Maximum-likelihood estimation for frequency-modulated continuous-wave laser ranging using photon-counting detectors.

We analyze the minimum achievable mean-square error in frequency-modulated continuous-wave range estimation of a single stationary target when photon-counting detectors are employed. Starting from the probability density function for the photon-arrival times in photodetectors with subunity quantum efficiency, dark counts, and dead time, we derive the Cramér-Rao bound and highlight three important asymptotic regimes. We then derive the maximum-likelihood (ML) estimator for arbitrary frequency modulation. Simulation of the ML estimator shows that its performance approaches the standard quantum limit only when the mean received photons are between two thresholds. We provide analytic approximations to these thresholds for linear frequency modulation. We also compare the ML estimator's performance to conventional Fourier transform (FT) frequency estimation, showing that they are equivalent if the reference arm is much stronger than the target return, but that when the reference field is weak the FT estimator is suboptimal by approximately a factor of √2 in root-mean-square error. Finally, we report on a proof-of-concept experiment in which the ML estimator achieves this theoretically predicted improvement over the FT estimator.