## Quantum derivation of *K*-distributed noise for finite 〈*N*〉

JOSA A, Vol. 5, Issue 5, pp. 730-734 (1988)

http://dx.doi.org/10.1364/JOSAA.5.000730

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### Abstract

Semiclassical derivations of the fluctuations of light beams have relied on limiting procedures in which the average number, 〈*N*〉, of scattering elements, photons, or superposed wave packets approaches infinity. We show that the fluctuations of thermal light having a Bose–Einstein photon distribution and of light with an amplitude distribution based on the modified Bessel functions, *K _{α}*

_{−1}, which has been found useful in describing light scattered from or through turbulent media, may be derived with a quantum-mechanical analysis as the superposition of a random number,

*N*, of single-photon eigenstates with finite 〈

*N*〉. The analysis also provides the

*P*representation for

*K*-distributed noise. Generalizations of

*K*noise are proposed. The factor-of-2 increase in the photon-number second factorial moment related to photon clumping in the Hanbury Brown–Twiss effect for thermal (Gaussian) fields is shown to arise generally in these random superposition models, even for non-Gaussian fields.

© 1988 Optical Society of America

**History**

Original Manuscript: June 29, 1987

Manuscript Accepted: December 30, 1987

Published: May 1, 1988

**Citation**

Edward B. Rockower, "Quantum derivation of K-distributed noise for finite 〈N〉," J. Opt. Soc. Am. A **5**, 730-734 (1988)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-5-5-730

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### References

- R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev.131, 2766–2788 (1963);R. Loudon, The Quantum Theory of Light, 2nd ed. (Clarendon, Oxford, 1983);E. B. Rockower, N. B. Abraham, S. R. Smith, “Evolution of the quantum statistics of light,” Phys. Rev. A 17, 1100–1112 (1978). [CrossRef]
- E. Jakeman, P. N. Pusey, “Significance of K distributions in scattering experiments,” Phys. Rev. Lett. 40, 546–550 (1978);G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,” J. Opt. Soc. Am. 69, 796–798 (1979). [CrossRef]
- E. Jakeman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980). [CrossRef]
- See, for instance, C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer-Verlag, New York, 1985).
- R. L. Phillips, L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71, 1440–1445 (1981);“Universal statistical model for irradiance fluctuations in a turbulent medium,” J. Opt. Soc. Am. 72, 864–870 (1982);“I-K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2, 160–163 (1985). [CrossRef]
- See, for instance, R. Loudon, The Quantum Theory of Light (Oxford U. Press, Oxford, 1983);J. R. Klauder, E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968), pp. 21–26.
- W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
- I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 686.
- Note that there is a limitation on how large p2may be and still yield a valid characteristic function for the contribution from a single scattering element. This is discussed further in E. B. Rockower, “Calculating the quantum characteristic function and the number-generating function in quantum optics,” Phys. Rev. A (to be published).
- This procedure could also be performed for the semiclassical theory of K-distributed noise.

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