Abstract

A scintillation index formulation for annular beams in strong turbulence is developed that is also valid in moderate and weak turbulence. In our derivation, a modified Rytov solution is employed to obtain the small-scale and large-scale scintillation indices of annular beams by utilizing the amplitude spatial filtering of the atmospheric spectrum. Our solution yields only the on-axis scintillation index for the annular beam and correctly reduces to the existing strong turbulence results for the Gaussian beam—thus plane and spherical wave scintillation indices—and also correctly yields the existing weak turbulence annular beam scintillations. Compared to collimated Gaussian beam, plane, and spherical wave scintillations, collimated annular beams seem to be advantageous in the weak regime but lose this advantage in strongly turbulent atmosphere. It is observed that the contribution of annular beam scintillations comes mainly from the small-scale effects. At a fixed primary beam size, the scintillations of thinner collimated annular beams compared to thicker collimated annular beams are smaller in moderate turbulence but larger in strong turbulence; however, thinner annular beams of finite focal length have a smaller scintillation index than the thicker annular beams in strong turbulence. Decrease in the focal length decreases the annular beam scintillations in strong turbulence. Examining constant area annular beams, smaller primary sized annular structures have larger scintillations in moderate but smaller scintillations in strong turbulence.

© 2010 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol.2 (Academic, 1978).
  3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
    [CrossRef]
  4. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
    [CrossRef]
  5. G. Ya. Patrushev, “Fluctuations of the field of a wave beam on reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 8, 1315–1318 (1978).
    [CrossRef]
  6. V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, and R. Sh. Tsvik, “Focused-laser-beam scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. 64, 516–518 (1974).
    [CrossRef]
  7. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
    [CrossRef]
  8. R. L. Fante, “Comparison of theories for intensity fluctuations in strong turbulence,” Radio Sci. 11, 215–220 (1976).
    [CrossRef]
  9. K. S. Gochelashvili, V. G. Pevgov, and V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere,” Sov. J. Quantum Electron. 4, 632–637 (1974).
    [CrossRef]
  10. S. I. Belousov and I. G. Yakushkin, “Strong fluctuations of fields of optical beams in randomly inhomogeneous media,” Sov. J. Quantum Electron. 10, 301–304 (1980).
    [CrossRef]
  11. Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Effect of beam types on the scintillations: A review,” (invited), Proc. SPIE 7200, 720002-1 (2009).
  12. Y. Baykal, “Log-amplitude and phase fluctuations of higher-order annular laser beams in a turbulent medium,” J. Opt. Soc. Am. A 22, 672–679 (2005).
    [CrossRef]
  13. F. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media,” Proc. SPIE 5160, 86–97 (2004).
    [CrossRef]
  14. H. T. Eyyuboğlu and Y. Baykal, “Scintillations of cos-Gaussian and annular beams,” J. Opt. Soc. Am. A 24, 156–162 (2007).
    [CrossRef]
  15. S. A. Arpali, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of higher order cos-Gaussian, cosh-Gaussian and annular beams,” J. Mod. Opt. 55, 227–239 (2008).
    [CrossRef]

2009

Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Effect of beam types on the scintillations: A review,” (invited), Proc. SPIE 7200, 720002-1 (2009).

2008

S. A. Arpali, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of higher order cos-Gaussian, cosh-Gaussian and annular beams,” J. Mod. Opt. 55, 227–239 (2008).
[CrossRef]

2007

2005

2004

F. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media,” Proc. SPIE 5160, 86–97 (2004).
[CrossRef]

1999

1980

S. I. Belousov and I. G. Yakushkin, “Strong fluctuations of fields of optical beams in randomly inhomogeneous media,” Sov. J. Quantum Electron. 10, 301–304 (1980).
[CrossRef]

1978

G. Ya. Patrushev, “Fluctuations of the field of a wave beam on reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 8, 1315–1318 (1978).
[CrossRef]

1976

R. L. Fante, “Comparison of theories for intensity fluctuations in strong turbulence,” Radio Sci. 11, 215–220 (1976).
[CrossRef]

1974

K. S. Gochelashvili, V. G. Pevgov, and V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere,” Sov. J. Quantum Electron. 4, 632–637 (1974).
[CrossRef]

V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, and R. Sh. Tsvik, “Focused-laser-beam scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. 64, 516–518 (1974).
[CrossRef]

Al-Habash, M. A.

Andrews, L. C.

F. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media,” Proc. SPIE 5160, 86–97 (2004).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

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

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

Arpali, S. A.

S. A. Arpali, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of higher order cos-Gaussian, cosh-Gaussian and annular beams,” J. Mod. Opt. 55, 227–239 (2008).
[CrossRef]

Banakh, V. A.

Baykal, Y.

Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Effect of beam types on the scintillations: A review,” (invited), Proc. SPIE 7200, 720002-1 (2009).

S. A. Arpali, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of higher order cos-Gaussian, cosh-Gaussian and annular beams,” J. Mod. Opt. 55, 227–239 (2008).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Scintillations of cos-Gaussian and annular beams,” J. Opt. Soc. Am. A 24, 156–162 (2007).
[CrossRef]

Y. Baykal, “Log-amplitude and phase fluctuations of higher-order annular laser beams in a turbulent medium,” J. Opt. Soc. Am. A 22, 672–679 (2005).
[CrossRef]

Belousov, S. I.

S. I. Belousov and I. G. Yakushkin, “Strong fluctuations of fields of optical beams in randomly inhomogeneous media,” Sov. J. Quantum Electron. 10, 301–304 (1980).
[CrossRef]

Cai, Y.

Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Effect of beam types on the scintillations: A review,” (invited), Proc. SPIE 7200, 720002-1 (2009).

Eyyuboglu, H. T.

Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Effect of beam types on the scintillations: A review,” (invited), Proc. SPIE 7200, 720002-1 (2009).

S. A. Arpali, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of higher order cos-Gaussian, cosh-Gaussian and annular beams,” J. Mod. Opt. 55, 227–239 (2008).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Scintillations of cos-Gaussian and annular beams,” J. Opt. Soc. Am. A 24, 156–162 (2007).
[CrossRef]

Fante, R. L.

R. L. Fante, “Comparison of theories for intensity fluctuations in strong turbulence,” Radio Sci. 11, 215–220 (1976).
[CrossRef]

Gochelashvili, K. S.

K. S. Gochelashvili, V. G. Pevgov, and V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere,” Sov. J. Quantum Electron. 4, 632–637 (1974).
[CrossRef]

Hopen, C. Y.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol.2 (Academic, 1978).

Khmelevtsov, S. S.

Krekov, G. M.

Mironov, V. L.

Pevgov, V. G.

K. S. Gochelashvili, V. G. Pevgov, and V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere,” Sov. J. Quantum Electron. 4, 632–637 (1974).
[CrossRef]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

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

Sh. Tsvik, R.

Shishov, V. I.

K. S. Gochelashvili, V. G. Pevgov, and V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere,” Sov. J. Quantum Electron. 4, 632–637 (1974).
[CrossRef]

Tatarski, V. I.

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

Vetelino, F. S.

F. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media,” Proc. SPIE 5160, 86–97 (2004).
[CrossRef]

Ya. Patrushev, G.

G. Ya. Patrushev, “Fluctuations of the field of a wave beam on reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 8, 1315–1318 (1978).
[CrossRef]

Yakushkin, I. G.

S. I. Belousov and I. G. Yakushkin, “Strong fluctuations of fields of optical beams in randomly inhomogeneous media,” Sov. J. Quantum Electron. 10, 301–304 (1980).
[CrossRef]

J. Mod. Opt.

S. A. Arpali, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of higher order cos-Gaussian, cosh-Gaussian and annular beams,” J. Mod. Opt. 55, 227–239 (2008).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. SPIE

Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Effect of beam types on the scintillations: A review,” (invited), Proc. SPIE 7200, 720002-1 (2009).

F. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media,” Proc. SPIE 5160, 86–97 (2004).
[CrossRef]

Radio Sci.

R. L. Fante, “Comparison of theories for intensity fluctuations in strong turbulence,” Radio Sci. 11, 215–220 (1976).
[CrossRef]

Sov. J. Quantum Electron.

K. S. Gochelashvili, V. G. Pevgov, and V. I. Shishov, “Saturation of fluctuations of the intensity of laser radiation at large distances in a turbulent atmosphere,” Sov. J. Quantum Electron. 4, 632–637 (1974).
[CrossRef]

S. I. Belousov and I. G. Yakushkin, “Strong fluctuations of fields of optical beams in randomly inhomogeneous media,” Sov. J. Quantum Electron. 10, 301–304 (1980).
[CrossRef]

G. Ya. Patrushev, “Fluctuations of the field of a wave beam on reflection in a turbulent atmosphere,” Sov. J. Quantum Electron. 8, 1315–1318 (1978).
[CrossRef]

Other

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

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol.2 (Academic, 1978).

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

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Scintillation index in strong turbulence for plane, spherical, Gaussian, and collimated annular beams versus square root of the Rytov plane wave scintillation index.

Fig. 2
Fig. 2

Scintillation index in strong turbulence for plane, spherical, and collimated annular beams with the large- and small-scale scintillation indices versus square root of the Rytov plane wave scintillation index at selected values of source sizes.

Fig. 3
Fig. 3

Scintillation index in strong turbulence for annular beams of finite focal length versus square root of the Rytov plane wave scintillation index at a constant primary source size α s 1 and selected values of secondary source sizes.

Fig. 4
Fig. 4

Scintillation index in strong turbulence for collimated annular versus square root of the Rytov plane wave scintillation index at a constant primary source size α s 1 and selected values of secondary source sizes.

Fig. 5
Fig. 5

Scintillation index in strong turbulence for Gaussian, collimated annular beams with the large- and small-scale scintillation indices versus square root of the Rytov plane wave scintillation index at selected dual values of source sizes.

Fig. 6
Fig. 6

Scintillation index in strong turbulence for annular beams of finite focal length versus square root of the Rytov plane wave scintillation index at selected values of source focusing parameters.

Fig. 7
Fig. 7

Scintillation index in strong turbulence for collimated annular beams versus square root of the Rytov plane wave scintillation index at selected values of source sizes of constant area.

Fig. 8
Fig. 8

Scintillation index in strong turbulence for Gaussian, collimated annular beams with the large- and small-scale scintillation indices versus square root of the Rytov plane wave scintillation index at selected values of source sizes.

Equations (16)

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m 2 = exp ( m LS 2 + m SS 2 ) 1 ,
u s ( s ) = u s ( s x , s y ) = l = 1 2 A l exp [ ( 0.5 k α l s x 2 ) ( 0.5 k α l s y 2 ) ] , t aking A 1 = A 2 = 1 ,
= exp [ ( 0.5 k α 1 s x 2 ) ( 0.5 k α 1 s y 2 ) ] exp [ ( 0.5 k α 2 s x 2 ) ( 0.5 k α 2 s y 2 ) ] ,
m 2 = 4 π Re { 0 L d η 0 κ d κ 0 2 π d θ [ G 1 ( L , η , κ , θ ) + G 2 ( L , η , κ , θ ) ] Φ n ( κ ) } ,
G 1 ( L , η , κ , θ ) = D 2 ( L ) l 1 = 1 2 l 2 = 1 2 A l 1 A l 2 k 2 ( 1 + i α l 1 L ) ( 1 + i α l 2 L ) × exp [ i ( L η ) 2 k ( 1 + i α l 1 η 1 + i α l 1 L + 1 + i α l 2 η 1 + i α l 2 L ) κ 2 ] ,
G 2 ( L , η , κ , θ ) = | D ( L ) | 2 l 1 = 1 2 l 2 = 1 2 A l 1 A l 2 * k 2 ( 1 + i α l 1 L ) ( 1 i α l 2 * L ) × exp [ i ( L η ) 2 k ( 1 + i α l 1 η 1 + i α l 1 L 1 i α l 2 * η 1 i α l 2 * L ) κ 2 ] ,
D ( L ) = l = 1 2 A l 1 ( 1 + i α l L ) .
Φ n , e ( κ ) = 0.033 C n 2 κ 11 3 G ( κ , l 0 , L 0 ) ,
G ( κ , l 0 , L 0 ) = G x ( κ , l 0 , L 0 ) + G y ( κ , l 0 ) = f ( κ l 0 ) g ( κ L 0 ) exp ( κ 2 κ x 2 ) + κ 11 3 ( κ 2 + κ y 2 ) 11 6 ,
m LS 2 = 8.7021 k 2 C n 2 Re { D 2 ( L ) l 1 = 1 2 l 2 = 1 2 A l 1 A l 2 ( 1 + i α l 1 L ) ( 1 + i α l 2 L ) × 0 L d η [ i ( L η ) 2 k ( 1 + i α l 1 η 1 + i α l 1 L + 1 + i α l 2 η 1 + i α l 2 L ) + 1 2 ( 1 κ x l 1 2 + 1 κ x l 2 2 ) ] 5 6 | D ( L ) | 2 l 1 = 1 2 l 2 = 1 2 A l 1 A l 2 * ( 1 + i α l 1 L ) ( 1 i α l 2 * L ) 0 L d η [ i ( L η ) 2 k ( 1 + i α l 1 η 1 + i α l 1 L 1 i α l 2 * η 1 i α l 2 * L ) + 1 2 ( 1 κ x l 1 2 + 1 κ x l 2 2 ) ] 5 6 } ,
m SS 2 = 1.385 C n 2 k 2 Re { | D ( L ) | 2 l 1 = 1 2 l 2 = 1 2 A l 1 A l 2 * ( 1 + i α l 1 L ) ( 1 i α l 2 * L ) t = 0 ( 1 ) t Γ ( t + 5 6 ) ( κ y l 1 κ y l 2 ) t 5 6 0 L d η [ i ( L η ) 2 k × ( 1 + i α l 1 η 1 + i α l 1 L 1 i α l 2 * η 1 i α l 2 * L ) ] t D 2 ( L ) l 1 = 1 2 l 2 = 1 2 A l 1 A l 2 ( 1 + i α l 1 L ) ( 1 + i α l 2 L ) t = 0 ( 1 ) t Γ ( t + 5 6 ) ( κ y l 1 κ y l 2 ) t 5 6 0 L d η [ i ( L η ) 2 k ( 1 + i α l 1 η 1 + i α l 1 L + 1 + i α l 2 η 1 + i α l 2 L ) ] t } ,
κ x l 2 = k η x l L , l = 1 , 2 ,
κ Y l 2 = k η Y l L l = 1 , 2 ,
1 η x l = ( 1 3 1 2 Θ ¯ + 1 5 Θ ¯ 2 ) 6 7 ( σ R m l ) 12 7 + 1.12 ( 1 3 1 2 Θ ¯ + 1 5 Θ ¯ 2 1 + 2.17 Θ ¯ ) 6 7 σ R 12 5 ,
η Y l = 3 ( σ R m l ) 12 5 + 2.07 σ R 12 5 ,
Θ ¯ = [ 1 ( k 2 α s l 4 ) + 1 F l 2 ] L 2 L F l ( 1 L F l ) 2 + [ L ( k α s l 2 ) ] 2 ,

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