Abstract

A theoretical analysis and experimental verification of the sensitivity limits of frequency-modulated continuous-wave (FMCW) ladar in the limit of a strong local oscillator is presented. The single-photon sensitivity of coherent heterodyne detection in this shot-noise dominated limit is verified to extend to linearly chirped waveforms. An information theoretic analysis is presented to estimate the information efficiency of received photons for the task of locating the range to single and multiple targets. It is found that the optimum receive signal level is proportional to the logarithm of the number of resolvable range locations and the maximum theoretical photon information efficiency for FMCW ranging with coherent fields is log(e)1.44 bits per received photon.

© 2013 Optical Society of America

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References

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2013 (1)

2012 (1)

2010 (1)

B. I. Erkmen, B. E. Moision, and K. M. Birnbaum, “A review of the information capacity of single-mode free-space optical communication,” Proc. SPIE 7587, 75870N (2010).
[CrossRef]

2009 (2)

2007 (1)

K. W. Holman, D. G. Kocher, and S. Kaushik, “MIT/LL development of broadband linear frequency chirp for high resolution ladar,” Proc. SPIE 6572, 65720J (2007).
[CrossRef]

2005 (1)

2000 (1)

W. F. Buell, B. A. Shadwick, and R. W. Farley, “Bayesian spectrum analysis for laser vibrometry processing,” Proc. SPIE 4035, 444–455 (2000).
[CrossRef]

1998 (1)

P. J. Winzer and W. R. Leeb, “Coherent lidar at low signal powers: basic considerations on optical heterodyning,” J. Mod. Opt. 45, 1549–1555 (1998).
[CrossRef]

1997 (2)

1989 (1)

R. Menzies and R. Hardesty, “Coherent doppler lidar for measurements of wind fields,” Proc. IEEE 77, 449–462 (1989).
[CrossRef]

1948 (1)

C. E. Shannon, “The mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, (1948).

Abramowitz, M.

M. Abramowitz, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover,1972).

Babbitt, W. R.

Barber, Z. W.

Beck, S. M.

Berg, T.

Birnbaum, K.

S. Dolinar, K. Birnbaum, B. Erkmen, and B. Moision, “On approaching the ultimate limits of photon-efficient and bandwidth-efficient optical communication,” in 2011 International Conference on Space Optical Systems and Applications (ICSOS), (IEEE, 2011), pp. 269–278.

Birnbaum, K. M.

B. I. Erkmen, B. E. Moision, and K. M. Birnbaum, “A review of the information capacity of single-mode free-space optical communication,” Proc. SPIE 7587, 75870N (2010).
[CrossRef]

Buck, J. R.

Buell, W. F.

Chinn, S. R.

Crouch, S.

Dahl, J.

Dickinson, R. P.

Dolinar, S.

S. Dolinar, K. Birnbaum, B. Erkmen, and B. Moision, “On approaching the ultimate limits of photon-efficient and bandwidth-efficient optical communication,” in 2011 International Conference on Space Optical Systems and Applications (ICSOS), (IEEE, 2011), pp. 269–278.

Erkmen, B.

S. Dolinar, K. Birnbaum, B. Erkmen, and B. Moision, “On approaching the ultimate limits of photon-efficient and bandwidth-efficient optical communication,” in 2011 International Conference on Space Optical Systems and Applications (ICSOS), (IEEE, 2011), pp. 269–278.

Erkmen, B. I.

B. I. Erkmen, Z. W. Barber, and J. Dahl, “Maximum-likelihood estimation for frequency-modulated continuous-wave laser ranging using photon-counting detectors,” Appl. Opt. 52, 2008–2018 (2013).
[CrossRef]

B. I. Erkmen, B. E. Moision, and K. M. Birnbaum, “A review of the information capacity of single-mode free-space optical communication,” Proc. SPIE 7587, 75870N (2010).
[CrossRef]

Farley, R. W.

W. F. Buell, B. A. Shadwick, and R. W. Farley, “Bayesian spectrum analysis for laser vibrometry processing,” Proc. SPIE 4035, 444–455 (2000).
[CrossRef]

Fujimoto, J. G.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Hardesty, R.

R. Menzies and R. Hardesty, “Coherent doppler lidar for measurements of wind fields,” Proc. IEEE 77, 449–462 (1989).
[CrossRef]

Haus, H.

H. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, 2000).

Hobbs, P. C.

Hobbs, P. C. D.

P. C. D. Hobbs, Building Electro-Optical Systems, Making It All Work (Wiley, 2000).

Holman, K. W.

K. W. Holman, D. G. Kocher, and S. Kaushik, “MIT/LL development of broadband linear frequency chirp for high resolution ladar,” Proc. SPIE 6572, 65720J (2007).
[CrossRef]

Kaushik, S.

K. W. Holman, D. G. Kocher, and S. Kaushik, “MIT/LL development of broadband linear frequency chirp for high resolution ladar,” Proc. SPIE 6572, 65720J (2007).
[CrossRef]

Kaylor, B.

Kocher, D. G.

K. W. Holman, D. G. Kocher, and S. Kaushik, “MIT/LL development of broadband linear frequency chirp for high resolution ladar,” Proc. SPIE 6572, 65720J (2007).
[CrossRef]

Kozlowski, D. A.

Leeb, W. R.

P. J. Winzer and W. R. Leeb, “Coherent lidar at low signal powers: basic considerations on optical heterodyning,” J. Mod. Opt. 45, 1549–1555 (1998).
[CrossRef]

Leyva, V.

Marechal, N. J.

Menzies, R.

R. Menzies and R. Hardesty, “Coherent doppler lidar for measurements of wind fields,” Proc. IEEE 77, 449–462 (1989).
[CrossRef]

Moision, B.

S. Dolinar, K. Birnbaum, B. Erkmen, and B. Moision, “On approaching the ultimate limits of photon-efficient and bandwidth-efficient optical communication,” in 2011 International Conference on Space Optical Systems and Applications (ICSOS), (IEEE, 2011), pp. 269–278.

Moision, B. E.

B. I. Erkmen, B. E. Moision, and K. M. Birnbaum, “A review of the information capacity of single-mode free-space optical communication,” Proc. SPIE 7587, 75870N (2010).
[CrossRef]

Rakuljic, G.

Reibel, R. R.

Roos, P. A.

Satyan, N.

Shadwick, B. A.

W. F. Buell, B. A. Shadwick, and R. W. Farley, “Bayesian spectrum analysis for laser vibrometry processing,” Proc. SPIE 4035, 444–455 (2000).
[CrossRef]

Shannon, C. E.

C. E. Shannon, “The mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, (1948).

Swanson, E. A.

Vasilyev, A.

Winzer, P. J.

P. J. Winzer and W. R. Leeb, “Coherent lidar at low signal powers: basic considerations on optical heterodyning,” J. Mod. Opt. 45, 1549–1555 (1998).
[CrossRef]

Wright, T. J.

Yariv, A.

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

C. E. Shannon, “The mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, (1948).

J. Mod. Opt. (1)

P. J. Winzer and W. R. Leeb, “Coherent lidar at low signal powers: basic considerations on optical heterodyning,” J. Mod. Opt. 45, 1549–1555 (1998).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Proc. IEEE (1)

R. Menzies and R. Hardesty, “Coherent doppler lidar for measurements of wind fields,” Proc. IEEE 77, 449–462 (1989).
[CrossRef]

Proc. SPIE (3)

W. F. Buell, B. A. Shadwick, and R. W. Farley, “Bayesian spectrum analysis for laser vibrometry processing,” Proc. SPIE 4035, 444–455 (2000).
[CrossRef]

K. W. Holman, D. G. Kocher, and S. Kaushik, “MIT/LL development of broadband linear frequency chirp for high resolution ladar,” Proc. SPIE 6572, 65720J (2007).
[CrossRef]

B. I. Erkmen, B. E. Moision, and K. M. Birnbaum, “A review of the information capacity of single-mode free-space optical communication,” Proc. SPIE 7587, 75870N (2010).
[CrossRef]

Other (5)

P. C. D. Hobbs, Building Electro-Optical Systems, Making It All Work (Wiley, 2000).

J. W. Goodman, Statistical Optics (Wiley, 2000).

M. Abramowitz, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover,1972).

H. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, 2000).

S. Dolinar, K. Birnbaum, B. Erkmen, and B. Moision, “On approaching the ultimate limits of photon-efficient and bandwidth-efficient optical communication,” in 2011 International Conference on Space Optical Systems and Applications (ICSOS), (IEEE, 2011), pp. 269–278.

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Figures (9)

Fig. 1.
Fig. 1.

Schematic of FMCW ladar measurement setup. The chirp laser operated at a wavelength of 1539 nm and had an output power of 40 mW. All fiber components are single mode and polarization maintaining. The homebuilt autobalanced detector has a QE of 0.798, an effective transimpedance gain of 7.14 kOhm, a dark noise of 140dBm/Hz in RF power, and 40 MHz bandwidth. The digitizer was set to an input impedance of 50 Ohm on the 0.4 V range.

Fig. 2.
Fig. 2.

Sample FMCW ladar spectra range profile for three different LO powers. The noise floor of the digitizer card used to capture the data is at 115dBm. The range profiles have been averaged 50 times to better reveal static structure in noise of the range profiles. The spike at 0.75 MHz is from an actual target with return signal level of approximately 20 photons during the chirp. Small stray peaks such as that around 1.8 MHz are due to unwanted reflections in the LO path and can be eliminated with care.

Fig. 3.
Fig. 3.

(a) Average of 8000 FMCW range profiles for a signal photon flux of two photons per integration period normalized to the background level. The normalized power of 2.4 indicates an average of 1.4 measured photoelectrons. The red line at the bottom is the dark noise of the balanced detector. (b)–(d) Example histograms of the normalized power measured in the target bin for nominal signal flux of approximately 1, 5, and 30 photons per measurement period, respectively. The measured photoelectron flux is consistently 70% of the nominal photon flux. The dotted lines are the corresponding analytic probability density functions.

Fig. 4.
Fig. 4.

Histogram of 8000 trials of maximum energy range estimator for a target location at bin 149 among 328 resolved range bin with an average of 1.4 signal photoelectrons per integration period. The errors of the estimator are equally spread among the range bins.

Fig. 5.
Fig. 5.

Probability that estimator correctly finds the target location for 328 range bins as a function of the mean signal photoelectron level. The open circles are experimental values calculated from the 8000 trials for each of several signal levels.

Fig. 6.
Fig. 6.

Theoretical (line) and experimental estimate (circles) of the PIE as a function of mean signal photoelectron level measured in bpp for shot noise limited detection and M=328 discrete range bins.

Fig. 7.
Fig. 7.

Contour map of the theoretical PIE as a function of expected received signal photoelectrons and the logarithm base 2 of the number of range bins. The labels indicate the PIE measured in bpp for the respective contour. The thick straight line on the figure is drawn through the maximum PIE for a given number of range bins.

Fig. 8.
Fig. 8.

Maximum PIE for task of single target FMCW range location as a function of number of discrete range bins using full entropy calculations (solid line) and for the simplified analytically calculable model described Eq. (11). The PIE approaches a maximum of 1.44 bpp very slowly, making bpp greater than 1 impractical.

Fig. 9.
Fig. 9.

Experimentally (markers) and theoretically (lines) estimated PIE for locating two targets with different relative reflectivity as expressed as the fraction of the total signal power received by one of the targets (i.e., a value of 0.5 indicates the targets have equal reflectivity). The PIE can only be increased modestly over the single target task (small dots).

Equations (12)

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id(t)=2ηqTNLONsigcos(2πκτt+ϕ)+is(t),
σt2=2ηq2BNLO/T,
pk(y|σ2,mk2)=1σ2exp(y+mk2σ2)I0(2mkσ2y),
σ2=RG2ηq2NLOT
mk2=12RG22η2q2NLONsigT,
CNR=y¯y¯0y¯0=mk2σ2=ηNsig.
SNR=y¯2σy2=(1+mk2/σ2)21+2mk2/σ2=(1+ηNsig)21+2ηNsig.
Pτ=P(r=rτ|Ns,M)=0(1ex)M1pk(x|σ2=1,mk2=Ns)dx,
0=ddx(C(1ex)M1ex)0=C((M1)(1ex)M2e2x(1ex)M1ex)0=Mex1Nthresh=ln(M),
H(X)=M(1Mlog2(1/M))=log2(M),
H(X|Y)=Pτlog2(Pτ)(Pτ1)log2(Pτ1M1),
PIElog2(M)ln(M)+22ln(M)log2(e)[122/ln(M)],

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