Optical heterodyne signal power obtained from finite sized sources of radiation

A general expression for the optical heterodyne signal power from a finite-sized radiation source is derived that is formally valid for an arbitrary degree of coherence of the source. The signal power is given explicitly in terms of those variables that describe the source of radiation (e.g., total radiated power, brightness distribution, and source size). Finite-sized incoherent and coherent sources are examined in detail, and analytic expressions are derived for various geometries of experimental interest. For a finitesized incoherent source, we derive a generalization of Siegman's results for an extended incoherent source, which states that the effective or capture area of the receiver integrated over its field of view is equal to the square of the optical wavelength. In addition, we derive an analytic expression for the heterodyne signal power for a coherent Gaussian laser source and a Gaussian local oscillator. In contrast to the case of an incoherent source, in which the heterodyne signal power increases in proportion to receiver area, the heterodyne signal power from a coherent source tends asymptotically to a constant with increasing receiver area. This implies that, in a shot-noise-limited system, the SNR tends to a constant or decreases inversely as the receiver area for incoherent or coherent sources, respectively.