer, F. K. Fatemi, and P. D. Kunz, “Quantum-limited atomic receiver in the electrically small regime,” Phys. Rev. Lett. 121(11), 110502 (2018). [CrossRef]
23. Y.-Y. Jau and T. Carter, “Vapor-cell-based atomic electrometry for detection frequencies below 1 kHz,” Phys. Rev. Appl. 13(5), 054034 (2020). [CrossRef]
24. Y. C. Jiao, X. Han, J. Fan, G. Raithel, J. M. Zhao, and S. T. Jia, “Atom-based receiver for amplitude-modulated baseband signals in high-frequency radio communication,” Appl. Phys. Express 12(12), 126002 (2019). [CrossRef]
25. E. Paradis, G. Raithel, and D. A. Anderson, “Atomic measurements of high-intensity VHF-band radio-frequency fields with a Rydberg vapor-cell detector,” Phys. Rev. A. 100(1), 013420 (2019). [CrossRef]
26. J.A. Sedlacek, A. Schwettmann, H. Kubler, and J.P. Shaffer, “Atom-based vector microwave electrometry using rubidium Rydberg atoms in a vapor cell,” Phys. Rev. Lett. 111(6), 063001 (2013). [CrossRef]
27. M. Menchetti, L. W. Bussey, D. Gilks, T. Whitley, C. Constantinou, and K. Bongs, “Digitally encoded RF to optical data transfer using excited Rb without the use of a local oscillator,” J. Appl. Phys. 133(1), 014401 (2023). [CrossRef]
28. M.T. Simons, M.D. Kautz, J.A. Gordon, and C.L. Holloway, “Uncertainties in Rydberg Atom-based RF E-field Measurements,” 2018 International Symposium on Electromagnetic Compatibility-EMC EUROPE (2018). [CrossRef]
29. J. H. Wang, “An_examination of the theory and practices of planar near-field measurement,” IEEE Trans. Antennas Propag. 36(6), 746–753 (1988). [CrossRef]
30. E. Martini, O. Breinbjerg, and S. Maci, “Reduction of truncation errors in planar near-field aperture antenna measurements using the gerchberg-papoulis algorithm,” IEEE Trans. Antennas Propag. 56(11), 3485–3493 (2008). [CrossRef]
31. Y. Ding, Y. Lin, D. M. Fu, and Q. Z. Liu, “Analysis and simulation of system phase errors in planar near-field measurements on ultra-low sidelobe antennas,” 2010 IEEE International Conference on Ultra-Wideband(ICUW) (2010).
