In the heterodyne receiver considered here, a coherent optical signal and incoherent gaussian background radiation impinge on a photosensitive surface together with the beam from a laser functioning as a local oscillator. The moment-generating function for the output current is derived. When the laser beam is much stronger than either signal or background, the output current consists of an IF signal and gaussian noise. The effective signal-to-noise ratio that determines the detectability of the signal is calculated as a function of the signal and local-oscillator fields at the photosurface. An integral equation is given for the local-oscillator field that maximizes the signal-to-noise ratio; it involves the signal field and the coefficient of partial coherence of the background. For a uniform background, the local-oscillator field must be a replica of the signal field.
CARL W. HELSTROM, "Detectability of Coherent Optical Signals in a Heterodyne Receiver," J. Opt. Soc. Am. 57, 353-360 (1967)