Abstract

Statistical characteristics of radiation intensity in the cross section of laser beams propagating in the turbulent atmosphere are analyzed by using numerical simulations. It is shown that for arbitrary values of the propagation distance, Rytov parameter, beam type, and position of the analyzed point in the beam cross section, the probability density function (PDF) of radiation intensity is fully determined by the scintillation index and the average intensity. In the case of moderate and weak intensity fluctuations characterized by the scintillation index smaller than unity, the probability density function is determined by the gamma distribution. For the case of strong fluctuations (with the scintillation index larger than unity), a new analytical expression for PDF is proposed, which well approximates PDFs obtained in numerical experiments under different conditions of propagation of different-type beams.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

V. P. Aksenov and V. V. Kolosov, “Probability density of field and intensity fluctuations of structured light in a turbulent atmosphere,” J. Opt. 21(3), 035605 (2019).
[Crossref]

2016 (2)

S. L. Lachinova and M. A. Vorontsov, “Giant irradiance spikes in laser beam propagation in volume turbulence: analysis and impact,” J. Opt. 18(2), 025608 (2016).
[Crossref]

V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, “Properties of vortex beams formed by an array of fibre lasers and their propagation in a turbulent atmosphere,” Quantum Electron. 46(8), 726–732 (2016).
[Crossref]

2015 (1)

V. P. Aksenov and V. V. Kolosov, “Scintillations of optical vortex in randomly inhomogeneous medium,” Photonics Res. 3(2), 44–47 (2015).
[Crossref]

2012 (2)

2011 (1)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

2009 (1)

2007 (1)

2005 (1)

V. V. Dudorov, G. A. Filimonov, and V. V. Kolosov, “Algorithm for formation of an infinite random turbulent screen,” Proc. SPIE 6160, 61600R (2005).
[Crossref]

2001 (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[Crossref]

1997 (1)

1994 (1)

1989 (1)

1988 (1)

1987 (1)

1985 (1)

1982 (1)

1976 (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. 10(2), 129–160 (1976).
[Crossref]

1975 (1)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63(5), 790–811 (1975).
[Crossref]

Aksenov, V. P.

V. P. Aksenov and V. V. Kolosov, “Probability density of field and intensity fluctuations of structured light in a turbulent atmosphere,” J. Opt. 21(3), 035605 (2019).
[Crossref]

V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, “Properties of vortex beams formed by an array of fibre lasers and their propagation in a turbulent atmosphere,” Quantum Electron. 46(8), 726–732 (2016).
[Crossref]

V. P. Aksenov and V. V. Kolosov, “Scintillations of optical vortex in randomly inhomogeneous medium,” Photonics Res. 3(2), 44–47 (2015).
[Crossref]

V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, “Probability distribution of intensity fluctuations of vortex laser beams in the turbulent atmosphere,” arXiv:1802.03172 (2018).

Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[Crossref]

Andrews, D. L.

D. L. Andrews, Structured Light and its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Academic, 2008).

Andrews, L.

Andrews, L. C.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
[Crossref]

Barrios, R.

Bracher, C.

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63(5), 790–811 (1975).
[Crossref]

Churnside, J. H.

Clare, B. A.

Cowley, W. G.

Dainty, J. C.

J. W. Goodman, “Statistical properties of laser speckle patterns,” Chap. 2 in Laser Speckle and Related Phenomena, J. C. Dainty, Ed., Springer-Verlag, New York (1975).

Dios, F.

Dudorov, V. V.

V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, “Properties of vortex beams formed by an array of fibre lasers and their propagation in a turbulent atmosphere,” Quantum Electron. 46(8), 726–732 (2016).
[Crossref]

V. V. Dudorov, G. A. Filimonov, and V. V. Kolosov, “Algorithm for formation of an infinite random turbulent screen,” Proc. SPIE 6160, 61600R (2005).
[Crossref]

V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, “Probability distribution of intensity fluctuations of vortex laser beams in the turbulent atmosphere,” arXiv:1802.03172 (2018).

Feit, M. D.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. 10(2), 129–160 (1976).
[Crossref]

Filimonov, G. A.

V. V. Dudorov, G. A. Filimonov, and V. V. Kolosov, “Algorithm for formation of an infinite random turbulent screen,” Proc. SPIE 6160, 61600R (2005).
[Crossref]

Flatté, S. M.

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. 10(2), 129–160 (1976).
[Crossref]

Frehlich, R. G.

Gochelashvily, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63(5), 790–811 (1975).
[Crossref]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” Chap. 2 in Laser Speckle and Related Phenomena, J. C. Dainty, Ed., Springer-Verlag, New York (1975).

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, “Similarity Relations and their Experimental Verification for Strong Intensity Fluctuations of Laser Radiation,” in Laser Beam Propagation in the Atmosphere, J.W. Strohbehn, ed. (Springer-Verlag, 1978).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 2000).

Grant, K. J.

Gudimetla, V. S. R.

Gurvich, A. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, “Similarity Relations and their Experimental Verification for Strong Intensity Fluctuations of Laser Radiation,” in Laser Beam Propagation in the Atmosphere, J.W. Strohbehn, ed. (Springer-Verlag, 1978).

Hill, R. J.

Holmes, J. F.

Kashkarov, S. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, “Similarity Relations and their Experimental Verification for Strong Intensity Fluctuations of Laser Radiation,” in Laser Beam Propagation in the Atmosphere, J.W. Strohbehn, ed. (Springer-Verlag, 1978).

Kolosov, V. V.

V. P. Aksenov and V. V. Kolosov, “Probability density of field and intensity fluctuations of structured light in a turbulent atmosphere,” J. Opt. 21(3), 035605 (2019).
[Crossref]

V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, “Properties of vortex beams formed by an array of fibre lasers and their propagation in a turbulent atmosphere,” Quantum Electron. 46(8), 726–732 (2016).
[Crossref]

V. P. Aksenov and V. V. Kolosov, “Scintillations of optical vortex in randomly inhomogeneous medium,” Photonics Res. 3(2), 44–47 (2015).
[Crossref]

V. V. Dudorov, G. A. Filimonov, and V. V. Kolosov, “Algorithm for formation of an infinite random turbulent screen,” Proc. SPIE 6160, 61600R (2005).
[Crossref]

V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, “Probability distribution of intensity fluctuations of vortex laser beams in the turbulent atmosphere,” arXiv:1802.03172 (2018).

Konyaev, P. A.

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. Vol. 4 Wave Propagation through Random Media (Springer, 1988).

Lachinova, S. L.

S. L. Lachinova and M. A. Vorontsov, “Giant irradiance spikes in laser beam propagation in volume turbulence: analysis and impact,” J. Opt. 18(2), 025608 (2016).
[Crossref]

Lukin, V. P.

Lyke, S. D.

Mackintosh, J. L.

Martin, J. M.

Mclaren, J. R. W.

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. 10(2), 129–160 (1976).
[Crossref]

Mudge, K. A.

Nakagami, M.

M. Nakagami, “The m distribution—a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, 1960).

Padgett, M. J.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

Phillips, R. L.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
[Crossref]

Pokasov, V. V.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, “Similarity Relations and their Experimental Verification for Strong Intensity Fluctuations of Laser Radiation,” in Laser Beam Propagation in the Atmosphere, J.W. Strohbehn, ed. (Springer-Verlag, 1978).

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63(5), 790–811 (1975).
[Crossref]

Roggemann, M. C.

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. Vol. 4 Wave Propagation through Random Media (Springer, 1988).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 2000).

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63(5), 790–811 (1975).
[Crossref]

Tatarskii, V. I.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. Vol. 4 Wave Propagation through Random Media (Springer, 1988).

Thomas, J. C.

van der Vaart, A. W.

A. W. van der Vaart, Asymptotic Statistics (Cambridge University Press, 1998).
[Crossref]

Vetelino, F. S.

Voelz, D. G.

Vorontsov, M. A.

S. L. Lachinova and M. A. Vorontsov, “Giant irradiance spikes in laser beam propagation in volume turbulence: analysis and impact,” J. Opt. 18(2), 025608 (2016).
[Crossref]

Wang, G.

Yao, A. M.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

Young, C.

Adv. Opt. Photonics (1)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

Appl. Opt. (5)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. 10(2), 129–160 (1976).
[Crossref]

J. Opt. (2)

V. P. Aksenov and V. V. Kolosov, “Probability density of field and intensity fluctuations of structured light in a turbulent atmosphere,” J. Opt. 21(3), 035605 (2019).
[Crossref]

S. L. Lachinova and M. A. Vorontsov, “Giant irradiance spikes in laser beam propagation in volume turbulence: analysis and impact,” J. Opt. 18(2), 025608 (2016).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[Crossref]

Opt. Express (1)

Photonics Res. (1)

V. P. Aksenov and V. V. Kolosov, “Scintillations of optical vortex in randomly inhomogeneous medium,” Photonics Res. 3(2), 44–47 (2015).
[Crossref]

Proc. IEEE (1)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance in turbulent media,” Proc. IEEE 63(5), 790–811 (1975).
[Crossref]

Proc. SPIE (1)

V. V. Dudorov, G. A. Filimonov, and V. V. Kolosov, “Algorithm for formation of an infinite random turbulent screen,” Proc. SPIE 6160, 61600R (2005).
[Crossref]

Quantum Electron. (1)

V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, “Properties of vortex beams formed by an array of fibre lasers and their propagation in a turbulent atmosphere,” Quantum Electron. 46(8), 726–732 (2016).
[Crossref]

Other (9)

D. L. Andrews, Structured Light and its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Academic, 2008).

M. Nakagami, “The m distribution—a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, 1960).

J. W. Goodman, “Statistical properties of laser speckle patterns,” Chap. 2 in Laser Speckle and Related Phenomena, J. C. Dainty, Ed., Springer-Verlag, New York (1975).

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. Vol. 4 Wave Propagation through Random Media (Springer, 1988).

A. W. van der Vaart, Asymptotic Statistics (Cambridge University Press, 1998).
[Crossref]

V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, “Probability distribution of intensity fluctuations of vortex laser beams in the turbulent atmosphere,” arXiv:1802.03172 (2018).

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pokasov, “Similarity Relations and their Experimental Verification for Strong Intensity Fluctuations of Laser Radiation,” in Laser Beam Propagation in the Atmosphere, J.W. Strohbehn, ed. (Springer-Verlag, 1978).

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 2000).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
[Crossref]

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Figures (8)

Fig. 1.
Fig. 1. Cross section of (a) average intensity I(r) and (b) scintillation index σI(r) for beams propagating through turbulence. Line (1) is for the Laguerre–Gaussian beam at $\beta _0^2 = 0.1$, l = 1, z = 0.1zd, zd = k0a2/2 is diffraction Rayleigh length, line (2) is for the Laguerre–Gaussian beam at $\beta _0^2 = 1$, l = 2, z = 2zd, and line (3) is for the Gaussian beam at $\beta _0^2 = 0.1$, l = 0, $z = 0.1{z_d}$.
Fig. 2.
Fig. 2. Probability density function: (a) at the axis (r = 0) of the Gaussian beam (Eq. (4) with l = 0); (b) at the point of maximum (r = 0.47a) of the mean intensity distribution of the Laguerre–Gaussian beam (Eq. (4) with l = 1). The values of σI(r) are 0.31 and 0.30, respectively.
Fig. 3.
Fig. 3. Probability density function: (a) at the axis (r = 0) Laguerre–Gaussian beam; (b) at the periphery (at the point r = 1.9a) of the Laguerre–Gaussian beam. The values of σI(r) are 1.04, and 1.02, respectively.
Fig. 4.
Fig. 4. Probability density function: (a) for the Gaussian beam at the point r = 1.9a; (b) for the Laguerre–Gaussian beam at the point r = 1.4a. The values of σI(r) are 0.67 and 0.68, respectively.
Fig. 5.
Fig. 5. Probability density distributions: (a) for the Gaussian beam (l = 0) at the point r = 1.2a, σI(r) = 1.17, m = 0.748; (b) for the Laguerre-Gaussian beam (l = 1) at the point r = 0.08 a, σI(r) = 1.19, m = 0.724 (b).
Fig. 6.
Fig. 6. Probability density distributions: (a) for the Gaussian beam (l = 0) at the point z = 0.1zd, r = 0.2a, σI(r) = 1.50, <I> = 0.18, m = 0.514; (b) for the Laguerre—Gaussian beam (l = 1) at the point z = 2.0zd, r = 3.28 a, σI(r) = 1.65, <I> = 0.013, m = 0.453.
Fig. 7.
Fig. 7. Probability density distributions for the Gaussian beam (l = 0): (a) at the distance z = zd at the point r = 6.6a, σI(r) = 1.84, <I> = 0.0028, m = 0.395; (b) at the point r = 9.5 a, σI(r) = 2.1, <I> = 0.0018, m = 0.339.
Fig. 8.
Fig. 8. Probability density distributions of intensity normalized to its average value: numerical simulation for the Gaussian beam (l = 0) at the point r = 6.6a (black circles); numerical simulation for the Laguerre—Gaussian beam at the distance z = zd at the point r = 2.2 a (red circles); fractional exponential PDF model with m = 0.395 (blue curve). σI(r) = 1.84 (Gaussian beam), σI(r) = 1.82 (Laguerre—Gaussian beam).

Equations (19)

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Φ n ( κ ) = 0.033 C n 2 exp ( κ 2 / κ 2 κ a 2 κ a 2 ) ( κ 2 + κ 0 2 ) 11 / 6 [ 1 + 1.802 κ κ a 0.254 ( κ κ a ) 7 / 6 ] ,
F N ( I ) = 1 N i = 1 N θ ( I I i ) ,
F ( I ) = 0 I P ( x ) d x .
E ( r , θ , z = 0 ) = ( 2 r a ) | l | exp ( r 2 a 2 ) exp [ i l θ ] ,
P L N ( I ) = 1 I ( r ) ξ 2 π exp [ ( ln I ( r ) μ ) 2 / ( ln I ( r ) μ ) 2 2 ξ 2 2 ξ 2 ] ,
μ = ln ( I ( r ) 1 + σ I 2 ) , ξ 2 = ln ( 1 + σ I 2 ) ,
σ I 2 ( r ) = I ( r ) 2 I ( r ) 2 I ( r ) 2
P E ( I ) = 1 I ( r ) exp ( I I ( r ) )
P G G ( I ) = 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) I ( I I ( r ) ) ( α + β ) / 2 K α β ( 2 α β I I ( r ) ) ,
σ I 2 ( r ) = 1 α + 1 β + 1 α β .
P G ( I ) = k k I k 1 Γ ( k ) I ( r ) k exp ( k I I ( r ) )
k = 1 σ I 2 ( r ) .
P C h ( I n ) = 1 2 π σ z 0 d z z 2 exp [ I n z ( ln z + σ z 2 / 2 ) 2 2 σ z 2 ] ,
σ z 2 = ln ( σ I 2 / σ I 2 2 + 1 / 1 2 2 2 + 1 / 1 2 2 ) .
P F ( I ) = Γ ( 2 / m ) Γ 2 ( 1 / m ) m I ( r ) exp [ ( Γ ( 2 / m ) Γ ( 1 / m ) ) m ( I I ( r ) ) m ] ,
σ I 2 ( r ) + 1 = Γ ( 1 / m ) Γ ( 3 / m ) Γ 2 ( 2 / m ) .
K ν ( z ) = π 2 I ν ( z ) I ν ( z ) sin ν π ,
I ν ( z ) = k = 0 1 k ! Γ ( ν + k + 1 ) ( z 2 ) ν + 2 k ,
P G G ( I ) = 1 Γ ( α ) Γ ( β ) 1 I [ Γ ( α β ) ( α β ) β ( I I ) β + Γ ( β α ) ( α β ) α ( I I ) α ] , I 0.

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