Antenna beam pattern

An antenna beam pattern gives an antenna's directional response as a function of observation angle. A normalized power pattern divides radiation intensity by its peak; a field-amplitude pattern is its square root under a consistent impedance convention. The beam solid angle integrates the normalized power pattern over direction, so ideal directivity is $4\pi /{\Omega }_A$. A short dipole antenna has a power pattern proportional to ${\sin }^2\theta$, zero along its axis and maximum broadside. A narrow main lobe can coexist with sidelobes, so beamwidth alone does not specify the complete response.

For a uniformly illuminated circular aperture of diameter D, normalized received power is $[2J_1(x)/x{]}^2$ with $x=\pi D\sin \theta /\lambda$. The first null has $\sin \theta =1.21967\lambda /D$ and the half-power point has $\sin \theta \simeq 0.5145\lambda /D$. A pointing tolerance must name the tolerated power loss; the first-null angle is not a good low-loss tolerance. These estimates require an ideal aperture pattern and are only approximate for a dish a few wavelengths across.

For a narrow angular power pattern $\exp [-(u^2+w^2)/(2{\alpha }^2)]$ about an equatorial beam centre, the solid angle measure is locally $dudw$. Extending the small lobe to the plane gives its beam solid angle ${\Omega }_A\simeq 2\pi {\alpha }^2$. Both sides of the azimuthal lobe must be included. A lossless antenna then has peak antenna gain $G_0\simeq 2/{\alpha }^2$ and peak antenna effective area $A_0\simeq {\lambda }^2/(2\pi {\alpha }^2)$.