7266 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 65, NO. 12, DECEMBER 2017 A Novel Far-Field Transformation via Complex Source Beams for Antenna Near-Field Measurements on Arbitrary Surfaces Hsi-Tseng Chou , Fellow, IEEE , Prabhakar H. Pathak, Life Fellow, IEEE , Shih-Chung Tuan, Senior Member, IEEE, and Robert J. Burkholder, Fellow, IEEE Abstract—This paper presents a general technique for an efficient and accurate near-field (NF) to far-field (FF) trans- formation based on the complex source beam (CSB) method. This CSB-based NF–FF transformation is applicable to all three conventional NF measurement techniques, namely, to the planar, cylindrical, and spherical as well as to an arbitrary measurement surface. In this technique, a set of CSBs launched radially out from the antenna under test (AUT) is used to represent its NF. The weights of these CSB radiation basis functions are found by matching them to the measured AUT NF on the measurement surface. Once the CSB weights are found, the same CSBs remain continuously valid everywhere from NF to FF. It is noted that due to the occurrence of a natural Gaussian-like field taper away from each beam axis, only a relatively few CSBs are typically needed to represent the AUT fields at any point in its far zone in comparison to the conventional plane wave or vector wave function expansions for a given size of AUT. The remaining CSBs can be discarded with negligible error. This property among a few others leads to some advantages over conventional NF–FF methods. Index Terms— Antenna measurements, antenna radiation patterns, beams, complex source, near fields, near-field to far-field (NF–FF) transformation. I. I NTRODUCTION T HIS paper presents a novel, accurate, and conceptually simple complex source beam (CSB)-based approach to perform near-field to far-field (NF–FF) transformations [1]–[7] required in NF measurement systems. Also, this CSB approach is directly valid for arbitrary NF measurement surfaces and is thus not restricted only to conventional planar [2], cylin- drical [3], or spherical [4] surfaces. Basically, the measured Manuscript received December 6, 2016; revised August 27, 2017; accepted September 27, 2017. Date of publication October 2, 2017; date of current version November 30, 2017. This work was supported by the Ministry of Science and Technology, Taiwan. (Corresponding author: Hsi-Tseng Chou.) H.-T. Chou is with the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan (e-mail: chouht@ntu.edu.tw). P. H. Pathak is with the ECE Department, The Ohio State University, Columbus, OH 43212 USA, and also with the EE Department, University of South Florida, Tampa, FL 33620 USA (e-mail: pathakph@gmail.com). S.-C. Tuan is with the Department of Communication Engineering, Ori- ental Institute of Technology, New Taipei City 22061, Taiwan (e-mail: fo012@mail.oit.edu.tw). R. J. Burkholder is with Electroscience Laboratory, The Ohio State Univer- sity, Columbus, OH 43212 USA (e-mail: burkholder.1@osu.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2017.2758420 NF of the AUT is represented as a convergent sum of over- lapping and generally relatively well-collimated CSB basis functions [8]–[11]. Each of the CSB fields, which serve as the basis functions for representing the AUT radiated fields, remains valid everywhere from the NF to FF, and it is noted that they are exact solutions of Maxwell’s equations [12]–[14]. Within the CSB paraxial region, which lies in the neighbor- hood of its axial propagation direction, the CSB reduces to a Gaussian beam (GB) [12]–[14]. These CBSs are launched radially out from the AUT [9]–[11]. The expansion coeffi- cients or amplitudes of these CSB field basis functions are found numerically via a matrix approach by matching the CSB expansion to the measured NF of the AUT. It is noted that due to the occurrence of a natural Gaussian-like field taper exhibited by each CSB away from its beam axis, only a relatively few CSBs are typically needed to represent the AUT fields at any point in its far zone in comparison to the conventional plane wave or vector wave function (VWF) expansions for a given size of AUT. The remaining CSBs can be discarded with negligible error. This property and its simplicity lead to some advantages of the CSB approach over conventional NF–FF methods. NF measurement systems are now widely used to predict the FF radiation characteristics of antennas via an NF–FF transformation [1]–[7]. Depending on the size of an antenna, a variety of NF measurement system configurations have been developed to effectively obtain its radiated FF. In particular, the planar, cylindrical, and spherical NF measurement sys- tems [1] are most commonly used. The latter measures the NFs on these surfaces enclosing the AUT. In general, most NF–FF transformation techniques are based on the use of a free space equivalence theorem [15] which requires a closed surface on which the NF values must be defined. In practice, the planar and cylindrical surfaces are not closed but are truncated. Often the spherical scan surfaces are also truncated. The FF of the AUT is obtained from the NF planar, cylindrical, and spherical measurements by mathematically transforming the measured NF data to the FF via the use of conventional planar, cylindrical, and spherical wave functions [2], respectively. For example, in the planar scan case, a fast Fourier transforma- tion (FFT) scheme [1], [2] is currently used to evaluate the plane wave spectral (PWS) expansion representation of the NF to obtain the FF. On the other hand, cylindrical and 0018-926X © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.