Cramer–Rao Bound for Frequency Estimation of Spectral Interference and Its Shot Noise-Limited Behavior

Interference frequency estimation is essential in spectral-domain interferometric sensing and imaging, and its performance determines system sensitivity. To date, an objective and practical criterion is still absent for the proper evaluation of fundamental sensitivity limit in a given system. Here, we report the derivation of Cramer-Rao bound (CRB) for unbiased estimators of spectral interference frequency, which imposes a theoretical limit on measurement sensitivity. Results based on a shot noise-limited model are presented. A more complete model, including dark and read noises, is also discussed and its approximate CRB is obtained. Poisson statistics were used in both cases. Asymptotic behaviors and simplified forms of CRB are studied for fringe visibility approaching 0 or 1. Further, we show that the current Fourier transform-based estimation algorithm achieves CRB only for low visibility, but is inferior by as much as √2 times for high visibility. This performance gap may potentially permit a sensitivity gain if better algorithms can be devised. Finally, we introduce a practical means to accurately estimate CRB from experimentally acquired spectral data. The results are verified in both simulation and experiments. Although spectral interference is discussed here, the same derivation can be applied to spatial or temporal interference as well, provided they can be similarly modelled.