Abstract

Photoelectron-counting distributions are obtained for stochastic light that is caused to scintillate by passage through a random medium. The result is applied specifically to transmission of amplitude-stabilized radiation, with and without independent additive background, and of chaotic radiation, through the turbulent atmosphere. The counting distribution is found to broaden markedly and its peak occurs at decreasing count numbers for increasing turbulence. The cases studied here are of particular interest for low-level direct optical communications and radar using single-mode lasers, multimode lasers, thermal sources, and scattering targets. The results are obtained by extending the usual formulation of photoelectron counting; a product of two random variables, the source intensity and the effect of the turbulent medium, rather than the usual single stochastic irradiance, is considered. Plots of the counting distributions for various degrees of turbulence and for several signal-to-noise ratios are presented. A possible explanation is given for the observed decrease of the log-amplitude variance to values below the saturation value in long path length and high turbulence experiments.

© 1970 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill, New York, 1961).
  2. D. L. Fried, J. Opt. Soc. Am. 57, 169 (1967).
    [Crossref]
  3. D. L. Fried, G. E. Mevers, and M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967).
    [Crossref]
  4. D. L. Fried and R. A. Schmeltzer, Appl. Opt. 6, 1729 (1967).
    [Crossref] [PubMed]
  5. G. R. Ochs and R. S. Lawrence, J. Opt. Soc. Am. 59, 226 (1969).
    [Crossref]
  6. G. R. Ochs, R. R. Bergman, and J. R. Snyder, J. Opt. Soc. Am. 59, 231 (1969).
    [Crossref]
  7. P. H. Deitz and N. J. Wright, J. Opt. Soc. Am. 59, 527 (1969).
    [Crossref]
  8. G. E. Mevers, M. P. Keister, and D. L. Fried, J. Opt. Soc. Am. 59, 491A (1969).
  9. D. A. deWolf, J. Opt. Soc. Am. 59, 1455 (1969).
    [Crossref]
  10. J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
    [Crossref]
  11. W. K. Pratt, Laser Communications Systems (Wiley, New York, 1969), p. 145.
  12. L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958).
    [Crossref]
  13. J. A. Armstrong and A. W. Smith, in Progress in Optics VI, edited by E. Wolf (North-Holland, Amsterdam, 1967), p. 213
  14. M. C. Teich and P. Diament, J. Appl. Phys. 40, 625 (1969); Phys. Letters 30A, 93 (1969); J. Appl. Phys. 41, 415 (1970).
    [Crossref]
  15. R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, Phys. Rev. 178, 612 (1969).
    [Crossref]
  16. E. Jakeman and E. R. Pike, J. Phys. A2, Ser. 2, 115 (1969)
  17. J. Peřina and R. Horák, J. Phys. A2, Ser. 2, 702 (1969).
  18. P. Diament and M. C. Teich, J. Opt. Soc. Am. 60, 682 (1970).
    [Crossref]
  19. M. C. Teich, in Semiconductors and Semimetals, edited by R. K. Willardson and A. C. Beer (Academic, New York, 1970), Vol. 5 (Infrared Detectors), p. 361. See also Proc. IEEE 57, 786 (1969) and Appl. Phys. Letters 14, 201 (1969).
    [Crossref]
  20. G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
    [Crossref]

1970 (1)

1969 (9)

P. H. Deitz and N. J. Wright, J. Opt. Soc. Am. 59, 527 (1969).
[Crossref]

M. C. Teich and P. Diament, J. Appl. Phys. 40, 625 (1969); Phys. Letters 30A, 93 (1969); J. Appl. Phys. 41, 415 (1970).
[Crossref]

R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, Phys. Rev. 178, 612 (1969).
[Crossref]

E. Jakeman and E. R. Pike, J. Phys. A2, Ser. 2, 115 (1969)

J. Peřina and R. Horák, J. Phys. A2, Ser. 2, 702 (1969).

G. E. Mevers, M. P. Keister, and D. L. Fried, J. Opt. Soc. Am. 59, 491A (1969).

G. R. Ochs and R. S. Lawrence, J. Opt. Soc. Am. 59, 226 (1969).
[Crossref]

G. R. Ochs, R. R. Bergman, and J. R. Snyder, J. Opt. Soc. Am. 59, 231 (1969).
[Crossref]

D. A. deWolf, J. Opt. Soc. Am. 59, 1455 (1969).
[Crossref]

1968 (1)

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[Crossref]

1967 (4)

1958 (1)

L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958).
[Crossref]

Alley, C. O.

R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, Phys. Rev. 178, 612 (1969).
[Crossref]

Armstrong, J. A.

J. A. Armstrong and A. W. Smith, in Progress in Optics VI, edited by E. Wolf (North-Holland, Amsterdam, 1967), p. 213

Bédard, G.

G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
[Crossref]

Bergman, R. R.

Chang, J. C.

G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
[Crossref]

Chang, R. F.

R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, Phys. Rev. 178, 612 (1969).
[Crossref]

Deitz, P. H.

Detenbeck, R. W.

R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, Phys. Rev. 178, 612 (1969).
[Crossref]

deWolf, D. A.

Diament, P.

P. Diament and M. C. Teich, J. Opt. Soc. Am. 60, 682 (1970).
[Crossref]

M. C. Teich and P. Diament, J. Appl. Phys. 40, 625 (1969); Phys. Letters 30A, 93 (1969); J. Appl. Phys. 41, 415 (1970).
[Crossref]

Fried, D. L.

Horák, R.

J. Peřina and R. Horák, J. Phys. A2, Ser. 2, 702 (1969).

Jakeman, E.

E. Jakeman and E. R. Pike, J. Phys. A2, Ser. 2, 115 (1969)

Keister, M. P.

G. E. Mevers, M. P. Keister, and D. L. Fried, J. Opt. Soc. Am. 59, 491A (1969).

D. L. Fried, G. E. Mevers, and M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967).
[Crossref]

Korenman, V.

R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, Phys. Rev. 178, 612 (1969).
[Crossref]

Lawrence, R. S.

Mandel, L.

G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
[Crossref]

L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958).
[Crossref]

Mevers, G. E.

G. E. Mevers, M. P. Keister, and D. L. Fried, J. Opt. Soc. Am. 59, 491A (1969).

D. L. Fried, G. E. Mevers, and M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967).
[Crossref]

Ochs, G. R.

Perina, J.

J. Peřina and R. Horák, J. Phys. A2, Ser. 2, 702 (1969).

Pike, E. R.

E. Jakeman and E. R. Pike, J. Phys. A2, Ser. 2, 115 (1969)

Pratt, W. K.

W. K. Pratt, Laser Communications Systems (Wiley, New York, 1969), p. 145.

Schmeltzer, R. A.

Smith, A. W.

J. A. Armstrong and A. W. Smith, in Progress in Optics VI, edited by E. Wolf (North-Holland, Amsterdam, 1967), p. 213

Snyder, J. R.

Strohbehn, J. W.

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[Crossref]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill, New York, 1961).

Teich, M. C.

P. Diament and M. C. Teich, J. Opt. Soc. Am. 60, 682 (1970).
[Crossref]

M. C. Teich and P. Diament, J. Appl. Phys. 40, 625 (1969); Phys. Letters 30A, 93 (1969); J. Appl. Phys. 41, 415 (1970).
[Crossref]

M. C. Teich, in Semiconductors and Semimetals, edited by R. K. Willardson and A. C. Beer (Academic, New York, 1970), Vol. 5 (Infrared Detectors), p. 361. See also Proc. IEEE 57, 786 (1969) and Appl. Phys. Letters 14, 201 (1969).
[Crossref]

Wright, N. J.

Appl. Opt. (1)

J. Appl. Phys. (1)

M. C. Teich and P. Diament, J. Appl. Phys. 40, 625 (1969); Phys. Letters 30A, 93 (1969); J. Appl. Phys. 41, 415 (1970).
[Crossref]

J. Opt. Soc. Am. (8)

J. Phys. (2)

E. Jakeman and E. R. Pike, J. Phys. A2, Ser. 2, 115 (1969)

J. Peřina and R. Horák, J. Phys. A2, Ser. 2, 702 (1969).

Phys. Rev. (2)

R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, Phys. Rev. 178, 612 (1969).
[Crossref]

G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
[Crossref]

Proc. IEEE (1)

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[Crossref]

Proc. Phys. Soc. (London) (1)

L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958).
[Crossref]

Other (4)

J. A. Armstrong and A. W. Smith, in Progress in Optics VI, edited by E. Wolf (North-Holland, Amsterdam, 1967), p. 213

W. K. Pratt, Laser Communications Systems (Wiley, New York, 1969), p. 145.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill, New York, 1961).

M. C. Teich, in Semiconductors and Semimetals, edited by R. K. Willardson and A. C. Beer (Academic, New York, 1970), Vol. 5 (Infrared Detectors), p. 361. See also Proc. IEEE 57, 786 (1969) and Appl. Phys. Letters 14, 201 (1969).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Photoelectron counting distributions for amplitude-stabilized light, after transmission through the atmosphere. The level of turbulence is indicated by the logarithmic-irradiance standard deviation σ. The mean count is N = 50 for all cases shown.

Fig. 2
Fig. 2

Photoelectron-counting distributions for light from a chaotic source after transmission through the atmosphere. The level of turbulence is indicated by the logarithmic-irradiance standard deviation σ. The mean count is N = 50 for all cases shown.

Fig. 3
Fig. 3

Photoelectron-counting distributions for combinations of amplitude-stabilized light after passage through a turbulent atmosphere (σ = 1) with independent, noninterfering, Poisson-distributed noise counts. The combined mean count is N = 20 for each SNR.

Equations (20)

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

p ( n ) = ( E n / n ! ) e - E ,
E = t t + T α ( t ) I ( t ) d t
p 0 ( n , N ) = ( N n / n ! ) e - N .
p 0 ( n , N ) = N n / ( N + 1 ) n + 1 ,
P ( K ) = exp ( - [ ln ( K / N ) + 1 2 σ 2 ] 2 / 2 σ 2 ) / ( 2 π ) 1 2 σ K ,
p ( n , σ , N ) = 0 p 0 ( n , K ) P ( K ) d K ,
q m ( n , N ) = m ln p 0 ( n , N ) / ( ln N ) m
ln M = ln N - 1 2 σ 2 + σ 2 q 1 ( n , M ) .
x = [ ln ( K / M ) ] / σ
P ( K ) d K = exp [ - 1 2 ( x + σ q 1 ) 2 ] d x / ( 2 π ) 1 2
p 0 ( n , K ) = p 0 ( n , M ) exp ( x σ q 1 + 1 2 x 2 σ 2 q 2 ) exp R ( x ) ,
p ( n , σ , N ) = p 0 ( n , M ) exp [ - 1 2 σ 2 q 1 2 ( n , M ) ] [ 1 - σ 2 q 2 ( n , M ) ] 1 2 .
q 1 ( n , N ) = n - N ,             q 2 ( n , N ) = - N ,
ln M = ln N + σ 2 ( n - 1 2 - M )
p ( n , σ , N ) = M n e - M exp [ - 1 2 σ 2 ( M - n ) 2 ] n ! [ 1 + σ 2 M ] 1 2 .
q 1 ( n , N ) = ( n - N ) / ( N + 1 ) ,
q 2 ( n , N ) = - ( n + 1 ) N / ( N + 1 ) 2 .
ln M = ln N - 1 2 σ 2 + σ 2 ( n - M ) / ( M + 1 )
p ( n , σ , N ) = [ M / ( M + 1 ) ] n exp ( - 1 2 σ 2 [ ( M - n ) / ( M + 1 ) ] 2 ) [ ( M + 1 ) 2 + σ 2 ( n + 1 ) M ] 1 2 .
p S + N ( n , σ , N ) = m = 0 n p S [ m , σ , N y / ( 1 + y ) ] p N [ n - m , N / ( 1 + y ) ] ,