# Near-Field Spherical Scanning Antenna Measurements: Probe Deconvolution and Sensitivity

Authors: Ronald C. Wittmann; Michael H. Francis
Journal: International Journal of Antennas and Propagation
 (2014)

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## Abstract

We define and calculate sensitivity for several actual and simulated probes. Probe sensitivity can have a significant impact on the measurement uncertainty associated with probe deconvolution in near-field spherical-scanning, antenna measurements.


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## Body

## 1. Introduction

In near-field planar scanning, we know that if the receiving pattern of the probe has a null in a given direction, then the far-field pattern of the test antenna cannot be determined reliably in that direction [1]. That is, the probe must be sensitive to plane-wave modes of interest. On the other hand, in near-field spherical scanning, there is no direct relationship between the probe and the test antenna patterns in a given direction. Nevertheless, we argue that the probe must be sensitive to all spherical-wave modes needed to represent the radiated fields.This paper presents a small selection of results from a larger study of probe-sensitivity issues in external and internal spherical-scanning measurements. We calculate and discuss probe (modal) sensitivity as an aid in the choice of probes and measurement parameters (this is an expanded version of a paper presented at the 2012 Antenna Measurements Techniques Association Symposium [2]).


## 2. External Scanning

In external spherical scanning, we determine the radiated fields outside of the measurement sphere that are due to sources confined within the measurement sphere (see Figure1). This is commonly called spherical scanning without the qualifier “external.” The theory presented here is a synopsis with little detail beyond that needed for understanding this paper. A complete discussion may be found in [3].Figure 1
External spherical scanning.The far-field radiation of an antenna can be characterized by(1)Er~r→∞tr^exp⁡⁡ikrikra0.
This formula embodies the linear relationship between the radiated electric field Er and the excitation a0. The transmission function tr^ may be expanded as
(2)tr^=∑n=1N∑m=-nn‍tnm1Xnm1r^+tnm2Xnm2r^,
where Xnm1 and Xnm2=ir^×Xnm1 are vector spherical harmonics [4, Chapter 16], which depend only on the direction r^, and the tnml are (unknown) modal coefficients.The antenna under test (AUT) is characterized by measurement with a probe that moves over a spherical surface of radiusr, enclosing the AUT. To account for polarization, two measurements are made for each probe location r. These measurements correspond to probe orientations differing by a rotation of 90° about the probe axis. We follow common practice and use a “special symmetric” or “μ=±1” probe. For such probes, the polarization measurements are the components of a transverse “measurement vector” wr that also may be expanded in spherical harmonics:
(3)wra0=∑n=1N∑m=-nn‍Tnm1rXnm1r^+Tnm2rXnm2r^.The relationship between the (known)Tnml and the tnml is given by the probe-correction equations
(4)Tnm1rTnm2r=Mnrtnm1tnm2,
where
(5)Mn=i4π2n+1×σn,-11-σn11σn,-12-σn12-σn,-11-σn11-σn,-12-σn12.
The translated probe coefficients σnml are basically the responses of the probe, located at rz^, to each of the multipoles that compose the basis of vector spherical waves. Symbolically,
(6)σnm1r=P·mnmrz^,σnm2r=P·nnmrz^,
where the operator P denotes the probe and mnmr and nnmr represent magnetic and electric multipoles, respectively, that generate outgoing waves at infinity. For a special symmetric probe, σnμl=0 unless μ=±1.We consider linearly polarized probes with symmetries(7)σn,-11=-σn11,σn,-12=σn12
that are actually satisfied by many practical probes. In this case Mn is diagonal and
(8)Tnm1r=αn1rtnm1,Tnm2r=αn2rtnm2,
where the probe sensitivity is defined as
(9)αnl=16π2n+1σn1l.
In the general case, appropriate sensitivities may be defined as the singular values of Mn [5]. Of course, quantities corresponding to our definition of probe sensitivity are calculated, directly or indirectly, in any standard version of probe-corrected, near- to far-field transformation software.Whenαnl is small, tnml is generally the ratio of two small numbers. As the signal falls towards the noise floor, the relative uncertainty in Tnml and thus tnml can become extremely large. On the other hand, enhanced sensitivity to some modes is not necessarily beneficial. When the full dynamic range of the receiver is available, it is generally better to have uniform sensitivity over the modes of interest. The condition number
(10)car=max⁡n,l⁡αnlrmin⁡n,l⁡αnlr
is a good measure of merit; the optimum value is cα=1.


## 3. Internal Scanning

In internal spherical scanning, we determine the radiated fields within the measurement sphere that are due to sources located outside of the measurement sphere (see Figure2). In the neighborhood of the coordinate origin (centered in the measurement sphere),
(11)Er=a04π∫tk^exp⁡⁡ik·rdk^,
where tk^ is the “transmitting function” of the sources. In principle, knowledge of tk^ allows compensation, say, of compact range measurements, for the effects of nonuniform illumination [6–8].Figure 2
Internal spherical scanning.Equations (2)–(10) also apply to internal scanning, with the exception that (6) becomes
(12)σnm1r=P·mnm1rz^,σnm2r=P·nnm1rz^,
where mnm1r and nnm1r represent magnetic and electric multipoles, respectively, that remain finite at the origin. This “small” change has a profound effect on the nature of probe correction in the interior versus exterior case.


## 4. Processing and Uncertainty

Data typically are collected on a uniform grid inθ and φ with sample increments chosen to satisfy the Nyquist sampling theorem
(13)Δθ,Δφ≤2π2N+1.
In the NIST algorithm, the angular measurements are transformed to a Fourier representation by use of the fast Fourier transform (FFT). Orthogonality integral relations are then employed to project the Fourier representation onto the basis of spherical harmonics, producing the representation (3). Finally, probe effects are removed through deconvolution to yield the far-field pattern (2).The FFT is unitary so that the root mean square (RMS) error in the Fourier representation is the same as the RMS error in the measured data. The projection operation also will not increase the RMS error; in fact, there will be a reduction if some of the error signal lies in the null space of the projection. For the probe deconvolution, assuming that the uncertainty in sensitivityΔαn is 0, we have
(14)Δtt≤cαΔTT
(see (A.3)), where, for example,
(15)t2=∑n=1N∑m=-nn‍tnm12+tnm22.
Thus, the use of a poor probe can lead to significantly expanded uncertainties.We can also account for the fact that the sensitivities are not known precisely. WithΔTn=0(16)Δtt≤β.
(See (A.5)) Here
(17)β≈max⁡n,lΔαnlαnl.
Equations (14) and (16) may be combined in quadrature to give an overall view of the effect of probe deconvolution on measurement uncertainty:
(18)Δtt2≲cαΔTT2+β2.
Although illustrative and easy to derive, (18) gives a conservative estimate of uncertainty that can be refined.NIST software calculates with double-precision accuracy to ensure that computational errors are negligible compared to measurement errors.


## 5. Examples

We consider a mode cut-off ofN=100. This is sufficient for the representation of radiated fields when kr0≲100. In the external case, r0 is the radius of the minimum sphere of the test antenna. In the internal case, r0 is the radius of the test zone. For practical reasons, the measurement radius rm is usually chosen to be 3 or more wavelengths greater than r0. The radius rm with krm=105 is just outside the sphere r0=100/k. The radius rm with krm=200 is greater than or equal to twice r0.Actual probes used in this study are shown in Figure3. These all operate at Ku band, 12.4–18 GHz.(a) SN 330. (b) SN 401. (c) SN 1057. (d) SN 331.
(a)
(b)
(c)
(d)The plots are normalized so that the peak sensitivity in any figure has the value 1.


### 5.1. External Case

Ideal dipole probes provide a direct measurement of components of the electric and magnetic field. Although practical probes only approximate ideal probes at best, ideal probes are effectively assumed in instances when “no probe correction” is applied. Also, practical probes become increasingly ideal asrm is increased. Figures 4–6 show sensitivities for a “Huygens probe,” which is an ideal probe that consists of a crossed electric and magnetic dipole. This is a maximum-directivity probe of order 1 with a directivity D=4.77 dB [9, Section  2.3.4]. Figure 4 plots αnl as a function of krm for several values of n. At any given radius, there is very little variation in sensitivity with n. Otherwise, we observe the expected decrease in sensitivity as rm increases. Figures 5 and 6 show αnl as a function of n for krm=105 and 200. When krm=105, cα=1.2 and at krm=200, cα=1.0. The trend for conditioning to improve as rm increases is consistent with an asymptotic analysis of the sensitivity of linearly polarized probes [3, Appendix  C].Figure 4
Huygens probe, external case.Figure 5
Huygens probe, external case,krm=105.Figure 6
Huygens probe, external case,krm=200.Figures7–9 feature the NIST SN 330 μ=±1 probe with gain G=10.5 dB [3, Appendix  B]. Figure 7 plots αnl as a function of krm for several values of n. Compared to the Huygens probe, there is considerable variation in sensitivity with n for smaller values of krm. Figures 8 and 9 show αnl as a function of n for krm=105 and 200. When krm=105, cα=4.6, and at krm=200, cα=1.4.Figure 7
SN 330, external case.Figure 8
SN 330, external case,krm=105.Figure 9
SN 330, external case,krm=200.Figures10–12 show the NIST SN 1057 dual-port μ=±1 probe with gain G=15 dB. Figure 10 plots αnl as a function of krm for several values of n. Figures 11 and 12 give αnl as a function of n for krm=105 and 200. When krm=105, cα=19, and at krm=200, cα=3.1.Figure 10
SN 1057, external case.Figure 11
SN 1057, external case,krm=105.Figure 12
SN 1057, external case,krm=200.Figures13–15 feature the NIST SN 401 μ=±1 probe with gain G=22.7 dB. This probe resembles SN 330, except that a flare has been added to increase the directivity. Figure 13 plots αnl as a function of krm for several values of n. Variation in sensitivity with n for smaller values of krm is problematic. Figures 14 and 15 show αnl as a function of n for krm=120 and 500. When krm=120, cα=200 and at krm=500, cα=2.8. In order to obtain similar conditioning, this probe must be used at considerably larger values of rm than those for the lower directivity probes discussed here.Figure 13
SN 401, external case.Figure 14
SN 401, external case,krm=120.Figure 15
SN 401, external case,krm=500.


### 5.2. Internal Case

In the internal case, conditioning is significantly affected by the fact that radial functions (spherical Bessel functions) have zeros. Figure16 plots αnl as a function of krm for several values of n, when the probe is an ideal electric dipole (D=1.76 dB). The variation in sensitivity with n is extreme and it seems unlikely that any choice of one or several values of rm would allow reliable determination of all of the modes. Figure 17 plots αnl as a function of n for krm=104.6. At this radius, cα=830.Figure 16
Ideal electric dipole, internal case.Figure 17
Ideal electric dipole,krm=104.6.Fortunately, the Huygens probe (D=4.77 dB) is much better behaved, as illustrated in Figures 18 and 19. Figure 18 plots αnl as a function of krm for several values of n and indicates a tendency for conditioning to improve as rm increases. Figure 19 shows αnl as a function of n for krm=106. At this radius, cα=2.8.Figure 18
Huygens probe, internal case.Figure 19
Huygens probe, internal case,krm=106.Figures20 and 21 feature the NIST SN 330 probe (G=10.5 dB). Figure 20 plots αnl as a function of krm for several values of n. Again, conditioning appears to improve as measurement radius is increased. Figure 21 shows αnl as a function of n for krm=110, for which cα=4.2.Figure 20
SN 330, internal case.Figure 21
SN 330, internal case,krm=110.Figures22 and 23 show the NIST SN 1057 dual-port probe (G=15 dB). Figure 22 plots αnl as a function of krm for several values of n. Again, conditioning appears to improve as measurement radius is increased. Figure 23 shows αnl as a function of n for krm=110, for which cα=20.Figure 22
SN 1057, internal case.Figure 23
SN 1057, internal case,krm=110.Figures24 and 25 feature the NIST SN 401 probe (G=22.7 dB). Figure 24 plots αnl as a function of krm for several values of n and indicates a significant range of sensitivity with n. Figure 25 shows αnl as a function of n for krm=110, for which cα=2200. This is certainly unacceptable for many applications.Figure 24
SN 401, internal case.Figure 25
SN 401, internal case,krm=110.Figure26 compares NIST SN 330 (G=10.5 dB), SN 331 (G=7.9 dB), and SN 401 (G=22.7 dB) at krm=110. SN 331 is a version of SN 330 with a smaller aperture. With all things equal, SN 401 has better sensitivity for multipole indexes less than n≈20 but is significantly less sensitive for larger values of n. Because of its higher gain, however, the SN 401 must be used at lower transmitted power levels. Improving the signal-to-noise ratio by increasing the transmitted power will tend to compensate for the reduced sensitivity of the lower-gain probes when n<20.Figure 26
Comparison, internal case,krm=110.


## 5.1. External Case

Ideal dipole probes provide a direct measurement of components of the electric and magnetic field. Although practical probes only approximate ideal probes at best, ideal probes are effectively assumed in instances when “no probe correction” is applied. Also, practical probes become increasingly ideal asrm is increased. Figures 4–6 show sensitivities for a “Huygens probe,” which is an ideal probe that consists of a crossed electric and magnetic dipole. This is a maximum-directivity probe of order 1 with a directivity D=4.77 dB [9, Section  2.3.4]. Figure 4 plots αnl as a function of krm for several values of n. At any given radius, there is very little variation in sensitivity with n. Otherwise, we observe the expected decrease in sensitivity as rm increases. Figures 5 and 6 show αnl as a function of n for krm=105 and 200. When krm=105, cα=1.2 and at krm=200, cα=1.0. The trend for conditioning to improve as rm increases is consistent with an asymptotic analysis of the sensitivity of linearly polarized probes [3, Appendix  C].Figure 4
Huygens probe, external case.Figure 5
Huygens probe, external case,krm=105.Figure 6
Huygens probe, external case,krm=200.Figures7–9 feature the NIST SN 330 μ=±1 probe with gain G=10.5 dB [3, Appendix  B]. Figure 7 plots αnl as a function of krm for several values of n. Compared to the Huygens probe, there is considerable variation in sensitivity with n for smaller values of krm. Figures 8 and 9 show αnl as a function of n for krm=105 and 200. When krm=105, cα=4.6, and at krm=200, cα=1.4.Figure 7
SN 330, external case.Figure 8
SN 330, external case,krm=105.Figure 9
SN 330, external case,krm=200.Figures10–12 show the NIST SN 1057 dual-port μ=±1 probe with gain G=15 dB. Figure 10 plots αnl as a function of krm for several values of n. Figures 11 and 12 give αnl as a function of n for krm=105 and 200. When krm=105, cα=19, and at krm=200, cα=3.1.Figure 10
SN 1057, external case.Figure 11
SN 1057, external case,krm=105.Figure 12
SN 1057, external case,krm=200.Figures13–15 feature the NIST SN 401 μ=±1 probe with gain G=22.7 dB. This probe resembles SN 330, except that a flare has been added to increase the directivity. Figure 13 plots αnl as a function of krm for several values of n. Variation in sensitivity with n for smaller values of krm is problematic. Figures 14 and 15 show αnl as a function of n for krm=120 and 500. When krm=120, cα=200 and at krm=500, cα=2.8. In order to obtain similar conditioning, this probe must be used at considerably larger values of rm than those for the lower directivity probes discussed here.Figure 13
SN 401, external case.Figure 14
SN 401, external case,krm=120.Figure 15
SN 401, external case,krm=500.


## 5.2. Internal Case

In the internal case, conditioning is significantly affected by the fact that radial functions (spherical Bessel functions) have zeros. Figure16 plots αnl as a function of krm for several values of n, when the probe is an ideal electric dipole (D=1.76 dB). The variation in sensitivity with n is extreme and it seems unlikely that any choice of one or several values of rm would allow reliable determination of all of the modes. Figure 17 plots αnl as a function of n for krm=104.6. At this radius, cα=830.Figure 16
Ideal electric dipole, internal case.Figure 17
Ideal electric dipole,krm=104.6.Fortunately, the Huygens probe (D=4.77 dB) is much better behaved, as illustrated in Figures 18 and 19. Figure 18 plots αnl as a function of krm for several values of n and indicates a tendency for conditioning to improve as rm increases. Figure 19 shows αnl as a function of n for krm=106. At this radius, cα=2.8.Figure 18
Huygens probe, internal case.Figure 19
Huygens probe, internal case,krm=106.Figures20 and 21 feature the NIST SN 330 probe (G=10.5 dB). Figure 20 plots αnl as a function of krm for several values of n. Again, conditioning appears to improve as measurement radius is increased. Figure 21 shows αnl as a function of n for krm=110, for which cα=4.2.Figure 20
SN 330, internal case.Figure 21
SN 330, internal case,krm=110.Figures22 and 23 show the NIST SN 1057 dual-port probe (G=15 dB). Figure 22 plots αnl as a function of krm for several values of n. Again, conditioning appears to improve as measurement radius is increased. Figure 23 shows αnl as a function of n for krm=110, for which cα=20.Figure 22
SN 1057, internal case.Figure 23
SN 1057, internal case,krm=110.Figures24 and 25 feature the NIST SN 401 probe (G=22.7 dB). Figure 24 plots αnl as a function of krm for several values of n and indicates a significant range of sensitivity with n. Figure 25 shows αnl as a function of n for krm=110, for which cα=2200. This is certainly unacceptable for many applications.Figure 24
SN 401, internal case.Figure 25
SN 401, internal case,krm=110.Figure26 compares NIST SN 330 (G=10.5 dB), SN 331 (G=7.9 dB), and SN 401 (G=22.7 dB) at krm=110. SN 331 is a version of SN 330 with a smaller aperture. With all things equal, SN 401 has better sensitivity for multipole indexes less than n≈20 but is significantly less sensitive for larger values of n. Because of its higher gain, however, the SN 401 must be used at lower transmitted power levels. Improving the signal-to-noise ratio by increasing the transmitted power will tend to compensate for the reduced sensitivity of the lower-gain probes when n<20.Figure 26
Comparison, internal case,krm=110.


## 6. Conclusions

For any given probe, sensitivities generally become more uniform as the measurement radius is increased. More directive probes must be used at larger measurement radii than less directive probes if the same condition number is desired.The topic of probe sensitivity in spherical scanning seems not to have received much attention to date; yet, sensitivity can have a very significant impact on measurement accuracy. Our results are based on a limited number of examples; however, sensitivity information is readily available to anyone who processes spherical-scanning data. We strongly encourage metrologists to explore probe-sensitivity issues before making near-field antenna measurements. Optimization of sensitivity should be a consideration in the design of new probes.



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