Single-Cut Phaseless Near-Field Measurements for Fast Antenna Testing

Single-cut techniques allow for fast antenna characterization by measuring and transforming to far-field individual pattern cuts instead of the full sphere. The cut fields are expanded in a reduced set of cylindrical coefficients, which can be used to accurately compute the far field in the main pattern cuts for antennas with separable aperture distributions. This paper introduces a fast single-cut near-field to far-field transformation method using amplitude-only data. The technique is based on the measurement of the near-field magnitude in two concentric cuts and starts an iterative propagation process to retrieve their phases. This technique becomes a fast tool for antenna characterization when one is interested in a few cardinal plane cuts of the antenna and only magnitude measurements are available. The proposed single-cut phaseless technique is tested using simulated and measured data, and shows its potential for fast and more reliable amplitude-only measurements.


I. INTRODUCTION
Spherical near-field antenna measurements [1]- [4] have been established as the most accurate techniques for general antenna testing. For a proper near-field to far-field transformation, the amplitude and phase of the complex near-field must be sampled in the complete sphere with angular increments inversely proportional to the AUT electrical size, which can lead to expensive and time-consuming measurement processes. In recent years, multiple efforts have been made to address such limitations. Achieving fast and amplitude-only spherical measurements becomes convenient for testing high antenna volumes [5] and reducing equipment and operating costs, in both industrial and research scenarios.
A straightforward method to reduce antenna measurement times is to exploit the natural symmetry exhibited by many antennas [6]. Perhaps the most drastic approach in this sense has been the single-cut transformation [7]- [10]. This technique is based on the fact that most antenna measurements are conducted to retrieve the radiation pattern on a few cardinal cuts, typically the E and H planes. Using the Cylindrical Wave Expansion (CWE) [11], it is possible to perform near-field to far-field transformations of the antenna individual pattern cuts, which tremendously saves time because instead of measuring the complete sphere, only a few cuts are acquired.
In parallel, phaseless near-field antenna measurements have received increasing interest in recent years [12]. The lack of phase information must be compensated by other means, which results in different phaseless techniques, such as interferometry [13], holography [14] and multiple scan surfaces [15]- [22]. The latter option has the advantage that it does not rely on extra hardware. After measuring the near-field on two surfaces, several nonlinear algorithms [23] can be applied to retrieve the phase and the far-field pattern.
As a contribution toward fast, cheap and reliable antenna measurements, this paper introduces a single-cut near-field technique using amplitude-only information. The proposed method is based on measuring the AUT near-field amplitude in two concentric cuts of different radii. The measured fields can be expanded in a truncated set of "single-cut waves", which involves much fewer degrees of freedom then the standard 3-dimensional (3D) measurements. This facilitates the phase retrieval process, which is implemented by iteratively propagating the field between the cuts, as is usually done in traditional phaseless techniques [17], [18], [21], [22] .

II. THEORETICAL BACKGROUND
A. Single-cut near-field to far-field transformation The cylindrical near-field measurement scenario of Fig. 1 with an AUT at the coordinate system origin is considered. The near-field of the antenna is described in cylindrical coordinates (ρ, ϕ, z). The tangential components of the electric field radiated by the antenna can be represented as a superposition of elementary cylindrical waves [11]. However, in single-cut techniques, the measurement is only performed in a z = 0 cut, which can be considered a cylinder of infinitesimal height [10]: where δ() is the Dirac delta function and ⃗ E(ρ, ϕ) is the field measured on the z = 0 cut. Inserting (1) into the CWE formulas leads to the following transform pairs: where the terms b n and a n are single-cut wave coefficients (SCWC) and are closely related to the cylindrical wave coefficients but in one dimension. Finally, (2-3) can be asymptotically evaluated for ρ → ∞ to retrieve the far-field pattern from SCWC [10]. Eqs. (2)(3)(4)(5) can be used to perform near-field to far-field transformation of fields measured over a single-cut assuming the fields over a single-cut are independent from the other. In a general case, this assumption is not satisfied because all near-field cuts affect any far-field cut. However, many antennas exhibit a separable aperture field, which implies that the 2D variations of the equivalent aperture fields can be split as the product of two 1D variations [8]. This property enables accurate single-cut near-field to far-field transformations on the E and H planes. In these cuts, the contributions from the rest of the cuts tend to cancel each other, which results in low transformation errors [8].

B. Single-cut phaseless algorithm
When only the field magnitude is measured, an alternative algorithm must be devised to retrieve the SCWC of the AUT. This can be solved by measuring the near-field amplitude signals on two cuts of different radii. Following an analogous approach to phaseless spherical measurements [17], an iterative propagation algorithm was selected to retrieve the phase. Although alternative nonlinear optimization algorithms have been proposed, our preliminary studies for spherical geometry [22] have shown that they tend to reconstruct the far-field with less accuracy.
The phase retrieval algorithm starts with the measured magnitudes on each cut. Then, an iterative algorithm [17] is started and propagates the electric field from one cut to the other. Each time the field is propagated to one cut, its amplitude is replaced by the one that has been measured. This process retrieves the phase of the field on both cuts. When the retrieved phase stagnates to a point with negligible changes between consecutive iterations, convergence to the true solution or a local minimum has been achieved. This convergence occurs before 1000 iterations in all simulations that we have performed.
Finally, the quality of the retrieved solution is evaluated with the normalized RMS error of the SCWC: whereã n andb n are the retrieved coefficients after the iterative process, and a n and b n are the true coefficients. Of course, this complex metric can only be computed when the SCWC of the AUT are a priori known, so it is not applicable to a real phaseless measurement scenario. However, it will be used in the simulation experiments of the next section.

III. PHASELESS SINGLE-CUT TRANSFORMATION RESULTS
Now, the phaseless single-cut technique is tested using a dipole-based simulation model. By arranging electric and magnetic dipoles in array configurations, AUTs of arbitrary size can be simulated with analytical equations to emulate aperture antennas. The near-field of a D y × D z = 10λ × 10λ uniform array of Huygens sources is simulated on two concentric cuts with radii ρ 1 and ρ 2 . The antenna is centered in the coordinate system origin over the YZ plane with the main beam pointing in thex axis direction, so it is acquired by the measurement cuts. In all experiments, an oversampling factor of 2 with respect to a standard single-cut (or cylindrical) measurement is employed, since this has been reported [20] to improve the accuracy of phaseless techniques.

A. Influence of measurement radii
A set of simulations was performed by sweeping ρ 1 and ρ 2 between 10% and 100 % of the AUT Rayleigh distance, ρ f ar = 182λ. This normalization by ρ f ar has been proposed in previous publications for spherical phaseless measurements [20] because it makes the obtained error plots independent from the AUT electrical size. For each combination of radii, the iterative phase retrieval algorithm introduced in the previous section is evaluated for 1000 iterations.
Then, the metric ε comp is calculated considering the SCWC obtained from a complex single-cut transformation as a reference. Fig. 2 depicts the resulting errors, where some combinations of measurement radii perform better than others. Both relative separation between measurement cuts and also the absolute values are important. In general, the algorithm tends to obtain very low error values as long as ρ 1 is approximately 15-30% of ρ f ar and ρ 2 has a separation of at least another 25%. There is a region of irregular behavior for low ρ 1 values, which has also been observed in previous publications for the spherical case [20], [21], but no explanation has been found.

B. Influence of AUT size and shape
A new set of simulations with different AUT sizes is performed in which the ρ 2 cut is swept between 0.1ρ f ar and ρ f ar . The other one is kept fixed to ρ 1 = 0.35ρ f ar because, as shown in Fig. 2, it is convenient that one of the cuts stays relatively near the AUT. A 5λ × 5λ array with a Taylor distribution, and 15λ × 5λ and 1λ × 15λ uniform arrays are simulated for this test. Fig. 3 depicts the obtained ε comp errors of the retrieved solution, which shows a similar behavior with respect to the measurement radii as in the previous experiment. This validates the normalization approach to make the results independent from the antenna electrical size. The curves exhibit a somewhat irregular behavior with notably different error levels, but the general trend is a decreasing error with respect to the increasing separation between the cuts. This shows the robustness of the single-cut phaseless technique against antennas of different types and aspect ratios, in contrast with our previous investigations [20] on full spherical phaseless measurements where it was not possible to retrieve the phase of linear arrays.

C. Influence of the AUT offset and steering
The presence of beam steering and antenna offsets has been reported to degrade the accuracy of phaseless measurements [20], [21]. Therefore, a new set of simulations is performed to assess the influence of these parameters. The 10λ × 10λ array is now fed with a linear phase taper to steer the beam in different ϕ 0 angles over the measured cut: 20 • , 40 • and 90 • . Additional simulations are performed by shifting the antenna from the coordinate system origin 5λ and 10λ (50% and 100% of the AUT size respectively) along thex axis to emulate offset mounted antenna measurements [24]. The obtained errors from the parametric sweep are depicted in Fig. 4, showing a nonlinear behavior due to the strong asymmetry created by the offset. This is translated into more irregular error plots, but the algorithm still can retrieve the phase for appropriate AUTprobe separations.

D. Far-field pattern analysis
Now the translation of the ε comp errors to the transformed radiation pattern is analyzed using the simulation data of the experiment in Fig. 3. The retrieved far-fields from the 10λ × 10λ uniform array are depicted in Figs. 5 and 6 for ρ 2 values of 0.24ρ f ar , 0.31ρ f ar and 0.32ρ f ar , which correspond to ϵ comp levels of −45, −20 and −13 dB, respectively. The farfield pattern obtained from a single-cut complex measurement has also been depicted for use as a reference. The pattern amplitude is consistent with the reference even for high values of complex error. Meanwhile, the phase exhibits more discrepancies. This is an inherent characteristic of the iterative propagation technique and also observed in previous experiments for the spherical case [22], where relatively accurate far-field pattern magnitudes were obtained, even when the complex error metrics were high.

IV. PHASELESS ANTENNA MEASUREMENTS
Now, the single-cut phaseless near-field to far-field transformation technique is tested using measured data. The measurements were performed in the anechoic chamber of the Technical University of Madrid (UPM). The spherical range used consists of a roll-over-azimuth positioner with a translation stage to perform spherical measurements with radii of 3-5.5 m.

A. 20 GHz parabolic reflector
The first antenna to be tested is a 20-GHz reflector antenna of 53 cm diameter with linear polarization and approximately 40 dBi of gain. Two complex spherical near-field measurements were performed at 3 m and 5.5 m, each with two orthogonal orientations of a conical horn probe. An angular step of 1º is selected to obtain some level of oversampling to improve the quality of the phase retrieval. Using the full spherical data, the far-field for both spheres is computed with the commercial software SNIFT [25]. The average of both patterns is used as a reference for the phaseless measurements.
As discussed in section III, the measurement radii play a significant role in the accuracy of single-cut phaseless measurements. The selected radii for this experiment correspond to 8% and 15% of the Rayleigh distance, respectively, which is somewhat near the optimal region of Fig. 2. Furthermore, both measurement radii in electrical dimensions are larger than 200λ, where the single-cut approximations show small errors [10].
Two sets of phaseless single-cut measurements are generated by extracting the ϕ = 0 • and ϕ = 45 • cuts of the spherical near-field data for both polarizations and spheres. One thousand iterations of the phaseless single-cut technique are performed for each of cut pair. The retrieved far-field patterns are depicted in Figs. 7 and 8, along with the Equivalent Error Signal (EES) [20]. Both patterns exhibit good consistency with the reference, but significant differences are appreciated in the ϕ = 45 • cut. This is a well-know limitation of single-cut measurements [10]. Nevertheless, the quality of the reconstruction is remarkable considering the challenges of working with amplitude-only data.

B. DTU-ESA mmVAST antenna
The second AUT is the DTU-ESA millimeter-wave validation standard antenna (mmVAST) [26], which is an offset single-reflector antenna with an astigmatic paraboloid having different focal lengths in the orthogonal offset and transverse planes. The resulting radiation pattern makes mmVAST a challenging test for near-field postprocessing techniques. The antenna has an aperture dimension of 230 mm x 230 mm and the measurements are performed at 37.8 GHz in its circular polarization configuration.
The AUT is measured again in two spheres of radii 3 m and 5.5 m with 0.5 • angular increments and two orthogonal probe orientations. From each sphere, two orthogonal cuts corresponding to the antenna main planes ϕ = 0 • and ϕ = 90 • are extracted. The field amplitude on these cuts is used for the phase retrieval algorithm, which is evaluated for 1000 iterations. The full spherical field is processed with SNIFT again to obtain a true reference far-field.
Figs. 9 and 10 depict the far-field patterns in the principal planes retrieved by both approaches. For the ϕ = 0 • plane, the accuracy of the phaseless single-cut transformation is remarkable with EES levels below −40 dB for both circular polarisations. Meanwhile, the asymmetry of this cut degrades the separability of the aperture. Such degradation is translated in a worse reconstruction of the ϕ = 90 • plane with EES values growing up to −20 dB. Table I shows a summary of the measurement time and accuracy of different measurement methods conducted for both AUTs, in its complex/phaseless and spherical/single-cut variants. The accuracy is assessed by integrating the EES over the entire angular range considering the complex spherical measurement as a reference. The measurement time savings of the single-cut approach become drastic with only a small price to pay in the accuracy. The phaseless variant takes more time because it requires two cuts, but it is still negligible with respect to a full sphere scanning. The spherical phaseless results have been gathered from our previous publication [20], which were also obtained using the well-known iterative propagation phaseless algorithm.

C. Final discussion
For the 20 GHz reflector antenna experiment, the singlecut and spherical measurements show similar accuracy when working with phaseless data (around −50 dB of EES). Having full sphere near-field data allows for an exact postprocessing but the phase retrieval introduces an error. On the other hand, the phase retrieval of the single-cut measurement has  been simplified thanks to a reduced number of unknowns, but it suffers from the lack of separability in the single-cut approximation. In the end both sources of errors limit the maximum accuracy to approximately −50 dB.
The phase retrieval of the mmVAST is more challenging due to the asymmetric pattern, which limits the accuracy of the spherical measurement to −37 dB. Here, the benefits of using a single-cut transformation become evident, since the reduced dimensionality enables the phaseless algorithm to reach an EES level of −43 dB. In this case, the lack of separability introduces an error of −53.5 dB when working with complex cut data. However this error is lower than that obtained in both single-cut and sphere phaseless techniques; thus, it becomes a secondary issue.  These results demonstrate the potential capabilities of single-cut measurements in phaseless scenarios as a more reliable and faster tool than phaseless spherical measurements. The errors due to the lack of separability and phase reference prevent this technique from being used as a general and reliable tool in high-accuracy applications. However, this technique is very suitable for industrial applications where it is convenient to sacrifice some accuracy to maximize the antenna measurement speed.
The use of single-cut approaches to reduce measurement time in phaseless scenarios has been proposed before [27]. However, in those cases, the phase retrieval was still solved in a 2D domain after generating virtual measurement surfaces using the separability of the radiated fields. With the technique introduced in this paper, the number of unknowns involved in the phaseless problem is drastically reduced, which minimizes the chances of falling into local minima or false solutions [28].

V. CONCLUSION
This paper has introduced a new single-cut phaseless nearfield to far-field transformation technique. The use of singlecut wave coefficients as a 1D field basis minimizes the number of unknowns during the phase retrieval process. As a result, the presence of local minima and false solutions typically encountered in traditional phaseless techniques is mitigated. This has been validated with numerical examples of different antenna types, orientations and offsets, which show high robustness when measuring near-field magnitudes in two cuts with sufficient separation. The proposed algorithm has been tested with antenna measurements, and obtains obtaining good consistency with full spherical complex measurements and drastic time savings.