Abstract

Based on a standard binomial model of sub-Poissonian photocounting statistics, we analyze the discrimination performance between the two possibilities that light has been potentially absorbed or not. For that purpose, we study with numerical simulations the behavior of different information-theory-based measures of the contrast and show that the Chernoff measure allows one to obtain a useful contrast characterization that has simple physical interpretation and that helps in analyzing the benefit of using sub-Poissonian light to improve detection tasks.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  29. A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153-1166 (2002).
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    [CrossRef]

2008 (3)

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” Europhys. Lett. 81, 44001 (2008).
[CrossRef]

J. Fade, N. Treps, C. Fabre, and P. Réfrégier, “Optimal precision of parameter estimation in images with local sub-Poissonian quantum fluctuations,” Eur. Phys. J. D 50, 215-227 (2008).
[CrossRef]

A. Sergé, N. Bertaux, H. Rigneault, and D. Marguet, “Dynamic multiple-target tracing probes spatiotemporal cartography of cell membranes,” Nat. Methods 5, 687-694 (2008).
[CrossRef] [PubMed]

2006 (2)

A. S. Van de Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323-2363 (2006).
[CrossRef]

M. T. L. Hsu, V. Delaubert, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, “A quantum study of multibit phase coding for optical storage,” IEEE J. Quantum Electron. 42, 1001-1007 (2006).
[CrossRef]

2005 (1)

S. Bonneau, M. Dahan, and L. Cohen, “Single quantum dot tracking based on perceptual grouping using minimal paths in a spatiotemporal volume,” IEEE Trans. Image Process. 14, 1384-1395 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (2)

T. M. Brown, “Expected detection and false alarm rates for transiting jovian planets,” Astrophys. J. 593, L125-L128 (2003).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940-943 (2003).
[CrossRef] [PubMed]

2002 (3)

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, 176-183 (2002).
[CrossRef]

N. Treps, U. Andersen, B. Buchler, P. K. Lam, H.-A. Bachor, A. Maître, and C. Fabre, “Crossing the standard quantum limit for high sensitivity measurements in optical images using non classical light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153-1166 (2002).
[CrossRef]

2000 (1)

1999 (1)

M. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539-1589 (1999).
[CrossRef]

1992 (1)

C. A. J. Putman, B. G. de Grooth, N. F. van Hulst, and J. Greve, “A detailed analysis of the optical beam deflection technique for use in atomic force microscopy,” J. Appl. Phys. 72, 6-12 (1992).
[CrossRef]

1989 (1)

M. Basseville, “Distance measures for signal processing and pattern recognition,” Springer Proc. Phys. 18, 349-369 (1989).

1985 (1)

D. Stoler, B. Saleh, and M. Teich, “Binomial states of the quantized radiation field,” J. Mod. Opt. 32, 345-355 (1985).

1984 (1)

1982 (2)

1979 (1)

1967 (1)

T. Kailath, “The divergence and Bhattacharyya distance measures in signal selection,” IEEE Trans. Commun. 15, 52-60 (1967).
[CrossRef]

1964 (1)

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, A316-A334 (1964).
[CrossRef]

1947 (1)

U. Fano, “Ionization yield of radiations. II. The fluctuations of the number of ions,” Phys. Rev. 72, 26-29 (1947).
[CrossRef]

Andersen, U.

N. Treps, U. Andersen, B. Buchler, P. K. Lam, H.-A. Bachor, A. Maître, and C. Fabre, “Crossing the standard quantum limit for high sensitivity measurements in optical images using non classical light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

Bachor, H. A.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” Europhys. Lett. 81, 44001 (2008).
[CrossRef]

M. T. L. Hsu, V. Delaubert, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, “A quantum study of multibit phase coding for optical storage,” IEEE J. Quantum Electron. 42, 1001-1007 (2006).
[CrossRef]

H. A. Bachor, A Guide to Experiments in Quantum Optics (Wiley, 2003).

Bachor, H.-A.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940-943 (2003).
[CrossRef] [PubMed]

N. Treps, U. Andersen, B. Buchler, P. K. Lam, H.-A. Bachor, A. Maître, and C. Fabre, “Crossing the standard quantum limit for high sensitivity measurements in optical images using non classical light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

Basseville, M.

M. Basseville, “Distance measures for signal processing and pattern recognition,” Springer Proc. Phys. 18, 349-369 (1989).

Bertaux, N.

A. Sergé, N. Bertaux, H. Rigneault, and D. Marguet, “Dynamic multiple-target tracing probes spatiotemporal cartography of cell membranes,” Nat. Methods 5, 687-694 (2008).
[CrossRef] [PubMed]

Bonneau, S.

S. Bonneau, M. Dahan, and L. Cohen, “Single quantum dot tracking based on perceptual grouping using minimal paths in a spatiotemporal volume,” IEEE Trans. Image Process. 14, 1384-1395 (2005).
[CrossRef] [PubMed]

Bowen, W. P.

M. T. L. Hsu, V. Delaubert, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, “A quantum study of multibit phase coding for optical storage,” IEEE J. Quantum Electron. 42, 1001-1007 (2006).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940-943 (2003).
[CrossRef] [PubMed]

Braat, J. J. M.

A. S. Van de Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323-2363 (2006).
[CrossRef]

Brambilla, E.

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, 176-183 (2002).
[CrossRef]

Brown, T. M.

T. M. Brown, “Expected detection and false alarm rates for transiting jovian planets,” Astrophys. J. 593, L125-L128 (2003).
[CrossRef]

Buchler, B.

N. Treps, U. Andersen, B. Buchler, P. K. Lam, H.-A. Bachor, A. Maître, and C. Fabre, “Crossing the standard quantum limit for high sensitivity measurements in optical images using non classical light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

Chipman, J. W.

R. A. Goodman and J. W. Chipman, Handbook of Optics (McGraw Hill, 1994), Chap. 30, pp. 30.3-30.23.

Cohen, L.

S. Bonneau, M. Dahan, and L. Cohen, “Single quantum dot tracking based on perceptual grouping using minimal paths in a spatiotemporal volume,” IEEE Trans. Image Process. 14, 1384-1395 (2005).
[CrossRef] [PubMed]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 1991).
[CrossRef]

Dahan, M.

S. Bonneau, M. Dahan, and L. Cohen, “Single quantum dot tracking based on perceptual grouping using minimal paths in a spatiotemporal volume,” IEEE Trans. Image Process. 14, 1384-1395 (2005).
[CrossRef] [PubMed]

de Grooth, B. G.

C. A. J. Putman, B. G. de Grooth, N. F. van Hulst, and J. Greve, “A detailed analysis of the optical beam deflection technique for use in atomic force microscopy,” J. Appl. Phys. 72, 6-12 (1992).
[CrossRef]

Delaubert, V.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” Europhys. Lett. 81, 44001 (2008).
[CrossRef]

M. T. L. Hsu, V. Delaubert, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, “A quantum study of multibit phase coding for optical storage,” IEEE J. Quantum Electron. 42, 1001-1007 (2006).
[CrossRef]

Delyon, G.

Fabre, C.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” Europhys. Lett. 81, 44001 (2008).
[CrossRef]

J. Fade, N. Treps, C. Fabre, and P. Réfrégier, “Optimal precision of parameter estimation in images with local sub-Poissonian quantum fluctuations,” Eur. Phys. J. D 50, 215-227 (2008).
[CrossRef]

M. T. L. Hsu, V. Delaubert, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, “A quantum study of multibit phase coding for optical storage,” IEEE J. Quantum Electron. 42, 1001-1007 (2006).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940-943 (2003).
[CrossRef] [PubMed]

N. Treps, U. Andersen, B. Buchler, P. K. Lam, H.-A. Bachor, A. Maître, and C. Fabre, “Crossing the standard quantum limit for high sensitivity measurements in optical images using non classical light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

C. Fabre, J. B. Fouet, and A. Maître, “Quantum limits in the measurement of very small displacements in optical images,” Opt. Lett. 25, 76-78 (2000).
[CrossRef]

Fade, J.

J. Fade, N. Treps, C. Fabre, and P. Réfrégier, “Optimal precision of parameter estimation in images with local sub-Poissonian quantum fluctuations,” Eur. Phys. J. D 50, 215-227 (2008).
[CrossRef]

Fano, U.

U. Fano, “Ionization yield of radiations. II. The fluctuations of the number of ions,” Phys. Rev. 72, 26-29 (1947).
[CrossRef]

Fouet, J. B.

Gagliardi, R. M.

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

Gatti, A.

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, 176-183 (2002).
[CrossRef]

Goodman, R. A.

R. A. Goodman and J. W. Chipman, Handbook of Optics (McGraw Hill, 1994), Chap. 30, pp. 30.3-30.23.

Goudail, F.

Greve, J.

C. A. J. Putman, B. G. de Grooth, N. F. van Hulst, and J. Greve, “A detailed analysis of the optical beam deflection technique for use in atomic force microscopy,” J. Appl. Phys. 72, 6-12 (1992).
[CrossRef]

Grosse, N.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940-943 (2003).
[CrossRef] [PubMed]

Hsu, M. T. L.

M. T. L. Hsu, V. Delaubert, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, “A quantum study of multibit phase coding for optical storage,” IEEE J. Quantum Electron. 42, 1001-1007 (2006).
[CrossRef]

Jain, A.

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153-1166 (2002).
[CrossRef]

Kailath, T.

T. Kailath, “The divergence and Bhattacharyya distance measures in signal selection,” IEEE Trans. Commun. 15, 52-60 (1967).
[CrossRef]

Karp, S.

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

Kelley, P. L.

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, A316-A334 (1964).
[CrossRef]

Kleiner, W. H.

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, A316-A334 (1964).
[CrossRef]

Kolobov, M.

M. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539-1589 (1999).
[CrossRef]

Kolobov, M. I.

M. I. Kolobov, Quantum Imaging (Springer, 2006).

Lam, P. K.

M. T. L. Hsu, V. Delaubert, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, “A quantum study of multibit phase coding for optical storage,” IEEE J. Quantum Electron. 42, 1001-1007 (2006).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940-943 (2003).
[CrossRef] [PubMed]

N. Treps, U. Andersen, B. Buchler, P. K. Lam, H.-A. Bachor, A. Maître, and C. Fabre, “Crossing the standard quantum limit for high sensitivity measurements in optical images using non classical light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

Lugiato, L. A.

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, 176-183 (2002).
[CrossRef]

Maître, A.

N. Treps, U. Andersen, B. Buchler, P. K. Lam, H.-A. Bachor, A. Maître, and C. Fabre, “Crossing the standard quantum limit for high sensitivity measurements in optical images using non classical light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

C. Fabre, J. B. Fouet, and A. Maître, “Quantum limits in the measurement of very small displacements in optical images,” Opt. Lett. 25, 76-78 (2000).
[CrossRef]

Mandel, L.

Mansuripur, M.

M. Mansuripur, Handbook of Optics (McGraw Hill, 1994), Chap. 31, pp. 31.1-31.32.

Marguet, D.

A. Sergé, N. Bertaux, H. Rigneault, and D. Marguet, “Dynamic multiple-target tracing probes spatiotemporal cartography of cell membranes,” Nat. Methods 5, 687-694 (2008).
[CrossRef] [PubMed]

Miller, M. I.

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153-1166 (2002).
[CrossRef]

Moulin, P.

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153-1166 (2002).
[CrossRef]

Paul, H.

H. Paul, “Photon antibunching,” Rev. Mod. Phys. 54, 1061-1102 (1982).
[CrossRef]

Pereira, S. F.

A. S. Van de Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323-2363 (2006).
[CrossRef]

Perina, J.

Putman, C. A. J.

C. A. J. Putman, B. G. de Grooth, N. F. van Hulst, and J. Greve, “A detailed analysis of the optical beam deflection technique for use in atomic force microscopy,” J. Appl. Phys. 72, 6-12 (1992).
[CrossRef]

Ramchandran, K.

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153-1166 (2002).
[CrossRef]

Réfrégier, P.

J. Fade, N. Treps, C. Fabre, and P. Réfrégier, “Optimal precision of parameter estimation in images with local sub-Poissonian quantum fluctuations,” Eur. Phys. J. D 50, 215-227 (2008).
[CrossRef]

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” Europhys. Lett. 81, 44001 (2008).
[CrossRef]

F. Goudail, P. Réfrégier, and G. Delyon, “Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images,” J. Opt. Soc. Am. A 21, 1231-1240 (2004).
[CrossRef]

Rigneault, H.

A. Sergé, N. Bertaux, H. Rigneault, and D. Marguet, “Dynamic multiple-target tracing probes spatiotemporal cartography of cell membranes,” Nat. Methods 5, 687-694 (2008).
[CrossRef] [PubMed]

Saleh, B.

D. Stoler, B. Saleh, and M. Teich, “Binomial states of the quantized radiation field,” J. Mod. Opt. 32, 345-355 (1985).

Saleh, B. E. A.

Sergé, A.

A. Sergé, N. Bertaux, H. Rigneault, and D. Marguet, “Dynamic multiple-target tracing probes spatiotemporal cartography of cell membranes,” Nat. Methods 5, 687-694 (2008).
[CrossRef] [PubMed]

Stoler, D.

D. Stoler, B. Saleh, and M. Teich, “Binomial states of the quantized radiation field,” J. Mod. Opt. 32, 345-355 (1985).

Teich, M.

D. Stoler, B. Saleh, and M. Teich, “Binomial states of the quantized radiation field,” J. Mod. Opt. 32, 345-355 (1985).

Teich, M. C.

Therrien, C. W.

C. W. Therrien, Decision, Estimation, and Classification (Wiley, 1989).

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 1991).
[CrossRef]

Treps, N.

J. Fade, N. Treps, C. Fabre, and P. Réfrégier, “Optimal precision of parameter estimation in images with local sub-Poissonian quantum fluctuations,” Eur. Phys. J. D 50, 215-227 (2008).
[CrossRef]

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” Europhys. Lett. 81, 44001 (2008).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940-943 (2003).
[CrossRef] [PubMed]

N. Treps, U. Andersen, B. Buchler, P. K. Lam, H.-A. Bachor, A. Maître, and C. Fabre, “Crossing the standard quantum limit for high sensitivity measurements in optical images using non classical light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

Van de Nes, A. S.

A. S. Van de Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323-2363 (2006).
[CrossRef]

van Hulst, N. F.

C. A. J. Putman, B. G. de Grooth, N. F. van Hulst, and J. Greve, “A detailed analysis of the optical beam deflection technique for use in atomic force microscopy,” J. Appl. Phys. 72, 6-12 (1992).
[CrossRef]

Astrophys. J. (1)

T. M. Brown, “Expected detection and false alarm rates for transiting jovian planets,” Astrophys. J. 593, L125-L128 (2003).
[CrossRef]

Eur. Phys. J. D (1)

J. Fade, N. Treps, C. Fabre, and P. Réfrégier, “Optimal precision of parameter estimation in images with local sub-Poissonian quantum fluctuations,” Eur. Phys. J. D 50, 215-227 (2008).
[CrossRef]

Europhys. Lett. (1)

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” Europhys. Lett. 81, 44001 (2008).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. T. L. Hsu, V. Delaubert, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, “A quantum study of multibit phase coding for optical storage,” IEEE J. Quantum Electron. 42, 1001-1007 (2006).
[CrossRef]

IEEE Trans. Commun. (1)

T. Kailath, “The divergence and Bhattacharyya distance measures in signal selection,” IEEE Trans. Commun. 15, 52-60 (1967).
[CrossRef]

IEEE Trans. Image Process. (1)

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Figures (5)

Fig. 1
Fig. 1

ROC curves obtained with various sub-Poissonian binomial photocounts with Fano factors F 1 { 0.01 , 0.1 , 0.5 , 0.9 } , with N 0 { 20 , 2.10 2 , 2.10 3 , 2.10 4 } , and with a value of the transmission coefficient τ such that the Chernoff measure between the two hypotheses is C * = 0.393 (◻ symbols) or C * = 1.2 (○ symbols). In star (∗) symbols (resp. cross (×) symbols), the ROC curves are obtained with Poisson photocounts with mean values m 1 { 10 , 10 2 , 10 4 } and parameter τ such that C * = 0.393 (resp. C * = 1.2 ). Each ROC curve is numerically evaluated from R = 10 6 outcomes of the discrimination task.

Fig. 2
Fig. 2

Comparison of the ROC curves obtained for intermediate contrast ( τ = 0.5 ) with sub-Poissonian photocounts with mean value m 1 = 60 and Fano factor F 1 = 0.5 (◆ symbols) and with Poisson photocounts of higher intensity m 1 F 1 (○ symbols), m 1 F 2 (◻ symbols), m 1 F ¯ (∗ symbols), m 1 F R (▼ symbols), m 1 F C (● symbols), or m 1 F B (△ symbols). The dotted curves are only guides for the eyes. Each ROC curve has been numerically evaluated from R = 10 6 outcomes of the detection task.

Fig. 3
Fig. 3

ROC curves plotted with ◆ symbols are obtained with binomial sub-Poissonian photocounts of Fano factors F 1 = 0.5 and F 1 = 0.1 and with m 1 = 10 3 . The ROC curve obtained with Poisson photocounts of mean value m 1 is plotted with the dashed curve while the continuous curves correspond to Poisson photocounts of mean value m 1 F 1 for F 1 = 0.5 and 0.1. Each ROC curve has been numerically evaluated from R = 10 4 outcomes of the detection task.

Fig. 4
Fig. 4

Contour plot of the ratio of the average Fano factor F ¯ to the effective Fano factor F C derived from the Chernoff measure as a function of F 1 and τ.

Fig. 5
Fig. 5

ROC curves obtained with sub-Poissonian binomial photocounts are plotted with ◆ symbols for F 1 = 0.1 or F 1 = 0.5 . The ROC curves obtained with Poisson photocounts of mean value m 1 F C (resp. m 1 F ¯ ) are plotted with ○ symbols (resp. ∗ symbols). The dotted curves are only guides for the eyes. Each ROC curve has been numerically evaluated from R = 10 6 outcomes of the detection task

Equations (18)

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P N ( n ) = e m p [ m p ] n n ! ,
F = N 2 N 2 N = var ( N ) m p ,
N B ( N 0 , η ) P N ( n ) = ( N 0 n ) η n ( 1 η ) N 0 n .
P N ( n H 2 ) P N ( n H 1 ) H 1 H 2 λ ,
C ( s ) = ln { n = 0 + [ P N ( n H 1 ) ] s [ P N ( n H 2 ) ] 1 s } ,
B = C ( 1 2 ) = ln { n = 0 + P N ( n H 1 ) P N ( n H 2 ) } ,
F B ( F 1 , τ ) = F 1 1 2 ( 1 τ ) 2 ln [ ( 1 F 1 ) τ + F 1 α ( F 1 , τ ) ] ,
R ( m 1 m 2 ) 2 var ( N 1 ) + var ( N 2 ) = m 1 F 1 ( 1 τ ) 2 1 + α ( F 1 , τ ) τ .
F R ( F 1 , τ ) = F 1 1 + α ( F 1 , τ ) τ 1 + τ ,
τ K % lim = 1 2 F 1 1 F 1 K % .
lim τ 0 F C = F 1 1 ln ( F 1 ) .
C ( s ) = ln { n = 0 N 0 ( N 0 n ) ( η 1 s η 2 1 s ) n ( F 1 s F 2 1 s ) N 0 n } = N 0 ln { η 1 [ τ ] 1 s + F 1 [ α ( F 1 , τ ) ] 1 s } .
s * = 1 1 ln z ln { F 1 1 F 1 ln [ τ ] ln [ α ( F 1 , τ ) ] } .
C * = C ( s * ) = m 1 F 1 1 ln [ ( 1 F 1 ) τ 1 s * + F 1 α ( F 1 , τ ) 1 s * ] ,
B = C ( 1 2 ) = m 1 F 1 1 ln [ ( 1 F 1 ) τ + F 1 α ( F 1 , τ ) ] .
C p ( s ) = ln { n = 0 [ e m p m p n n ! ] s [ e τ m p ( τ m p ) n n ! ] 1 s } = s m p + τ m p ( 1 s ) m p e ( 1 s ) ln τ .
C p * = C p ( s p * ) = m p { 1 τ 1 ln τ [ 1 ln ( τ 1 ln τ ) ] } .
F B ( F 1 , τ ) = F ¯ = F 1 ( 1 + 1 F 1 2 F 1 ε ) .

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