Abstract

Cumulative distributions of the number of photoelectrons ejected during a fixed interval can be computed by numerical contour integration in the complex plane when the light incident upon the detector is a combination of coherent light and incoherent background light with arbitrary spectral density. The integrand involves the probability-generating function of the distribution, and a method for computing it in terms of the solution of a certain integral equation is described. The method is related to those for the estimation of a stochastic process in the presence of white noise. An approximation valid for large values of the time–bandwidth product is also described.

© 1985 Optical Society of America

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References

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  1. L. Mandel, “Fluctuations of photon beams: the distribution of the photoelectrons,” Proc. Phys. Soc. London 74, 233–243 (1959).
    [CrossRef]
  2. R. J. Glauber, “Optical coherence and photon statistics,” in Quantum Optics and Electronics, C. DeWitt, A. Blandin, C. Cohen-Tannoudji, eds. (Gordon and Breach, New York, 1965), pp. 65–185; see Lecture XVII, pp. 176–185.
  3. J. Peřina, R. Horák, “On the quantum statistics of the superposition of coherent and chaotic fields,”J. Phys. A 2, 702–712 (1969).
    [CrossRef]
  4. S. Karp, J. R. Clark, “Photon counting: a problem in classical noise theory,”IEEE Trans. Inf. Theory IT-16, 672–680 (1970).
    [CrossRef]
  5. A. K. Jaiswal, C. L. Mehta, “Photon counting statistics of harmonic signal mixed with thermal light. I. Single photoelectron counting,” Phys. Rev. A 2, 168–172 (1970).
    [CrossRef]
  6. G. Lachs, “Approximate photocount statistics for coherent and chaotic radiation of arbitrary spectral shape,” J. Appl. Phys. 42, 602–609 (1971).
    [CrossRef]
  7. R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), pp. 80–83.
  8. B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978), pp. 206–210.
  9. R. Barakat, J. Blake, “Theory of photoelectron counting statistics, an essay,” Phys. Reports 60, 225–340 (1980).
    [CrossRef]
  10. E. Jakeman, E. R. Pike, “Statistics of heterodyne detection of Gaussian light,”J. Phys. A 2, 115–125 (1969).
    [CrossRef]
  11. C. W. Helstrom, “The distribution of photoelectric counts from partially polarized Gaussian light,” Proc. Phys. Soc. London 83, 777–782 (1964).
    [CrossRef]
  12. P. J. Bickel, K. A. Doksum, Mathematical Statistics (Holden-Day, San Francisco, Calif., 1977), Sec. 9.6, pp. 378–389.
  13. A. J. F. Siegert, “A systematic approach to a class of problems in the theory of noise and other random phenomena. Part II, Examples,”IRE Trans. Inf. Theory IT-3, 38–44 (1957).
    [CrossRef]
  14. G. Bédard, “Photon counting statistics of Gaussian light,” Phys. Rev. 151, 1038–1039 (1966).
    [CrossRef]
  15. S. R. Laxpati, G. Lachs, “Closed-form solutions for the photocount statistics of superposed coherent and chaotic radiation,” J. Appl. Phys. 43, 4773–4776 (1972).
    [CrossRef]
  16. C. W. Helstrom, “Comment: Distribution of quadratic forms in normal random variables—evaluation by numerical integration,” SIAM J. Sci. Stat. Comput. 4, 353–356 (1983).
    [CrossRef]
  17. R. W. Hornbeck, Numerical Methods (Quantum, New York, 1975), pp. 69–71.
  18. H. A. Spang, “A review of minimization techniques for nonlinear functions,”SIAM Rev. 4, 343–365 (1962).
    [CrossRef]
  19. G. F. Carrier, M. Krook, C. E. Pearson, Functions of a Complex Variable (McGraw-Hill, New York, 1966), pp. 257ff.
  20. C. W. Helstrom, “Evaluating the detectability of Gaussian stochastic signals by steepest descent integration,”IEEE Trans. Aerosp. Electron. Syst. AES-19, 428–437 (1983).
    [CrossRef]
  21. S. O. Rice, “Efficient evaluation of integrals of analytic functions by the trapezoidal rule,” Bell Sys. Tech. J. 52, 707–722 (1973).
  22. H. E. Daniels, “Saddlepoint approximations in statistics,” Ann. Math. Statist. 25, 631–650 (1954).
    [CrossRef]
  23. C. W. Helstrom, “Approximate evaluation of detection probabilities in radar and optical communications,”IEEE Trans. Aerosp. Electron. Syst. AES-14, 630–640 (1978).
    [CrossRef]
  24. D. Slepian, T. Kadota, “Four integral equations of detection theory,” SIAM J. Appl. Math. 17, 1102–1117 (1969).
    [CrossRef]
  25. A. B. Baggeroer, “A state-variable approach to the solution of Fredholm integral equations,”IEEE Trans. Inf. Theory IT-15, 557–570 (1969).
    [CrossRef]
  26. A. B. Baggeroer, “State variable analysis procedures,” appendix in H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1971), Vol. II, pp. 286–327.
  27. C. W. Helstrom, Statistical Theory of Signal Detection, 2nd ed. (Pergamon, London, 1968).
  28. F. Schweppe, “Evaluation of likelihood functions for Gaussian signals,”IEEE Trans. Inf. Theory IT-11, 61–70 (1965).
    [CrossRef]
  29. T. Kailath, B. Lévy, L. Ljung, M. Morf, “Time-invariant implementations of Gaussian signal detectors,”IEEE Trans. Inf. Theory IT-24, 469–477 (1977).
  30. R. Hestenes, E. Stiefel, “Methods of conjugate gradients for solving linear systems,”J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
    [CrossRef]
  31. R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction theory,” Trans. ASME Ser. D 83, 95–107 (1961).
    [CrossRef]
  32. H. L. Van Trees, ed., Detection, Estimation, and Modulation Theory (Wiley, New York, 1971), Vol. III, App. pp. 565–604.
  33. T. Kailath, “Some new algorithms for recursive estimation in constant linear systems,”IEEE Trans. Inf. Theory IT-19, 750–760 (1973).
    [CrossRef]
  34. U. Grenander, H. O. Pollak, D. Slepian, “The distribution of quadratic forms in normal variates: a small sample theory with applications to spectral analysis,”J. Soc. Ind. Appl. Math. 7, 374–401 (1959).
    [CrossRef]

1983 (2)

C. W. Helstrom, “Comment: Distribution of quadratic forms in normal random variables—evaluation by numerical integration,” SIAM J. Sci. Stat. Comput. 4, 353–356 (1983).
[CrossRef]

C. W. Helstrom, “Evaluating the detectability of Gaussian stochastic signals by steepest descent integration,”IEEE Trans. Aerosp. Electron. Syst. AES-19, 428–437 (1983).
[CrossRef]

1980 (1)

R. Barakat, J. Blake, “Theory of photoelectron counting statistics, an essay,” Phys. Reports 60, 225–340 (1980).
[CrossRef]

1978 (1)

C. W. Helstrom, “Approximate evaluation of detection probabilities in radar and optical communications,”IEEE Trans. Aerosp. Electron. Syst. AES-14, 630–640 (1978).
[CrossRef]

1977 (1)

T. Kailath, B. Lévy, L. Ljung, M. Morf, “Time-invariant implementations of Gaussian signal detectors,”IEEE Trans. Inf. Theory IT-24, 469–477 (1977).

1973 (2)

S. O. Rice, “Efficient evaluation of integrals of analytic functions by the trapezoidal rule,” Bell Sys. Tech. J. 52, 707–722 (1973).

T. Kailath, “Some new algorithms for recursive estimation in constant linear systems,”IEEE Trans. Inf. Theory IT-19, 750–760 (1973).
[CrossRef]

1972 (1)

S. R. Laxpati, G. Lachs, “Closed-form solutions for the photocount statistics of superposed coherent and chaotic radiation,” J. Appl. Phys. 43, 4773–4776 (1972).
[CrossRef]

1971 (1)

G. Lachs, “Approximate photocount statistics for coherent and chaotic radiation of arbitrary spectral shape,” J. Appl. Phys. 42, 602–609 (1971).
[CrossRef]

1970 (2)

S. Karp, J. R. Clark, “Photon counting: a problem in classical noise theory,”IEEE Trans. Inf. Theory IT-16, 672–680 (1970).
[CrossRef]

A. K. Jaiswal, C. L. Mehta, “Photon counting statistics of harmonic signal mixed with thermal light. I. Single photoelectron counting,” Phys. Rev. A 2, 168–172 (1970).
[CrossRef]

1969 (4)

J. Peřina, R. Horák, “On the quantum statistics of the superposition of coherent and chaotic fields,”J. Phys. A 2, 702–712 (1969).
[CrossRef]

E. Jakeman, E. R. Pike, “Statistics of heterodyne detection of Gaussian light,”J. Phys. A 2, 115–125 (1969).
[CrossRef]

D. Slepian, T. Kadota, “Four integral equations of detection theory,” SIAM J. Appl. Math. 17, 1102–1117 (1969).
[CrossRef]

A. B. Baggeroer, “A state-variable approach to the solution of Fredholm integral equations,”IEEE Trans. Inf. Theory IT-15, 557–570 (1969).
[CrossRef]

1966 (1)

G. Bédard, “Photon counting statistics of Gaussian light,” Phys. Rev. 151, 1038–1039 (1966).
[CrossRef]

1965 (1)

F. Schweppe, “Evaluation of likelihood functions for Gaussian signals,”IEEE Trans. Inf. Theory IT-11, 61–70 (1965).
[CrossRef]

1964 (1)

C. W. Helstrom, “The distribution of photoelectric counts from partially polarized Gaussian light,” Proc. Phys. Soc. London 83, 777–782 (1964).
[CrossRef]

1962 (1)

H. A. Spang, “A review of minimization techniques for nonlinear functions,”SIAM Rev. 4, 343–365 (1962).
[CrossRef]

1961 (1)

R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction theory,” Trans. ASME Ser. D 83, 95–107 (1961).
[CrossRef]

1959 (2)

U. Grenander, H. O. Pollak, D. Slepian, “The distribution of quadratic forms in normal variates: a small sample theory with applications to spectral analysis,”J. Soc. Ind. Appl. Math. 7, 374–401 (1959).
[CrossRef]

L. Mandel, “Fluctuations of photon beams: the distribution of the photoelectrons,” Proc. Phys. Soc. London 74, 233–243 (1959).
[CrossRef]

1957 (1)

A. J. F. Siegert, “A systematic approach to a class of problems in the theory of noise and other random phenomena. Part II, Examples,”IRE Trans. Inf. Theory IT-3, 38–44 (1957).
[CrossRef]

1954 (1)

H. E. Daniels, “Saddlepoint approximations in statistics,” Ann. Math. Statist. 25, 631–650 (1954).
[CrossRef]

1952 (1)

R. Hestenes, E. Stiefel, “Methods of conjugate gradients for solving linear systems,”J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
[CrossRef]

Baggeroer, A. B.

A. B. Baggeroer, “A state-variable approach to the solution of Fredholm integral equations,”IEEE Trans. Inf. Theory IT-15, 557–570 (1969).
[CrossRef]

A. B. Baggeroer, “State variable analysis procedures,” appendix in H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1971), Vol. II, pp. 286–327.

Barakat, R.

R. Barakat, J. Blake, “Theory of photoelectron counting statistics, an essay,” Phys. Reports 60, 225–340 (1980).
[CrossRef]

Bédard, G.

G. Bédard, “Photon counting statistics of Gaussian light,” Phys. Rev. 151, 1038–1039 (1966).
[CrossRef]

Bickel, P. J.

P. J. Bickel, K. A. Doksum, Mathematical Statistics (Holden-Day, San Francisco, Calif., 1977), Sec. 9.6, pp. 378–389.

Blake, J.

R. Barakat, J. Blake, “Theory of photoelectron counting statistics, an essay,” Phys. Reports 60, 225–340 (1980).
[CrossRef]

Bucy, R. S.

R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction theory,” Trans. ASME Ser. D 83, 95–107 (1961).
[CrossRef]

Carrier, G. F.

G. F. Carrier, M. Krook, C. E. Pearson, Functions of a Complex Variable (McGraw-Hill, New York, 1966), pp. 257ff.

Clark, J. R.

S. Karp, J. R. Clark, “Photon counting: a problem in classical noise theory,”IEEE Trans. Inf. Theory IT-16, 672–680 (1970).
[CrossRef]

Daniels, H. E.

H. E. Daniels, “Saddlepoint approximations in statistics,” Ann. Math. Statist. 25, 631–650 (1954).
[CrossRef]

Doksum, K. A.

P. J. Bickel, K. A. Doksum, Mathematical Statistics (Holden-Day, San Francisco, Calif., 1977), Sec. 9.6, pp. 378–389.

Gagliardi, R. M.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), pp. 80–83.

Glauber, R. J.

R. J. Glauber, “Optical coherence and photon statistics,” in Quantum Optics and Electronics, C. DeWitt, A. Blandin, C. Cohen-Tannoudji, eds. (Gordon and Breach, New York, 1965), pp. 65–185; see Lecture XVII, pp. 176–185.

Grenander, U.

U. Grenander, H. O. Pollak, D. Slepian, “The distribution of quadratic forms in normal variates: a small sample theory with applications to spectral analysis,”J. Soc. Ind. Appl. Math. 7, 374–401 (1959).
[CrossRef]

Helstrom, C. W.

C. W. Helstrom, “Comment: Distribution of quadratic forms in normal random variables—evaluation by numerical integration,” SIAM J. Sci. Stat. Comput. 4, 353–356 (1983).
[CrossRef]

C. W. Helstrom, “Evaluating the detectability of Gaussian stochastic signals by steepest descent integration,”IEEE Trans. Aerosp. Electron. Syst. AES-19, 428–437 (1983).
[CrossRef]

C. W. Helstrom, “Approximate evaluation of detection probabilities in radar and optical communications,”IEEE Trans. Aerosp. Electron. Syst. AES-14, 630–640 (1978).
[CrossRef]

C. W. Helstrom, “The distribution of photoelectric counts from partially polarized Gaussian light,” Proc. Phys. Soc. London 83, 777–782 (1964).
[CrossRef]

C. W. Helstrom, Statistical Theory of Signal Detection, 2nd ed. (Pergamon, London, 1968).

Hestenes, R.

R. Hestenes, E. Stiefel, “Methods of conjugate gradients for solving linear systems,”J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
[CrossRef]

Horák, R.

J. Peřina, R. Horák, “On the quantum statistics of the superposition of coherent and chaotic fields,”J. Phys. A 2, 702–712 (1969).
[CrossRef]

Hornbeck, R. W.

R. W. Hornbeck, Numerical Methods (Quantum, New York, 1975), pp. 69–71.

Jaiswal, A. K.

A. K. Jaiswal, C. L. Mehta, “Photon counting statistics of harmonic signal mixed with thermal light. I. Single photoelectron counting,” Phys. Rev. A 2, 168–172 (1970).
[CrossRef]

Jakeman, E.

E. Jakeman, E. R. Pike, “Statistics of heterodyne detection of Gaussian light,”J. Phys. A 2, 115–125 (1969).
[CrossRef]

Kadota, T.

D. Slepian, T. Kadota, “Four integral equations of detection theory,” SIAM J. Appl. Math. 17, 1102–1117 (1969).
[CrossRef]

Kailath, T.

T. Kailath, B. Lévy, L. Ljung, M. Morf, “Time-invariant implementations of Gaussian signal detectors,”IEEE Trans. Inf. Theory IT-24, 469–477 (1977).

T. Kailath, “Some new algorithms for recursive estimation in constant linear systems,”IEEE Trans. Inf. Theory IT-19, 750–760 (1973).
[CrossRef]

Kalman, R. E.

R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction theory,” Trans. ASME Ser. D 83, 95–107 (1961).
[CrossRef]

Karp, S.

S. Karp, J. R. Clark, “Photon counting: a problem in classical noise theory,”IEEE Trans. Inf. Theory IT-16, 672–680 (1970).
[CrossRef]

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), pp. 80–83.

Krook, M.

G. F. Carrier, M. Krook, C. E. Pearson, Functions of a Complex Variable (McGraw-Hill, New York, 1966), pp. 257ff.

Lachs, G.

S. R. Laxpati, G. Lachs, “Closed-form solutions for the photocount statistics of superposed coherent and chaotic radiation,” J. Appl. Phys. 43, 4773–4776 (1972).
[CrossRef]

G. Lachs, “Approximate photocount statistics for coherent and chaotic radiation of arbitrary spectral shape,” J. Appl. Phys. 42, 602–609 (1971).
[CrossRef]

Laxpati, S. R.

S. R. Laxpati, G. Lachs, “Closed-form solutions for the photocount statistics of superposed coherent and chaotic radiation,” J. Appl. Phys. 43, 4773–4776 (1972).
[CrossRef]

Lévy, B.

T. Kailath, B. Lévy, L. Ljung, M. Morf, “Time-invariant implementations of Gaussian signal detectors,”IEEE Trans. Inf. Theory IT-24, 469–477 (1977).

Ljung, L.

T. Kailath, B. Lévy, L. Ljung, M. Morf, “Time-invariant implementations of Gaussian signal detectors,”IEEE Trans. Inf. Theory IT-24, 469–477 (1977).

Mandel, L.

L. Mandel, “Fluctuations of photon beams: the distribution of the photoelectrons,” Proc. Phys. Soc. London 74, 233–243 (1959).
[CrossRef]

Mehta, C. L.

A. K. Jaiswal, C. L. Mehta, “Photon counting statistics of harmonic signal mixed with thermal light. I. Single photoelectron counting,” Phys. Rev. A 2, 168–172 (1970).
[CrossRef]

Morf, M.

T. Kailath, B. Lévy, L. Ljung, M. Morf, “Time-invariant implementations of Gaussian signal detectors,”IEEE Trans. Inf. Theory IT-24, 469–477 (1977).

Pearson, C. E.

G. F. Carrier, M. Krook, C. E. Pearson, Functions of a Complex Variable (McGraw-Hill, New York, 1966), pp. 257ff.

Perina, J.

J. Peřina, R. Horák, “On the quantum statistics of the superposition of coherent and chaotic fields,”J. Phys. A 2, 702–712 (1969).
[CrossRef]

Pike, E. R.

E. Jakeman, E. R. Pike, “Statistics of heterodyne detection of Gaussian light,”J. Phys. A 2, 115–125 (1969).
[CrossRef]

Pollak, H. O.

U. Grenander, H. O. Pollak, D. Slepian, “The distribution of quadratic forms in normal variates: a small sample theory with applications to spectral analysis,”J. Soc. Ind. Appl. Math. 7, 374–401 (1959).
[CrossRef]

Rice, S. O.

S. O. Rice, “Efficient evaluation of integrals of analytic functions by the trapezoidal rule,” Bell Sys. Tech. J. 52, 707–722 (1973).

Saleh, B. E. A.

B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978), pp. 206–210.

Schweppe, F.

F. Schweppe, “Evaluation of likelihood functions for Gaussian signals,”IEEE Trans. Inf. Theory IT-11, 61–70 (1965).
[CrossRef]

Siegert, A. J. F.

A. J. F. Siegert, “A systematic approach to a class of problems in the theory of noise and other random phenomena. Part II, Examples,”IRE Trans. Inf. Theory IT-3, 38–44 (1957).
[CrossRef]

Slepian, D.

D. Slepian, T. Kadota, “Four integral equations of detection theory,” SIAM J. Appl. Math. 17, 1102–1117 (1969).
[CrossRef]

U. Grenander, H. O. Pollak, D. Slepian, “The distribution of quadratic forms in normal variates: a small sample theory with applications to spectral analysis,”J. Soc. Ind. Appl. Math. 7, 374–401 (1959).
[CrossRef]

Spang, H. A.

H. A. Spang, “A review of minimization techniques for nonlinear functions,”SIAM Rev. 4, 343–365 (1962).
[CrossRef]

Stiefel, E.

R. Hestenes, E. Stiefel, “Methods of conjugate gradients for solving linear systems,”J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
[CrossRef]

Ann. Math. Statist. (1)

H. E. Daniels, “Saddlepoint approximations in statistics,” Ann. Math. Statist. 25, 631–650 (1954).
[CrossRef]

Bell Sys. Tech. J. (1)

S. O. Rice, “Efficient evaluation of integrals of analytic functions by the trapezoidal rule,” Bell Sys. Tech. J. 52, 707–722 (1973).

IEEE Trans. Aerosp. Electron. Syst. (2)

C. W. Helstrom, “Evaluating the detectability of Gaussian stochastic signals by steepest descent integration,”IEEE Trans. Aerosp. Electron. Syst. AES-19, 428–437 (1983).
[CrossRef]

C. W. Helstrom, “Approximate evaluation of detection probabilities in radar and optical communications,”IEEE Trans. Aerosp. Electron. Syst. AES-14, 630–640 (1978).
[CrossRef]

IEEE Trans. Inf. Theory (5)

F. Schweppe, “Evaluation of likelihood functions for Gaussian signals,”IEEE Trans. Inf. Theory IT-11, 61–70 (1965).
[CrossRef]

T. Kailath, B. Lévy, L. Ljung, M. Morf, “Time-invariant implementations of Gaussian signal detectors,”IEEE Trans. Inf. Theory IT-24, 469–477 (1977).

A. B. Baggeroer, “A state-variable approach to the solution of Fredholm integral equations,”IEEE Trans. Inf. Theory IT-15, 557–570 (1969).
[CrossRef]

T. Kailath, “Some new algorithms for recursive estimation in constant linear systems,”IEEE Trans. Inf. Theory IT-19, 750–760 (1973).
[CrossRef]

S. Karp, J. R. Clark, “Photon counting: a problem in classical noise theory,”IEEE Trans. Inf. Theory IT-16, 672–680 (1970).
[CrossRef]

IRE Trans. Inf. Theory (1)

A. J. F. Siegert, “A systematic approach to a class of problems in the theory of noise and other random phenomena. Part II, Examples,”IRE Trans. Inf. Theory IT-3, 38–44 (1957).
[CrossRef]

J. Appl. Phys. (2)

S. R. Laxpati, G. Lachs, “Closed-form solutions for the photocount statistics of superposed coherent and chaotic radiation,” J. Appl. Phys. 43, 4773–4776 (1972).
[CrossRef]

G. Lachs, “Approximate photocount statistics for coherent and chaotic radiation of arbitrary spectral shape,” J. Appl. Phys. 42, 602–609 (1971).
[CrossRef]

J. Phys. A (2)

J. Peřina, R. Horák, “On the quantum statistics of the superposition of coherent and chaotic fields,”J. Phys. A 2, 702–712 (1969).
[CrossRef]

E. Jakeman, E. R. Pike, “Statistics of heterodyne detection of Gaussian light,”J. Phys. A 2, 115–125 (1969).
[CrossRef]

J. Res. Nat. Bur. Stand. (1)

R. Hestenes, E. Stiefel, “Methods of conjugate gradients for solving linear systems,”J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
[CrossRef]

J. Soc. Ind. Appl. Math. (1)

U. Grenander, H. O. Pollak, D. Slepian, “The distribution of quadratic forms in normal variates: a small sample theory with applications to spectral analysis,”J. Soc. Ind. Appl. Math. 7, 374–401 (1959).
[CrossRef]

Phys. Reports (1)

R. Barakat, J. Blake, “Theory of photoelectron counting statistics, an essay,” Phys. Reports 60, 225–340 (1980).
[CrossRef]

Phys. Rev. (1)

G. Bédard, “Photon counting statistics of Gaussian light,” Phys. Rev. 151, 1038–1039 (1966).
[CrossRef]

Phys. Rev. A (1)

A. K. Jaiswal, C. L. Mehta, “Photon counting statistics of harmonic signal mixed with thermal light. I. Single photoelectron counting,” Phys. Rev. A 2, 168–172 (1970).
[CrossRef]

Proc. Phys. Soc. London (2)

L. Mandel, “Fluctuations of photon beams: the distribution of the photoelectrons,” Proc. Phys. Soc. London 74, 233–243 (1959).
[CrossRef]

C. W. Helstrom, “The distribution of photoelectric counts from partially polarized Gaussian light,” Proc. Phys. Soc. London 83, 777–782 (1964).
[CrossRef]

SIAM J. Appl. Math. (1)

D. Slepian, T. Kadota, “Four integral equations of detection theory,” SIAM J. Appl. Math. 17, 1102–1117 (1969).
[CrossRef]

SIAM J. Sci. Stat. Comput. (1)

C. W. Helstrom, “Comment: Distribution of quadratic forms in normal random variables—evaluation by numerical integration,” SIAM J. Sci. Stat. Comput. 4, 353–356 (1983).
[CrossRef]

SIAM Rev. (1)

H. A. Spang, “A review of minimization techniques for nonlinear functions,”SIAM Rev. 4, 343–365 (1962).
[CrossRef]

Trans. ASME Ser. D (1)

R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction theory,” Trans. ASME Ser. D 83, 95–107 (1961).
[CrossRef]

Other (9)

H. L. Van Trees, ed., Detection, Estimation, and Modulation Theory (Wiley, New York, 1971), Vol. III, App. pp. 565–604.

A. B. Baggeroer, “State variable analysis procedures,” appendix in H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1971), Vol. II, pp. 286–327.

C. W. Helstrom, Statistical Theory of Signal Detection, 2nd ed. (Pergamon, London, 1968).

G. F. Carrier, M. Krook, C. E. Pearson, Functions of a Complex Variable (McGraw-Hill, New York, 1966), pp. 257ff.

R. W. Hornbeck, Numerical Methods (Quantum, New York, 1975), pp. 69–71.

P. J. Bickel, K. A. Doksum, Mathematical Statistics (Holden-Day, San Francisco, Calif., 1977), Sec. 9.6, pp. 378–389.

R. J. Glauber, “Optical coherence and photon statistics,” in Quantum Optics and Electronics, C. DeWitt, A. Blandin, C. Cohen-Tannoudji, eds. (Gordon and Breach, New York, 1965), pp. 65–185; see Lecture XVII, pp. 176–185.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), pp. 80–83.

B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978), pp. 206–210.

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Figures (2)

Fig. 1
Fig. 1

Cumulative distribution of photoelectron counts, Lorentz spectral density, m = μT = 5, n0 = 10. Curves for ns = 4, 8 indexed with the value of ΔT. Δ is the offset frequency of the coherent component. + indicates results of Toeplitz approximation (Section 4).

Fig. 2
Fig. 2

Cumulative distribution of photoelectron counts, Lorentz spectral density, m = T = 1.3195, n0 = 1400. Curves for ns = 1000, 2000 indexed with the value of ΔT. Δ is the offset frequency of the coherent component.

Tables (3)

Tables Icon

Table 1 Probability Distribution of Photoelectron Counts by Fourier Transformationa

Tables Icon

Table 2 Cumulative Distribution of Photoelectron Counts by Saddle-Point Integrationa

Tables Icon

Table 3 Cumulative Distribution of Photoelectron Counts by Saddle-Point Integration

Equations (78)

Equations on this page are rendered with MathJax. Learn more.

p k = P r ( n = k ) = I k e - I / k ! ,
h ( z ) = k = 0 p k z k = e I ( z - 1 ) .
v ( t ) = Re V ( t ) e i Ω t ,             0 < t < T ,
I = 1 2 η 0 T V ( t ) 2 d t ,
V ( t ) = S ( t )
V b ( t ) = V ( t ) - S ( t )
G ( t 1 - t 2 ) = 1 2 V b ( t 1 ) V b * ( t 2 ) .
Γ ( ω ) = - G ( t ) e - i ω t d t ,
n 0 = η G ( 0 ) T ,
n s = 1 2 η 0 T S ( t ) 2 d t .
V ( t ) = k = 1 v k f k ( t ) ,
λ k f k ( t ) = T - 1 0 T γ ( t - s ) f k ( s ) d s ,             γ ( τ ) = G ( τ ) / G ( 0 ) .
k = 1 λ k = 1.
h ( z ) = [ D ( x ) ] - 1 exp { n s ( z - 1 ) k = 1 σ k 2 1 + λ k x } , x = n 0 ( 1 - z ) ,
D ( x ) = k = 1 ( 1 + λ k x )
σ k = 0 T f k * ( t ) σ ( t ) d t ,             σ ( t ) = C s - 1 S ( t ) , C s 2 = 0 T S ( t ) 2 d t ,
0 T σ ( t ) 2 d t = k = 1 σ k 2 = 1.
q k - = Pr ( n < k ) = m = 0 k - 1 p m
q k + = Pr ( n k ) = m = k k - 1 p m = 1 - q k - .
p k = 0 2 π h ( e i θ ) e - i k θ d θ / 2 π .
p k = N - 1 m = 1 N h ( z m ) exp ( - 2 π i k m / N ) , z m = exp ( 2 π i m / N ) ,
Γ ( ω ) = 2 μ G ( 0 ) ω 2 + μ 2 ,             γ ( τ ) = e - μ τ .
h ( z ) = [ D ( x ) ] - 1 = 4 g e m [ ( g + 1 ) 2 e m g - ( g - 1 ) 2 e - m g ] - 1 , g = ( 1 + 2 x / m ) 1 / 2 ,             x = n 0 ( 1 - z ) ,             m = μ T ,
x k = 1 + ( n 0 λ k ) - 1 ,             1 < x 1 < x 2 < ,
p m = C z - ( m + 1 ) h ( z ) d z / 2 π i .
q k - = C 1 - z - k z - 1 h ( z ) d z / 2 π i .
q k - = C - z - k h ( z ) 1 - z d z 2 π i .
q k + = 1 - q k - = C + z - k h ( z ) z - 1 d z 2 π i .
Φ ( z ) = ln h ( z ) - k ln z - ln [ ± ( z - 1 ) ] ,
Φ ( x 0 ) = 0 ,
x 0 x 0 - Φ ( x 0 ) Φ ( x 0 ) .
Φ ( z ) = m = 1 { n s σ m 2 ( z - 1 ) 1 + n 0 λ m ( 1 - z ) - ln [ 1 + n 0 λ m ( 1 - z ) ] } - k ln z - ln [ ± ( z - 1 ) ] ,
Im Φ ( x + i ) = 0
Φ ˜ ( z ) = ln h ( z ) - k ln z = Φ ( z ) + ln [ ± ( z - 1 ) ] ,
κ = Φ ˜ ( x ˜ 0 ) 3 Φ ˜ ( x ˜ 0 ) .
x = x 0 + ½ κ y 2 ,
q k ± = π - 1 0 Re [ e Φ ( z ) ( 1 - i κ y ) ] d y , z = x 0 + ½ κ y 2 + i y ,             x 0 = x 0 ± .
Δ y = [ Φ ( x 0 ) ] - 1 / 2 .
λ K + 1 = 1 - m = 1 K λ m ,             σ K + 1 2 = 1 - m = 1 K σ m 2 ,
h ( z ) = [ D ( x ) ] - 1 exp { n s ( z - 1 ) × [ 1 - 0 T 0 T σ * ( t ) ψ ( t , u ; x ; T ) σ ( u ) d t d u ] } ,
ψ ( t , u ; x ; T ) = x n = 1 λ n f n ( t ) f n * ( u ) 1 + λ n x
ψ ( t , u ; x ; τ ) + x T 0 τ ψ ( t , v ; x ; τ ) γ ( v - u ) d v = x T - 1 γ ( t - u ) ,             0 ( t , u ) τ ,
D ( x ) = exp [ 0 x d y y 0 T ψ ( t , t ; y ; T ) d t ] ,
ψ ( t , u ; x ; τ ) = x T { g [ ( g + 1 ) 2 e μ g τ - ( g - 1 ) 2 e - μ g τ ] } - 1 × [ ( g + 1 ) e μ g t + ( g - 1 ) e - μ g t ] [ ( g + 1 ) e μ g ( τ - u ) + ( g - 1 ) e - μ g ( τ - u ) ] , g = ( 1 + 2 x m ) 1 / 2 ,             m = μ T ,             0 < t < u < τ .
ψ ( t , u ; x ; τ ) = ψ * ( u , t ; x * ; τ )
S ( t ) = S e i Δ t ,             σ ( t ) = T - 1 / 2 e i Δ t ,
h ( z ) = [ D ( x ) ] - 1 exp ( n s ( z - 1 ) { 1 + δ 2 g 2 + δ 2 + 4 x [ ( g + 1 ) ( g - δ 2 ) e m g - ( g - 1 ) ( g + δ 2 ) e - m g - 2 g [ ( 1 - δ 2 ) cos m δ - 2 δ sin m δ ] m 2 ( g 2 + δ 2 ) 2 [ ( g + 1 ) 2 e m g - ( g - 1 ) 2 e - m g ] } ) , x = n 0 ( 1 - z ) ,             g = ( 1 + 2 x m ) 1 / 2 ,             m = μ T ,             δ = Δ / μ .
D ( x ) = exp [ 0 T ψ ¯ ( τ , τ ; x ) d τ ] ,
ψ ¯ ( τ , u ; x ) = ψ ( t , u ; x ; τ ) .
ψ ¯ ( τ , u ; x ) + x T - 1 0 τ ψ ¯ ( τ , v ; x ) γ ( v - u ) d v = x T - 1 γ ( τ - u ) ,             0 u τ T .
ψ ¯ ( τ , u ; x ) = 2 x T ( g + 1 ) e μ g u + ( g - 1 ) e - μ g u ( g + 1 ) 2 e μ g τ - ( g - 1 ) 2 e - μ g τ , 0 u τ ,             g = ( 1 + 2 x m ) 1 / 2 ,             m = μ T .
J ( x ; τ ) = 0 τ 0 τ σ * ( t ) ψ ( t , u ; x ; τ ) σ ( u ) d t d u .
J ( x ; τ ) τ = J τ ( x ; τ ) = 0 τ 0 τ σ * ( t ) ψ τ ( t , u ; x ; τ ) σ ( u ) d t d u + σ * ( τ ) 0 τ ψ ( τ , u ; x ; τ ) σ ( u ) d u + σ ( τ ) × 0 τ σ * ( t ) ψ ( t , τ ; x , τ ) d t .
ψ τ ( t , u ; x ; τ ) = - ψ ( t , τ ; x ; τ ) ψ ( τ , u ; x ; τ ) = - ψ ¯ ( τ , u ; x ) ψ ¯ * ( τ , t ; x * ) ,
J τ ( x ; τ ) = - σ ˜ * ( τ ; x * ) σ ˜ ( τ , x ) + σ * ( τ ) σ ˜ ( τ ; x ) + σ ( τ ) σ ˜ * ( τ ; x * ) ,
σ ˜ ( τ ; x ) = 0 τ ψ ¯ ( τ , u ; x ) σ ( u ) d u .
1 = 0 T σ ( τ ) 2 d τ ,
h ( z ) = [ D ( x ) ] - 1 exp { n s ( z - 1 ) 0 T [ σ ˜ * ( τ ; x * ) - σ * ( τ ) ] [ σ ˜ ( τ ; x ) - σ ( τ ) ] d τ } ,
( τ + u ) ψ ¯ ( τ , u ; x ) = - ψ ¯ ( τ ; 0 , x ) ψ ¯ * ( τ , τ - u ; x * ) .
ψ ¯ ( t j + Δ t , u k + Δ t ; x ) ψ ¯ ( t j , u k ) - ψ ¯ ( t j , 0 ; x ) ψ ¯ * ( t j , t j - u k ; x * ) Δ t ,             t j = j Δ t ,             u k = k Δ t
½ W ( t 1 ) W * ( t 2 ) = x T γ ( t 1 - t 2 ) .
½ N ( t 1 ) N * ( t 2 ) = δ ( t 1 - t 2 ) ,
W ^ ( τ ) = 0 τ ψ ¯ ( τ , u ; x ) Y ( u ) d u ,             0 < τ < T
½ W ^ ( τ ) - W ( τ ) 2 .
ψ ( t , u ; x ; T ) ψ ˜ ( t - u ; x ; T ) ,
Ψ ˜ ( ω ; x ) = x T - 1 Γ ˜ ( ω ) 1 + x T - 1 Γ ˜ ( ω ) ,
Γ ˜ ( ω ) = - γ ( s ) e - i ω s d s
D ( x ) exp { T - ln [ 1 + x T - 1 Γ ˜ ( ω ) ] d ω / 2 π } .
Γ ˜ ( ω ) = M ( ω ) m = 1 n [ ( ω - c m ) 2 + d m 2 ] - 1 ,
1 + x T - 1 Γ ˜ ( ω ) = m = 1 n ( ω - a m ) 2 + b m 2 ( ω - c m ) 2 + d m 2 .
D ( x ) exp [ T m = 1 n ( b m - d m ) ] = exp [ T Im m = 1 n ( α m - β m ) ] ,
D ( x ) exp [ m ( g - 1 ) ] ,             m = μ T ,             g = ( 1 + 2 x / m ) 1 / 2 .
- - σ * ( t ) ψ ˜ ( t - u ; x ; T ) σ ( u ) d t d u = x T - Σ ( ω ) 2 1 + x T - 1 Γ ˜ ( ω ) d ω 2 π ,
Σ ( ω ) = - σ ( t ) e i ω t d t
h ( z ) exp { n s ( z - 1 ) - Σ ( ω ) 2 1 + x T - 1 Γ ˜ ( ω ) d ω 2 π - T - ln [ 1 + x T - 1 Γ ˜ ( ω ) ] d ω / 2 π } ,             x = n 0 ( 1 - z ) .
T - 1 0 T 0 T ψ ( t , u ; x ; T ) e i Δ ( t - u ) d t d u - ψ ˜ ( s ; x ; T ) e - i Δ s d s = Ψ ˜ ( Δ ; x ) ,
h ( z ) exp { n s ( z - 1 ) 1 + x T - 1 Γ ˜ ( Δ ) - T × - ln [ 1 + x T - 1 Γ ˜ ( ω ) ] d ω 2 π } ,             x = n 0 ( 1 - z ) .
n s ( z - 1 ) 1 + δ 2 g 2 + δ 2

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