Abstract

A laser beam was passed through a turbulent mixture of hot and cold water. The optical path contained no solid scattering particles. It was, therefore, similar to an atmospheric path containing only clear-air turbulence. The field irradiance was accurately described by a truncated three-parameter log-normal probability density. The statistics of the integral of the field irradiance over an area comparable in size to one coherence area was also always characterized by a three-parameter log-normal density. This probability density, when used in the Mandel formula for the single-photon-counting probability, correctly predicted the observed results. The parameters of the irradiance density could be varied by changing the temperature of the hot- and cold-water mixture, or the position of the optical path. The observed photon-counting probabilities could not be obtained by use of a two-parameter log-normal irradiance density (more commonly called the log-normal density) in the Mandel formula.

© 1975 Optical Society of America

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References

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  1. R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523 (1970). Review paper, containing extensive references to works published until 1970.
    [Crossref]
  2. J. W. Strohbehn, in Progress in Optics, IX, edited by E. Wolf (North–Holland, Amsterdam, 1971), p. 75 (review paper).
  3. J. W. Strohbehn, Proc. IEEE 56, 1301 (1968) (review paper).
    [Crossref]
  4. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (translated from Russian) (McGraw–Hill, New York, 1961).
  5. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, IPST Catalog No. 5319 (translated from Russian) (National Technical Information Service, U. S. Dept. of Commerce, Springfield, Va., 1971).
  6. L. A. Chernov, Wave Propagation in a Randon Medium (translated from Russian) (McGraw–Hill, New York, 1960).
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    [Crossref]
  8. D. A. de Wolf, J. Opt. Soc. Am. 59, 1455 (1969).
    [Crossref]
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  10. D. A. de Wolf, J. Opt. Soc. Am. 63, 1249 (1973).
    [Crossref]
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  12. K. Furutsu, J. Opt. Soc. Am. 62, 240 (1972).
    [Crossref]
  13. V. I. Klyatskin and V. I. Tatarskii, Zh. Eksp. Teor. Fiz. 58, 624 (1970) [Sov. Phys.–JETP 31, 335 (1970)].
  14. V. I. Klyatskin, Zh. Eksp. Teor. Fiz. 57, 952 (1969) [Sov. Phys. –JETP 30, 520 (1970)].
  15. V. I. Klyatskin, Zh. Eksp. Teor. Fiz. 60, 1300 (1971) [Sov. Phys.–JETP 33, 703 (1971)].
  16. G. R. Ochs, R. R. Bergman, and J. R. Snyder, J. Opt. Soc. Am. 59, 231 (1969).
    [Crossref]
  17. G. R. Ochs and R. S. Lawrence, J. Opt. Soc. Am. 59, 226 (1969).
    [Crossref]
  18. J. R. Kerr, J. Opt. Soc. Am. 62, 1040 (1972).
    [Crossref]
  19. J. R. Dunphy and J. R. Kerr, J. Opt. Soc. Am. 63, 981 (1973).
    [Crossref]
  20. M. E. Gracheva and A. S. Gurvich, Izv. Vyssh. Ucheb. Zaved. Radiofiz. 8, 717 (1965).
  21. M. E. Gracheva, Izv. Vyssh. Ucheb. Zaved. Radiofiz. 10, 775 (1967).
  22. M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, Izv. Vyssh. Ucheb. Zaved. Radiofiz. 13, 55 (1970).
  23. P. H. Deitz and N. J. Wright, J. Opt. Soc. Am. 59, 527 (1969).
    [Crossref]
  24. M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, Similarity Correlations and Their Experimental Verification in the Case of Strong Intensity Fluctuations of Laser Radiation[Akademiia Nauk. SSSR, Moscow (1973)] (translated from Russian), (The Aerospace Corp. Library Services, Literature Research Group, Los Angeles, Calif., Translation No. LRG-73-T-28).
  25. J. W. Strohbehn and T-i. Wang, J. Opt. Soc. Am. 62, 1061 (1972).
    [Crossref]
  26. T-i. Wang and J. W. Strohbehn, J. Opt. Soc. Am. 64, 583 (1974).
    [Crossref]
  27. T-i. Wang and J. W. Strohbehn, J. Opt. Soc. Am. 64, 994 (1974).
    [Crossref]
  28. D. L. Fried, G. E. Mevers, and M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967).
    [Crossref]
  29. J. R. Kerr and J. R. Dunphy, J. Opt. Soc. Am. 63, 1 (1973).
    [Crossref]
  30. J. A. Armstrong and A. W. Smith, in Progress in Optics, VI, edited by E. Wolf (North-Holland, Amsterdam, 1967), p. 213.
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  32. L. Mandel, Proc. Phys. Soc. (Lond.) 72, 1037 (1958).
    [Crossref]
  33. L. Mandel, in Progress in Optics, II, edited by E. Wolf (North–Holland, Amsterdam, 1963), p. 181.
    [Crossref]
  34. J. Aitchison and J. A. C. Brown, The Lognormal Distribution (Cambridge U. P., Cambridge, 1957).
  35. P. Diament and M. C. Teich, J. Opt. Soc. Am. 60, 1489 (1970).
    [Crossref]
  36. B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).
  37. Reference 34, Ch. 9.
  38. R. J. Glauber, in Physics of Quantum Electronics, Conference Proceedings, edited by P. L. Kelley, B. Lax, and P. E. Tannenwald (McGraw–Hill, New York, 1966), p. 788.
  39. J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968).
  40. R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, Phys. Rev. 178, 612 (1969).
    [Crossref]
  41. E. Jakeman, E. R. Pike, and S. Swain, J. Phys. A (Lond.) 4, 517 (1971).
  42. H. Tennekes and J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, Mass., 1972).

1974 (3)

1973 (4)

1972 (3)

1971 (2)

E. Jakeman, E. R. Pike, and S. Swain, J. Phys. A (Lond.) 4, 517 (1971).

V. I. Klyatskin, Zh. Eksp. Teor. Fiz. 60, 1300 (1971) [Sov. Phys.–JETP 33, 703 (1971)].

1970 (4)

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, Izv. Vyssh. Ucheb. Zaved. Radiofiz. 13, 55 (1970).

V. I. Klyatskin and V. I. Tatarskii, Zh. Eksp. Teor. Fiz. 58, 624 (1970) [Sov. Phys.–JETP 31, 335 (1970)].

R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523 (1970). Review paper, containing extensive references to works published until 1970.
[Crossref]

P. Diament and M. C. Teich, J. Opt. Soc. Am. 60, 1489 (1970).
[Crossref]

1969 (6)

1968 (2)

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

D. A. de Wolf, J. Opt. Soc. Am. 58, 461 (1968).
[Crossref]

1967 (2)

M. E. Gracheva, Izv. Vyssh. Ucheb. Zaved. Radiofiz. 10, 775 (1967).

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

1965 (1)

M. E. Gracheva and A. S. Gurvich, Izv. Vyssh. Ucheb. Zaved. Radiofiz. 8, 717 (1965).

1958 (1)

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

Aitchison, J.

J. Aitchison and J. A. C. Brown, The Lognormal Distribution (Cambridge U. P., Cambridge, 1957).

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.

Bergman, R. R.

Brown, J. A. C.

J. Aitchison and J. A. C. Brown, The Lognormal Distribution (Cambridge U. P., Cambridge, 1957).

Carnahan, B.

B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Chang, R. F.

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

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Randon Medium (translated from Russian) (McGraw–Hill, New York, 1960).

de Wolf, D. A.

Deitz, P. H.

Detenbeck, R. W.

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

Diament, P.

Dunphy, J. R.

Fried, D. L.

Furutsu, K.

Glauber, R. J.

R. J. Glauber, in Physics of Quantum Electronics, Conference Proceedings, edited by P. L. Kelley, B. Lax, and P. E. Tannenwald (McGraw–Hill, New York, 1966), p. 788.

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, Izv. Vyssh. Ucheb. Zaved. Radiofiz. 13, 55 (1970).

M. E. Gracheva, Izv. Vyssh. Ucheb. Zaved. Radiofiz. 10, 775 (1967).

M. E. Gracheva and A. S. Gurvich, Izv. Vyssh. Ucheb. Zaved. Radiofiz. 8, 717 (1965).

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, Similarity Correlations and Their Experimental Verification in the Case of Strong Intensity Fluctuations of Laser Radiation[Akademiia Nauk. SSSR, Moscow (1973)] (translated from Russian), (The Aerospace Corp. Library Services, Literature Research Group, Los Angeles, Calif., Translation No. LRG-73-T-28).

Gurvich, A. S.

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, Izv. Vyssh. Ucheb. Zaved. Radiofiz. 13, 55 (1970).

M. E. Gracheva and A. S. Gurvich, Izv. Vyssh. Ucheb. Zaved. Radiofiz. 8, 717 (1965).

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, Similarity Correlations and Their Experimental Verification in the Case of Strong Intensity Fluctuations of Laser Radiation[Akademiia Nauk. SSSR, Moscow (1973)] (translated from Russian), (The Aerospace Corp. Library Services, Literature Research Group, Los Angeles, Calif., Translation No. LRG-73-T-28).

Jakeman, E.

E. Jakeman, E. R. Pike, and S. Swain, J. Phys. A (Lond.) 4, 517 (1971).

Kallistratova, M. A.

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, Izv. Vyssh. Ucheb. Zaved. Radiofiz. 13, 55 (1970).

Kashkarov, S. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, Similarity Correlations and Their Experimental Verification in the Case of Strong Intensity Fluctuations of Laser Radiation[Akademiia Nauk. SSSR, Moscow (1973)] (translated from Russian), (The Aerospace Corp. Library Services, Literature Research Group, Los Angeles, Calif., Translation No. LRG-73-T-28).

Keister, M. P.

Kerr, J. R.

Klauder, J. R.

J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968).

Klyatskin, V. I.

V. I. Klyatskin, Zh. Eksp. Teor. Fiz. 60, 1300 (1971) [Sov. Phys.–JETP 33, 703 (1971)].

V. I. Klyatskin and V. I. Tatarskii, Zh. Eksp. Teor. Fiz. 58, 624 (1970) [Sov. Phys.–JETP 31, 335 (1970)].

V. I. Klyatskin, Zh. Eksp. Teor. Fiz. 57, 952 (1969) [Sov. Phys. –JETP 30, 520 (1970)].

Korenman, V.

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

Lawrence, R. S.

R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523 (1970). Review paper, containing extensive references to works published until 1970.
[Crossref]

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

Lumley, J. L.

H. Tennekes and J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, Mass., 1972).

Luther, H. A.

B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Mandel, L.

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

L. Mandel, in Progress in Optics, II, edited by E. Wolf (North–Holland, Amsterdam, 1963), p. 181.
[Crossref]

Mevers, G. E.

Ochs, G. R.

Pike, E. R.

E. Jakeman, E. R. Pike, and S. Swain, J. Phys. A (Lond.) 4, 517 (1971).

Pokasov, V. V.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, Similarity Correlations and Their Experimental Verification in the Case of Strong Intensity Fluctuations of Laser Radiation[Akademiia Nauk. SSSR, Moscow (1973)] (translated from Russian), (The Aerospace Corp. Library Services, Literature Research Group, Los Angeles, Calif., Translation No. LRG-73-T-28).

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.

T-i. Wang and J. W. Strohbehn, J. Opt. Soc. Am. 64, 994 (1974).
[Crossref]

T-i. Wang and J. W. Strohbehn, J. Opt. Soc. Am. 64, 583 (1974).
[Crossref]

J. W. Strohbehn and T-i. Wang, J. Opt. Soc. Am. 62, 1061 (1972).
[Crossref]

R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523 (1970). Review paper, containing extensive references to works published until 1970.
[Crossref]

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

J. W. Strohbehn, in Progress in Optics, IX, edited by E. Wolf (North–Holland, Amsterdam, 1971), p. 75 (review paper).

Sudarshan, E. C. G.

J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968).

Swain, S.

E. Jakeman, E. R. Pike, and S. Swain, J. Phys. A (Lond.) 4, 517 (1971).

Tatarskii, V. I.

V. I. Klyatskin and V. I. Tatarskii, Zh. Eksp. Teor. Fiz. 58, 624 (1970) [Sov. Phys.–JETP 31, 335 (1970)].

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (translated from Russian) (McGraw–Hill, New York, 1961).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, IPST Catalog No. 5319 (translated from Russian) (National Technical Information Service, U. S. Dept. of Commerce, Springfield, Va., 1971).

Teich, M. C.

Tennekes, H.

H. Tennekes and J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, Mass., 1972).

Wang, T-i.

Wilkes, J. O.

B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Wright, N. J.

Izv. Vyssh. Ucheb. Zaved. Radiofiz. (3)

M. E. Gracheva and A. S. Gurvich, Izv. Vyssh. Ucheb. Zaved. Radiofiz. 8, 717 (1965).

M. E. Gracheva, Izv. Vyssh. Ucheb. Zaved. Radiofiz. 10, 775 (1967).

M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, Izv. Vyssh. Ucheb. Zaved. Radiofiz. 13, 55 (1970).

J. Opt. Soc. Am. (17)

J. Phys. A (Lond.) (1)

E. Jakeman, E. R. Pike, and S. Swain, J. Phys. A (Lond.) 4, 517 (1971).

Phys. Rev. (1)

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

Proc. IEEE (2)

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

R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523 (1970). Review paper, containing extensive references to works published until 1970.
[Crossref]

Proc. Phys. Soc. (Lond.) (1)

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

Zh. Eksp. Teor. Fiz. (3)

V. I. Klyatskin and V. I. Tatarskii, Zh. Eksp. Teor. Fiz. 58, 624 (1970) [Sov. Phys.–JETP 31, 335 (1970)].

V. I. Klyatskin, Zh. Eksp. Teor. Fiz. 57, 952 (1969) [Sov. Phys. –JETP 30, 520 (1970)].

V. I. Klyatskin, Zh. Eksp. Teor. Fiz. 60, 1300 (1971) [Sov. Phys.–JETP 33, 703 (1971)].

Other (14)

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (translated from Russian) (McGraw–Hill, New York, 1961).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, IPST Catalog No. 5319 (translated from Russian) (National Technical Information Service, U. S. Dept. of Commerce, Springfield, Va., 1971).

L. A. Chernov, Wave Propagation in a Randon Medium (translated from Russian) (McGraw–Hill, New York, 1960).

L. Mandel, in Progress in Optics, II, edited by E. Wolf (North–Holland, Amsterdam, 1963), p. 181.
[Crossref]

J. Aitchison and J. A. C. Brown, The Lognormal Distribution (Cambridge U. P., Cambridge, 1957).

J. W. Strohbehn, in Progress in Optics, IX, edited by E. Wolf (North–Holland, Amsterdam, 1971), p. 75 (review paper).

B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Reference 34, Ch. 9.

R. J. Glauber, in Physics of Quantum Electronics, Conference Proceedings, edited by P. L. Kelley, B. Lax, and P. E. Tannenwald (McGraw–Hill, New York, 1966), p. 788.

J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968).

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

Handbook of Chemistry and Physics, 48th Edition, edited by R. C. Weast (The Chemical Rubber Co., Cleveland, Ohio, 1967), p. E-159.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, Similarity Correlations and Their Experimental Verification in the Case of Strong Intensity Fluctuations of Laser Radiation[Akademiia Nauk. SSSR, Moscow (1973)] (translated from Russian), (The Aerospace Corp. Library Services, Literature Research Group, Los Angeles, Calif., Translation No. LRG-73-T-28).

H. Tennekes and J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, Mass., 1972).

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Figures (5)

FIG. 1
FIG. 1

Photon-counting probability P(n, T) as a function of n, for the measured values E{n} = 1.66, H2 = 1.34, H3 = 8.83, H4 = 63.9, and H5 = 608. T = 25 μs. The solid and dashed curve are theoretical values as computed from Mandel formula under assumptions of a truncated three-parameter or a two-parameter log-normal irradiance probability density, respectively. Computed values were: for solid curve, H4 = 66.4, H5 = 723, σ ln I 2 = 1.52, σ2 = 0.85; for dashed curve, H3 = 11.8, H4 = 163, H5 = 4920, σ ln I 2 = 0.85. The field was sampled over an area of 1.82 × 10−2 mm2, small compared to the field coherence area. The total number of samples was 107. Water temperature was 63 °C; water-flow rate was 2.5 cm3/s.

FIG. 2
FIG. 2

Photon-counting probability P(n, T) as a function of n, obtained with a field-sampling area of 4.87 × 10−1 mm2, comparable to the field coherence area. T = 25 μs. Measured parameters of the counting probability were E{n} = 1.81, H2 = 1.07, H3 = 6.73, H4 = 45.8, H5 = 415. Theoretical values of these parameters obtained from Mandel formula by use of a truncated three-parameter (solid curve) or two-parameter (dashed curve) integrated irradiance probability density were–solid curve, H4 = 49.1, H5 = 552, σ ln W 2 = 1.15, σ2 = 0.73; dashed curve, H3 = 7.84, H4 = 77.2, H5 = 1430, σ ln W 2 = 0.73. Total number of samples was 107. Water temperature was 63 °C; water-flow rate was 2.5 cm3/s.

FIG. 3
FIG. 3

Experimental values of the normalized second factorial moment of the photon-counting probability are plotted as functions of T for two different turbulence conditions. The beam position below the cold-water injection tube was -- ○ data, 3 cm; × data, 1 cm. The two curves with the largest H2 values were obtained with sampling areas of 9.93 × 10−2 mm2, a value small compared to the field coherence area. Those with the lowest H2 values were obtained with sampling areas of 2.48 mm2 a value comparable to the field coherence area. Water temperature: 67 °C; cold-water injection rate: 4 cm3/s.

FIG. 4
FIG. 4

Second-order irradiance-correlation function E{I(t)I(t + τ)}/E2{I} is plotted as a function of delay, τ. The field was sampled over an area small compared to a coherence area. The solid curve is simply a smooth line drawn through the data points.

FIG. 5
FIG. 5

Second through fifth normalized factorial moments of the counting distribution are plotted as functions of field-sampling area for fixed turbulence conditions. All measured data are represented by circles. The curves labeled 3-P, 2-P, and R are theoretical values computed under the assumption of a truncated three-parameter, two-parameter, or Rice–Nakagami irradiance probability density. Points indicated by ⊙ are those for which the truncated three-parameter probability density is identical to the two-parameter log-normal probability density. The Rice–Nakagami probability density cannot exist for H2 > 1. Hot-water temperature: 63 °C; cold-water injection rate: 2.5 cm3/s; beam position: 3 cm below the injection tube. T = 25 μs.

Tables (1)

Tables Icon

TABLE I Typical values of σ2, σ ln I 2, and σ ln I 2, for three different turbulence conditions.

Equations (50)

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

P ( n , T ) = 0 W n n ! e - W p ( W ) d W .
u = c S 0 T I ( r , t ) d S d t ;
E { n ( n - 1 ) ( n - 2 ) ( n - k + 1 ) } = E { W k } .
H k E { n ( n - 1 ) ( n - 2 ) ( n - k + 1 ) } E k { n } - 1 = E { W k } E k { W } - 1 = E { I k } E k { I } - 1.
p ( I ) = 1 ( 2 π σ ln I 2 ) 1 / 2 I exp [ - ( ln I - μ I ) 2 / 2 σ ln I 2 ] ,
σ ln W 2 = σ ln I 2 ,
μ W = ln ( α c S T ) + μ I ,
ln W - μ W = ln I - μ I ,
E { W k } = exp ( k μ W + 1 2 k 2 σ ln W 2 ) .
H k + 1 = E { W k } E k { W } = exp [ 1 2 k ( k - 1 ) σ ln W 2 ] ,             k = 2 , 3 , 4 ,
σ ln W 2 = ln E { W 2 } E 2 { W } = ln ( H 2 + 1 ) ,
μ W = ln ( E { W } ) - 1 2 σ ln W 2 = ln E { n } ( H 2 + 1 ) 1 / 2 .
P ( n , T ) = 0 W n - 1 n ! ( 2 π σ ln W 2 ) 1 / 2 e - W × exp [ - ( ln W - μ W ) 2 / 2 σ ln W 2 ] d W .
σ ln I 2 E { ( ln I - E { ln I } ) 2 } .
σ 2 ln E { I 2 } E 2 { I } .
p ( x ) = 1 ( 2 π σ ln x 2 ) 1 / 2 ( x - τ ) × exp [ - ( ln ( x - τ ) - μ ) 2 / 2 σ ln x 2 ] ,
x = x - τ ,             μ = E { ln ( x - τ ) } ,     σ ln x 2 = E { ( ln ( x - τ ) - μ ) 2 } .
p ( I ) = A I + I 0 exp [ - ( ln ( I + I 0 ) - μ I ) 2 / 2 σ ln I 2 ] ,
A = 1 ( 2 π σ ln I 2 ) 1 / 2 erfc ( y ) ,
y = ln I 0 - μ I ( 2 σ ln I 2 ) 1 / 2 ,
E { W k } = m = 0 k ( - 1 ) k - m ( k m ) W 0 k - m erfc [ y - m ( 1 2 σ ln W 2 ) 1 / 2 ] erfc ( y ) × exp [ m μ W + 1 2 m 2 σ ln W 2 ] ,
y = ln W 0 - μ W ( 2 σ ln W 2 ) 1 / 2 .
erfc [ y - ( 1 2 σ ln W 2 ) 1 / 2 ] erfc ( y ) exp [ μ W + 1 2 σ ln W 2 ] - W 0 - E { W } = 0 ,
erfc [ y - 2 ( 1 2 σ ln W 2 ) 1 / 2 ] erfc ( y ) exp [ 2 μ W + 2 σ ln W 2 ] - W 0 2 - 2 W 0 E { W } - E { W 2 } = 0 ,
erfc [ y - 3 ( 1 2 σ ln W 2 ) 1 / 2 ] erfc ( y ) exp [ 3 μ W + 9 2 σ ln W 2 ] - W 0 3 - 3 W 0 2 E { W } - 3 W 0 E { W 2 } - E { W 3 } = 0.
E { W k } = E { ( W + W 0 ) k } = exp ( k μ W + 1 2 k 2 σ ln W 2 ) ,             k = 1 , 2 , 3 , .
E { W k } = m = 0 k ( - 1 ) k - m ( k m ) W 0 k - m exp [ m μ W + 1 2 m 2 σ ln W 2 ] .
( H 3 - 3 H 2 ) x 3 - 3 H 2 2 x 2 - H 2 3 = 0 ,
W 0 = ( x - 1 ) E { n } ,
σ ln W 2 = ln H 2 + x 2 x 2 ,
μ W = ln ( E { n } + W 0 ) - 1 2 σ ln W 2 .
P ( n , T ) = 0 A W n n ! ( W + W 0 ) e - W × exp [ - ( ln ( W + W 0 ) - μ W ) 2 / 2 σ ln W 2 ] d W
E { ln k W } = 0 ln k W A W + W 0 × exp [ - ( ln ( W + W 0 ) - μ W ) 2 / 2 σ ln W 2 ] d W
L k ln ( H k + 1 ) ln ( H 2 + 1 ) ,             k = 2 , 3 , 4 .
L k = 1 2 k ( k - 1 ) .
p ( I ) = 1 E { I } - I c exp ( - I + I c E { I } - I c ) I 0 ( 2 I c I E { I } - I c ) ,
P ( n , T ) = ( E { n } - n 0 ) n ( 1 + E { n } - n 0 ) n + 1 exp ( - n 0 1 + E { n } - n 0 ) × L n ( - n 0 ( E { n } - n 0 ) ( 1 + E { n } - n 0 ) ) .
n 0 = ( 2 E 2 { n } + E { n } - E { n 2 } ) 1 / 2 .
E { W k } = 2 ( 2 π σ ln W 2 ) 1 / 2 erfc ( y ) × W 0 W k - 1 exp [ - ( ln W - μ W ) 2 / 2 σ ln W 2 ] d W ,
x = ln W - μ W ( 2 σ ln W 2 ) 1 / 2 - k ( 1 2 σ ln W 2 ) 1 / 2 ,
E { W k } = exp [ k μ W + 1 2 k 2 σ ln W 2 ] erfc ( y ) 2 π z e - x 2 d x ,
z ln W 0 - μ W ( 2 σ ln W 2 ) - k ( 1 2 σ ln W 2 ) 1 / 2 y - k ( 1 2 σ ln W 2 ) 1 / 2 .
( 2 / π ) z e - x 2 d x erfc ( z ) ,
E { W k } = erfc [ y - k ( 1 2 σ ln W 2 ) 1 / 2 ] erfc ( y ) exp [ k μ W + 1 2 k 2 σ ln W 2 ] .
E { W 2 } ( E { W } ) 2 = E { W 2 } + 2 W 0 E { W } + W 0 2 ( E { W } + W 0 ) 2 = e σ ln W 2 ,
E { W 3 } ( E { W } ) 3 = E { W 3 } + 3 W 0 E { W 2 } + 3 W 0 2 E { W } + W 0 3 ( E { W } + W 0 ) 3 = e 3 σ ln W 2 .
H 2 + ( 1 + W 0 / E { W } ) 2 ( 1 + W 0 / E { W } ) 2 = e σ ln W 2 .
H 3 + 3 H 2 ( 1 + W 0 / E { W } ) - 3 H 2 + ( 1 + W 0 / E { W } ) 3 ( 1 + W 0 / E { W } ) 3 = e 3 σ ln W 2 .
( H 3 - 3 H 2 ) x 3 - 3 H 2 2 x 2 - H 2 3 = 0
μ W = ln ( E { W } + W 0 ) - 1 2 σ ln W 2 .