Abstract

Level crossing statistics is applied to the complex problem of atmospheric turbulence-induced beam wander for laser propagation from ground to space. A comprehensive estimate of the single-axis wander angle temporal autocorrelation function and the corresponding power spectrum is used to develop, for the first time to our knowledge, analytic expressions for the mean angular level crossing rate and the mean duration of such crossings. These results are based on an extension and generalization of a previous seminal analysis of the beam wander variance by Klyatskin and Kon. In the geometrical optics limit, we obtain an expression for the beam wander variance that is valid for both an arbitrarily shaped initial beam profile and transmitting aperture. It is shown that beam wander can disrupt bidirectional ground-to-space laser communication systems whose small apertures do not require adaptive optics to deliver uniform beams at their intended target receivers in space. The magnitude and rate of beam wander is estimated for turbulence profiles enveloping some practical laser communication deployment options and suggesting what level of beam wander effects must be mitigated to demonstrate effective bidirectional laser communication systems.

© 2011 Optical Society of America

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References

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  1. F. Heine, H. Kämpfner, R. Czichy, and R. Meyer, “Optical intersatellite communication operational,” in Proceedings of the Military Communications Conference (MILCOM), 2010 (IEEE, 2010), pp. 1583–1587.
    [CrossRef]
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    [CrossRef]
  3. V. I. Klyatskin and A. I. Kon, “On the displacement of spatially bounded light beams in a turbulent atmosphere in the Markovian-random-process approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
    [CrossRef]
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    [CrossRef]
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  12. R. R. Beland, “Propagation through atmospheric optical turbulence,” The Infrared & Electro-Optical Systems Handbook, Vol.  2, F.G.Smith, ed. (SPIE Press, 1993).
  13. W. Bradford, “Maui4: a 24 hour Haleakala turbulence profile,” presented at the Advanced Maui Optical and Space Surveillance Technologies Conference, Wailea, Maui, Hawaii, USA, 14–17 September 2010. This model is a generalization of the AMOS night model discussed in , adapted by the authors to match median values of r0, and the isoplanatic angle for the Haleakala site based on the most reliable data available.
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    [CrossRef] [PubMed]
  15. Note that, for elevation angles greater than 20 degrees, r0 is greater than 7 cm for α less than about 6.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2009 (1)

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE 7330, 73300Q(2009).
[CrossRef]

2006 (1)

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

1995 (1)

1994 (1)

K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communication systems impaired by pointing jitter and turbulence,” Opt. Eng. 33, 3748–3757(1994).
[CrossRef]

1990 (1)

1977 (1)

1973 (2)

1972 (1)

V. I. Klyatskin and A. I. Kon, “On the displacement of spatially bounded light beams in a turbulent atmosphere in the Markovian-random-process approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[CrossRef]

1971 (1)

1967 (1)

1948 (1)

S. O. Rice, “Statistical properties of a sine wave plus random noise,” Bell Syst. Tech. J. 27, 109–157 (1948).

Andrews, L. C.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Beckmann, P.

P. Beckmann, Probability in Communication Engineering (Harcourt Brace & World, 1967).

Beland, R. R.

R. R. Beland, “Propagation through atmospheric optical turbulence,” The Infrared & Electro-Optical Systems Handbook, Vol.  2, F.G.Smith, ed. (SPIE Press, 1993).

Bradford, W.

W. Bradford, “Maui4: a 24 hour Haleakala turbulence profile,” presented at the Advanced Maui Optical and Space Surveillance Technologies Conference, Wailea, Maui, Hawaii, USA, 14–17 September 2010. This model is a generalization of the AMOS night model discussed in , adapted by the authors to match median values of r0, and the isoplanatic angle for the Haleakala site based on the most reliable data available.

Bufton, J. L.

Chiba, T.

Churnside, J. H.

Czichy, R.

F. Heine, H. Kämpfner, R. Czichy, and R. Meyer, “Optical intersatellite communication operational,” in Proceedings of the Military Communications Conference (MILCOM), 2010 (IEEE, 2010), pp. 1583–1587.
[CrossRef]

Fields, R.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE 7330, 73300Q(2009).
[CrossRef]

Fried, D. L.

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006), Section 4.3.1.

Hansen, B.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE 7330, 73300Q(2009).
[CrossRef]

Heine, F.

F. Heine, H. Kämpfner, R. Czichy, and R. Meyer, “Optical intersatellite communication operational,” in Proceedings of the Military Communications Conference (MILCOM), 2010 (IEEE, 2010), pp. 1583–1587.
[CrossRef]

Jordan, J.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE 7330, 73300Q(2009).
[CrossRef]

Kahle, R.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE 7330, 73300Q(2009).
[CrossRef]

Kämpfner, H.

F. Heine, H. Kämpfner, R. Czichy, and R. Meyer, “Optical intersatellite communication operational,” in Proceedings of the Military Communications Conference (MILCOM), 2010 (IEEE, 2010), pp. 1583–1587.
[CrossRef]

Kiasaleh, K.

K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communication systems impaired by pointing jitter and turbulence,” Opt. Eng. 33, 3748–3757(1994).
[CrossRef]

Klyatskin, V. I.

V. I. Klyatskin and A. I. Kon, “On the displacement of spatially bounded light beams in a turbulent atmosphere in the Markovian-random-process approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[CrossRef]

Kon, A. I.

V. I. Klyatskin and A. I. Kon, “On the displacement of spatially bounded light beams in a turbulent atmosphere in the Markovian-random-process approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[CrossRef]

Lataitis, R. J.

Lunde, C.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE 7330, 73300Q(2009).
[CrossRef]

Meyer, R.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE 7330, 73300Q(2009).
[CrossRef]

F. Heine, H. Kämpfner, R. Czichy, and R. Meyer, “Optical intersatellite communication operational,” in Proceedings of the Military Communications Conference (MILCOM), 2010 (IEEE, 2010), pp. 1583–1587.
[CrossRef]

Mironov, V. I.

Muehlnikel, G.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE 7330, 73300Q(2009).
[CrossRef]

Nosov, V. V.

Parenti, R. R.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Rice, S. O.

S. O. Rice, “Statistical properties of a sine wave plus random noise,” Bell Syst. Tech. J. 27, 109–157 (1948).

Sasiela, R. J.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Scheel, W.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE 7330, 73300Q(2009).
[CrossRef]

Shelton, J. D.

Sterr, U.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE 7330, 73300Q(2009).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961), Section 12.4.

Wicker, J.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE 7330, 73300Q(2009).
[CrossRef]

Wolfram, S.

S. Wolfram, “Mathematica,” Version 7 (Cambridge University, 2008).

Wong, R.

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE 7330, 73300Q(2009).
[CrossRef]

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

S. O. Rice, “Statistical properties of a sine wave plus random noise,” Bell Syst. Tech. J. 27, 109–157 (1948).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Eng. (2)

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communication systems impaired by pointing jitter and turbulence,” Opt. Eng. 33, 3748–3757(1994).
[CrossRef]

Proc. SPIE (1)

R. Fields, C. Lunde, R. Wong, J. Wicker, J. Jordan, B. Hansen, G. Muehlnikel, W. Scheel, U. Sterr, R. Kahle, and R. Meyer, “NFIRE-to-TerraSAR-X laser communication results: satellite pointing, disturbances, and other attributes consistent with successful performance,” Proc. SPIE 7330, 73300Q(2009).
[CrossRef]

Radiophys. Quantum Electron. (1)

V. I. Klyatskin and A. I. Kon, “On the displacement of spatially bounded light beams in a turbulent atmosphere in the Markovian-random-process approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[CrossRef]

Other (8)

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006), Section 4.3.1.

S. Wolfram, “Mathematica,” Version 7 (Cambridge University, 2008).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961), Section 12.4.

R. R. Beland, “Propagation through atmospheric optical turbulence,” The Infrared & Electro-Optical Systems Handbook, Vol.  2, F.G.Smith, ed. (SPIE Press, 1993).

W. Bradford, “Maui4: a 24 hour Haleakala turbulence profile,” presented at the Advanced Maui Optical and Space Surveillance Technologies Conference, Wailea, Maui, Hawaii, USA, 14–17 September 2010. This model is a generalization of the AMOS night model discussed in , adapted by the authors to match median values of r0, and the isoplanatic angle for the Haleakala site based on the most reliable data available.

P. Beckmann, Probability in Communication Engineering (Harcourt Brace & World, 1967).

Note that, for elevation angles greater than 20 degrees, r0 is greater than 7 cm for α less than about 6.

F. Heine, H. Kämpfner, R. Czichy, and R. Meyer, “Optical intersatellite communication operational,” in Proceedings of the Military Communications Conference (MILCOM), 2010 (IEEE, 2010), pp. 1583–1587.
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Beam wander correlation coefficient for a collimated beam as a function of normalized time, τ / ( D / V ) , for both constant turbulence and wind conditions and various values of a 0 = 4 L / k D 2 .

Fig. 2
Fig. 2

Single-axis beam wander temporal power spectrum for collimated beams as a function of normalized angular frequency, ω / ( V / D ) , for both constant turbulence and wind conditions and various values of a 0 = 4 L / k D 2 .

Fig. 3
Fig. 3

Normalized quasi-frequency, v 0 / ( V / D ) , as a function of a 0 = 4 L / k D 2 for collimated beams, constant turbulence, and wind speed conditions.

Fig. 4
Fig. 4

Maui3 turbulence profile as a function of height above the Haleakala site.

Fig. 5
Fig. 5

Standard deviation of single-axis beam wander angle as a function of elevation angle for an aperture diameter of 7 cm for the nighttime Maui3 and HV- 5 / 7 turbulence profiles.

Fig. 6
Fig. 6

Effective normal wind speed to the LOS as a function of height above the Haleakala site for various values of elevation angle.

Fig. 7
Fig. 7

The beam wander correlation coefficient as a function of time delay for a nominal LEO satellite nighttime engagement above the Haleakala site, for elevation angles of 20 and 45 degrees.

Fig. 8
Fig. 8

Quasi-frequency as a function of elevation angle for a nominal LEO satellite nighttime engagement above the Haleakala site.

Fig. 9
Fig. 9

Mean duration of a positive angular excursion of 6, 8, and 10 μrad as a function of elevation angle for a nominal LEO satellite nighttime engagement above the Haleakala site.

Fig. 10
Fig. 10

Normalized single-axis beam wander power spectrum as a function of frequency for elevation angles of 20 and 45 deg and a nominal LEO satellite nighttime engagement above the Haleakala site.

Fig. 11
Fig. 11

Normalized variance of irradiance as a function of the 1 / e 2 transmitter aperture radius for zenith propagation at 500 nm , and the HV- 5 / 7 profile.

Fig. 12
Fig. 12

PDF of irradiance, as obtained from Eq. (A11), as a function of irradiance normalized to its mean compared to the log- normal PDF of conventional Rytov scintillation theory for a Gaussian aperture radius of 10 cm . For this example, θ 0 = 2.47 μrad , σ = 1.41 μrad , and σ ln I 2 = 0.062 .

Equations (41)

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θ ( t ) = 1 L r I ( r , t ) d 2 r I ( r , t ) d 2 r ,
p [ θ x , y ( t 1 ) , θ x , y ( t 2 ) ] = 1 2 π σ 2 ( 1 μ 2 ) exp [ θ x , y 2 ( t 1 ) + θ x , y 2 ( t 2 ) 2 μ θ x , y ( t 1 ) θ x , y ( t 2 ) 2 ( 1 μ 2 ) ] ,
p ( θ 1 , θ 2 ) = θ 1 θ 2 ( 1 μ 2 ) σ 4 exp [ θ 1 2 + θ 2 2 2 ( 1 μ 2 ) σ 2 ] I 0 ( μ θ 1 θ 2 ( 1 μ 2 ) σ 2 ) ,
B W ( μ ) = 0 d θ 1 0 d θ 2 θ 1 θ 2 p ( θ 1 , θ 2 ) .
B W ( μ ) = [ 2 E ( μ 2 ) ( μ 2 1 ) K ( μ 2 ) ] σ 2 ,
R W ( τ ) = 2 E ( μ 2 ) ( μ 2 1 ) K ( μ 2 ) π / 2 2 π / 2 .
ν + ( θ ) = 1 2 1 2 π ν 0 θ σ exp ( θ 2 2 σ 2 ) ,
ν 0 = R ¨ w ( 0 ) ,
R ¨ w ( 0 ) = 2 4 π μ ¨ ( 0 ) .
Δ T + ( θ ) = 2 2 π σ θ ν + ( θ ) .
σ 2 = 2 π 2 0 L d s ( 1 s / L ) 2 0 d K K F W ( K , s ) ,
F W ( K , s ) = K 2 Φ n ( K , s ) exp { K 2 D 2 8 [ ( 1 s F ) 2 + ( 4 s k D 2 ) 2 ] } ,
B W ( τ ) = 2 π 2 0 L d s ( 1 s / L ) 2 0 d K K F W ( K , s ) J 0 ( K V ( s ) τ ) ,
σ 2 = 2.56 D 1 / 3 0 L d s C n 2 ( s ) ( 1 s / L ) 2 [ ( 1 s / F ) 2 + a 2 ( s ) ] 1 / 6 ,
μ W ( τ ) = 0 L d s C n 2 ( s ) ( 1 s / L ) 2 F 1 1 [ 1 6 ; 1 ; 2 V 2 ( s ) τ 2 D 2 ( [ 1 s / F ] 2 + a 2 ( s ) ) 1 / 6 ] [ ( 1 s / F ) 2 + a 2 ( s ) ] 1 / 6 0 L d s C n 2 ( s ) ( 1 s / L ) 2 [ ( 1 s / F ) 2 + a 2 ( s ) ] 1 / 6 ,
σ 2 = 2.56 D 1 / 3 0 L d s C n 2 ( s ) ( 1 s / L ) 2 | 1 s / F | 1 / 3 .
σ U 2 = 2 π 2 0 L d s ( 1 s L ) 2 0 d K K 3 Φ n ( K , s ) ( 2 J 1 ( K D U / 2 ) K D U / 2 ) 2 ,
σ 2 = 2.83 D U 1 / 3 0 L d s C n 2 ( s ) ( 1 s / L ) 2 .
D = 0.74 D U ,
σ 2 = 2 π 2 0 L d s ( 1 s / L ) 2 0 d K K 3 Φ n ( K , s ) | H ( K ) | 2 ,
H ( K ) = 1 P 0 d 2 r exp [ i K r ] I 0 ( r ) ,
S W ( ω ) = 0 L d s C n 2 ( s ) ( 1 s / L ) 2 S W ( ω ; s ) ,
S W ( ω ; s ) = 8 π V ( s ) 0 d K F W ( K 2 + ω 2 / V 2 ( s ) ) .
S W ( ω ; s ) = 2 π V ( s ) exp [ ( 1 + a 2 ( s ) ) D 2 ω 2 8 V 2 ( s ) ] ( Γ ( 1 3 ) Γ ( 1 6 ) ( 1 + a 2 ( s ) ) 3 / 2 F 1 1 [ 5 6 ; 4 3 ; ( 1 + a 2 ( s ) ) D 2 ω 2 8 V 2 ( s ) ] 12 π Γ ( 1 3 ) ( V ( s ) D ω ) 2 / 3 F 1 1 [ 1 2 ; 2 3 ; ( 1 + a 2 ( s ) ) D 2 ω 2 8 V 2 ( s ) ] ) ,
ν 0 = ( 4 3 ( 4 π ) 0 1 d x ( 1 x ) 2 [ ( 1 x L / F ) 2 + a 0 2 x 2 ] 7 / 6 0 1 d x ( 1 x ) 2 [ ( 1 x L / F ) 2 + a 0 2 x 2 ] 1 / 6 ) 1 / 2 V D
C n 2 ( h ) = { 10 ( 9.401 + 1.5913 h + 0 / 0606 h 2 ) , 3.048 h 4.2 10 ( 17.1273 + 0.0332 h + 0.0015 h 2 0.9061 exp [ 0.5 ( 15.0866 h 5.2977 ) 2 ] ) , 4.2 h 25 10 ( 17.1273 + 0.0332 h + 0.0015 h 2 0.9061 exp [ 0.5 ( 15.0866 h 5.2977 ) 2 ] ) exp [ h 25 5 ] , h 25 ,
ν 0 = 1 D ( 4 3 ( 4 π ) 0 h 0 d h C n 2 ( h ) ( 1 h / h 0 ) 2 V 2 ( h ) [ 1 + a 2 ( h ) ] 7 / 6 0 h 0 d h C n 2 ( h ) ( 1 h / h 0 ) 2 [ 1 + a 2 ( h ) ] 1 / 6 ) 1 / 2 ,
I = I 1 I 2 .
p I ( I ) = 0 d I 1 0 d I 2 p 1 ( I 1 ) p 2 ( I 2 ) δ ( I I 1 I 2 ) = 0 d I 1 p 1 ( I 1 ) p 2 ( I / I 1 ) / I 1 ,
P I ( I ) = 0 I p ( I ) d I .
I n = 0 d I 1 p 1 ( I 1 ) I 1 n × 0 d I 2 p 2 ( I 2 ) I 2 n .
I = I pk exp [ θ 2 2 θ 0 2 ] ,
p 1 ( I ) = { m I N m 1 , 0 I N 1 0 , otherwise ,
m = ( θ 0 σ ) 2 .
p 2 ( I 2 ) = 1 2 π σ ln I 2 1 I 2 exp [ ( ln I 2 + σ ln I 2 / 2 ) 2 σ ln I 2 2 ] , 0 I 2 ,
I n = m m + n exp ( 1 2 n ( n 1 ) σ ln I 2 ) , n = 1 , 2 ,
I = m m + 1 I pk
var [ I ] = ( m + 1 ) 2 m ( m + 2 ) exp ( σ ln I 2 ) 1 .
p I ( x ) = m 2 2 ( m + 1 ) ( m x m + 1 ) m 1 exp [ m ( m + 1 ) σ ln I 2 / 2 ] erfc ( ln [ m x m + 1 ] + ( 2 m + 1 ) σ ln I 2 / 2 2 σ ln I 2 ) ,
P I ( x ) = 1 2 ( 1 + erf [ 2 ζ ( x ) σ ln I ] + exp [ m ( m + 1 ) σ ln I 2 / 2 ] ( m x m + 1 ) m erfc [ m σ ln I 2 + 2 ζ ( x ) σ ln I ] ) ,
ζ ( x ) = 1 2 ln ( m x m + 1 ) + σ ln I 2 4 ,

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