Antenna power pattern, solid angles, associated efficiencies

The power pattern, $P_\ensuremath{\mathrm{ant}}$( $\theta_{},\phi_{}$,$\nu$), is a measure of the response of the antenna to radiation as a function of the angles $\theta$ and $\phi$. It is a normalized (maximum value unity), dimensionless quantity. It can depend on the frequency of observation.

The beam area or beam solid angle (or pattern solid angle) is

$\displaystyle \boxed{%
\ensuremath{\Omega_\ensuremath{\mathrm{ant}}}(\ensurema...
...},\phi_{}},\ensuremath{\nu}) \, d\ensuremath{\Omega_\ensuremath{\mathrm{}}}},
}$ (60)

where

$\Omega_\ensuremath{\mathrm{ant}}$ [ $\mathrm{sr}$] beam area,
$P_\ensuremath{\mathrm{ant}}$ [ $\mathrm{dimensionless}$] normalized power pattern of antenna,
$d\Omega$ [ $\mathrm{sr}$] infinitesimal solid angle of sky $(= \sin \theta\,d\theta\,d\phi)$.

The forward beam and main beam solid angles are the integral of the power pattern over $2\pi$ and the main lobe, respectively, i.e.,

$\displaystyle \boxed{%
\ensuremath{\Omega_\ensuremath{\mathrm{fb}}}(\ensuremat...
...},\phi_{}},\ensuremath{\nu}) \, d\ensuremath{\Omega_\ensuremath{\mathrm{}}}}.
}$ (61)

The forward and beam efficiencies are then defined as

$\displaystyle \boxed{%
\ensuremath{F_\ensuremath{\mathrm{eff}}}= \frac{\ensure...
...\ensuremath{\mathrm{mb}}}}{\ensuremath{\Omega_\ensuremath{\mathrm{ant}}}}. %
}$ (62)