## 1. Introduction

[2] The spherical near-field measurement is an extensively used method for accurate characterization of the radiated fields of antennas [*Hansen*, 1988]. The near field of the antenna under test (AUT) is measured in a number of points on a spherical surface enclosing the AUT by a probe. In the near field of the AUT, the ratio between the signal received by the probe and the incident tangential field at the probe is not constant but a function of the measurement direction. This ratio, that is necessary to know for an accurate far-field determination [*Yaghjian*, 1986], is inherently found by applying a probe correction.

[3] The probe correction for spherical near-field measurements was first formulated by *Wacker* [1974, 1975] and *Jensen* [1975]. Since then this probe correction technique has become a generally accepted technique [*Hansen*, 1988]. Several suggestions for other probe correction techniques have been later presented by *Larsen* [1980], but to the authors' knowledge none of these techniques have been formulated in detail, reportedly tested nor reportedly shown to work. It is noted that, according to *Hansen* [1988, chapter 4], one of the suggested techniques for the probe correction, the straightforward method of forming a system of linear equations and numerically inverting the matrix, is inefficient and not practical.

[4] The well-known probe correction [*Hansen*, 1988] is practical for first-order (μ = ±1) probes for which the azimuthal variation of the receiving pattern of the probe is restricted to a simple sine-cosine variation [*Wacker*, 1974, 1975; *Jensen*, 1975]. In this case, the full-sphere measurement of the AUT must be performed only twice, with different orientations of the probe. Moreover, if the probe is dual polarized, the probe rotation may be fully avoided, and one full-sphere measurement of the AUT is then sufficient. This probe correction technique is in principle also applicable to probes which are not just first-order probes and which thus possess patterns with more complicated azimuthal variations. This requires, however, that the full-sphere measurement of the AUT be performed for more than two orientations of the probe, and it thus leads to an overall longer measurement time. Since the measurement time is generally the limiting factor in practical antenna measurement projects, this is highly undesirable.

[5] A typical example of a first-order probe is a conical horn fed by a circular-cylindrical waveguide [*Larsen and Hansen*, 1979]. These probes have been used extensively for spherical near-field antenna measurements for years [*Hansen*, 1988]. Because of a relatively narrow bandwidth of the circular-cylindrical waveguide and the need for antenna measurements over a wide frequency range, a large set of probes is required. Conical horns may also be impractically heavy and large at low frequencies.

[6] The purpose of this paper is to introduce a specific high-order probe correction technique, namely an odd-order probe correction technique, for spherical near-field antenna measurements. An odd-order (μ odd) probe is defined and the probe correction technique for odd-order probes is formulated. As is the case with the existing first-order probe correction technique [*Hansen*, 1988], the odd-order probe correction technique presented here requires only two probe orientations during the measurement.

[7] Odd-order probes, in particular, are important for antenna measurements, because a wide and significant range of realistic antennas belongs to this class. A good example of an odd-order probe is an open-ended rectangular waveguide operating with the TE_{10} waveguide mode.

[8] In section 2 the background theory is presented and an odd-order probe is defined. In section 3 the probe correction technique for odd-order probes is derived. Section 4 documents the validation of the odd-order probe correction technique. Finally, conclusions are given in section 5.