This paper defines the time-invariant detection scheme which yields the largest average signal-to-noise ratio in the heterodyne detection of a randomly distorted optical signal. It is shown that the detection scheme may be realized by properly shaping both the isophase surface and the irradiance distribution of the local-oscillator beam. Applied to the case of an atmospherically distorted optical plane wave it is shown that although the optimum local-oscillator beam differs significantly from a plane wave, the increase of average signal-to-noise ratio, over the plane-wave case as reported by Fried, is negligible (less than 10%).
J. P. MORELAND and S. A. COLLINS,JR., "Optical Heterodyne Detection of a Randomly Distorted Signal Beam," J. Opt. Soc. Am. 59, 10-11 (1969)