## Computation of photoelectron counting distributions by numerical contour integration

JOSA A, Vol. 2, Issue 5, pp. 674-682 (1985)

http://dx.doi.org/10.1364/JOSAA.2.000674

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### 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

**History**

Original Manuscript: July 23, 1984

Manuscript Accepted: January 10, 1985

Published: May 1, 1985

**Citation**

Carl W. Helstrom, "Computation of photoelectron counting distributions by numerical contour integration," J. Opt. Soc. Am. A **2**, 674-682 (1985)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-2-5-674

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### References

- L. Mandel, “Fluctuations of photon beams: the distribution of the photoelectrons,” Proc. Phys. Soc. London 74, 233–243 (1959). [CrossRef]
- 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.
- J. Peřina, R. Horák, “On the quantum statistics of the superposition of coherent and chaotic fields,”J. Phys. A 2, 702–712 (1969). [CrossRef]
- 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]
- G. Lachs, “Approximate photocount statistics for coherent and chaotic radiation of arbitrary spectral shape,” J. Appl. Phys. 42, 602–609 (1971). [CrossRef]
- 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.
- R. Barakat, J. Blake, “Theory of photoelectron counting statistics, an essay,” Phys. Reports 60, 225–340 (1980). [CrossRef]
- E. Jakeman, E. R. Pike, “Statistics of heterodyne detection of Gaussian light,”J. Phys. A 2, 115–125 (1969). [CrossRef]
- C. W. Helstrom, “The distribution of photoelectric counts from partially polarized Gaussian light,” Proc. Phys. Soc. London 83, 777–782 (1964). [CrossRef]
- P. J. Bickel, K. A. Doksum, Mathematical Statistics (Holden-Day, San Francisco, Calif., 1977), Sec. 9.6, pp. 378–389.
- 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]
- G. Bédard, “Photon counting statistics of Gaussian light,” Phys. Rev. 151, 1038–1039 (1966). [CrossRef]
- 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]
- 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]
- R. W. Hornbeck, Numerical Methods (Quantum, New York, 1975), pp. 69–71.
- H. A. Spang, “A review of minimization techniques for nonlinear functions,”SIAM Rev. 4, 343–365 (1962). [CrossRef]
- G. F. Carrier, M. Krook, C. E. Pearson, Functions of a Complex Variable (McGraw-Hill, New York, 1966), pp. 257ff.
- 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]
- S. O. Rice, “Efficient evaluation of integrals of analytic functions by the trapezoidal rule,” Bell Sys. Tech. J. 52, 707–722 (1973).
- H. E. Daniels, “Saddlepoint approximations in statistics,” Ann. Math. Statist. 25, 631–650 (1954). [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]
- 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]
- 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).
- 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).
- R. Hestenes, E. Stiefel, “Methods of conjugate gradients for solving linear systems,”J. Res. Nat. Bur. Stand. 49, 409–436 (1952). [CrossRef]
- R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction theory,” Trans. ASME Ser. D 83, 95–107 (1961). [CrossRef]
- H. L. Van Trees, ed., Detection, Estimation, and Modulation Theory (Wiley, New York, 1971), Vol. III, App. pp. 565–604.
- T. Kailath, “Some new algorithms for recursive estimation in constant linear systems,”IEEE Trans. Inf. Theory IT-19, 750–760 (1973). [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]

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