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Beam
1. Power gain
The beam or power gain of a radio antenna is nothing but directional gain, the gain that varies toward different direction written as G(θ,ϕ)G(θ,ϕ). It is normalized to be 11 at the center of the field of view of an antenna.
So beam can be defined as the direction-dependent gain of a directional antenna with respect to the gain of an isotropic antenna:
G(θ,ϕ)=10log10GGiG(θ,ϕ)=10log10GGi
where GiGi is the gain of an isotropic antenna. This value is given in units of decibel (dB). By definition, the gain integrated over the surface of a sphere has to be unity:
∫sphereG dΩ=1∫sphereG dΩ=1
which gives rise to the definition of beam solid angle:
ΩA=4πG0ΩA=4πG0
where G0G0 is the peak gain. For example, the gain pattern or radiation pattern of a short dipole
G=32sin2θ=G0sin2θG=32sin2θ=G0sin2θ
which can be derived from the formula of power for a dipole antenna. Here G0=3/2=1.5=1.76G0=3/2=1.5=1.76 dB is the peak gain because this would be the gain at θ=90∘θ=90∘.
2. Effective area
For a receiving antenna, we use effective area instead of power gain for defining the beam. It is defined by the fact that flux is nothing but power per unit area. So the collecting area of a radio telescope
Ae=2PSAe=2PS
where the factor 22 is there because an antenna detects only half of the incident light from an unpolarized source. This can be derived using the following thought experiment.
Two cavities are in thermodynamic equilibrium, one has a resistor, the other an antenna. The antenna is connected to the resistor via a wire, so current of a specific frequency range can pass, but em wave cannot. The power received by the antenna
Pν=12∫4πAe(θ,ϕ)BνdΩPν=12∫4πAe(θ,ϕ)BνdΩ
which can be calculated using Planck's law and Nyquist formula. Planck’s law gives the radiation from a blackbody, a random mess of particles in thermal equilibrium. And the Nyquist formula gives the radiation from a warm a resistor, one that has been heated. All heated resistors radiate. The power of a warm resistor
Pν=kThν/kTehν/kT−1Pν=kThν/kTehν/kT−1
and the intensity of a blackbody
Bν=2kTλ2hν/kTehν/kT−1Bν=2kTλ2hν/kTehν/kT−1