Radiation pattern

The radiation pattern of an antenna (or any localised source) is the angular distribution of the far-field radiated power per unit solid angle, dP/dΩ(θ, φ), usually plotted normalised to its maximum on a polar or 3D surface. For an oscillating electric dipole — the Hertzian model — the pattern is dP/dΩ ∝ sin²θ, the "doughnut" with zeros along the dipole axis and maximum broadside, independent of azimuthal angle φ. For a centre-fed dipole of arbitrary length the pattern becomes the squared modulus of [cos(kL cosθ/2) − cos(kL/2)]/sinθ, which for a half-wave dipole pinches the sin²θ doughnut into a slightly narrower lobe with gain 1.64. For a continuous current distribution the pattern is the magnitude-squared Fourier transform of the current evaluated at the radiation wavevector k = (ω/c)r̂ — a result that is the antenna-theory analogue of the aperture-diffraction theorem in Fraunhofer optics.

The pattern encodes both the angular selectivity and the angular nulls of an antenna, both of which are operationally important. The directivity is the ratio 4π · dP/dΩ_max / P_rad, which combined with efficiency η yields the antenna gain G = η · directivity. The half-power beamwidth (HPBW) — the angular width between the two −3 dB points — sets the angular resolution in radio-astronomy beam-formed observations. Sidelobe levels determine how much stray signal couples in from unwanted directions. For arrays of radiating elements the pattern is the element pattern multiplied by the array factor, a result that makes phased-array beam-steering (changing the array factor by changing the inter-element phase without moving the antenna) computationally tractable. In astrophysics, the effective radiation pattern of a relativistic beamed source (a pulsar, a blazar) contracts with γ and sweeps past the observer at the rotation frequency to produce the characteristic pulsed-emission light curves.