An ab-initio derivation to discuss the heterodyne versus direct detection decision problem for astronomical infrared interferometry

A consistent and explicit spectral comparison between heterodyne (HD) and direct detection (DD) derived from first principles including the atmospheric transmission and low beam-filling factors could not be found yet in literature but is needed for decisions in technology planification for future infrared interferometry facilities which are e.g. focused on planet formation. This task requires both, high sensitivity continuum and Doppler-resolved emission and absorption line detection in the mid-IR range (N- and Q-bands) at lower source temperatures (300-1000 K). The signal-to-noise ratios (SNRs) are derived for both schemes within the same semi-classical theory, which consists of classical mode theory for coupling to an antenna and occupation of these modes by quanta of three radiation fields, the thermal signal, the thermal background, and for HD also the coherent local oscillator (LO). The effects of very small beam filling factors (interferometry) and atmospheric absorption/emission could be consistently incorporated this way, as well as quantum-noise propagation which allows in HD the consideration of balanced mixers with cross-correlation (CC). Especially, the transition from pre- to post-detection SNRs was considered meticulously. We do this all because the usually cited SNR-expressions were derived for a too simple and unrealistic case, and moreover contain some wrong assumptions. We introduce a novel HD scheme for astronomical interferometry gaining an order of magnitude in sensitivity against conventional HD and calculate that it should trespass the sensitivity of DD interferometry in the N- and Q-bands for a spectral resolution of R=10000, and should do also for R=300 with doable technical improvements. This result encourages to develop broad-band heterodyne technologies for future mid-infrared interferometry facilities and for new instruments at existing facilities.