Abstract

It is well known that photoelectric measurements yield information about the statistical behavior of fluctuating light beams. The probability of n photoelectrons being ejected, in a fixed time interval, from the photosensitive surface of the photoelectric detector, upon which the light beam is normally incident, is a linear (Poisson) transform of the probability density for the intensity of the beam. With the help of some plausible assumptions, the present analysis provides solutions to the problem of inverting the Poisson transform, thus determining the probability density for the intensity from experimentally obtained photocounting distributions. The effectiveness of the method is demonstrated by an actual inversion of a typical experimental counting distribution. The technique is of particular interest in connection with efforts to understand the statistical behavior of optical fields, especially of laser fields.

© 1967 Optical Society of America

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  1. L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958); Proc. Phys. Soc. (London) 74, 233 (1959); Proc. Phys. Soc. (London) 81, 1104 (1963).
    [Crossref]
  2. L. Mandel, in Progress in Optics, Vol. II, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1963), p. 181.
    [Crossref]
  3. F. A. Johnson, T. P. McLean, and E. R. Pike, in Physics of Quantum Electronics, P. L. Kelley, B. Lax, and P. E. Tannenwald, Eds. (McGraw-Hill Book Co., New York, 1965), p. 706.
  4. C. Freed and H. A. Haus, in Physics of Quantum Electronics, P. L. Kelley, B. Lax, and P. E. Tannenwald, Eds. (McGraw-Hill Book Co., New York, 1965), p. 715.
  5. F. T. Arecchi, Phys. Rev. Letters 15, 912 (1965).
    [Crossref]
  6. C. Freed and H. A. Haus, Phys. Rev. Letters 15, 943 (1965).
    [Crossref]
  7. C. Freed and H. A. Haus, IEEE J. Quantum Electronics QE-2, 190 (1966).
    [Crossref]
  8. A. W. Smith and J. A. Armstrong, Phys. Letters 19, 650 (1966); see also Phys. Rev. Letters 16, 1169 (1966).
    [Crossref]
  9. F. T. Arecchi, A. Berné, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966).
    [Crossref]
  10. F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. Letters 16, 589 (1966).
    [Crossref]
  11. W. Martienssen and E. Spiller, Phys. Rev. Letters 16, 531 (1966).
    [Crossref]
  12. P. J. Magill and R. P. Soni, Phys. Rev. Letters 16, 911 (1966).
    [Crossref]
  13. S. Fray, F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. 153, 357 (1967); see also F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Opt. Acta 14, 35 (1967).
    [Crossref]
  14. E. Wolf and C. L. Mehta, Phys. Rev. Letters 13, 705 (1964).
    [Crossref]
  15. L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
    [Crossref]
  16. G. Bédard, Proc. Phys. Soc. (London) 90, 131 (1967).
    [Crossref]
  17. See, for example, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Co., New York, 1963), p. 947.
  18. See, for example, I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), p. 1038.
  19. G. Lachs, Phys. Rev. 138, B1012 (1965).
    [Crossref]
  20. R. J. Glauber, in Physics of Quantum Electronics, P. L. Kelley, B. Lax, and P. E. Tannenwald, Eds. (McGraw-Hill Book Co., New York, 1965), p. 788.
  21. See, for example, M. G. Kendall and A. Stuart, The Advanced Theory of Statistics (Hafner Publ. Co., New York, 1958), Vol. I, p. 148.
  22. W. P. Elderton, Frequency Curves and Correlation (Cambridge Univ. Press, London, 1938).

1967 (2)

S. Fray, F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. 153, 357 (1967); see also F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Opt. Acta 14, 35 (1967).
[Crossref]

G. Bédard, Proc. Phys. Soc. (London) 90, 131 (1967).
[Crossref]

1966 (6)

C. Freed and H. A. Haus, IEEE J. Quantum Electronics QE-2, 190 (1966).
[Crossref]

A. W. Smith and J. A. Armstrong, Phys. Letters 19, 650 (1966); see also Phys. Rev. Letters 16, 1169 (1966).
[Crossref]

F. T. Arecchi, A. Berné, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966).
[Crossref]

F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. Letters 16, 589 (1966).
[Crossref]

W. Martienssen and E. Spiller, Phys. Rev. Letters 16, 531 (1966).
[Crossref]

P. J. Magill and R. P. Soni, Phys. Rev. Letters 16, 911 (1966).
[Crossref]

1965 (4)

F. T. Arecchi, Phys. Rev. Letters 15, 912 (1965).
[Crossref]

C. Freed and H. A. Haus, Phys. Rev. Letters 15, 943 (1965).
[Crossref]

G. Lachs, Phys. Rev. 138, B1012 (1965).
[Crossref]

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

1964 (1)

E. Wolf and C. L. Mehta, Phys. Rev. Letters 13, 705 (1964).
[Crossref]

1958 (1)

L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958); Proc. Phys. Soc. (London) 74, 233 (1959); Proc. Phys. Soc. (London) 81, 1104 (1963).
[Crossref]

Arecchi, F. T.

F. T. Arecchi, A. Berné, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966).
[Crossref]

F. T. Arecchi, Phys. Rev. Letters 15, 912 (1965).
[Crossref]

Armstrong, J. A.

A. W. Smith and J. A. Armstrong, Phys. Letters 19, 650 (1966); see also Phys. Rev. Letters 16, 1169 (1966).
[Crossref]

Bédard, G.

G. Bédard, Proc. Phys. Soc. (London) 90, 131 (1967).
[Crossref]

Berné, A.

F. T. Arecchi, A. Berné, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966).
[Crossref]

Bulamacchi, P.

F. T. Arecchi, A. Berné, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966).
[Crossref]

Elderton, W. P.

W. P. Elderton, Frequency Curves and Correlation (Cambridge Univ. Press, London, 1938).

Feshbach, H.

See, for example, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Co., New York, 1963), p. 947.

Fray, S.

S. Fray, F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. 153, 357 (1967); see also F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Opt. Acta 14, 35 (1967).
[Crossref]

Freed, C.

C. Freed and H. A. Haus, IEEE J. Quantum Electronics QE-2, 190 (1966).
[Crossref]

C. Freed and H. A. Haus, Phys. Rev. Letters 15, 943 (1965).
[Crossref]

C. Freed and H. A. Haus, in Physics of Quantum Electronics, P. L. Kelley, B. Lax, and P. E. Tannenwald, Eds. (McGraw-Hill Book Co., New York, 1965), p. 715.

Glauber, R. J.

R. J. Glauber, in Physics of Quantum Electronics, P. L. Kelley, B. Lax, and P. E. Tannenwald, Eds. (McGraw-Hill Book Co., New York, 1965), p. 788.

Gradshteyn, I. S.

See, for example, I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), p. 1038.

Haus, H. A.

C. Freed and H. A. Haus, IEEE J. Quantum Electronics QE-2, 190 (1966).
[Crossref]

C. Freed and H. A. Haus, Phys. Rev. Letters 15, 943 (1965).
[Crossref]

C. Freed and H. A. Haus, in Physics of Quantum Electronics, P. L. Kelley, B. Lax, and P. E. Tannenwald, Eds. (McGraw-Hill Book Co., New York, 1965), p. 715.

Johnson, F. A.

S. Fray, F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. 153, 357 (1967); see also F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Opt. Acta 14, 35 (1967).
[Crossref]

F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. Letters 16, 589 (1966).
[Crossref]

F. A. Johnson, T. P. McLean, and E. R. Pike, in Physics of Quantum Electronics, P. L. Kelley, B. Lax, and P. E. Tannenwald, Eds. (McGraw-Hill Book Co., New York, 1965), p. 706.

Jones, R.

S. Fray, F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. 153, 357 (1967); see also F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Opt. Acta 14, 35 (1967).
[Crossref]

F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. Letters 16, 589 (1966).
[Crossref]

Kendall, M. G.

See, for example, M. G. Kendall and A. Stuart, The Advanced Theory of Statistics (Hafner Publ. Co., New York, 1958), Vol. I, p. 148.

Lachs, G.

G. Lachs, Phys. Rev. 138, B1012 (1965).
[Crossref]

Magill, P. J.

P. J. Magill and R. P. Soni, Phys. Rev. Letters 16, 911 (1966).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958); Proc. Phys. Soc. (London) 74, 233 (1959); Proc. Phys. Soc. (London) 81, 1104 (1963).
[Crossref]

L. Mandel, in Progress in Optics, Vol. II, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1963), p. 181.
[Crossref]

Martienssen, W.

W. Martienssen and E. Spiller, Phys. Rev. Letters 16, 531 (1966).
[Crossref]

McLean, T. P.

S. Fray, F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. 153, 357 (1967); see also F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Opt. Acta 14, 35 (1967).
[Crossref]

F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. Letters 16, 589 (1966).
[Crossref]

F. A. Johnson, T. P. McLean, and E. R. Pike, in Physics of Quantum Electronics, P. L. Kelley, B. Lax, and P. E. Tannenwald, Eds. (McGraw-Hill Book Co., New York, 1965), p. 706.

Mehta, C. L.

E. Wolf and C. L. Mehta, Phys. Rev. Letters 13, 705 (1964).
[Crossref]

Morse, P. M.

See, for example, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Co., New York, 1963), p. 947.

Pike, E. R.

S. Fray, F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. 153, 357 (1967); see also F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Opt. Acta 14, 35 (1967).
[Crossref]

F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. Letters 16, 589 (1966).
[Crossref]

F. A. Johnson, T. P. McLean, and E. R. Pike, in Physics of Quantum Electronics, P. L. Kelley, B. Lax, and P. E. Tannenwald, Eds. (McGraw-Hill Book Co., New York, 1965), p. 706.

Ryzhik, I. M.

See, for example, I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), p. 1038.

Smith, A. W.

A. W. Smith and J. A. Armstrong, Phys. Letters 19, 650 (1966); see also Phys. Rev. Letters 16, 1169 (1966).
[Crossref]

Soni, R. P.

P. J. Magill and R. P. Soni, Phys. Rev. Letters 16, 911 (1966).
[Crossref]

Spiller, E.

W. Martienssen and E. Spiller, Phys. Rev. Letters 16, 531 (1966).
[Crossref]

Stuart, A.

See, for example, M. G. Kendall and A. Stuart, The Advanced Theory of Statistics (Hafner Publ. Co., New York, 1958), Vol. I, p. 148.

Wolf, E.

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

E. Wolf and C. L. Mehta, Phys. Rev. Letters 13, 705 (1964).
[Crossref]

IEEE J. Quantum Electronics (1)

C. Freed and H. A. Haus, IEEE J. Quantum Electronics QE-2, 190 (1966).
[Crossref]

Phys. Letters (1)

A. W. Smith and J. A. Armstrong, Phys. Letters 19, 650 (1966); see also Phys. Rev. Letters 16, 1169 (1966).
[Crossref]

Phys. Rev. (2)

S. Fray, F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. 153, 357 (1967); see also F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Opt. Acta 14, 35 (1967).
[Crossref]

G. Lachs, Phys. Rev. 138, B1012 (1965).
[Crossref]

Phys. Rev. Letters (7)

E. Wolf and C. L. Mehta, Phys. Rev. Letters 13, 705 (1964).
[Crossref]

F. T. Arecchi, Phys. Rev. Letters 15, 912 (1965).
[Crossref]

C. Freed and H. A. Haus, Phys. Rev. Letters 15, 943 (1965).
[Crossref]

F. T. Arecchi, A. Berné, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966).
[Crossref]

F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. Letters 16, 589 (1966).
[Crossref]

W. Martienssen and E. Spiller, Phys. Rev. Letters 16, 531 (1966).
[Crossref]

P. J. Magill and R. P. Soni, Phys. Rev. Letters 16, 911 (1966).
[Crossref]

Proc. Phys. Soc. (London) (2)

L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958); Proc. Phys. Soc. (London) 74, 233 (1959); Proc. Phys. Soc. (London) 81, 1104 (1963).
[Crossref]

G. Bédard, Proc. Phys. Soc. (London) 90, 131 (1967).
[Crossref]

Rev. Mod. Phys. (1)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

Other (8)

R. J. Glauber, in Physics of Quantum Electronics, P. L. Kelley, B. Lax, and P. E. Tannenwald, Eds. (McGraw-Hill Book Co., New York, 1965), p. 788.

See, for example, M. G. Kendall and A. Stuart, The Advanced Theory of Statistics (Hafner Publ. Co., New York, 1958), Vol. I, p. 148.

W. P. Elderton, Frequency Curves and Correlation (Cambridge Univ. Press, London, 1938).

L. Mandel, in Progress in Optics, Vol. II, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1963), p. 181.
[Crossref]

F. A. Johnson, T. P. McLean, and E. R. Pike, in Physics of Quantum Electronics, P. L. Kelley, B. Lax, and P. E. Tannenwald, Eds. (McGraw-Hill Book Co., New York, 1965), p. 706.

C. Freed and H. A. Haus, in Physics of Quantum Electronics, P. L. Kelley, B. Lax, and P. E. Tannenwald, Eds. (McGraw-Hill Book Co., New York, 1965), p. 715.

See, for example, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Co., New York, 1963), p. 947.

See, for example, I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), p. 1038.

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Figures (1)

Fig. 1
Fig. 1

Probability density for the intensity of a single-mode laser operating below the threshold of oscillation, corresponding to a typical photocount distribution.

Tables (3)

Tables Icon

Table I Parameters of the calculated probability density for laser below threshold.

Tables Icon

Table II Number of samples S(n) vs photoelectron count n for laser below threshold.

Tables Icon

Table III Comparison of the probability density P ˜(W) predicted by the present analysis and the probability density PM(W) proposed by Freed and Haus7 in connection with the photocount data under consideration.

Equations (40)

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p ( n ) = 0 W n n ! exp [ - W ] P ( W ) d W .
W = α 0 T I ( t ) d t ;
M ( n ) = a b W n f ( W ) ρ ( W ) d W ,
P ( W ) = k = 0 c k L k ( W ) .
c k = r = 0 k ( - 1 ) r ( k r ) p ( r ) .
P ( W ) = k = 0 r = 0 k ( - 1 ) r ( k r ) p ( r ) L k ( W ) .
p ( m ) = k = 0 m ( - 1 ) k ( m k ) c k .
p ( n ) = n n / [ n + 1 ] n + 1 ,
c k = [ n + 1 ] - k - 1 .
P ( W ) = ( 1 / W ) exp [ - W / W ] ,
n = 0 n ! L n α ( x ) L n α ( y ) z n Γ ( n + α + 1 ) = ( x y z ) - α / 2 ( 1 - z ) I α ( 2 ( x y z ) 1 2 1 - z ) × exp [ - z ( x + y ) 1 - z ] ,
p ( n ) = ( n n / n ! ) exp ( - n ) .
c k = exp ( - n ) L k ( n ) ,
P ( W ) = δ ( W - W ) ,
p ( n ) = n T n [ n T + 1 ] n + 1 L n ( - n c n T ( n T + 1 ) ) × exp [ - n c n T + 1 ] ,
c k = [ n T + 1 ] - k - 1 L k ( n c n T + 1 ) exp [ - n c n T + 1 ] .
P ( W ) = 1 n T exp [ - ( W + n c n T ) ] I 0 ( 2 ( W n c ) 1 2 n T ) ,
p M ( n ) = j = 1 M a j exp ( - b j n ) .
c k = j = 1 M a j [ 1 - exp ( - b j ) ] k .
P ( W ) = j = 1 M a j exp [ b j - { exp ( b j ) - 1 } W ] .
n [ k ] = n = 0 n ( n - 1 ) ( n - k + 1 ) p ( n )
μ k = 0 W k P ( W ) d W
n [ k ] = μ k .
d f d x = [ x - A a 0 + a 1 x + a 2 x 2 ] f ( x ) .
p ˜ ( n ) = 0 W n n ! exp ( - W ) P ˜ ( W ) d W .
d P ˜ d W = [ ( W - A ) / j = 0 N a j W j ] P ˜ ( W ) ,
[ j = 0 N a j W j ] d P ˜ d W = ( W - A ) P ˜ ( W ) ,
j = 0 N ( k + j ) a j n [ k + j - 1 ] + n [ k + 1 ] - A n [ k ] - a 0 P ˜ ( 0 ) δ k 0 = 0 , ( k = 0 , 1 , ) .
χ 2 = N k { [ p ( k ) - p ˜ ( k ) ] 2 p ˜ ( k ) } ;
P ˜ ( W ) = Z γ a exp ( - Z ) / a Γ ( γ a ) , 0 < Z < = 0 Z < 0 ,
Z = γ ( W - W + μ 2 γ ) ,
γ = 2 μ 2 / μ 3 , a = μ 2 γ - γ - 1 ,
p ˜ ( n ) = 1 n ! [ γ γ + 1 ] γ a + 1 exp ( - y ) × r = 0 n ( n r ) Γ ( γ a + r + 1 ) γ a Γ ( γ a ) y n - r ( γ + 1 ) r ,
y = W - μ 2 γ .
P M ( W ) = ( 1 / n T ) exp [ - ( W - n b ) / n T ] , W n b = 0 , W < n b ,
P ( W ) = k = 0 c k L k ( W ) ,
L k ( W ) = r = 0 k ( k r ) ( - W ) r r ! ,
0 exp ( - W ) L n ( W ) L m ( W ) = δ n m .
c k = 0 exp ( - W ) P ( W ) L k ( W ) d W .
c k = r = 0 k ( - 1 ) r ( k r ) p ( r ) .