Approximate formulas for the far field and gain of open-ended rectangular waveguide

Approximate formulas are derived for the far field and gain of standard, open-ended, rectangular waveguide probes operating within their recommended usable bandwidth. (Such probes are commonly used in making near-field antenna measurements.) The derivation assumes first-order azimuthal dependence for the fields, and an
E
-plan pattern given by the traditional Stratton-Chu integration of the transverse electric (
TE_{10}
) mode. The
H
-plane pattern is estimated by two different methods. The first method uses a purely
E
-field integration across the end of the waveguide. The second, more accurate method approximates the fringe currents at the shorter edges of the guide by isotropically radiating line sources. The amplitude of the line sources is determined by equating the total power radiated into free space to the net input power to the waveguide. Comparisons with measurements indicate that for
X
-band and larger waveguide probes, both methods predict on-axis gain to about 0.2 dB accuracy. The second method predicts far-field power patterns to about 2 dB accuracy in the region
90\deg
off boresight and with rapidly increasing accuracy toward boresight.