Abstract

An intuitive model for the scintillation index of a partially coherent beam is developed in which essentially the only critical parameter is the properly defined Fresnel number equal to the ratio of the “working” aperture area to the area of the Fresnel zone. The model transpired from and is supported by numerical simulations using Rytov method for weak fluctuations regime and Tatarskii turbulence spectrum with inner scale. The ratio of the scintillation index of a partially coherent beam to that of a plane wave displays a characteristic minimum, the magnitude of which and its distance from the transmitter are easily explained using the intuitive model. A theoretical asymptotic is found for the scintillation index of a source with decreasing coherence at this minimum.

© 2014 Optical Society of America

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References

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  1. G. Gbur, “Partially coherent beam propagation in atmospheric turbulence,” J. Opt. Soc. Am. A 31, 2038–2045 (2014).
    [Crossref]
  2. A. Efimov, K. Velizhanin, and G. Gelikonov, “Simultaneous scintillation measurements of coherent and partially coherent beams in an open atmosphere experiment,” Proc. SPIE 8971, 8971051 (2014).
  3. I. I. Kim, H. Hakakha, P. Adhikari, E. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proc, SPIE 2990, 102–113 (1997).
    [Crossref]
  4. G. P. Berman and A. A. Chumak, “Infuence of phase-diffuser dynamics on scintillations of laser radiation in earths atmosphere: Long-distance propagation,” Phys. Rev. A 79, 0638481 (2009).
    [Crossref]
  5. A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in the Turbulent Atmosphere (Nauka, 1976).
  6. V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 626–629 (1983).
  7. Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Science 18, 551–556 (1983).
    [Crossref]
  8. J. Pearson, Computation of hypergeometric functions, (Master Thesis, University of Oxford, 2009).

2014 (2)

G. Gbur, “Partially coherent beam propagation in atmospheric turbulence,” J. Opt. Soc. Am. A 31, 2038–2045 (2014).
[Crossref]

A. Efimov, K. Velizhanin, and G. Gelikonov, “Simultaneous scintillation measurements of coherent and partially coherent beams in an open atmosphere experiment,” Proc. SPIE 8971, 8971051 (2014).

2009 (1)

G. P. Berman and A. A. Chumak, “Infuence of phase-diffuser dynamics on scintillations of laser radiation in earths atmosphere: Long-distance propagation,” Phys. Rev. A 79, 0638481 (2009).
[Crossref]

1997 (1)

I. I. Kim, H. Hakakha, P. Adhikari, E. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proc, SPIE 2990, 102–113 (1997).
[Crossref]

1983 (2)

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 626–629 (1983).

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Science 18, 551–556 (1983).
[Crossref]

Adhikari, P.

I. I. Kim, H. Hakakha, P. Adhikari, E. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proc, SPIE 2990, 102–113 (1997).
[Crossref]

Banakh, V. A.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 626–629 (1983).

Baykal, Y.

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Science 18, 551–556 (1983).
[Crossref]

Berman, G. P.

G. P. Berman and A. A. Chumak, “Infuence of phase-diffuser dynamics on scintillations of laser radiation in earths atmosphere: Long-distance propagation,” Phys. Rev. A 79, 0638481 (2009).
[Crossref]

Buldakov, V. M.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 626–629 (1983).

Chumak, A. A.

G. P. Berman and A. A. Chumak, “Infuence of phase-diffuser dynamics on scintillations of laser radiation in earths atmosphere: Long-distance propagation,” Phys. Rev. A 79, 0638481 (2009).
[Crossref]

Efimov, A.

A. Efimov, K. Velizhanin, and G. Gelikonov, “Simultaneous scintillation measurements of coherent and partially coherent beams in an open atmosphere experiment,” Proc. SPIE 8971, 8971051 (2014).

Gbur, G.

Gelikonov, G.

A. Efimov, K. Velizhanin, and G. Gelikonov, “Simultaneous scintillation measurements of coherent and partially coherent beams in an open atmosphere experiment,” Proc. SPIE 8971, 8971051 (2014).

Gurvich, A. S.

A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in the Turbulent Atmosphere (Nauka, 1976).

Hakakha, H.

I. I. Kim, H. Hakakha, P. Adhikari, E. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proc, SPIE 2990, 102–113 (1997).
[Crossref]

Khmelevtsov, S. S.

A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in the Turbulent Atmosphere (Nauka, 1976).

Kim, I. I.

I. I. Kim, H. Hakakha, P. Adhikari, E. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proc, SPIE 2990, 102–113 (1997).
[Crossref]

Kon, A. I.

A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in the Turbulent Atmosphere (Nauka, 1976).

Korevaar, E.

I. I. Kim, H. Hakakha, P. Adhikari, E. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proc, SPIE 2990, 102–113 (1997).
[Crossref]

Majumdar, A. K.

I. I. Kim, H. Hakakha, P. Adhikari, E. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proc, SPIE 2990, 102–113 (1997).
[Crossref]

Mironov, V. L.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 626–629 (1983).

A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in the Turbulent Atmosphere (Nauka, 1976).

Pearson, J.

J. Pearson, Computation of hypergeometric functions, (Master Thesis, University of Oxford, 2009).

Plonus, M. A.

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Science 18, 551–556 (1983).
[Crossref]

Velizhanin, K.

A. Efimov, K. Velizhanin, and G. Gelikonov, “Simultaneous scintillation measurements of coherent and partially coherent beams in an open atmosphere experiment,” Proc. SPIE 8971, 8971051 (2014).

Wang, S. J.

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Science 18, 551–556 (1983).
[Crossref]

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

Opt. Spectrosc. (1)

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 626–629 (1983).

Phys. Rev. A (1)

G. P. Berman and A. A. Chumak, “Infuence of phase-diffuser dynamics on scintillations of laser radiation in earths atmosphere: Long-distance propagation,” Phys. Rev. A 79, 0638481 (2009).
[Crossref]

Proc, SPIE (1)

I. I. Kim, H. Hakakha, P. Adhikari, E. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proc, SPIE 2990, 102–113 (1997).
[Crossref]

Proc. SPIE (1)

A. Efimov, K. Velizhanin, and G. Gelikonov, “Simultaneous scintillation measurements of coherent and partially coherent beams in an open atmosphere experiment,” Proc. SPIE 8971, 8971051 (2014).

Radio Science (1)

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Science 18, 551–556 (1983).
[Crossref]

Other (2)

J. Pearson, Computation of hypergeometric functions, (Master Thesis, University of Oxford, 2009).

A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in the Turbulent Atmosphere (Nauka, 1976).

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Figures (4)

Fig. 1
Fig. 1 Scintillation index ratio for plane wave (blue), spherical wave (green), Gaussian beam (red) and PCB (black) for turbulence spectra without (left) and with (right) inner scale l 0 = 5 mm. Beam intensity radius is 3 mm for curves “1”, 3 cm for “2”, and 10 cm for “3”. Coherence radius rc = 3 mm, except for the dashed black curve, where rc = 1 mm.
Fig. 2
Fig. 2 Scintillation index ratio for PCB as a function of propagation distance (left) and normalized distance L/Ltr (right). Beam radius a is 1 cm (red curves), 3 cm (green), and 10 cm (blue), while the coherence radius rc is set to 1 mm (solid lines), 3 mm (dashed), and 10 mm (dot-dashed).
Fig. 3
Fig. 3 Conceptual illustration of the intuitive model: Detectors located at L 1 and L 2 receive signals only from a fraction of the PCB aperture, determined by the NA of the “elementary emitters”, while the detector located at the transition length L 3 = Ltr “sees” all the “elementary emitters”.
Fig. 4
Fig. 4 Left: Scintillation index ratio of a collimated PCB (red circles) evaluated at L = Ltr for a large number of parameter pairs a and rc . The asymptotic (blue) and best fit (black) are also shown. Right: SI ratio for all L ⊂ {1, 105} meters and a large number of parameter pairs a and rc (colored curves) overlayed with a power-law fit (blue line).

Equations (14)

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L tr = 4 π r c a λ .
Ω w = ( γ w L L / k ) 2 = k L γ w 2 = k a w 2 L ,
σ PCB 2 σ R 2 = 2.52 ( r c / a ) 7 / 6 , L = L tr , r c / a 1
σ pl 2 = 7.07 σ R 2 0 1 { [ ( 1 ξ ) 2 + Q m 2 ] 5 / 12 cos ( 5 6 arctan [ Q m ( 1 ξ ) ] ) Q m 5 / 6 } d ξ ,
σ sp 2 = 7.07 σ R 2 0 1 { [ ξ 2 ( 1 ξ ) 2 + Q m 2 ] 5 / 12 cos ( 5 6 arctan [ Q m ξ ( 1 ξ ) ] ) Q m 5 / 6 } d ξ ,
σ G 2 = 7.07 σ R 2 Re 0 1 { [ Q m 1 + i γ ( 1 ξ ) ] 5 / 6 [ Q m 1 Im ( γ ) ( 1 ξ ) ] 5 / 6 } d ξ ,
Γ ( r 1 , r 2 ) = φ 0 ( r 1 ) φ 0 ( r 2 ) ¯ = φ 0 2 exp { r 1 2 + r 2 2 2 a 2 ( r 1 r 2 ) 2 4 r c 2 + i k 2 F 0 ( r 1 2 r 2 2 ) } ,
σ PCB 2 = 7.07 g u 5 / 3 σ R 2 Re 0 1 { [ Ω u ( 1 ξ ) 2 + i ( 1 ξ ) [ Ω 2 ( 1 f ) ( 1 ξ f ) + k u 2 ξ ] + g u 2 Q m ] 5 / 6 [ Ω u ( 1 ξ ) 2 + g u 2 Q m ] 5 / 6 } d ξ ,
σ PCB 2 σ R 2 = 2.67 Ω 5 / 6 ( 1 + k u 2 2 g u 2 ) 5 / 6 { 16 11 Re [ ( 1 + 2 i Ω ( 1 f ) 1 + k u 2 ) 5 / 6 F 1 2 ( 5 6 , 1 ; 17 6 ; z ) ] 1 }
z = 2 g u 2 ( f i k u / Ω ) + ( 1 k u ) 2 ( k u i Ω ( 1 f ) ) 2 g u 2 + ( 1 k u ) 2 ( k u i Ω ( 1 f ) )
F 1 2 ( a , b ; c ; z ) = Γ ( c ) Γ ( c a b ) Γ ( c a ) Γ ( c b ) F 1 2 ( a , b ; a + b c + 1 ; 1 z )
F 1 2 ( a , b ; c ; z ) = j = 0 ( a ) j ( b ) j ( c ) j z j j ! ,
( μ ) 0 = 1 ; ( μ ) j = μ ( μ + 1 ) ( μ + j 1 ) , j = 1 , 2 ,
lim ξ 0 σ PCB 2 σ R 2 = 2.52 ξ 7 / 6 , at L = L tr = 4 π a r c λ .

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