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Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Editor: Joseph N. Mait
  • Vol. 52, Iss. 10 — Apr. 1, 2013
  • pp: 2008–2018

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

Baris I. Erkmen, Zeb W. Barber, and Jason Dahl  »View Author Affiliations


Applied Optics, Vol. 52, Issue 10, pp. 2008-2018 (2013)
http://dx.doi.org/10.1364/AO.52.002008


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Abstract

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.

© 2013 Optical Society of America

OCIS Codes
(000.5490) General : Probability theory, stochastic processes, and statistics
(010.3640) Atmospheric and oceanic optics : Lidar
(030.5260) Coherence and statistical optics : Photon counting
(270.5290) Quantum optics : Photon statistics
(040.1345) Detectors : Avalanche photodiodes (APDs)

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: October 11, 2012
Revised Manuscript: January 22, 2013
Manuscript Accepted: February 15, 2013
Published: March 21, 2013

Citation
Baris I. Erkmen, Zeb W. Barber, and Jason Dahl, "Maximum-likelihood estimation for frequency-modulated continuous-wave laser ranging using photon-counting detectors," Appl. Opt. 52, 2008-2018 (2013)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-52-10-2008


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References

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  27. The (appropriately normalized) mean of the two estimators at this flux level are separated by approximately 1 Hz, whereas the standard deviation of the estimates is on the order of 10 Hz. Thus, the contribution of the bias term to the MSE is negligible as long as the true range to the target is in the vicinity of the mean estimates generated by the two estimators. We believe that both estimators having significant bias at the highest flux level is unlikely.
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