Gaussian beam solid angle

For a narrow axisymmetric Gaussian power pattern $P(\theta )/P(0)=e^{-{\theta }^2/(2{\sigma }^2)}$, the beam solid angle is approximatelyThe approximation replaces $\sin \theta$ by $\theta$ and extends a negligible angular tail. If $\Theta$ is the full half-power beamwidth, then $\Theta =2\sqrt{2\log 2}\sigma$ and ${\Omega }_A\simeq \pi {\Theta }^2/(4\log 2)\approx 1.133{\Theta }^2$, with angles in radians. This is a power-pattern convention; a Gaussian field-amplitude width uses a different factor.

For $\Theta =0.1$ radians the solid angle is approximately $0.01133$ steradians, giving peak antenna directivity $4\pi /{\Omega }_A\approx 1109$. Rotating a calibrated source or receiver and measuring the whole angular pattern tests the approximation: significant sidelobes contribute extra solid angle that a main-beam width cannot detect.^[ref]