This method works because the uncompensated far-field pattern, as measured by the near-field range, is the product of the probe and AUT's true far-field patterns. The AUT's true far-field pattern can be determined by dividing the uncompensated far-field pattern by the probe's far-field pattern.
Since the probe's cross-pol component is sometimes significant, it must be taken into account. The uncompensated far-field equation are thus a function of both the probe's principle and cross-pol components.
The uncompensated far-field is:
Ep = True principle AUT response
EC = True cross-pol AUT response
EPU =Uncompensated principle AUT response
ECU =Uncompensated cross-pol AUT response
Epp1,2 = Probe-1,2 principle response
Ecp1,2 = Probe-1,2 cross-pol response
Note that single probe oriented in two different
polarization can be used instead of two probes. When the probe
is linearly polarized Epp2() = Epp1(+90) and Ecp1(+90).
Solving for the true AUT far-field (Ep,Ec) gives:
with 8 = Epp1*Epp2 - Ecp1*Ecp2
In many cases the probe's cross-pol effect is negligible and can be ignored. This is true of an open-ended waveguide near broadside and along the principle cuts. Ignoring probe cross-pol effects simplifies the probe compensation equations:
It should be noted that the AUT and near-field probe coordinate systems are slightly different as shown in Figure 2. This means that the +el side of the AUT pattern must be corrected by the -el side of the near-field probe pattern and the +az side by the +az side. This fact places a restriction of principle-cut patten symmetry on the Probe-square-root technique.
At 0 (broadside) the receiver signal is maximum between the probes. At 1 there is a loss of 10 dB (5 dB from each pattern). When the far-field probe pattern is derived from near-field measurements the pattern at any angle, other than on-axis, will have received twice the signal loss that it should have. Taking the square root of the pattern level (dividing the dB levels by two) will compensate correctly. Taking the square root of the probe pattern can be though of as self-compensation.
The phase is treated in a similar way by dividing the phase angles by two. It should be noted however, that if the far-field pattern goes through a null, due to a side lobe, the phase change through the null must be taken into account. In low directivity probe patterns such as open-ended waveguides (OEWG), there are usually no side lobes within the near-field scan limits and so the phase is easier to calculate.
Figures 4A and 4B show the near-field range set-up for the Probe-square-root test. Figures 5 and 6 show the comparison between near-field and far-field-measured probe patterns of a wr-137 C-Band OEWG. Their agreement is excellent along the principle cuts out to 75 degrees. Figures 7,8 and 9 are near-field and far-field comparisons of a WR-90 X-Band OEWG on a different set of ranges. Even though the far-field range reflections were much higher this time, the agreement is still excellent in both the principle and slant-45 degree cuts.
As an additional comparison, Figure 10 is a contour plot from an OEWG model based on NBS equations (Yaghjian - 1983). Neglecting small ripples it agrees well with Figure 11 which is a contour generated with the probe-square-root technique from the same measurements ad Figure 5 and 6. The agreement is good out to 50 degrees at which point our implementation of the OEWG model departs from theory.
The limitations of this method have not been fully investigated but some restrictions have been noted.
To date, only principle-pol comparisons have been made and these show excellent agreement with principle-plane as well as off-axis cuts. The cross-pol patterns have not yet been checked. To extract them from the near-field measurements requires additional steps and assumptions in the probe-square-root method. These will be treated in a future paper on this subject.
1. Yaghjian, A. D. "Approximate Formulas for
the Far-Field and Gain Open-Ended Rectangular Waveguide".
NBSIR 83-1689 National Bureau of Standards, Boulder, CO. 80303