Beam solid angle (${\Omega }_A$)

The beam solid angle of an antenna is the integral of its normalized radiated-power pattern over the complete sphere. If $U(\theta ,\varphi )$ is radiated power per unit solid angle and $U_{\max }$ is its maximum, thenIt is measured in steradians and includes sidelobes and backlobes, not just a chosen main-beam boundary. Since total radiated power is $P_{\mathrm{rad}}=\int Ud\Omega$, peak antenna directivity satisfiesThe definition relates concentration of radiation to the entire pattern, rather than to a single width measurement.^[ref]

For a narrow, axisymmetric Gaussian power pattern $U/U_{\max }=e^{-{\theta }^2/(2{\sigma }^2)}$, replace $\sin \theta$ by $\theta$ near the axis and extend the negligible tail to infinity. The resulting Gaussian beam solid angle is ${\Omega }_A\simeq 2\pi {\sigma }^2$. Its full half-power width is ${\theta }_{\mathrm{HP}}=2\sqrt{2\ln 2}\sigma$, so ${\Omega }_A\simeq \pi {\theta }_{\mathrm{HP}}^2/(4\ln 2)$. These approximations require a narrow beam with no significant additional lobes; a broad pattern or strong sidelobes requires the full integral.

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]