Abstract

The propagation of a multi-Gaussian beam in turbulent atmosphere in a slant path is studied. The analytical expression for the average intensity of a general multi-Gaussian beam is derived. As special cases the average intensities of a two- and a four-Gaussian beam are investigated and numerically calculated. The investigation reveals that at lower altitude and with large σ the intensity distribution at the receiver plane can have a shape (multiple peaks) similar to that at the source plane. But with increase in altitude or decrease in σ, the multiple peaks gradually disappear and evolve into the profile of a fundamental Gaussian beam. From the comparisons between the different propagations we can see that the beam spreading due to wavelength and initial waist width in a slant path is much slower than that in a horizontal path.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
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2007 (5)

Y. Cai and D. Ge, "Analytical formula for a decentered elliptical Gaussian beam propagating in a turbulent atmosphere," Opt. Commun. 271, 509-516 (2007).
[CrossRef]

X. Chu, Y. Ni, and G. Zhou, "Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere," Appl. Phys. B 87, 547-552 (2007).
[CrossRef]

X. Chu, Y. Ni, and G. Zhou, "Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere," Opt. Commun. 274, 274-280 (2007).
[CrossRef]

Y. Zhang, Y. Song, Z. Chen, J. Ji, and Z. Shi, "Virtual sources for a cosh-Gaussian beam," Opt. Lett. 32, 292-294 (2007).
[CrossRef] [PubMed]

H. T. Eyyuboglu, Y. Baykal, and Y. Li, "Scintillation characteristics of cosh-Gaussian beams," Appl. Opt. 46, 1099-1106 (2007).
[CrossRef] [PubMed]

2006 (7)

Y. Cai and S. He, "Propagation of various dark hollow beams in turbulent atmosphere," Opt. Express 14, 1353-1367 (2006).
[CrossRef] [PubMed]

H. T. Eyyuboglu, C. Arpali, and Y. Baykal, "Flat topped beams and their characteristics in turbulent media," Opt. Express 14, 4196-4207 (2006).
[CrossRef] [PubMed]

Y. Baykal and H. T. Eyyuboglu, "Scintillation index of flat-topped Gaussian beams," Appl. Opt. 45, 3793-3797 (2006).
[CrossRef] [PubMed]

C. Arpali, C. Yazicioglu, H. T. Eyyuboglu, S. A. Arpali, and Y. Baykal, "Simulator for general-type beam propagation in turbulent atmosphere," Opt. Express 14, 8918-8928 (2006).
[CrossRef] [PubMed]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, "Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects," Opt. Eng. 45, 076001-076012 (2006).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405 (2006).
[CrossRef]

Y. Cai, "Propagation of various flat-topped beams in a turbulent atmosphere," J. Opt. A, Pure Appl. Opt. 8, 537-545 (2006).
[CrossRef]

2005 (5)

2004 (1)

2003 (2)

2001 (1)

ITU-R Document 3J/31-E, "On propagation data and prediction methods required for the design of space-to-earth and earth-to-space optical communication systems," Presented at the Radio-Communication Study Group meeting, Budapest, 2001.

1987 (1)

1979 (1)

1974 (1)

Andrews, L. C.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, "Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects," Opt. Eng. 45, 076001-076012 (2006).
[CrossRef]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, "Beam wander effects on the scintillation index of a focused beam," Proc. SPIE 5793, 28-37 (2005).
[CrossRef]

Arpali, C.

Arpali, S. A.

Baykal, Y.

Cai, Y.

Y. Cai and D. Ge, "Analytical formula for a decentered elliptical Gaussian beam propagating in a turbulent atmosphere," Opt. Commun. 271, 509-516 (2007).
[CrossRef]

Y. Cai, "Propagation of various flat-topped beams in a turbulent atmosphere," J. Opt. A, Pure Appl. Opt. 8, 537-545 (2006).
[CrossRef]

Y. Cai and S. He, "Propagation of various dark hollow beams in turbulent atmosphere," Opt. Express 14, 1353-1367 (2006).
[CrossRef] [PubMed]

Camm, J.

Chen, Z.

Chu, X.

X. Chu, Y. Ni, and G. Zhou, "Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere," Opt. Commun. 274, 274-280 (2007).
[CrossRef]

X. Chu, Y. Ni, and G. Zhou, "Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere," Appl. Phys. B 87, 547-552 (2007).
[CrossRef]

Davidson, F. M.

Dogariu, A.

Eyyuboglu, H. T.

Ge, D.

Y. Cai and D. Ge, "Analytical formula for a decentered elliptical Gaussian beam propagating in a turbulent atmosphere," Opt. Commun. 271, 509-516 (2007).
[CrossRef]

Hanson, S. G.

He, S.

Itzkan, I.

Ji, J.

Li, Y.

Ni, Y.

X. Chu, Y. Ni, and G. Zhou, "Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere," Opt. Commun. 274, 274-280 (2007).
[CrossRef]

X. Chu, Y. Ni, and G. Zhou, "Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere," Appl. Phys. B 87, 547-552 (2007).
[CrossRef]

Parenti, R. R.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, "Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects," Opt. Eng. 45, 076001-076012 (2006).
[CrossRef]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, "Beam wander effects on the scintillation index of a focused beam," Proc. SPIE 5793, 28-37 (2005).
[CrossRef]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, "Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects," Opt. Eng. 45, 076001-076012 (2006).
[CrossRef]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, "Beam wander effects on the scintillation index of a focused beam," Proc. SPIE 5793, 28-37 (2005).
[CrossRef]

Plonus, M. A.

Ricklin, J. C.

Sasiela, R. J.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, "Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects," Opt. Eng. 45, 076001-076012 (2006).
[CrossRef]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, "Beam wander effects on the scintillation index of a focused beam," Proc. SPIE 5793, 28-37 (2005).
[CrossRef]

Sermutlu, E.

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405 (2006).
[CrossRef]

Shi, Z.

Shirai, T.

Song, Y.

Wallace, J.

Wang, S. C. H.

Wolf, E.

Yazicioglu, C.

Yura, H. T.

Zhang, Y.

Zhou, G.

X. Chu, Y. Ni, and G. Zhou, "Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere," Opt. Commun. 274, 274-280 (2007).
[CrossRef]

X. Chu, Y. Ni, and G. Zhou, "Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere," Appl. Phys. B 87, 547-552 (2007).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. B (1)

X. Chu, Y. Ni, and G. Zhou, "Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere," Appl. Phys. B 87, 547-552 (2007).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

Y. Cai, "Propagation of various flat-topped beams in a turbulent atmosphere," J. Opt. A, Pure Appl. Opt. 8, 537-545 (2006).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Opt. Commun. (4)

Y. Cai and D. Ge, "Analytical formula for a decentered elliptical Gaussian beam propagating in a turbulent atmosphere," Opt. Commun. 271, 509-516 (2007).
[CrossRef]

H. T. Eyyuboglu, "Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere," Opt. Commun. 245, 37-47 (2005).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405 (2006).
[CrossRef]

X. Chu, Y. Ni, and G. Zhou, "Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere," Opt. Commun. 274, 274-280 (2007).
[CrossRef]

Opt. Eng. (1)

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, "Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects," Opt. Eng. 45, 076001-076012 (2006).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Proc. SPIE (1)

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, "Beam wander effects on the scintillation index of a focused beam," Proc. SPIE 5793, 28-37 (2005).
[CrossRef]

Other (1)

ITU-R Document 3J/31-E, "On propagation data and prediction methods required for the design of space-to-earth and earth-to-space optical communication systems," Presented at the Radio-Communication Study Group meeting, Budapest, 2001.

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Figures (4)

Fig. 1
Fig. 1

Cross section of the average normalized intensity distribution of a two-Gaussian beam for propagation in turbulent atmosphere in a slant path. Solid, dashed, and dotted curves denote σ = 1 , 0.8, and 0.5, respectively. ( a ) H = 30 km ( b ) H = 50 km ( c ) H = 100 km .

Fig. 2
Fig. 2

Cross section of the average normalized intensity distribution of a two-Gaussian beam for different propagations, where L = 20000 3 m , ζ = π 6 . Solid, dashed, and dotted curves denote propagation in a slant path, in a horizontal path, and in free space, respectively. ( a ) λ = 3.8 × 10 6 m , w 0 = 0.5 m , ( b ) λ = 1.06 × 10 6 m , w 0 = 0.5 m , ( c ) λ = 1.06 × 10 6 m , w 0 = 0.2 m .

Fig. 3
Fig. 3

Cross section of the average normalized intensity distribution of a four-Gaussian beam for propagation in turbulent atmosphere in a slant path. Solid, dashed, and dotted curves denote σ = 1 , 0.8, and 0.5, respectively. ( a ) H = 30 km , ( b ) H = 50 km , ( c ) H = 100 km .

Fig. 4
Fig. 4

Cross section of the average normalized intensity distribution of four-Gaussian beam for different propagations, where L = 20000 3 m , ζ = π 6 . Solid, dashed, and dotted curves denote propagation in a slant path, in a horizontal path, and in free space, respectively. (a) λ = 3.8 × 10 6 m , w 0 = 0.5 m ; (b) λ = 1.06 × 10 6 m , w 0 = 0.5 m ; (c) λ = 1.06 × 10 6 m , w 0 = 0.2 m .

Equations (22)

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u ( x , y , 0 ) = i N u i ( x , y , 0 ) ,
u i ( x , y , 0 ) = exp [ ( x a i ) 2 + ( y b i ) 2 w 0 2 ] .
I ( p , q , L ) = k 2 ( 2 π L ) 2 u ( x , y , 0 ) u * ( ξ , η , 0 ) × exp { i k 2 L [ ( p x ) 2 + ( q y ) 2 ( p ξ ) 2 ( q η ) 2 ] } × exp [ ψ ( x , y , p , q ) + ψ * ( ξ , η , p , q ) ] d x d y d ξ d η ,
exp [ ψ ( x , y , p , q ) + ψ * ( ξ , η , p , q ) ] = exp ( D w 2 ) = exp { [ ( x ξ ) 2 + ( y η ) 2 ] 5 6 ρ 0 5 3 } ,
ρ 0 = ( 0.545 C ¯ n 2 k 2 L ) 3 5 ,
C ¯ n 2 = 1 H 0 H C n 2 ( h ) d h ,
L = H sec ζ , z = h sec ζ .
C n 2 ( h ) = 8.148 × 10 56 V 2 h 10 exp ( h 1000 ) + 2.7 × 10 16 exp ( h 1500 ) + C 0 exp ( h 100 ) ,
Γ i j ( p , q , L ) = k 2 ( 2 π L ) 2 u i ( x , y , 0 ) u j * ( ξ , η , 0 ) × exp ( i k 2 L [ ( p x ) 2 + ( q y ) 2 ( p ξ ) 2 ( q η ) 2 ] ) × exp [ ψ ( x , y , p , q ) + ψ * ( ξ , η , p , q ) ] d x d y d ξ d η ,
I ( p , q , L ) = i = 1 N j = 1 N Γ i j ( p , q , L ) .
w = 1 + τ 1 + τ 2 w 0 .
Γ i j ( p , q , L ) = w 0 2 w 2 exp { 2 w 2 [ ( p a i + a j 2 ) 2 + ( q b i + b j 2 ) 2 ] } × exp { i k w 0 2 τ 1 2 w 2 L [ ( a i a j ) ( a i + a j 2 p ) + ( b i b j ) ( b i + b j 2 q ) ] } × exp { τ 2 2 w 2 [ ( a i a j ) 2 + ( b i b j ) 2 ] + 2 w 2 [ ( a i a j 2 ) 2 + ( b i b j 2 ) 2 ] } .
Γ i i ( p , q , L ) = w 0 2 w 2 exp { 2 w 2 [ ( p a i ) 2 + ( q b i ) 2 ] } ,
Γ i j ( p , q , L ) = Γ j i * ( p , q , L ) ,
Γ i j ( p , q , L ) + Γ j i ( p , q , L ) = 2 Re [ Γ i j ( p , q , L ) ] .
I ( p , q , L ) = i = 1 N Γ i i ( p , q , L ) + 2 i = 1 N j = i + 1 N Re [ Γ i j ( p , q , L ) ] ,
Re [ Γ i j ( p , q , L ) ] = w 0 2 w 2 exp { 2 w 2 [ ( p a i + a j 2 ) 2 + ( q b i + b j 2 ) 2 ] } × cos { k w 0 2 τ 1 2 w 2 L [ ( a i a j ) ( a i + a j 2 p ) + ( b i b j ) ( b i + b j 2 q ) ] } × exp { τ 2 2 w 2 [ ( a i a j ) 2 + ( b i b j ) 2 ] + 2 w 2 [ ( a i a j 2 ) 2 + ( b i b j 2 ) 2 ] } .
I ( p , q , L ) = w 0 2 w 2 exp [ 2 ( p 2 + q 2 ) w 2 ] ,
I ( p , q , L ) = w 0 2 w 2 exp { 2 [ p 2 + ( q + b ) 2 ] w 2 } { 1 + exp ( 8 b w 2 q ) + 2 exp [ 2 b w 2 ( 2 q b τ 2 ) ] cos [ 2 b k τ 1 w 0 2 w 2 L q ] } .
I ( p , 0 , L ) = 2 w 0 2 w 2 { exp ( 2 b 2 w 2 ) + exp [ 2 ( τ 2 + 1 ) b 2 w 2 ] } exp [ 2 p 2 w 2 ] .
I N ( 0 , q , L ) = I ( 0 , q , L ) Max [ I ( 0 , y , 0 ) ] ,
I ( p , q , L ) = w 0 2 w 2 { exp [ 2 p 2 + ( q b ) 2 w 2 ] + exp [ 2 ( p b ) 2 + q 2 w 2 ] + exp [ 2 p 2 + ( q + b ) 2 w 2 ] + exp [ 2 ( p + b ) 2 + q 2 w 2 ] } + 2 w 0 2 w 2 exp [ 2 w 2 ( p 2 + q 2 + b 2 + b 2 τ 2 ) ] { cos [ 2 b k τ 1 w 0 2 w 2 L p ] + cos [ 2 b k τ 1 w 0 2 w 2 L q ] } + 2 w 0 2 w 2 exp [ 2 w 2 ( p 2 + q 2 + b 2 + b 2 τ 2 2 ) ] cos [ b k τ 1 w 0 2 w 2 L ( p q ) ] { exp [ 2 b w 2 ( p + q ) ] + exp [ 2 b w 2 ( p + q ) ] } + 2 w 0 2 w 2 exp [ 2 w 2 ( p 2 + q 2 + b 2 + b 2 τ 2 2 ) ] cos [ b k τ 1 w 0 2 w 2 L ( p + q ) ] { exp [ 2 b w 2 ( p q ) ] + exp [ 2 b w 2 ( p q ) ] } .

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