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, Kα−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

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References

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  1. 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]
  2. 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]
  3. E. Jakeman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
    [CrossRef]
  4. See, for instance, C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer-Verlag, New York, 1985).
  5. 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]
  6. 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.
  7. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
  8. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 686.
  9. 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).
  10. This procedure could also be performed for the semiclassical theory of K-distributed noise.

1981 (1)

1980 (1)

E. Jakeman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
[CrossRef]

1978 (1)

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]

Andrews, L. C.

Gardiner, C. W.

See, for instance, C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer-Verlag, New York, 1985).

Glauber, R. J.

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]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 686.

Jakeman, E.

E. Jakeman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
[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]

Loudon, R.

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.

Louisell, W. H.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).

Phillips, R. L.

Pusey, P. N.

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]

Rockower, E. B.

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).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 686.

J. Opt. Soc. Am. (1)

J. Phys. A (1)

E. Jakeman, “On the statistics of K-distributed noise,” J. Phys. A 13, 31–48 (1980).
[CrossRef]

Phys. Rev. Lett. (1)

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]

Other (7)

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]

See, for instance, C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer-Verlag, New York, 1985).

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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Equations (46)

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p ( A ) = 2 b Γ ( β ) ( b A 2 ) β K β 1 ( b A ) ,
lim N exp [ i ( u 1 1 + u 2 2 ) ] = ( 1 + u 2 2 4 β ) β ,
C ( ξ , ξ * ) = tr [ p exp ( ξ a + ) exp ( ξ * a ) ] = exp ( ξ a + ) exp ( ξ * a ) .
C 1 ( ξ , ξ * ) = 1 | exp ( ξ a + ) exp ( ξ * a ) | 1
C 1 ( ξ , ξ * ) = 0 | a exp ( ξ a + ) exp ( ξ * a ) a + | 0 ,
C 1 ( ξ , ξ * ) = 0 | ( a + ξ ) ( a + ξ * ) | 0
C 1 ( ξ , ξ * ) = 1 | ξ | 2 .
ρ 0 = ( 1 p ) 0 0 + p 1 1 ,
C 0 ( ξ , ξ * ) = ( 1 p ) 1 + p ( 1 | ξ | 2 )
C 0 ( ξ , ξ * ) = 1 p | ξ | 2 .
C ( ξ , ξ * ) N = ( 1 p | ξ | 2 ) N .
P N = ( N + β 1 N ) ( N / β ) N ( 1 + N / β ) N + β ,
G ( γ ) = N = 0 P N ( 1 γ ) N
G ( γ ) = ( 1 + N β γ ) β .
C ( ξ , ξ * ) = ( 1 + N β p | ξ | 2 ) β .
C ( ξ , ξ * ) = exp ( ξ α * ξ * α ) P ( α ) d 2 α .
C ( u , υ ) = exp ( 2 i υ x 2 i υ y ) P ( x , y ) d x d y .
P ( x , y ) = 4 ( 2 π ) 2 exp ( 2 iuy 2 i υ x ) C ( u , υ ) d u d υ .
P ( r , θ ) = 4 ( 2 π ) 2 0 0 2 π exp [ 2 i r ρ sin ( φ θ ) ] C ( ρ ) ρ d ρ d φ .
P ( r ) = 4 2 π 0 J 0 ( 2 ρ r ) C ( ρ ) ρ d ρ .
P ( r ) = 4 2 π 0 J 0 ( 2 ρ r ) ( 1 + N β p ρ 2 ) β ρ d ρ .
P ( α ) = 4 2 π ( β p N ) ( 1 + β ) / 2 ( | α | ) β 1 K β 1 [ ( β ρ N ) 1 / 2 2 | α | ] Γ ( β ) ,
f ( | α | ) d | α | = 0 2 π P ( α ) | α | d | α | d θ .
f ( | α | ) d | α | = 2 π P ( α ) | α | d | α | .
f ( A ) = 4 ( β p N ) ( 1 + β ) / 2 ( A ) β K β 1 [ ( β p N ) 1 / 2 2 A ] Γ ( β ) .
C 0 ( ξ , ξ * ) = ( 1 p 1 p 2 ) 1 + p 1 ( 1 | ξ | 2 ) + p 2 ( 1 | ξ | 2 ) 2
C 0 ( ξ , ξ * ) = 1 ( p 1 + 2 p 2 ) | ξ | 2 + p 2 | ξ | 4 .
C ( ξ , ξ * ) = { 1 + N β [ ( p 1 + 2 p 2 ) | ξ | 2 | ξ | 4 ] } β .
C 0 ( ξ , ξ * ) = m = 0 p m ( 1 | ξ | 2 ) m
C 0 ( ξ , ξ * ) = G m ( | ξ | 2 ) .
C ( ξ , ξ * ) = N = 0 p N { 1 G m ( | ξ | 2 ) } N ,
C ( ξ , ξ * ) = G N [ 1 G m ( | ξ | 2 ) ] ,
n = i = 0 N m i
G cl ( γ ) = G N [ 1 G m ( γ ) ] .
n = a + a = d 2 d ξ d ξ * C ( ξ , ξ * ) ξ = 0 = N m ;
n ( n 1 ) = 2 N ( N 1 ) m 2 + 2 N m ( m 1 ) ,
I 2 I 2 = n ( n 1 ) n 2 = 2 [ N ( N 1 ) N 2 + m ( m 1 ) N m 2 ] .
G N ( γ ) = exp ( N γ )
G m ( γ ) = 1 p γ .
C ( ξ , ξ * ) = exp ( p N | ξ | 2 ) .
C ( ξ , ξ * ) = { 1 + N β [ 1 G m ( | ξ | 2 ) ] } β .
n ( n 1 ) = 2 ( 1 + 1 / β ) N 2 m 2 + 2 N m ( m 1 )
I 2 I 2 = n ( n 1 ) n 2 = 2 ( 1 + 1 / β ) + 2 N m ( m 1 ) m 2 .
C ( ξ , ξ * ) = exp ( ξ α 0 * ξ * α 0 ) ( 1 + N β p | ξ | 2 ) β .
n ( n 1 ) = | α 0 | 4 + 4 p N | α 0 | 2 + 2 ( p N ) 2 β + 1 β ,
I 2 = A 4 + 2 β + 1 β p N A 2 + 2 ( p N ) 2 β + 1 β ,

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