Abstract

Based on the generalized beam formulation, we derive the scintillation index and selectively evaluate it for cos–Gaussian and annular beams propagating in weak atmospheric turbulence. Dependence of the scintillation index on propagation length, focusing and displacement parameters, wavelength of operation, and source size are individually investigated. From our graphical outputs, it is observed that a cos–Gaussian beam exhibits lower scintillations and thus has a tendency to be advantageous over a pure Gaussian beam particularly at lower propagation lengths. It is also found that at longer propagation lengths, this advantage switches to the side of the annular beam. Furthermore, the scintillation index of a focused annular beam will be below those of both Gaussian and cos–Gaussian beams starting at earlier propagation distances. When analyzed against source sizes, it is seen that cos–Gaussian beams will offer advantages at relatively large source sizes, while the reverse will be applicable for annular beams.

© 2006 Optical Society of America

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  1. 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]
  2. L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, 'Theory of optical scintillation: Gaussian-beam wave model,' Waves Random Media 11, 271-291 (2001).
    [CrossRef]
  3. V. A. Banakh and V. L. Mironov, 'Influence of the diffraction size of a transmitting aperture and the turbulence spectrum on the intensity fluctuations of laser radiation,' Sov. J. Quantum Electron. 8, 875-878 (1978).
    [CrossRef]
  4. W. B. Miller, J. C. Ricklin, and L. C. Andrews, 'Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,' J. Opt. Soc. Am. A 11, 2719-2726 (1994).
    [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. R. L. Fante, 'Comparison of theories for intensity fluctuations in strong turbulence,' Radio Sci. 11, 215-220 (1976).
    [CrossRef]
  7. 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]
  8. 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]
  9. 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]
  10. F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, 'Scintillation: theory vs. experiment,' in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
    [CrossRef]
  11. J. C. Ricklin and F. M. Davidson, 'Atmospheric optical communication with a Gaussian-Schell beam,' J. Opt. Soc. Am. A 20, 856-866 (2003).
    [CrossRef]
  12. O. Korotkova, 'Control of the intensity fluctuations of random electromagnetic beams on propagation in weak atmospheric turbulence,' in Free-Space Laser Communication Technologies XVIII, G. S. Mecherle, ed., Proc. SPIE 6105, 61050V (2006).
    [CrossRef]
  13. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  14. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.
  15. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
    [CrossRef]
  16. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
    [CrossRef]
  17. F. S. Vetelino and L. C. Andrews, 'Annular Gaussian beams inturbulent media,' in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 86-97 (2004).
    [CrossRef]
  18. D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, 'Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model,' in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B (2006).
    [CrossRef]
  19. Y. Baykal, 'Formulation of correlations for general-type beams in atmospheric turbulence,' J. Opt. Soc. Am. A 23, 889-893 (2006).
    [CrossRef]
  20. Y. Baykal and H. T. Eyyuboglu, 'Scintillation index of flat-topped-Gaussian beams,' Appl. Opt. 45, 3793-3797 (2006).
    [CrossRef] [PubMed]
  21. Y. Baykal, 'Beams with arbitrary field profiles in turbulence,' in SPIE Conference, XIII International Symposium on Atmospheric and Ocean Optics. Atmospheric Physics (Tomsk, Russia, July 2-7, 2006), invited paper.
    [PubMed]
  22. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).
  23. H. T. Eyyuboglu and Y. Baykal, 'Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,' Opt. Express 12, 4659-4674 (2004).
    [CrossRef] [PubMed]
  24. A. Ishimaru, 'Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere,' Radio Sci. 4, 295-305 (1969).
    [CrossRef]

2006 (4)

O. Korotkova, 'Control of the intensity fluctuations of random electromagnetic beams on propagation in weak atmospheric turbulence,' in Free-Space Laser Communication Technologies XVIII, G. S. Mecherle, ed., Proc. SPIE 6105, 61050V (2006).
[CrossRef]

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, 'Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model,' in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B (2006).
[CrossRef]

Y. Baykal, 'Formulation of correlations for general-type beams in atmospheric turbulence,' J. Opt. Soc. Am. A 23, 889-893 (2006).
[CrossRef]

Y. Baykal and H. T. Eyyuboglu, 'Scintillation index of flat-topped-Gaussian beams,' Appl. Opt. 45, 3793-3797 (2006).
[CrossRef] [PubMed]

2005 (1)

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, 'Scintillation: theory vs. experiment,' in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

2004 (2)

F. S. Vetelino and L. C. Andrews, 'Annular Gaussian beams inturbulent media,' in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 86-97 (2004).
[CrossRef]

H. T. Eyyuboglu and Y. Baykal, 'Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,' Opt. Express 12, 4659-4674 (2004).
[CrossRef] [PubMed]

2003 (1)

2001 (1)

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, 'Theory of optical scintillation: Gaussian-beam wave model,' Waves Random Media 11, 271-291 (2001).
[CrossRef]

1999 (1)

1994 (1)

1980 (1)

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 (2)

V. A. Banakh and V. L. Mironov, 'Influence of the diffraction size of a transmitting aperture and the turbulence spectrum on the intensity fluctuations of laser radiation,' Sov. J. Quantum Electron. 8, 875-878 (1978).
[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]

1976 (1)

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

1974 (2)

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]

1969 (1)

A. Ishimaru, 'Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere,' Radio Sci. 4, 295-305 (1969).
[CrossRef]

Al-Habash, M. A.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, 'Theory of optical scintillation: Gaussian-beam wave model,' Waves Random Media 11, 271-291 (2001).
[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]

Andrews, L. C.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, 'Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model,' in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B (2006).
[CrossRef]

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, 'Scintillation: theory vs. experiment,' in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

F. S. Vetelino and L. C. Andrews, 'Annular Gaussian beams inturbulent media,' in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 86-97 (2004).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, 'Theory of optical scintillation: Gaussian-beam wave model,' Waves Random Media 11, 271-291 (2001).
[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]

W. B. Miller, J. C. Ricklin, and L. C. Andrews, 'Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,' J. Opt. Soc. Am. A 11, 2719-2726 (1994).
[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]

Banakh, V. A.

V. A. Banakh and V. L. Mironov, 'Influence of the diffraction size of a transmitting aperture and the turbulence spectrum on the intensity fluctuations of laser radiation,' Sov. J. Quantum Electron. 8, 875-878 (1978).
[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]

Baykal, Y.

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]

Clare, B.

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, 'Scintillation: theory vs. experiment,' in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

Corbett, K.

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, 'Scintillation: theory vs. experiment,' in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

Cowan, D. C.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, 'Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model,' in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B (2006).
[CrossRef]

Davidson, F. M.

Eyyuboglu, H. T.

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]

Gradshteyn, I. S.

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

Grant, K.

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, 'Scintillation: theory vs. experiment,' in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

Hopen, C. Y.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, 'Theory of optical scintillation: Gaussian-beam wave model,' Waves Random Media 11, 271-291 (2001).
[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, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

Ishimaru, A.

A. Ishimaru, 'Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere,' Radio Sci. 4, 295-305 (1969).
[CrossRef]

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

Khmelevtsov, S. S.

Korotkova, O.

O. Korotkova, 'Control of the intensity fluctuations of random electromagnetic beams on propagation in weak atmospheric turbulence,' in Free-Space Laser Communication Technologies XVIII, G. S. Mecherle, ed., Proc. SPIE 6105, 61050V (2006).
[CrossRef]

Krekov, G. M.

Miller, W. B.

Mironov, V. L.

V. A. Banakh and V. L. Mironov, 'Influence of the diffraction size of a transmitting aperture and the turbulence spectrum on the intensity fluctuations of laser radiation,' Sov. J. Quantum Electron. 8, 875-878 (1978).
[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]

Patrushev, G. Ya.

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]

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, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, 'Theory of optical scintillation: Gaussian-beam wave model,' Waves Random Media 11, 271-291 (2001).
[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]

Recolons, J.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, 'Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model,' in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B (2006).
[CrossRef]

Ricklin, J. C.

Ryzhik, I. M.

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

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).

Tsvik, R. Sh.

Vetelino, F. S.

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, 'Scintillation: theory vs. experiment,' in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

F. S. Vetelino and L. C. Andrews, 'Annular Gaussian beams inturbulent media,' in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 86-97 (2004).
[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]

Young, C.

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, 'Scintillation: theory vs. experiment,' in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

Young, C. Y.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, 'Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model,' in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B (2006).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

Opt. Express (1)

Proc. SPIE (4)

F. S. Vetelino and L. C. Andrews, 'Annular Gaussian beams inturbulent media,' in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 86-97 (2004).
[CrossRef]

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, 'Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model,' in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B (2006).
[CrossRef]

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, 'Scintillation: theory vs. experiment,' in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

O. Korotkova, 'Control of the intensity fluctuations of random electromagnetic beams on propagation in weak atmospheric turbulence,' in Free-Space Laser Communication Technologies XVIII, G. S. Mecherle, ed., Proc. SPIE 6105, 61050V (2006).
[CrossRef]

Radio Sci. (2)

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

A. Ishimaru, 'Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere,' Radio Sci. 4, 295-305 (1969).
[CrossRef]

Sov. J. Quantum Electron. (4)

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]

V. A. Banakh and V. L. Mironov, 'Influence of the diffraction size of a transmitting aperture and the turbulence spectrum on the intensity fluctuations of laser radiation,' Sov. J. Quantum Electron. 8, 875-878 (1978).
[CrossRef]

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]

Waves Random Media (1)

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, 'Theory of optical scintillation: Gaussian-beam wave model,' Waves Random Media 11, 271-291 (2001).
[CrossRef]

Other (6)

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

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

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]

Y. Baykal, 'Beams with arbitrary field profiles in turbulence,' in SPIE Conference, XIII International Symposium on Atmospheric and Ocean Optics. Atmospheric Physics (Tomsk, Russia, July 2-7, 2006), invited paper.
[PubMed]

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

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Figures (6)

Fig. 1
Fig. 1

(a) Contour plot of overlaid cos–Gaussian and Gaussian beams. (b) Three-dimensional normalized intensity of cos–Gaussian beam belonging to (a).

Fig. 2
Fig. 2

Scintillation index of cos–Gaussian beam versus propagation distance at selected values of displacement parameters.

Fig. 3
Fig. 3

Scintillation index for flat-topped, collimated annular, Gaussian, and cos–Gaussian beams versus propagation distance.

Fig. 4
Fig. 4

Scintillation index for focused annular, Gaussian, and cos–Gaussian beams versus propagation distance.

Fig. 5
Fig. 5

Scintillation index of cos–Gaussian beam versus displacement distance at selected values of wavelengths of operation.

Fig. 6
Fig. 6

Scintillation index of Gaussian and cos–Gaussian beams versus source size at selected values of propagation lengths.

Equations (9)

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

u s ( s ) = u s ( s x , s y ) = l = 1 N ( n , m ) A l n , m exp ( i θ l n m ) H n ( a x l n s x + b x l n ) exp [ ( 0.5 k α x l n s x 2 + i V x l n s x ) ] H m ( a y l m s y + b y l m ) exp [ ( 0.5 k α y l m s y 2 + i V y l m s y ) ] ,
α x l n = 1 ( k α s x l n 2 ) + i F x l n , α y l m = 1 ( k α s y l m 2 ) + i F y l m ,
m 2 = 4 π Re { 0 L d η 0 κ d κ 0 2 π d θ [ G 1 ( p , L , η , κ , θ ) + G 2 ( p , L , η , κ , θ ) ] Φ n ( κ ) } ,
G 1 ( p , L , η , κ , θ ) = G N ( p , L , η , κ , θ ) G N ( p , L , η , κ , θ ) D 2 ( p , L ) ,
G 2 ( p , L , η , κ , θ ) = G N ( p , L , η , κ , θ ) G N * ( p , L , η , κ , θ ) D ( p , L ) 2 ,
G N ( p , L , η , κ , θ ) = l = 1 N A l e i θ l i k ( 1 + i α l L ) exp [ i ( V x l 2 + V y l 2 ) L 2 k ( 1 + i α l L ) ] exp [ i ( L η ) ( V x l cos θ + V y l sin θ ) κ k ( 1 + i α l L ) ] × exp [ 0.5 i ( L η ) ( 1 + i α l η ) κ 2 k ( 1 + i α l L ) ] ,
D ( p , L ) = l = 1 N A l e i θ l 1 ( 1 + i α l L ) exp [ i ( V x l 2 + V y l 2 ) L 2 k ( 1 + i α l L ) ] .
m 2 = 2.6056 C n 2 k 2 Re { D ( p , L ) 2 l 1 = 1 N l 2 = 1 N A l 1 A l 2 * e i ( θ l 1 θ l 2 ) 1 ( 1 + i α l 1 L ) 1 ( 1 i α l 2 * L ) exp { i ( V x l 1 2 + V y l 1 2 ) L 2 k ( 1 + i α l 1 L ) + i [ ( V x l 2 2 ) * + ( V y l 2 2 ) * ] L 2 k ( 1 i α l 2 L ) } [ 0 L d η 0 d κ κ 8 3 × I 0 ( { [ i ( L η ) V x l 1 k ( 1 + i α l 1 L ) i ( L η ) V x l 2 * k ( 1 i α l 2 * L ) ] 2 + [ i ( L η ) V y l 1 k ( 1 + i α l 1 L ) i ( L η ) V y l 2 * k ( 1 i α l 2 * L ) ] 2 } 0.5 κ ) exp { 0.5 ( L η ) [ i ( 1 + i α l 1 η ) k ( 1 + i α l 1 L ) i ( 1 i α l 2 * η ) k ( 1 i α l 2 * L ) ] κ 2 } ] D 2 ( p , L ) l 1 = 1 N l 2 = 1 N A l 1 A l 2 e i ( θ l 1 + θ l 2 ) 1 ( 1 + i α l 1 L ) 1 ( 1 + i α l 2 L ) × exp [ i ( V x l 1 2 + V y l 1 2 ) L 2 k ( 1 + i α l 1 L ) i ( V x l 2 2 + V y l 2 2 ) L 2 k ( 1 + i α l 2 L ) ] [ 0 L d η 0 d κ κ 8 3 I 0 ( { [ i ( L η ) V x l 1 k ( 1 + i α l 1 L ) i ( L η ) V x l 2 k ( 1 + i α l 2 L ) ] 2 + [ i ( L η ) V y l 1 k ( 1 + i α l 1 L ) i ( L η ) V y l 2 k ( 1 + i α l 2 L ) ] 2 } 0.5 κ ) exp { 0.5 ( L η ) [ i ( 1 + i α l 1 η ) k ( 1 + i α l 1 L ) + i ( 1 + i α l 2 η ) k ( 1 + i α l 2 L ) ] κ 2 } ] } ,
m 2 = 1.3028 C n 2 k 2 Re [ D ( p , L ) 2 l 1 = 1 N l 2 = 1 N r = 0 A l 1 A l 2 * exp [ i ( θ l 1 θ l 2 ) ] 1 ( 1 + i α l 1 L ) 1 ( 1 i α l 2 * L ) ( 0.25 ) r ( r ! ) 2 Γ ( r 5 6 ) exp { i ( V x l 1 2 + V y l 1 2 ) L 2 k ( 1 + i α l 1 L ) + i [ ( V x l 2 2 ) * + ( V y l 2 2 ) * ] L 2 k ( 1 i α l 2 * L ) } ( 0 L d η { [ i ( L η ) V x l 1 k ( 1 + i α l 1 L ) i ( L η ) V x l 2 * k ( 1 i α l 2 * L ) ] 2 + [ i ( L η ) V y l 1 k ( 1 + i α l 1 L ) i ( L η ) V y l 2 * k ( 1 i α l 2 * L ) ] 2 } r { 0.5 ( L η ) [ i ( 1 + i α l 1 η ) k ( 1 + i α l 1 L ) i ( 1 i α l 2 * η ) k ( 1 i α l 2 * L ) ] } r + 5 6 ) D 2 ( p , L ) l 1 = 1 N l 2 = 1 N r = 0 A l 1 A l 2 exp [ i ( θ l 1 + θ l 2 ) ] 1 ( 1 + i α l 1 L ) 1 ( 1 + i α l 2 * L ) ( 0.25 ) r ( r ! ) 2 Γ ( r 5 6 ) exp [ i ( V x l 1 2 + V y l 1 2 ) L 2 k ( 1 + i α l 1 L ) i ( V x l 2 2 + V y l 2 2 ) L 2 k ( 1 + i α l 2 L ) ] × ( 0 L d η { [ i ( L η ) V x l 1 k ( 1 + i α l 1 L ) i ( L η ) V x l 2 k ( 1 + i α l 2 L ) ] 2 + [ i ( L η ) V y l 1 k ( 1 + i α l 1 L ) i ( L η ) V y l 2 k ( 1 + i α l 2 L ) ] 2 } r { 0.5 ( L η ) [ i ( 1 + i α l 1 η ) k ( 1 + i α l 1 L ) + i ( 1 + i α l 2 η ) k ( 1 + i α l 2 L ) ] } r + 5 6 ) ] ,

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