= Gaussian beam solid angle
{c}
For a narrow axisymmetric Gaussian power pattern $P(\theta)/P(0)=e^{-\theta^2/(2\sigma^2)}$, the <beam solid angle> is approximately
$$
\Omega_A\simeq2\pi\int_0^\infty e^{-\theta^2/(2\sigma^2)}\theta\,d\theta=2\pi\sigma^2.
$$
The approximation replaces $\sin\theta$ by $\theta$ and extends a negligible angular tail. If $\Theta$ is the full half-power beamwidth, then $\Theta=2\sqrt{2\log2}\,\sigma$ and \b[$\Omega_A\simeq\pi\Theta^2/(4\log2)\approx1.133\Theta^2$], 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> $4\pi/\Omega_A\approx1109$. 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.<Antenna Models for Electromagnetic Compatibility Analyses by C. W. Wang and T. Keech>[Antenna radiation-pattern and directivity conventions]{ref}
Back to article page
