S. Hernandez-Marin, A. M. Wallace, and G. J. Gibson, “Bayesian analysis of lidar signals with multiple returns,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2170–2180 (2007).

[CrossRef]

T. J. Karr, “Atmospheric phase error in coherent laser radar,” IEEE Trans. Antennas Propag. 55, 1122–1133(2007).

[CrossRef]

F. Athley, “Threshold region performance of maximum likelihood direction of arrival estimators,” IEEE Trans. Signal Process. 53, 1359–1373 (2005).

[CrossRef]

P. Forster, P. Larzabal, and E. Boyer, “Threshold performance analysis of maximum likelihood DOA estimation,” IEEE Trans. Signal Process. 52, 3183–3191 (2004).

[CrossRef]

M.-C. Amann, T. Bosch, M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).

[CrossRef]

K. Asaka, Y. Hirano, K. Tatsumi, K. Kasahara, and T. Tajime, “A pseudo-random frequency modulation continuous wave coherent lidar using an optical field correlation detection method,” Opt. Rev. 5, 310–314 (1998).

[CrossRef]

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).

[CrossRef]

L. T. Masters, M. B. Mark, and B. D. Duncan, “Analysis of ladar range resolution enhancement by sinusoidal phase modulation,” Opt. Eng. 34, 3115–3121 (1995).

[CrossRef]

T. J. Green and J. H. Shapiro, “Maximum-likelihood laser radar range profiling with the expectation-maximization algorithm,” Opt. Eng. 31, 2343–2354 (1992).

[CrossRef]

Z. Y. Ou, C. K. Hong, and L. Mandel, “Relation between input and output states for a beam splitter,” Opt. Commun. 63, 118–122 (1987).

[CrossRef]

G. Vannucci and M. C. Teich, “Effects of rate variation on the counting statistics of dead-time-modified Poisson processes,” Opt. Commun. 25, 267–272 (1978).

[CrossRef]

M.-C. Amann, T. Bosch, M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).

[CrossRef]

K. Asaka, Y. Hirano, K. Tatsumi, K. Kasahara, and T. Tajime, “A pseudo-random frequency modulation continuous wave coherent lidar using an optical field correlation detection method,” Opt. Rev. 5, 310–314 (1998).

[CrossRef]

F. Athley, “Threshold region performance of maximum likelihood direction of arrival estimators,” IEEE Trans. Signal Process. 53, 1359–1373 (2005).

[CrossRef]

M.-C. Amann, T. Bosch, M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).

[CrossRef]

P. Forster, P. Larzabal, and E. Boyer, “Threshold performance analysis of maximum likelihood DOA estimation,” IEEE Trans. Signal Process. 52, 3183–3191 (2004).

[CrossRef]

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).

[CrossRef]

D. Dupuy, M. Lescure, and M. Cousineau, “A FMCW laser range-finder based on a delay line technique,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IEEE, 2001), pp. 1084–1088.

L. T. Masters, M. B. Mark, and B. D. Duncan, “Analysis of ladar range resolution enhancement by sinusoidal phase modulation,” Opt. Eng. 34, 3115–3121 (1995).

[CrossRef]

D. Dupuy, M. Lescure, and M. Cousineau, “A FMCW laser range-finder based on a delay line technique,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IEEE, 2001), pp. 1084–1088.

B. I. Erkmen and B. Moision, “Maximum likelihood time-of-arrival estimation of optical pulses via photon-counting photodetectors,” in Proceedings of IEEE International Symposium on Information Theory (IEEE, 2009), pp. 1909–1913.

P. Forster, P. Larzabal, and E. Boyer, “Threshold performance analysis of maximum likelihood DOA estimation,” IEEE Trans. Signal Process. 52, 3183–3191 (2004).

[CrossRef]

R. M. Gagliardi and S. Karp, Optical Communications (Wiley, 1976).

S. Hernandez-Marin, A. M. Wallace, and G. J. Gibson, “Bayesian analysis of lidar signals with multiple returns,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2170–2180 (2007).

[CrossRef]

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).

[CrossRef]

T. J. Green and J. H. Shapiro, “Maximum-likelihood laser radar range profiling with the expectation-maximization algorithm,” Opt. Eng. 31, 2343–2354 (1992).

[CrossRef]

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).

[CrossRef]

S. Hernandez-Marin, A. M. Wallace, and G. J. Gibson, “Bayesian analysis of lidar signals with multiple returns,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2170–2180 (2007).

[CrossRef]

K. Asaka, Y. Hirano, K. Tatsumi, K. Kasahara, and T. Tajime, “A pseudo-random frequency modulation continuous wave coherent lidar using an optical field correlation detection method,” Opt. Rev. 5, 310–314 (1998).

[CrossRef]

Z. Y. Ou, C. K. Hong, and L. Mandel, “Relation between input and output states for a beam splitter,” Opt. Commun. 63, 118–122 (1987).

[CrossRef]

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).

[CrossRef]

R. M. Gagliardi and S. Karp, Optical Communications (Wiley, 1976).

T. J. Karr, “Atmospheric phase error in coherent laser radar,” IEEE Trans. Antennas Propag. 55, 1122–1133(2007).

[CrossRef]

K. Asaka, Y. Hirano, K. Tatsumi, K. Kasahara, and T. Tajime, “A pseudo-random frequency modulation continuous wave coherent lidar using an optical field correlation detection method,” Opt. Rev. 5, 310–314 (1998).

[CrossRef]

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).

[CrossRef]

P. Forster, P. Larzabal, and E. Boyer, “Threshold performance analysis of maximum likelihood DOA estimation,” IEEE Trans. Signal Process. 52, 3183–3191 (2004).

[CrossRef]

M.-C. Amann, T. Bosch, M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).

[CrossRef]

D. Dupuy, M. Lescure, and M. Cousineau, “A FMCW laser range-finder based on a delay line technique,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IEEE, 2001), pp. 1084–1088.

Z. Y. Ou, C. K. Hong, and L. Mandel, “Relation between input and output states for a beam splitter,” Opt. Commun. 63, 118–122 (1987).

[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

L. T. Masters, M. B. Mark, and B. D. Duncan, “Analysis of ladar range resolution enhancement by sinusoidal phase modulation,” Opt. Eng. 34, 3115–3121 (1995).

[CrossRef]

L. T. Masters, M. B. Mark, and B. D. Duncan, “Analysis of ladar range resolution enhancement by sinusoidal phase modulation,” Opt. Eng. 34, 3115–3121 (1995).

[CrossRef]

B. I. Erkmen and B. Moision, “Maximum likelihood time-of-arrival estimation of optical pulses via photon-counting photodetectors,” in Proceedings of IEEE International Symposium on Information Theory (IEEE, 2009), pp. 1909–1913.

M.-C. Amann, T. Bosch, M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).

[CrossRef]

Z. Y. Ou, C. K. Hong, and L. Mandel, “Relation between input and output states for a beam splitter,” Opt. Commun. 63, 118–122 (1987).

[CrossRef]

M.-C. Amann, T. Bosch, M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).

[CrossRef]

T. J. Green and J. H. Shapiro, “Maximum-likelihood laser radar range profiling with the expectation-maximization algorithm,” Opt. Eng. 31, 2343–2354 (1992).

[CrossRef]

D. L. Snyder, Random Point Processes (Wiley, 1975).

K. Asaka, Y. Hirano, K. Tatsumi, K. Kasahara, and T. Tajime, “A pseudo-random frequency modulation continuous wave coherent lidar using an optical field correlation detection method,” Opt. Rev. 5, 310–314 (1998).

[CrossRef]

K. Asaka, Y. Hirano, K. Tatsumi, K. Kasahara, and T. Tajime, “A pseudo-random frequency modulation continuous wave coherent lidar using an optical field correlation detection method,” Opt. Rev. 5, 310–314 (1998).

[CrossRef]

G. Vannucci and M. C. Teich, “Effects of rate variation on the counting statistics of dead-time-modified Poisson processes,” Opt. Commun. 25, 267–272 (1978).

[CrossRef]

H. L. V. Trees, Detection, Estimation and Modulation Theory, Part 1 (Prentice Hall, 2001).

G. Vannucci and M. C. Teich, “Effects of rate variation on the counting statistics of dead-time-modified Poisson processes,” Opt. Commun. 25, 267–272 (1978).

[CrossRef]

S. Hernandez-Marin, A. M. Wallace, and G. J. Gibson, “Bayesian analysis of lidar signals with multiple returns,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2170–2180 (2007).

[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).

[CrossRef]

Z. W. Barber, W. R. Babbitt, B. Kaylor, R. R. Reibel, and P. A. Roos, “Accuracy of active chirp linearization for broadband frequency modulated continuous-wave ladar,” Appl. Opt. 49, 213–219 (2010).

[CrossRef]

L. A. Jiang and J. X. Luu, “Heterodyne detection with a weak local oscillator,” Appl. Opt. 47, 1486–1503 (2008).

[CrossRef]

T. J. Karr, “Atmospheric phase error in coherent laser radar,” IEEE Trans. Antennas Propag. 55, 1122–1133(2007).

[CrossRef]

S. Hernandez-Marin, A. M. Wallace, and G. J. Gibson, “Bayesian analysis of lidar signals with multiple returns,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2170–2180 (2007).

[CrossRef]

P. Forster, P. Larzabal, and E. Boyer, “Threshold performance analysis of maximum likelihood DOA estimation,” IEEE Trans. Signal Process. 52, 3183–3191 (2004).

[CrossRef]

F. Athley, “Threshold region performance of maximum likelihood direction of arrival estimators,” IEEE Trans. Signal Process. 53, 1359–1373 (2005).

[CrossRef]

G. Vannucci and M. C. Teich, “Effects of rate variation on the counting statistics of dead-time-modified Poisson processes,” Opt. Commun. 25, 267–272 (1978).

[CrossRef]

Z. Y. Ou, C. K. Hong, and L. Mandel, “Relation between input and output states for a beam splitter,” Opt. Commun. 63, 118–122 (1987).

[CrossRef]

T. J. Green and J. H. Shapiro, “Maximum-likelihood laser radar range profiling with the expectation-maximization algorithm,” Opt. Eng. 31, 2343–2354 (1992).

[CrossRef]

L. T. Masters, M. B. Mark, and B. D. Duncan, “Analysis of ladar range resolution enhancement by sinusoidal phase modulation,” Opt. Eng. 34, 3115–3121 (1995).

[CrossRef]

M.-C. Amann, T. Bosch, M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).

[CrossRef]

K. Asaka, Y. Hirano, K. Tatsumi, K. Kasahara, and T. Tajime, “A pseudo-random frequency modulation continuous wave coherent lidar using an optical field correlation detection method,” Opt. Rev. 5, 310–314 (1998).

[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

H. L. V. Trees, Detection, Estimation and Modulation Theory, Part 1 (Prentice Hall, 2001).

D. Dupuy, M. Lescure, and M. Cousineau, “A FMCW laser range-finder based on a delay line technique,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IEEE, 2001), pp. 1084–1088.

The square-bracketed term in Eq. (8) is approximated as εϕ/2 to obtain this result.

Here we continue to assume ψ0=0, consistent with our treatment in the previous section.

Throughout this paper, an ideal laser field refers to a paraxial and quasi-monochromatic optical field in a single spatial and polarization mode, which is also in a coherent state of the quantized field operator, such that it gives rise to Poisson statistics when an ideal photon-counting measurement is performed on it [9].

This characterization of the photodetector output implies that the detector has infinite electrical bandwidth, allowing the arrival times to be precisely resolvable. In practice, there is little loss in adopting this idealization if the photodetector impulse response is significantly narrower than the mean photoelectron interarrival time.

R. M. Gagliardi and S. Karp, Optical Communications (Wiley, 1976).

D. L. Snyder, Random Point Processes (Wiley, 1975).

Tensor-product coherent states incident on a beam splitter yield tensor-product coherent-state outputs [19]. Because the propagation paths in Fig. 1 can be modeled as a sequence of beam splitters (including the loss elements), the fields incident on the photodetectors will also be tensor-product coherent states. It is well known that in this case the semiclassical theory of photodetection and quantum measurement theory predict exactly the same statistics for the output photocurrent [9].

B. I. Erkmen and B. Moision, “Maximum likelihood time-of-arrival estimation of optical pulses via photon-counting photodetectors,” in Proceedings of IEEE International Symposium on Information Theory (IEEE, 2009), pp. 1909–1913.

The mean incident photon number, NI, is estimated from the mean registered photoelectron counts, ndet, by accounting for the incident photons lost to dead time. ndet is equal to the sum and integral of the mean arrival rates given in Eq. (A4) in Appendix A. To a good degree of approximation, ndet=ηNI(1+α(1−β2/2))/(1+2α+α2(1−β2/2)).

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.