Abstract

Log-amplitude and phase correlations of general-type beams are formulated in atmospheric turbulence. A general beam is described as the superposition of many sets of multimode contents, each mode being off-axis Hermite–Gaussian. Since the Rytov solution is utilized, the formulas are valid in the weakly turbulent regime. The results are presented in integral forms that should be numerically evaluated for the specific beam type of interest. Our general beam results correctly reduce to the existing solutions for the correlations of limiting-case beams such as higher-order single-mode, multimode, off-axis Hermite–Gaussian, Hermite–sinusoidal-Gaussian, higher-order-annular, flat-topped-Gaussian, and thus naturally fundamental mode, plane, and spherical waves. Scintillation index and phase fluctuations in atmospheric optical links employing such special beams will be examined in future using the results reported here.

© 2006 Optical Society of America

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References

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  1. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
  2. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
    [CrossRef]
  3. J. C. Ricklin and F. M. Davidson, "Atmospheric optical communication with a Gaussian Schell beam," J. Opt. Soc. Am. A 20, 856-866 (2003).
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  4. S. Wilks, K. Baker, J. Brase, C. Carrano, J. Morris, A. Ruggerio, and E. Stappaerts, "Adaptive-optics models address free-space communications," Laser Focus World 41, 93-96 (2005).
  5. Y. Baykal, "Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere," J. Opt. Soc. Am. A 21, 1290-1299 (2004).
    [CrossRef]
  6. 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]
  7. Y. Baykal, "Correlation and structure functions for multimode-laser-beam incidence in atmospheric turbulence," J. Opt. Soc. Am. A 4, 817-819 (1987).
    [CrossRef]
  8. Y. Baykal and H. T. Eyyuboglu, "Scintillation index of flat-topped-Gaussian beams," Appl. Opt. (to be published).
  9. A. Ishimaru, "Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere," Radio Sci. 4, 295-305 (1969).
    [CrossRef]
  10. I. S. Gradysteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

2005

S. Wilks, K. Baker, J. Brase, C. Carrano, J. Morris, A. Ruggerio, and E. Stappaerts, "Adaptive-optics models address free-space communications," Laser Focus World 41, 93-96 (2005).

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]

2004

2003

1987

1969

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

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

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

Baker, K.

S. Wilks, K. Baker, J. Brase, C. Carrano, J. Morris, A. Ruggerio, and E. Stappaerts, "Adaptive-optics models address free-space communications," Laser Focus World 41, 93-96 (2005).

Baykal, Y.

Brase, J.

S. Wilks, K. Baker, J. Brase, C. Carrano, J. Morris, A. Ruggerio, and E. Stappaerts, "Adaptive-optics models address free-space communications," Laser Focus World 41, 93-96 (2005).

Carrano, C.

S. Wilks, K. Baker, J. Brase, C. Carrano, J. Morris, A. Ruggerio, and E. Stappaerts, "Adaptive-optics models address free-space communications," Laser Focus World 41, 93-96 (2005).

Davidson, F. M.

Eyyuboglu, H. T.

Y. Baykal and H. T. Eyyuboglu, "Scintillation index of flat-topped-Gaussian beams," Appl. Opt. (to be published).

Gradysteyn, I. S.

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

Hopen, C. Y.

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]

Morris, J.

S. Wilks, K. Baker, J. Brase, C. Carrano, J. Morris, A. Ruggerio, and E. Stappaerts, "Adaptive-optics models address free-space communications," Laser Focus World 41, 93-96 (2005).

Phillips, R. L.

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, 1998).

Ricklin, J. C.

Ruggerio, A.

S. Wilks, K. Baker, J. Brase, C. Carrano, J. Morris, A. Ruggerio, and E. Stappaerts, "Adaptive-optics models address free-space communications," Laser Focus World 41, 93-96 (2005).

Ryzhik, I. M.

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

Stappaerts, E.

S. Wilks, K. Baker, J. Brase, C. Carrano, J. Morris, A. Ruggerio, and E. Stappaerts, "Adaptive-optics models address free-space communications," Laser Focus World 41, 93-96 (2005).

Wilks, S.

S. Wilks, K. Baker, J. Brase, C. Carrano, J. Morris, A. Ruggerio, and E. Stappaerts, "Adaptive-optics models address free-space communications," Laser Focus World 41, 93-96 (2005).

J. Opt. Soc. Am. A

Laser Focus World

S. Wilks, K. Baker, J. Brase, C. Carrano, J. Morris, A. Ruggerio, and E. Stappaerts, "Adaptive-optics models address free-space communications," Laser Focus World 41, 93-96 (2005).

Radio Sci.

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

Other

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

Y. Baykal and H. T. Eyyuboglu, "Scintillation index of flat-topped-Gaussian beams," Appl. Opt. (to be published).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

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

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Equations (30)

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u l n m inc ( s x , s y , z = 0 ) = A l n m exp ( i θ l n m ) H n ( a x l n s x + b x l n ) × H m ( a y l m s y + b y l m ) exp [ k 2 ( α x l n s x 2 + α y l m s y 2 ) ] exp [ i ( V x l n s x + 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 ;
u inc ( s x , s y , z = 0 ) = l = 1 N ( n , m ) u l n m inc ( s x , s y , z = 0 ) ,
u inc ( s x , s y , z = 0 ) = 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 ) × H m ( a y l m s y + b y l m ) exp [ k 2 ( α x l n s x 2 + α y l m s y 2 ) ] × exp [ i ( V x l n s x + V y l m s y ) ] .
u FS ( p , z ) = l = 1 N ( n , m ) k exp ( i k z ) A l n m exp ( i θ l n m ) 2 π i z × d 2 s H n ( a x l n s x + b x l n ) H m ( a y l m s y + b y l m ) exp [ k 2 ( α x l n s x 2 + α y l m s y 2 ) ] exp [ i ( V x l n s x + V y l m s y ) ] exp [ i k 2 z ( s p ) 2 ] ,
u FS ( p , z ) = l = 1 N ( n , m ) A l n m exp ( i k z ) exp ( i θ l n m ) [ 1 2 i a x l n 2 z k ( 1 + i α x l n z ) ] n 2 [ 1 2 i a y l m 2 z k ( 1 + i α y l m z ) ] m 2 × 1 ( 1 + i α x l n z ) 1 2 1 ( 1 + i α y l m z ) 1 2 exp [ i V x l n 2 z 2 k ( 1 + i α x l n z ) ] exp [ i V y l m 2 z 2 k ( 1 + i α y l m z ) ] exp [ k a x l n 2 ( 1 + i α x l n z ) p x 2 ] exp [ i V x l n ( 1 + i α x l n z ) p x ] exp [ k α y l m 2 ( 1 + i α y l m z ) p y 2 ] exp [ i V y l m ( 1 + i α y l m z ) p y ] × H n ( β 1 x l n + β 2 x l n p x ) H m ( β 1 y l m + β 2 y l m p y ) ,
β 2 x l n = a x l n ( 1 + i α x l n z ) 1 2 1 [ 1 i z ( 2 a x l n 2 k α x l n ) ] 1 2 ,
β 1 x l n = β 2 x l n [ a x l n V x l n z + k b x l n ( 1 + i α x l n z ) k a x l n ] .
u ( p , z ) = u FS ( p , z ) exp [ ψ ( p , z ) ] ,
ψ ( p , z ) = χ ( p , z ) + i S ( p , z ) = k 2 2 π u FS ( p , z ) × V d 3 r n 1 ( p , z ) u FS ( p , z ) exp ( i k r r ) r r
n 1 ( p x , p y , z ) = exp ( i κ x p x + i κ y p y ) d Z n ( κ x , κ y , z ) .
B χ ( p 1 , p 2 , L ) = π Re { 0 L d η 0 κ d κ 0 2 π d θ [ M 1 ( p 1 , p 2 , η , κ , θ ) + M 2 ( p 1 , p 2 , η , κ , θ ) ] Φ n ( κ ) } ,
κ = κ e i θ κ x = κ cos θ , κ y = κ sin θ ,
κ = κ = ( κ x 2 + κ y 2 ) 1 2 d 2 κ = d κ x d κ y = κ d κ d θ ,
M 1 ( p 1 , p 2 , L , η , κ , θ ) = N ( p 1 , L , η , κ , θ ) N ( p 2 , L , η , κ , θ ) D ( p 1 , L ) D ( p 2 , L ) ,
M 2 ( p 1 , p 2 , L , η , κ , θ ) = N ( p 1 , L , η , κ , θ ) N * ( p 2 , L , η , κ , θ ) D ( p 1 , L ) D * ( p 2 , L ) .
N ( p , L , η , κ , θ ) = { l = 1 N ( n , m ) A l n m exp ( i θ l n m ) i k ( 1 + i α x l n L ) ( 1 + i α y l m L ) exp [ i k p x 2 2 ( L η ) + i k p y 2 2 ( L η ) ] exp [ i k γ x l n p x 2 2 ( L η ) i V x l n p x ( 1 + i α x l n L ) ] exp [ i k γ y l m p y 2 2 ( L η ) i V y l m p y ( 1 + i α y l m L ) ] exp [ i V x l n 2 η 2 k ( 1 + i α x l n η ) ] exp [ i V y l m 2 η 2 k ( 1 + i α y l m η ) ] g 2 x l n g 2 y l m exp ( t 1 x l n ) exp ( t 1 y l m ) exp ( t 2 x l n κ cos θ ) exp ( t 2 y l m κ sin θ ) × exp ( i γ x l n p x κ cos θ ) exp ( i γ y l m p y κ sin θ ) exp ( b 3 x l n 2 κ 2 cos 2 θ ) exp ( b 3 y l m 2 κ 2 sin 2 θ ) H n ( g 4 x l n p x 1 + g 5 x l n κ cos θ + g 7 x l n ) H m ( g 4 y l m p y + g 5 y l m κ sin θ + g 7 y l m ) } ,
D ( p , L ) = l = 1 N ( n , m ) A l n m e i θ l n m g 2 x l n g 2 y l m 1 ( 1 + i α x l n L ) 1 2 1 ( 1 + i α y l m L ) 1 2 exp [ i V x l n 2 L 2 k ( 1 + i α x l n L ) ] exp [ i V y l m 2 L 2 k ( 1 + i α y l m L ) ] exp [ k α x l n 2 ( 1 + i α x l n L ) p x 2 ] exp [ i V x l n ( 1 + i α x l n L ) p x ] exp [ k α y l m 2 ( 1 + i α y l m L ) p y 2 ] exp [ i V y l m ( 1 + i α y l m L ) p y ] H n ( g 4 x l n p x + g 7 x l n ) H m ( g 4 y l m p y + g 7 y l m ) ,
g 2 x l n = { 1 i L [ 2 ( a x l n 2 k ) α x l n ] 1 + i α x l n L } n 2 ,
b 3 x l n = i γ x l n ( η L ) k ,
t 1 x l n = i γ x l n ( η L ) 2 k ( 1 + i α x l n η ) 2 V x l n 2 ,
t 2 x l n = i γ x l n ( η L ) k ( 1 + i α x l n η ) V x l n ,
γ x l n = ( 1 + i α x l n η ) ( 1 + i α x l n L ) ,
g 4 x l n = a x l n ( 1 + i α x l n L ) 1 2 { 1 i L [ 2 ( a x l n 2 k ) α x l n ] } 1 2 ,
g 5 x l n = ( η L ) k g 4 x l n ,
g 7 x l n = g 4 x l n [ L V x l n k + b x l n a x l n ( 1 + i α x l n L ) ] .
B S ( p 1 , p 2 , L ) = π Re { 0 L d η 0 κ d κ 0 2 π d θ [ M 1 ( p 1 , p 2 , η , κ , θ ) M 2 ( p 1 , p 2 , η , κ , θ ) ] Φ n ( κ ) } .
D χ S ( p 1 , p 2 , L ) = B χ S ( p 1 , p 1 , L ) + B χ S ( p 2 , p 2 , L ) 2 B χ S ( p 1 , p 2 , L ) .
D ψ ( p 1 , p 2 , L ) = D χ ( p 1 , p 2 , L ) + D S ( p 1 , p 2 , L ) .

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