A near-field measurement technique for the prediction of asymptotic far-field antenna patterns from data obtained from a modified cylindrical or plane... Antenna measurements - Measurement techniques - Electromagnetic measurements - Computational electromagnetics - Computational modeling - antenna radiation patterns - antenna testing - computational electromagnetics - conical antennas - near-field measurement technique - far-field antenna pattern - cylindrical-mode expansion - computational electromagnetics - range measurements - conical antenna - Antenna measurements - antenna radiation patterns - near field measurement - conic frustum - cylindrical

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Abstract A near-field measurement technique for the prediction of asymptotic far-field antenna patterns from data obtained from a modified cylindrical or plane-polar near-field measurement system is presented. This technique utilizes a simple change in facility alignment to enable near-field data to be taken over the surface of a conceptual right cone [1, 2], or right conic frustum [3, 4], thereby allowing existing facilities to characterize wide-angle antenna performance in situations where hitherto they would perhaps have been limited by truncation. This paper aims to introduce the measurement technique, and to describe the novel probe-corrected near-field-to-farfield transform algorithm. The algorithm is based upon a cylindrical-mode expansion of the measured fields. Preliminary results of both computational electromagnetic simulations and actual range measurements are presented. As this paper recounts the progress of ongoing research, it concludes with a discussion of the remaining outstanding issues, and presents an overview of the planned future work. Keywords: Antenna measurements; antenna radiation patterns; near field measurement; conic frustum, cylindrical

1. Introduction

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t is well known that far-field antenna parameters, such as the pattern, gain, directivity, beamwidth, etc., can be derived from near-field measurements. For such parameters, which are not obtained directly from measurements made in the near field, a transformation from one surface to another is necessitated. This transformation of monochromatic but otherwise arbitrary waves can be accomplished efficiently by representing the field as a summation of any elementary wave solution of Maxwell's equations. Here, the coefficients to these solutions are determined by matching the fields on the surface on which the fields are known, and by using mode orthogonality. Solving this modal expansion for the fields over a spherical surface of infinite radius, centered about the radiator, results in the far-field pattern. Generally, this is most effective when selecting a modal basis that is commensurate with the measurement geometry, i.e., by utilizing plane waves, cylindrical waves, or spherical waves, respectively, for the case where the measurements are taken over planar, cylindrical, or spherical surfaces. Although complete solutions of the complex vector wave equation are available for certain other systems of coordinates, until relatively recently these have received comparatively little attention in the published literature.

Since it is possible to construct a right-conical measurement system from any existing cylindrical system by merely tilting the linear stage (or, alternatively, by tilting the rotation axis), then this is also true for the conical case. The conical system can thus be conveniently fabricated using existing commercial off-the-shelf (COTS) positioning stages, providing a solution for the characterization of a class of antennas that currently can only be effectively served with spherical near-field scanning.

2. Overview of Measurement Technique and Transform Conceptually, the right-conical measurement system is perhaps most closely related to the well-documented, well-understood, cylindrical near-field scanning technique. However, here the axis of rotation of the antenna under test (AUT) and the linear translation stage that caries the probe are no longer constrained to be exactly parallel with one another. By taking samples incrementally on a raster grid, by varying the azimuthal angle and linear displacement, the near electric field can be sampled over the surface of a right cone. This is illustrated schematically in Figure 1.

The utility and viability of any measurement system depends not only on the availability of the requisite probe-compensating near-field-to-far-:-field transformation, but also on the ease with which a sufficiently accurate robotic positioning subsystem can be constructed. The planar, cylindrical, and spherical geometries have the inherent advantages that their respective robotic subsystems can be conveniently realized from combinations of readily available rotation and/or linear translation stages at an economical price.

If the near-field probe is rotated through 90° about its axis of rotation and this process is repeated, two orthogonal near electricfield components can be acquired. It is from these that the far-field pattern can be obtained. Previous transformation algorithms have been based on spherical or plane-wave expansions. In contrast, the near-field-to-far-field transform considered here is based upon a cylindrical-mode expansion [5]. However, one assumption that has been introduced is that the probe is aligned such that its axis of

IEEE Antennas and Propagation Magazine, Vol. 51, No.1, February 2009

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Figure 6b. The far-field pattern from the theoretical model: Eo·

IEEE Antennas and Propagation Magazine, Vol. 51, No.1, February 2009

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Once the cylindrical mode coefficients have been determined, the asymptotic far-field pattern can be obtained from a simple summation of modes:

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rotation is orthogonal to, and intersects with, the azimuthal (i.e., rotation) axis of the AUT. This has the benefit that the conventional cylindrical transmission-matrix formula can be retained and, when inverted, used to compensate the conical measurements without introducing any tangible practical limitations into the measurement process. To provide an illustration of how this new transform works, let us consider obtaining the far-field pattern of a given radiator from cylindrical near electric-field data, sampled using an infinitesimal Hertzian dipole probe. This last restriction is introduced purely to simplify and ease the pedagogy, and is not a fundamental constraint of the measurement technique. In summary, the electromagnetic fields outside an arbitrary test antenna, radiating into free space, can be expanded into a set of orthogonal cylindrical mode coefficients. These eigenfunctions can then be used to obtain the electric and magnetic fields everywhere in space outside of this conceptual cylindrical surface. Conveniently, these can be used to obtain explicit expressions for the asymptotic far-field pattern. When expressed in component from, the two sets of orthogonal cylindrical mode coefficients can be obtained from [5] 196

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Here, and as per the usual convention, the unimportant far-field spherical phase factor and the inverse r term have been suppressed. In practice, the number of cylindrical mode coefficients can be truncated to a finite number that equates approximately to halfwavelength sampling over the surface of a conceptual cylinder that is centered on the origin of the measurement coordinate system, and that encloses the majority of the current sources. The maximum mode index, N, is thus given by N = ko'i + 10, where 'i is the maximum radial extent (MRE) [7]. Also, the Fourier variable r can be limited to ±ko (equivalently, where I( = 0), as these are the highest-order propagating modes. As the sample spacing, i.e., resolution, is determined from the maximum value of r, we can write that bz = 1Z'/ko = 2/2, where A denotes the wavelength. For the conical case, the angular sample spacing is thus held fixed for all values of z, at an amount determined by the size of the maximum radial extent. Samples are taken at every half waveIEEE Antennas and Propagation Magazine, Vol. 51, No.1, February 2009

length along the linear scan axis, that is, over the surface of the cone (i.e., not along the rotational axis, where the two amounts differ by a factor of the cosine of the half cone angle). From the analysis of the cylindrical case, it is clear that the cylindrical mode coefficients of the measured data depend upon the measurement radii in a fundamental way. Unlike the conventional cylindrical case, it is thus not possible to represent the measured fields using just two sets of cylindrical mode coefficients. Instead, two sets must be used for every value of Po. In practice, this equates to computing a complete set of cylindrical mode coefficients for every ring of near-field data, as this is the only case for which the value of Po will be held fixed. By computing the complete far-field pattern for each radial cut sequentially, and then using the principle of linear superposition, the complete far-field pattern of the antenna can thus be constructed by essentially integrating over the set of near-field rings. Although not discussed herein, the probe-pattern correction that is inevitably required when taking real near-field data can be incorporated into this in a straightforward, but rigorous, way, by using the usual inversion of the cylindrical transmission formula.

3. Preliminary Simulated Results In order that the transformation algorithm outlined above could be verified, cylindrical and conical near-field measurement systems were simulated. The purpose of this was to allow the farfield patterns as obtained from the new transformation algorithm to be compared with the far-field pattern as obtained directly from the modeled data. To this end, a proprietary three-dimensional, fullwave computational electromagnetic (CEM) solver, employing the Finite-Difference Time-Domain (FDTD) Method, was used to solve for the electric and magnetic fields in a problem space encapsulating a radiator. In this case, a simple open-ended rectangular waveguide (OEWG) section, excited by the fundamental TE 10 mode, was modeled. As the amount of computer memory required to solve problems such as this is closely related to the electrical size of the problem space, the Kirchhoff-Huygens principle, which is in essence a direct integration of Maxwell's equations, was used to calculate the radiated fields outside of the problem space (c.f. the Stratton-Chu solution [8]). Using this method, almost any form of near-field antenna measurement system could be simulated with a high degree of accuracy, irrespective of how large it was. Figure 2 contains the tangential E¢ and Ez field amplitude components over the surface of a cylinder, shown in a three-dimensional "virtual reality" space. Figure 3 contains equivalent plots showing similar field components plotted over the surface of a truncated cone. In each of these simulations, the main beam of the antenna was aligned with the z axis of the plot, i.e., through the side of the cylinder or frustum. These simulated near-field measurements were then transfortned to the true far field, using the algorithm described above. They could then be compared with the predicted ideal patterns. The equivalent far-field patterns can be found presented in the form of false-color (i.e., checkerboard) intensity plots in Figures 4-6. From inspection, the degree of agreement attained was encouraging. The differences primarily resulted from the varying degrees of truncation within the simulated near-field data sets. This was evident in the broadening of the pattern in the elevation plane, IEEE Antennas and Propagation Magazine, Vol. 51, No.1, February 2009

the loss of field at large elevation angles, and the variation in the level of the "cross-polar" lobes: the far field pattern from the theoretical model was the only pattern free of truncation. As these preliminary results were viewed to be encouraging - particularly, the agreement attained between the cylindrical and conical patterns the work progressed to actual range measurements, which are detailed in the sections below.

4. Preliminary Measured Results The goal of this work was limited to demonstrating a proof of concept. The success of this measurement technique was assessed by evaluating the repeatability between successive preliminary measurements where a single parametric change had been introduced. To this end, actual range measurements were taken using an NSI-200V-5x5 planar/cylindrical near-field measurement system. This was used with a precision tilting fixture that allowed for a 0° or 30° half cone angle, which is the angle between the local gravity vector and the linear axis. The AUT was an X-band standard-gain hom (SGH), and multi-frequency data was taken from 8 to 12 GHz using an Agilent PNA-X-based RF subsystem. The acquisitions were made using standard NSI 2000 cylindrical data-acquisition software. This conical near-field system can be seen presented in Figure 7. Three different measurement cases were examined: Case 1 consisted of a conventional cylindrical near-field measurement. This was to be used as the baseline measurement against which other test cases could be compared. Case 2 involved tilting the vertical axis of the scanner through 30° so that an equatorial conical near-field measurement was made. This case was selected so that the basic conical near-field-to-far-field transform could be verified against the baseline cylindrical case. Case 3 (illustrated in Figure 7) was intended to take advantage of the greater elevation coverage in order to improve the wide-out pattern coverage. In this configuration, the AUT was tilted up by 30°, so that th~ boresight of the antenna was pointing towards the top of the conical measurement, in order that truncation would be lessened. The goal of this measurement campaign was to obtain far-field patterns from Cases 2 and 3 that agreed with those obtained from the baseline, i.e., Case 1.

The customized NSI-200V-5x5 planar/cylindrical measurement system, shown in Figure 7, was used to take near-field data for these three configurations. Plots of the measured amplitude of the principal polarization can be seen presented as two-dimensional false-color intensity plots in Figures 8, 9, and 10. Since a cylindrical measurement can be considered to be a special case of the conical case or vice-versa, the first - and possibly the simplest - test was to compare the conical transform against a reference cylindrical near-field-to-far-field transform to process the baseline Case 1 data set. Here, the well-regarded National Institute of Standards and Technology (NIST) cylindrical near-field-to-far-field transform was used as the reference transform. 197

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Cylindrical Measurement

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Here, it is evident that Figures 12, 13, and 15 were in encouraging agreement with one another, giving further confidence that the near-field-to-far-field transform and probe-pattern compensation were working correctly. It is worth noting that as the effects of the spatial filtering of the near-field probe depend upon the orientation of the probe and the AUT, Case 3 constituted a stringent test for the transformation process. The patterns could only agree if the near-field-to-far-field transform, probe-pattern-correction algorithm, probe pattern, and vector rotation were all implemented correctly. It is worth noting that due to the pattern rotation, the highelevation antenna pattern was not filtered. The sharp lobe located at Az = 00 , EI = 60 0 was the original pole in the far-field pattern, and, as such, this anomaly should be ignored. A similar comment is true for the lobe at Az = ±1800 , EI = -60 0 , which corresponded to the other pole.

Figure 7. The conical near-field measurement system. Figure 11 contains plots of the elevation cardinal cuts of the far-field antenna patterns, which were obtained from the NIST transform with (red trace) and without (blue trace) probe pattern correction. Similar plots (which use the same color convention) can be seen that were obtained from the novel conical near-fieldto-far-field transform. These can be seen to be in encouragingly good agreement with those patterns obtained from the NIST (i.e., reference) transform. Although not shown, the agreement obtained between the respective azimuth cardinal cuts was equally good. As is evident from inspection of the plots of the near-field measured patterns, the data sets were all truncated to some degree in the nominally vertical linear axis, which will inevitably lead to some leakage in the far-field pattern. Firstly, and as is the case with planar scanning, the first-order truncation effect will result in the error within the pattern being effectively infinitely large outside of some angular limit that can be approximated by geometry (c.f. an infinite-frequency Geometrical Optics approximation). Secondly, the holistic nature of the relationship that exists between the nearfield and far-field regions will result in the introduction of some ripple into the far-field pattern within even this angular range. Thus, when plotting these far-field patterns, fields outside of the first-order truncation range were omitted from the far-field falsecolor plots, as evidenced by the white areas. Here, Figure 12 contains the far-field pattern from Case 1, cylindrical measurement; Figure 13 contains the far-field pattern from Case 2, conical measurement; whilst Figure 14 contains the far-field pattern obtained from Case 3, conical measurement with tilted AUT. As Case 3 involved the measurement of an AUT that was not aligned with the axes of the range (i.e., the peak of the pattern was located at Az = 00 , EI = 30 0 ) , a vector isometric rotation was utilized to rotate the antenna pattern so that it could be compared with the other, nominally aligned cases. The results of this can be found presented in Figure 15. This constituted a full rotation of both the pattern and the polarization, so any differences that remained were not an artifact of any particular polarization basis used to plot the patterns. A detailed description of this correction technique can be found in [9]. 200

Since the success of this measurement technique was being assessed by evaluating the repeatability between successive measurements where a single parametric change was introduced, the measurements had to be very strictly controlled. Any additional changes that were introduced between successive measurements would affect the results. Unfortunately, as these measurements were preliminary in nature, they were not conducted within a screened anechoic environment. As such, the multipath within the measurements did differ between each measurement configuration, which degraded the agreement attained. This was particularly crucial as both the orientation of the AUT and the probe changed between measurements. Crucially, the radii of the cylindrical - and particularly, the conical - measurements was not accurately determined during these measurements. This is the most likely cause of the small discrepancies in the location of the sidelobes in the alignment-corrected Case 3 far-field patterns (i.e., Figure 15).

5. Discussion Ideally, the boresight of the AUT would be orientated so that it points directly through the tip of the cone, i.e., in a polar mode (c.f. the equatorial-mode measurements discussed herein), so that the undesirable effects of truncation are minimized. However, in practice any imperfection in the alignment of the conical system could result in the introduction of significant errors in the corresponding far-field pattern. This is a consequence of the fact that, naturally, the boresight direction of the AUT, and thus the region of greatest field intensity, will be directed towards the tip of the cone. This is where the set of radial conical linear cuts intersect, and where the alignment issues are most critical. Obviously, this can be eased by orientating the AUT so that it "looks" out through the side of the cone, thus avoiding the tip region, but this is perhaps an inelegant solution. One alternative that has been used with considerable success in the closely related poly-planar measurement technique is to us a flat-topped measurement surface. For the polyplanar case, a truncated pyramid, i.e., a pyramidal frustum, was employed to resolve this difficulty. However, here an analogous conical frustum would be used. This is a frustum created by slicing the top off a right cone, where the cut is made parallel to the base of the cone. Here, the cap that is used to replace the tip of the conic section constitutes a conventional plane-polar measurement. It is intended to displace the intersection between the individual cuts from the region of greatest field intensity to a less-sensitive location. In the event that the adjacent scans do not intersect perfectly, the resulting positional error will thus impact less on the far-field pattern. IEEE Antennas and Propagation Magazine, Vol. 51, No.1, February 2009

It is often preferable when taking near-field antenna measurements for a measurement geometry to be selected that is commensurate with the geometry of the AUT. This technique would thus be particularly well suited to the characterization of base-station antennas, or arrays installed behind tangent ogive radomes, such as those commonly employed with nose-mounted fire-control radars. This is an electrically large system that often presents the experimentalist with both electromagnetic and mechanical challenges.

6. Conclusions and Future Work This paper has recounted the use of a cylindrical mode expansion and an inversion of the cylindrical transmission fonnula as the basis of a novel probe-pattern-corrected conical near-field-to-farfield transform. This transform is for use with a conical near-field antenna-measurement system. The validity of this novel approach has been demonstrated through numerical simulation and empirical measurement. Finally, it should be noted that this paper recounts the progress of an ongoing research study. Consequently, several issues remain to be addressed. The planned future work is to include obtaining verification of the success of the right conic frustum measurement technique through further numerical simulation and actual range measurement.

7. Acknowledgements The authors wish to express their gratitude to A. C. Newell for his valuable comments in reviewing this paper.

IEEE Antennas and Propagation Magazine, Vol. 51, No.1, February 2009

8. References 1. S. F. Gregson, "Probe-Corrected Poly-Planar Near-Field Antenna Measurements," Doctoral Thesis, University of London, 2003. 2. D. Leatherwood, "Conical Near-Field Antenna Measurement System," Antenna Measurements and Techniques Association (AMTA) Symposium, St. Louis, November 2007. 3. S. F. Gregson, C. G. Parini, and J. McCormick, "Full Sphere Far-Field Antenna Patterns Obtained Using a Small Planar Scanner and a Poly-Planar Measurement Technique," AMTA, Austin Texas, October 2006. 4. S. F. Gregson and C. G. Parini, J. McCormick, "Proposal for a Novel Near-Field Antenna Measurement Technique Employing a Conic Frustum Geometry," LAPC, Loughborough, March 2008. 5. A. D. Yaghjian, ''Near-Field Antenna Measurements on a Cylindrical Surface: A Source Scattering Matrix Formulation," NBS Technical Note 696, 1977. 6. C. A. Balanis, Advanced Engineering Electromagnetics, New York, J. Wiley & Sons, 1989, pp. 936. 7. J. E. Hansen (ed.), Spherical Near-Field Antenna Measurements, London, Institution of Engineering and Technology, UK, Peter Peregrinus Ltd., 1988. 8. S. Silver, Microwave Antenna Theory and Design, First Edition, New York, McGraw Book Company Inc., 1947. 9. S. F. Gregson, J. McCormick, and C. G. Parini, Principles of Planar Near-Field Antenna Measurements, London, Institution of Engineering and Technology, 2007. @)

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