Abstract

The accuracy of optical measurements at low light levels is limited by the quantum noise of the source and by the random nature of the interaction with the measured object. The source noise may be reduced by use of nonclassical photon-number squeezed light. We consider the use of two photon-correlated beams (generated, for example, by spontaneous parametric downconversion) to measure the optical transmittance of an object. The photons of each beam obey a random Poisson process but are synchronized in time. One beam is used to probe the object, and the other is used as a reference providing information on the realization of the random arrival of photons at the object. The additional information available by such measurement may be exploited to improve the accuracy of the measurement. Various estimators, including the maximum-likelihood estimator, are considered, and their performance is evaluated and compared with the measurement based on a single-beam conventional (Poissonian) source and a maximally squeezed (fixed-photon-number) source. The performance advantage that is established depends on parameters such as the intensity of the source, the transmittance of the object, the quantum efficiency of the detectors, the background noise, and the degree of correlation of the photon numbers in the two beams.

© 1999 Optical Society of America

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References

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  1. B. E. A. Saleh, Photoelectron Statistics (Springer, Berlin, 1978).
  2. M. Rabbani, “Bayesian filtering of Poisson noise using local statistics,” IEEE Trans. Acoust., Speech, Signal Process. 36, 933–937 (1988).
    [CrossRef]
  3. R. E. Sequeira, J. A. Gubner, B. E. A. Saleh, “Quantum-limited image detection,” IEEE Trans. Image Process. 2, 18–26 (1993).
    [CrossRef]
  4. B. E. A. Saleh, “Quantum noise in optical processing,” in Real-Time Optical Processing, B. Javidi, J. Horner, eds. (Academic, New York, 1994), pp. 407–437.
  5. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995). Sect. 22.4.
  6. M. C. Teich, B. E. A. Saleh, “Photon bunching and antibunching,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), pp. 1–104.
  7. B. E. A. Saleh, M. C. Teich, “Information transmission with photon-number-squeezed light,” Proc. IEEE 80, 451–460 (1992).
    [CrossRef]
  8. S.-H. Youn, J.-H. Lee, J.-S. Chang, “Quantum-mechanical noise characteristics in doubly resonant optical parametric oscillator,” J. Opt. Soc. Am. B 11, 2282–2286 (1994).
    [CrossRef]
  9. B. R. Mollow, “Photon correlations in the parametric frequency splitting of light,” Phys. Rev. A 8, 2684–2694 (1973).
    [CrossRef]
  10. C. K. Hong, L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
    [CrossRef] [PubMed]
  11. N. Klyshko, Photons and Nonlinear Optics (Gordon & Breach, New York, 1988).
  12. A. J. Joobeur, B. E. A. Saleh, M. C. Teich, “Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion,” Phys. Rev. A 50, 3349–3361 (1994).
    [CrossRef] [PubMed]
  13. A. J. Joobeur, B. E. A. Saleh, T. S. Larchuk, M. C. Teich, “Coherence properties of entangled light beams generated by parametric down-conversion: theory and experiment,” Phys. Rev. A 53, 4360–4371 (1996).
    [CrossRef] [PubMed]
  14. P. R. Tapster, J. G. Rarity, J. S. Satchell, “Use of parametric down-conversion to generate sub-Poisson light,” Phys. Rev. A 37, 2963–2967 (1988).
    [CrossRef] [PubMed]
  15. J. G. Rarity, P. R. Tapster, E. Jakeman, “Observation of sub-Poisson light in parametric downconversion,” Opt. Commun. 62, 201–206 (1987).
    [CrossRef]
  16. J. G. Rarity, P. R. Tapster, J. A. Levenson, J. C. Farreau, I. Abram, J. Mertz, T. Debuisschert, A. Heidman, C. Fabre, E. Giacobino, “Quantum correlated twin beams,” Appl. Phys. B: Photophys. Laser Chem. 55, 250–257 (1992).
    [CrossRef]
  17. E. A. Perkins, R. J. Carr, J. G. Rarity, “A twin-beam fibre laser light scattering system,” Meas. Sci. Technol. 4, 215–220 (1993).
    [CrossRef]
  18. C. K. Hong, S. R. Friberg, L. Mandel, “Optical communication channel based on coincident photon pairs,” Appl. Opt. 24, 3877–3882 (1985).
    [CrossRef] [PubMed]
  19. L. Mandel, “Proposal for almost noise-free optical communication under conditions of high background,” J. Opt. Soc. Am. B 1, 108–110 (1984).
    [CrossRef]
  20. E. Jakeman, J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmission measurement,” Opt. Commun. 59, 219–223 (1986).
    [CrossRef]
  21. H. V. Poor, Introduction to Signal Detection and Estimation (Springer-Verlag, New York, 1988).
  22. L. Mandel, “Sub-Poissonian photon statistics in resonance fluorescence,” Opt. Lett. 4, 205–207 (1979).
    [CrossRef] [PubMed]
  23. H. Stark, J. W. Woods, Probability, Random Processes, and Estimation Theory for Engineers (Prentice-Hall, Upper Saddle River, N.J., 1994).
  24. B. E. A. Saleh, “Quantum imaging,” invited paper presented at the 1997 OSA Annual Meeting, Long Beach, Calif., 1997.
  25. B. Huttner, J. J. Baumberg, J. F. Ryan, “Detection of short pulses of non-classical light,” Opt. Commun. 90, 128–132 (1992).
    [CrossRef]
  26. J. K. Breslin, G. J. Milburn, “Conditional variance reduction by measurements on correlated field modes,” Phys. Rev. A 55, 1430–1436 (1997).
    [CrossRef]

1997 (1)

J. K. Breslin, G. J. Milburn, “Conditional variance reduction by measurements on correlated field modes,” Phys. Rev. A 55, 1430–1436 (1997).
[CrossRef]

1996 (1)

A. J. Joobeur, B. E. A. Saleh, T. S. Larchuk, M. C. Teich, “Coherence properties of entangled light beams generated by parametric down-conversion: theory and experiment,” Phys. Rev. A 53, 4360–4371 (1996).
[CrossRef] [PubMed]

1994 (2)

S.-H. Youn, J.-H. Lee, J.-S. Chang, “Quantum-mechanical noise characteristics in doubly resonant optical parametric oscillator,” J. Opt. Soc. Am. B 11, 2282–2286 (1994).
[CrossRef]

A. J. Joobeur, B. E. A. Saleh, M. C. Teich, “Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion,” Phys. Rev. A 50, 3349–3361 (1994).
[CrossRef] [PubMed]

1993 (2)

E. A. Perkins, R. J. Carr, J. G. Rarity, “A twin-beam fibre laser light scattering system,” Meas. Sci. Technol. 4, 215–220 (1993).
[CrossRef]

R. E. Sequeira, J. A. Gubner, B. E. A. Saleh, “Quantum-limited image detection,” IEEE Trans. Image Process. 2, 18–26 (1993).
[CrossRef]

1992 (3)

B. E. A. Saleh, M. C. Teich, “Information transmission with photon-number-squeezed light,” Proc. IEEE 80, 451–460 (1992).
[CrossRef]

J. G. Rarity, P. R. Tapster, J. A. Levenson, J. C. Farreau, I. Abram, J. Mertz, T. Debuisschert, A. Heidman, C. Fabre, E. Giacobino, “Quantum correlated twin beams,” Appl. Phys. B: Photophys. Laser Chem. 55, 250–257 (1992).
[CrossRef]

B. Huttner, J. J. Baumberg, J. F. Ryan, “Detection of short pulses of non-classical light,” Opt. Commun. 90, 128–132 (1992).
[CrossRef]

1988 (2)

P. R. Tapster, J. G. Rarity, J. S. Satchell, “Use of parametric down-conversion to generate sub-Poisson light,” Phys. Rev. A 37, 2963–2967 (1988).
[CrossRef] [PubMed]

M. Rabbani, “Bayesian filtering of Poisson noise using local statistics,” IEEE Trans. Acoust., Speech, Signal Process. 36, 933–937 (1988).
[CrossRef]

1987 (1)

J. G. Rarity, P. R. Tapster, E. Jakeman, “Observation of sub-Poisson light in parametric downconversion,” Opt. Commun. 62, 201–206 (1987).
[CrossRef]

1986 (1)

E. Jakeman, J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmission measurement,” Opt. Commun. 59, 219–223 (1986).
[CrossRef]

1985 (2)

C. K. Hong, S. R. Friberg, L. Mandel, “Optical communication channel based on coincident photon pairs,” Appl. Opt. 24, 3877–3882 (1985).
[CrossRef] [PubMed]

C. K. Hong, L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
[CrossRef] [PubMed]

1984 (1)

1979 (1)

1973 (1)

B. R. Mollow, “Photon correlations in the parametric frequency splitting of light,” Phys. Rev. A 8, 2684–2694 (1973).
[CrossRef]

Abram, I.

J. G. Rarity, P. R. Tapster, J. A. Levenson, J. C. Farreau, I. Abram, J. Mertz, T. Debuisschert, A. Heidman, C. Fabre, E. Giacobino, “Quantum correlated twin beams,” Appl. Phys. B: Photophys. Laser Chem. 55, 250–257 (1992).
[CrossRef]

Baumberg, J. J.

B. Huttner, J. J. Baumberg, J. F. Ryan, “Detection of short pulses of non-classical light,” Opt. Commun. 90, 128–132 (1992).
[CrossRef]

Breslin, J. K.

J. K. Breslin, G. J. Milburn, “Conditional variance reduction by measurements on correlated field modes,” Phys. Rev. A 55, 1430–1436 (1997).
[CrossRef]

Carr, R. J.

E. A. Perkins, R. J. Carr, J. G. Rarity, “A twin-beam fibre laser light scattering system,” Meas. Sci. Technol. 4, 215–220 (1993).
[CrossRef]

Chang, J.-S.

Debuisschert, T.

J. G. Rarity, P. R. Tapster, J. A. Levenson, J. C. Farreau, I. Abram, J. Mertz, T. Debuisschert, A. Heidman, C. Fabre, E. Giacobino, “Quantum correlated twin beams,” Appl. Phys. B: Photophys. Laser Chem. 55, 250–257 (1992).
[CrossRef]

Fabre, C.

J. G. Rarity, P. R. Tapster, J. A. Levenson, J. C. Farreau, I. Abram, J. Mertz, T. Debuisschert, A. Heidman, C. Fabre, E. Giacobino, “Quantum correlated twin beams,” Appl. Phys. B: Photophys. Laser Chem. 55, 250–257 (1992).
[CrossRef]

Farreau, J. C.

J. G. Rarity, P. R. Tapster, J. A. Levenson, J. C. Farreau, I. Abram, J. Mertz, T. Debuisschert, A. Heidman, C. Fabre, E. Giacobino, “Quantum correlated twin beams,” Appl. Phys. B: Photophys. Laser Chem. 55, 250–257 (1992).
[CrossRef]

Friberg, S. R.

Giacobino, E.

J. G. Rarity, P. R. Tapster, J. A. Levenson, J. C. Farreau, I. Abram, J. Mertz, T. Debuisschert, A. Heidman, C. Fabre, E. Giacobino, “Quantum correlated twin beams,” Appl. Phys. B: Photophys. Laser Chem. 55, 250–257 (1992).
[CrossRef]

Gubner, J. A.

R. E. Sequeira, J. A. Gubner, B. E. A. Saleh, “Quantum-limited image detection,” IEEE Trans. Image Process. 2, 18–26 (1993).
[CrossRef]

Heidman, A.

J. G. Rarity, P. R. Tapster, J. A. Levenson, J. C. Farreau, I. Abram, J. Mertz, T. Debuisschert, A. Heidman, C. Fabre, E. Giacobino, “Quantum correlated twin beams,” Appl. Phys. B: Photophys. Laser Chem. 55, 250–257 (1992).
[CrossRef]

Hong, C. K.

C. K. Hong, L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
[CrossRef] [PubMed]

C. K. Hong, S. R. Friberg, L. Mandel, “Optical communication channel based on coincident photon pairs,” Appl. Opt. 24, 3877–3882 (1985).
[CrossRef] [PubMed]

Huttner, B.

B. Huttner, J. J. Baumberg, J. F. Ryan, “Detection of short pulses of non-classical light,” Opt. Commun. 90, 128–132 (1992).
[CrossRef]

Jakeman, E.

J. G. Rarity, P. R. Tapster, E. Jakeman, “Observation of sub-Poisson light in parametric downconversion,” Opt. Commun. 62, 201–206 (1987).
[CrossRef]

E. Jakeman, J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmission measurement,” Opt. Commun. 59, 219–223 (1986).
[CrossRef]

Joobeur, A. J.

A. J. Joobeur, B. E. A. Saleh, T. S. Larchuk, M. C. Teich, “Coherence properties of entangled light beams generated by parametric down-conversion: theory and experiment,” Phys. Rev. A 53, 4360–4371 (1996).
[CrossRef] [PubMed]

A. J. Joobeur, B. E. A. Saleh, M. C. Teich, “Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion,” Phys. Rev. A 50, 3349–3361 (1994).
[CrossRef] [PubMed]

Klyshko, N.

N. Klyshko, Photons and Nonlinear Optics (Gordon & Breach, New York, 1988).

Larchuk, T. S.

A. J. Joobeur, B. E. A. Saleh, T. S. Larchuk, M. C. Teich, “Coherence properties of entangled light beams generated by parametric down-conversion: theory and experiment,” Phys. Rev. A 53, 4360–4371 (1996).
[CrossRef] [PubMed]

Lee, J.-H.

Levenson, J. A.

J. G. Rarity, P. R. Tapster, J. A. Levenson, J. C. Farreau, I. Abram, J. Mertz, T. Debuisschert, A. Heidman, C. Fabre, E. Giacobino, “Quantum correlated twin beams,” Appl. Phys. B: Photophys. Laser Chem. 55, 250–257 (1992).
[CrossRef]

Mandel, L.

Mertz, J.

J. G. Rarity, P. R. Tapster, J. A. Levenson, J. C. Farreau, I. Abram, J. Mertz, T. Debuisschert, A. Heidman, C. Fabre, E. Giacobino, “Quantum correlated twin beams,” Appl. Phys. B: Photophys. Laser Chem. 55, 250–257 (1992).
[CrossRef]

Milburn, G. J.

J. K. Breslin, G. J. Milburn, “Conditional variance reduction by measurements on correlated field modes,” Phys. Rev. A 55, 1430–1436 (1997).
[CrossRef]

Mollow, B. R.

B. R. Mollow, “Photon correlations in the parametric frequency splitting of light,” Phys. Rev. A 8, 2684–2694 (1973).
[CrossRef]

Perkins, E. A.

E. A. Perkins, R. J. Carr, J. G. Rarity, “A twin-beam fibre laser light scattering system,” Meas. Sci. Technol. 4, 215–220 (1993).
[CrossRef]

Poor, H. V.

H. V. Poor, Introduction to Signal Detection and Estimation (Springer-Verlag, New York, 1988).

Rabbani, M.

M. Rabbani, “Bayesian filtering of Poisson noise using local statistics,” IEEE Trans. Acoust., Speech, Signal Process. 36, 933–937 (1988).
[CrossRef]

Rarity, J. G.

E. A. Perkins, R. J. Carr, J. G. Rarity, “A twin-beam fibre laser light scattering system,” Meas. Sci. Technol. 4, 215–220 (1993).
[CrossRef]

J. G. Rarity, P. R. Tapster, J. A. Levenson, J. C. Farreau, I. Abram, J. Mertz, T. Debuisschert, A. Heidman, C. Fabre, E. Giacobino, “Quantum correlated twin beams,” Appl. Phys. B: Photophys. Laser Chem. 55, 250–257 (1992).
[CrossRef]

P. R. Tapster, J. G. Rarity, J. S. Satchell, “Use of parametric down-conversion to generate sub-Poisson light,” Phys. Rev. A 37, 2963–2967 (1988).
[CrossRef] [PubMed]

J. G. Rarity, P. R. Tapster, E. Jakeman, “Observation of sub-Poisson light in parametric downconversion,” Opt. Commun. 62, 201–206 (1987).
[CrossRef]

E. Jakeman, J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmission measurement,” Opt. Commun. 59, 219–223 (1986).
[CrossRef]

Ryan, J. F.

B. Huttner, J. J. Baumberg, J. F. Ryan, “Detection of short pulses of non-classical light,” Opt. Commun. 90, 128–132 (1992).
[CrossRef]

Saleh, B. E. A.

A. J. Joobeur, B. E. A. Saleh, T. S. Larchuk, M. C. Teich, “Coherence properties of entangled light beams generated by parametric down-conversion: theory and experiment,” Phys. Rev. A 53, 4360–4371 (1996).
[CrossRef] [PubMed]

A. J. Joobeur, B. E. A. Saleh, M. C. Teich, “Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion,” Phys. Rev. A 50, 3349–3361 (1994).
[CrossRef] [PubMed]

R. E. Sequeira, J. A. Gubner, B. E. A. Saleh, “Quantum-limited image detection,” IEEE Trans. Image Process. 2, 18–26 (1993).
[CrossRef]

B. E. A. Saleh, M. C. Teich, “Information transmission with photon-number-squeezed light,” Proc. IEEE 80, 451–460 (1992).
[CrossRef]

M. C. Teich, B. E. A. Saleh, “Photon bunching and antibunching,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), pp. 1–104.

B. E. A. Saleh, “Quantum imaging,” invited paper presented at the 1997 OSA Annual Meeting, Long Beach, Calif., 1997.

B. E. A. Saleh, “Quantum noise in optical processing,” in Real-Time Optical Processing, B. Javidi, J. Horner, eds. (Academic, New York, 1994), pp. 407–437.

B. E. A. Saleh, Photoelectron Statistics (Springer, Berlin, 1978).

Satchell, J. S.

P. R. Tapster, J. G. Rarity, J. S. Satchell, “Use of parametric down-conversion to generate sub-Poisson light,” Phys. Rev. A 37, 2963–2967 (1988).
[CrossRef] [PubMed]

Sequeira, R. E.

R. E. Sequeira, J. A. Gubner, B. E. A. Saleh, “Quantum-limited image detection,” IEEE Trans. Image Process. 2, 18–26 (1993).
[CrossRef]

Stark, H.

H. Stark, J. W. Woods, Probability, Random Processes, and Estimation Theory for Engineers (Prentice-Hall, Upper Saddle River, N.J., 1994).

Tapster, P. R.

J. G. Rarity, P. R. Tapster, J. A. Levenson, J. C. Farreau, I. Abram, J. Mertz, T. Debuisschert, A. Heidman, C. Fabre, E. Giacobino, “Quantum correlated twin beams,” Appl. Phys. B: Photophys. Laser Chem. 55, 250–257 (1992).
[CrossRef]

P. R. Tapster, J. G. Rarity, J. S. Satchell, “Use of parametric down-conversion to generate sub-Poisson light,” Phys. Rev. A 37, 2963–2967 (1988).
[CrossRef] [PubMed]

J. G. Rarity, P. R. Tapster, E. Jakeman, “Observation of sub-Poisson light in parametric downconversion,” Opt. Commun. 62, 201–206 (1987).
[CrossRef]

Teich, M. C.

A. J. Joobeur, B. E. A. Saleh, T. S. Larchuk, M. C. Teich, “Coherence properties of entangled light beams generated by parametric down-conversion: theory and experiment,” Phys. Rev. A 53, 4360–4371 (1996).
[CrossRef] [PubMed]

A. J. Joobeur, B. E. A. Saleh, M. C. Teich, “Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion,” Phys. Rev. A 50, 3349–3361 (1994).
[CrossRef] [PubMed]

B. E. A. Saleh, M. C. Teich, “Information transmission with photon-number-squeezed light,” Proc. IEEE 80, 451–460 (1992).
[CrossRef]

M. C. Teich, B. E. A. Saleh, “Photon bunching and antibunching,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), pp. 1–104.

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995). Sect. 22.4.

Woods, J. W.

H. Stark, J. W. Woods, Probability, Random Processes, and Estimation Theory for Engineers (Prentice-Hall, Upper Saddle River, N.J., 1994).

Youn, S.-H.

Appl. Opt. (1)

Appl. Phys. B: Photophys. Laser Chem. (1)

J. G. Rarity, P. R. Tapster, J. A. Levenson, J. C. Farreau, I. Abram, J. Mertz, T. Debuisschert, A. Heidman, C. Fabre, E. Giacobino, “Quantum correlated twin beams,” Appl. Phys. B: Photophys. Laser Chem. 55, 250–257 (1992).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process. (1)

M. Rabbani, “Bayesian filtering of Poisson noise using local statistics,” IEEE Trans. Acoust., Speech, Signal Process. 36, 933–937 (1988).
[CrossRef]

IEEE Trans. Image Process. (1)

R. E. Sequeira, J. A. Gubner, B. E. A. Saleh, “Quantum-limited image detection,” IEEE Trans. Image Process. 2, 18–26 (1993).
[CrossRef]

J. Opt. Soc. Am. B (2)

Meas. Sci. Technol. (1)

E. A. Perkins, R. J. Carr, J. G. Rarity, “A twin-beam fibre laser light scattering system,” Meas. Sci. Technol. 4, 215–220 (1993).
[CrossRef]

Opt. Commun. (3)

B. Huttner, J. J. Baumberg, J. F. Ryan, “Detection of short pulses of non-classical light,” Opt. Commun. 90, 128–132 (1992).
[CrossRef]

J. G. Rarity, P. R. Tapster, E. Jakeman, “Observation of sub-Poisson light in parametric downconversion,” Opt. Commun. 62, 201–206 (1987).
[CrossRef]

E. Jakeman, J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmission measurement,” Opt. Commun. 59, 219–223 (1986).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (6)

J. K. Breslin, G. J. Milburn, “Conditional variance reduction by measurements on correlated field modes,” Phys. Rev. A 55, 1430–1436 (1997).
[CrossRef]

A. J. Joobeur, B. E. A. Saleh, M. C. Teich, “Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion,” Phys. Rev. A 50, 3349–3361 (1994).
[CrossRef] [PubMed]

A. J. Joobeur, B. E. A. Saleh, T. S. Larchuk, M. C. Teich, “Coherence properties of entangled light beams generated by parametric down-conversion: theory and experiment,” Phys. Rev. A 53, 4360–4371 (1996).
[CrossRef] [PubMed]

P. R. Tapster, J. G. Rarity, J. S. Satchell, “Use of parametric down-conversion to generate sub-Poisson light,” Phys. Rev. A 37, 2963–2967 (1988).
[CrossRef] [PubMed]

B. R. Mollow, “Photon correlations in the parametric frequency splitting of light,” Phys. Rev. A 8, 2684–2694 (1973).
[CrossRef]

C. K. Hong, L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
[CrossRef] [PubMed]

Proc. IEEE (1)

B. E. A. Saleh, M. C. Teich, “Information transmission with photon-number-squeezed light,” Proc. IEEE 80, 451–460 (1992).
[CrossRef]

Other (8)

B. E. A. Saleh, Photoelectron Statistics (Springer, Berlin, 1978).

H. V. Poor, Introduction to Signal Detection and Estimation (Springer-Verlag, New York, 1988).

N. Klyshko, Photons and Nonlinear Optics (Gordon & Breach, New York, 1988).

B. E. A. Saleh, “Quantum noise in optical processing,” in Real-Time Optical Processing, B. Javidi, J. Horner, eds. (Academic, New York, 1994), pp. 407–437.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995). Sect. 22.4.

M. C. Teich, B. E. A. Saleh, “Photon bunching and antibunching,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), pp. 1–104.

H. Stark, J. W. Woods, Probability, Random Processes, and Estimation Theory for Engineers (Prentice-Hall, Upper Saddle River, N.J., 1994).

B. E. A. Saleh, “Quantum imaging,” invited paper presented at the 1997 OSA Annual Meeting, Long Beach, Calif., 1997.

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of single-beam measurement. A probe beam with photon flux λ is transmitted through an object with transmittance t, and the photon count N (in an interval T) is measured by using a detector (with quantum efficiency η) and a counter. The background-noise photon flux is μ. The measurement N is the sum of the detected probe beam photons Nds and the detected background photons Nn. A single-beam estimator uses the photon count N to generate an estimate tˆ of t.

Fig. 2
Fig. 2

Schematic diagram of measurement with photon-correlated beams. The signal beam (with photon flux λ) is used as a probe and transmitted through the object (with transmittance t), and the idler beam (also with photon flux λ) is used as a reference. The observed output Nss of the signal-channel counter is the sum of the detected photons Nds, in the duration T, resulting from the transmitted signal beam and the detected background photons Nns. The observed output Nii of the idler-channel counter is the sum of the detected photons Ndi resulting from the idler beam and the detected background photons Nns. The quantum efficiencies of the detectors in the signal and idler channels are ηs and ηi, respectively. A photon-correlated estimator uses the observations Nss and Nii to generate an estimate tˆ of t.

Fig. 3
Fig. 3

(a) Estimation error ϵ as a function of the transmittance parameter t for the various estimators: Poissonian single-beam ML estimator tˆP,ML (top solid curve), fixed-photon-number ML estimator tˆF,ML (bottom solid curve), photon-correlated-beams ML estimator tˆC,ML (dashed curve), count-ratio estimator tˆC,R (represented by the symbol × and overlaying the dashed curve), and count-difference estimator tˆC,D (dotted–dashed curve). The estimation parameters are n=10, ηs=ηi=1.0, μs=μi=0, and β=0. Note the complete overlap between the count-ratio estimator curve and the photon-correlated-beams ML estimator curve. (b) Improvement factor ρ as a function of t. The curve symbols are same as in (a).

Fig. 4
Fig. 4

(a) Estimation error ϵ as a function of the mean number of signal photons n for the various estimators: Poissonian single-beam ML estimator tˆP,ML (top solid curve), fixed-photon-number ML estimator tˆF,ML (bottom solid curve), photon-correlated-beams ML estimator tˆC,ML (dashed curve just above the bottom solid curve), count-ratio estimator tˆC,R (dotted curve just above the bottom solid curve), and count-difference estimator tˆC,D (dotted–dashed curve). The estimation parameters are t=0.5, ηs=ηi=1.0, μs=μi=0, and β=0. Note the complete overlap between the count-ratio estimator curve and the photon-correlated-beams ML estimator curve. (b) Improvement factor ρ as a function of n.

Fig. 5
Fig. 5

(a) Estimation error ϵ as a function of the transmittance parameter t for the various estimators: Poissonian single-beam ML estimator tˆP,ML (top solid curve), fixed-photon-number ML estimator tˆF,ML (bottom solid curve), photon-correlated-beams ML estimator tˆC,ML (dashed curve), count-ratio estimator tˆC,R (dotted curve), and count-difference estimator tˆC,D (dotted–dashed curve). The estimation parameters are n=20, ηs=ηi=0.7, μs=μi=5, and β=0. (b) Improvement factor ρ as a function of t.

Fig. 6
Fig. 6

(a) Estimation error ϵ as a function of the mean number of signal photons n for the various estimators: Poissonian single-beam ML estimator tˆP,ML (top solid curve), fixed-photon-number ML estimator tˆF,ML (bottom solid curve), photon-correlated-beams ML estimator tˆC,ML (dashed curve), count-ratio estimator tˆC,R (dotted curve), and count-difference estimator tˆC,D (dotted–dashed curve). The estimation parameters are T=1, t=0.8, ηs=ηi=0.7, μs=μi=5, and β=0. (b) Improvement factor ρ as a function of n.

Fig. 7
Fig. 7

(a) Estimation error ϵ as a function of the transmittance parameter t for the various estimators: Poissonian single-beam ML estimator tˆP,ML (top solid curve), fixed-photon-number ML estimator tˆF,ML (bottom solid curve), photon-correlated-beams ML estimator tˆC,ML (dashed curve), count-ratio estimator tˆC,R (dotted curve), and count-difference estimator tˆC,D (dotted–dashed curve). The estimation parameters are T=1, n=20, ηs=ηi=0.7, μs=μi=5, and β=0.2. (b) Improvement factor ρ as a function of t.

Fig. 8
Fig. 8

(a) Estimation error ϵ as a function of the mean number of signal photons n for the various estimators: Poissonian single-beam ML estimator tˆP,ML (top solid curve), fixed-photon-number ML estimator tˆF,ML (bottom solid curve), photon-correlated-beams ML estimator tˆC,ML (dashed curve), count-ratio estimator tˆC,R (dotted curve), and count-difference estimator tˆC,D (dotted–dashed curve). The estimation parameters are T=1, t=0.8, ηs=ηi=0.7, μs=μi=5, and β=0.2. (b) Improvement factor ρ as a function of n.

Equations (47)

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ϵ2=E[(tˆ-t)2]=k=0[tˆ(k)-t]2PN(k).
PNs(k)=δ(k-n),
Q(0)=E[(Ns-N¯s)2]N¯s-1,
PNds(k)=B(n, ηt; k),
B(n, ηt; k)=nk(ηt)k(1-ηt)n-k
PN(k)=i=0min(n, k)B(n, ηt, i)P(ημT; k-i).
ddtPN(k)=i=0min(n, k)it-η(n-i)1-tη×B(n, tη, i)P(ημT; k-i)=0.
PN(k)=P(ηtλT+ημT; k).
tˆP,ML=0if NηλTμλ1if NηλT1+μλNηλT-μλotherwise.
ϵP,ML2=1ηλTt+μλ.
PNt(k)=P((1-β)λT; k),
PNus(k)=PNui(k)=P(βλT; k).
PNssNii|Nt(k, l|n)
=i=0min(n, k)B(n, tηs; i)P(βλtηsT+ηsμsT; k-i)×j=0min(n, l)B(n, ηi; j)P(βληiT+ηiμiT; l-j).
PNssNii(k, l)
=n=0P(βλT; n)i=0min(n, k)B(n, tηs; i)×P(βλtηsT+ηsμsT; k-i)j=0min(n, l)B(n, ηi; j)×P(βληiT+ηiμiT; l-j).
ϵtˆ2=k=0l=0[tˆ(k, l)-t]2PNssNii(k, l).
Qc(n)=E[(Nss-N¯ss)2|Nt=n]N¯ss-1,
Qc(n)=βλTn+βλT-1.
n=0P(λT; n)i=0min(n, k)it-ηs(n-i)1-tηs-βληsT
+βληsT(k-i)μsηsT+βλtμsT×B(n, tη; i)P(βλtηT+ημT; k-i)
×j=0min(n, l)B(n, η; j)P(βληT+ημT; l-j)=0.
tˆC,R(Nss, Nii)=1αRNssNii-βRαR,
αR=ηsλT1-exp[-ηi(λ+μi)T](λ+μi)T+(1+β-ηi)E[Nii-1u(Nii)],
βR=ηsμsTE[Nii-1u(Nii)],
t{ηs-1-ηit[1-(1-ηi-1)2]}λ-1+o(λ-1).
tˆC,D(Nss, Nii)=1αD(Nss-Nii)-βDαD,
αD=ηsλT,
βD=ηsμsT-ηiμiT-ηiλT.
ϵC,D2=1η1Tt[1-2η2(1-β)]+η2η1λ-1+o(λ-1),
ρ=ϵϵP,ML.
limn ρtˆC,R2=1-ηs(2-ηi-1)t.
limn ρtˆC,D2=ηiηst+[1-2ηi(1-β)].
ENssNiiu(Nii)|Nii=Nii-1u(Nii)E(Nss|Nii).
E(Nss|Nii)=E[E(Nss|Nii, Nt)|Nii].
E(Nss|Nii, Nt)=E(Nss|Nt),
E(Nss|Nt)=tηsNt+ηs(μs+tβλ)T.
E(Nss|Nii)=E[E(Nss|Nt)|Nii]=ηs(μs+tβλ)T+tηsE(Nt|Nii),
ENssNiiu(Nii)=ηs(μs+tβλ)TE[Nii-1u(Nii)]+tηsE[Nii-1u(Nii)E(Nt|Nii)].
E(Nt|Nii=l)=n=0n P(Nt=n, Nii=l)P(Nii=l).
E(Nt|Nii=l)=n=0ni=0l P(Nt=n, Nii=l, M=i)P(Nii=l)=n=0ni=0l P(Nii=l|Nt=n, M=i)P(Nt=n)P(M=i)P(Nii=l).
P(Nii=l|Nt=n, M=i)
=u(n-l+i)nl-iηil-i(1-ηi)n-l+i.
P(Nt=n)=(βλ)n exp(-βλ)/n!,
P(M=i)={ηiT[(1-β)λ+μi]}i×exp{-ηiT[(1-β)λ+μi]}/i!
E(Nt|Nii=l)=(1-β)λ(1-ηi)T+lλ+μi.
ENssNiiu(Nii)=αRt+βR,

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