Coherent Detection Signal-to-Noise

General equations are derived for the signal-to-noise ratio of a coherent (heterodyne) receiver in terms of the distribution functions of the signal and local oscillator fields and the size and shape of the detector. The optimum local oscillator field distribution is identical to the signal field distribution over the detector surface. The special case of an Airy function signal distribution and a uniform local oscillator distribution on a circular detector has an optimum detector radius 72% of the radius of the Airy disk, which gives a signal-to-noise ratio of 0.72 ηPs/hνBi.f..