Abstract

The directionality of general beams propagating in free space and in atmospheric turbulence is studied. Based on the partial-coherence theory, the analytical expressions for the mean-squared width and the angular spread of general beams are derived by using the integral transform technique. It is shown that the mean-squared width and the angular spread depend not only on the weighting factors of all basis modes but also on the weighting factors of the corresponding mode coherence coefficients of the ith and the (i+2)th if the Hermite-Gaussian modes are adopted. It is found that under a certain condition there exist the equivalent general beams which may have the same directionality as a fully coherent Gaussian beam in free space and also in atmospheric turbulence. The result holds true, irrespective of the model of turbulence used.

© 2008 Optical Society of America

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References

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  1. A. E. Siegman, Laser (University Science Books Mill Valley, 1986).
  2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).
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    [CrossRef] [PubMed]
  4. E. Wolf and E. Collett, "Partially coherent sources which produce the same far-field intensity distribution as a laser," Opt. Commun. 25, 293-296 (1978).
    [CrossRef]
  5. P. De Santis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
    [CrossRef]
  6. J. D. Farina, L. M. Narducci, and E. Collett, "Generation of highly directional beams from a globally incoherent source," Opt. Commun. 32, 203-208 (1980).
    [CrossRef]
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    [CrossRef] [PubMed]
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  12. X. Ji, X. Chen, and B. Lü, "Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 25, 21-28 (2008).
    [CrossRef]
  13. A. Yang, E. Zhang, X. Ji, and B. Lü, "Angular spread of partially coherent Hermite-cosh-Gaussian beams propagating through atmospheric turbulence," Opt. Express,  16, 8366-8380 (2008).
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  14. K. M. Du, G. Herziger, P. Loosen, and F. Rühl, "Coherence and intensity moments of laser light," Opt. Quantum Electron. 24, 1081-1093 (1992).
    [CrossRef]
  15. A. Starikov and E. Wolf, "Coherent-mode representation of Gaussian Schell-model sources and their radiation fields," J. Opt. Soc. Am. 72, 923-928 (1982).
    [CrossRef]
  16. A. E. Siegman, "New developments in laser resonators," Proc. SPIE 1224, 2-14 (1990).
    [CrossRef]
  17. A. A. Tovar and L. W. Casperson, "Production and propagation of Hermite-sinusoidal-Gaussian laser beams," J. Opt. Soc. Am. A 15, 2425-2432 (1998).
    [CrossRef]

2008

2006

2005

2003

2002

1998

1992

K. M. Du, G. Herziger, P. Loosen, and F. Rühl, "Coherence and intensity moments of laser light," Opt. Quantum Electron. 24, 1081-1093 (1992).
[CrossRef]

1990

A. E. Siegman, "New developments in laser resonators," Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

1982

1980

J. D. Farina, L. M. Narducci, and E. Collett, "Generation of highly directional beams from a globally incoherent source," Opt. Commun. 32, 203-208 (1980).
[CrossRef]

1979

P. De Santis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

1978

E. Collett and E. Wolf, "Is complete spatial coherence necessary for the generation of highly directional light beams?," Opt. Lett. 2, 27-29 (1978).
[CrossRef] [PubMed]

E. Wolf and E. Collett, "Partially coherent sources which produce the same far-field intensity distribution as a laser," Opt. Commun. 25, 293-296 (1978).
[CrossRef]

Amarande, S.

Baykal, Y.

Cai, Y.

Casperson, L. W.

Chen, X.

Collett, E.

J. D. Farina, L. M. Narducci, and E. Collett, "Generation of highly directional beams from a globally incoherent source," Opt. Commun. 32, 203-208 (1980).
[CrossRef]

E. Wolf and E. Collett, "Partially coherent sources which produce the same far-field intensity distribution as a laser," Opt. Commun. 25, 293-296 (1978).
[CrossRef]

E. Collett and E. Wolf, "Is complete spatial coherence necessary for the generation of highly directional light beams?," Opt. Lett. 2, 27-29 (1978).
[CrossRef] [PubMed]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Dogariu, A.

Du, K. M.

K. M. Du, G. Herziger, P. Loosen, and F. Rühl, "Coherence and intensity moments of laser light," Opt. Quantum Electron. 24, 1081-1093 (1992).
[CrossRef]

Eyyuboglu, H. T.

Farina, J. D.

J. D. Farina, L. M. Narducci, and E. Collett, "Generation of highly directional beams from a globally incoherent source," Opt. Commun. 32, 203-208 (1980).
[CrossRef]

Gbur, G.

Gori, F.

P. De Santis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Guattari, G.

P. De Santis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

He, S.

Herziger, G.

K. M. Du, G. Herziger, P. Loosen, and F. Rühl, "Coherence and intensity moments of laser light," Opt. Quantum Electron. 24, 1081-1093 (1992).
[CrossRef]

Ji, X.

Loosen, P.

K. M. Du, G. Herziger, P. Loosen, and F. Rühl, "Coherence and intensity moments of laser light," Opt. Quantum Electron. 24, 1081-1093 (1992).
[CrossRef]

Lü, B.

Narducci, L. M.

J. D. Farina, L. M. Narducci, and E. Collett, "Generation of highly directional beams from a globally incoherent source," Opt. Commun. 32, 203-208 (1980).
[CrossRef]

Palma, C.

P. De Santis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Rühl, F.

K. M. Du, G. Herziger, P. Loosen, and F. Rühl, "Coherence and intensity moments of laser light," Opt. Quantum Electron. 24, 1081-1093 (1992).
[CrossRef]

Shirai, T.

Siegman, A. E.

A. E. Siegman, "New developments in laser resonators," Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

Starikov, A.

Tovar, A. A.

Wolf, E.

Yang, A.

Zhang, E.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

E. Wolf and E. Collett, "Partially coherent sources which produce the same far-field intensity distribution as a laser," Opt. Commun. 25, 293-296 (1978).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

J. D. Farina, L. M. Narducci, and E. Collett, "Generation of highly directional beams from a globally incoherent source," Opt. Commun. 32, 203-208 (1980).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

K. M. Du, G. Herziger, P. Loosen, and F. Rühl, "Coherence and intensity moments of laser light," Opt. Quantum Electron. 24, 1081-1093 (1992).
[CrossRef]

Proc. SPIE

A. E. Siegman, "New developments in laser resonators," Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

Other

A. E. Siegman, Laser (University Science Books Mill Valley, 1986).

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

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Figures (2)

Fig. 1.
Fig. 1.

w x (z) versus z. a: the corresponding fully coherent Gaussian beam with w 0 Gs=9.7mm, λ Gs=1.54µm; b: an equivalent ShG beam with w 0=4mm, γ=0.5, δ=6, λ=0.6328µm.

Fig. 2.
Fig. 2.

w x (z) versus z. a: the corresponding fully coherent Gaussian beam with w 0 Gs=7.5mm, λ Gs=1.54µm; b: an equivalent GSM beam with w 0=4mm, γ=1.28, α=3.06, λ=0.6328µm.

Equations (44)

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W ( 0 ) ( r 1 , r 2 ) = W ( 0 ) ( x 1 , x 2 ) W ( 0 ) ( y 1 , y 2 ) ,
W ( 0 ) ( x 1 , x 2 ) = i 1 = 0 + j 1 = 0 + λ i 1 , j 1 φ i 1 ( x 1 ) φ j 1 * ( x 2 ) ,
φ l ( x ) = [ 2 1 2 ( π 1 2 2 l w 0 h l ! ) ] 1 2 exp ( x 2 w 0 h 2 ) H l ( 2 1 2 x w 0 h ) , ( l = i 1 , j 1 ) ,
λ i 1 , j 1 = φ i 1 * ( x 1 ) W ( 0 ) ( x 1 , x 2 ) φ j 1 ( x 2 ) d x 1 d x 2 .
I ( r , z ) = [ k 2 π z ] 2 d 2 r 1 d 2 r 2 W ( 0 ) ( r 1 , r 2 )
× exp { [ i k ( 2 z ) ] [ ( r 1 2 r 2 2 ) 2 r · ( r 1 r 2 ) ] } exp [ ψ * ( r , r 1 , z ) + ψ ( r , r 2 , z ) ] m ,
exp [ ψ * ( r , r 1 , z ) + ψ ( r , r 2 , z ) ] m = exp { 4 π 2 k 2 z 0 1 0 κ Φ n ( κ ) [ 1 J 0 ( κ ξ r 2 r 1 ) ] d κ d ξ } ,
u = ( r 2 + r 1 ) 2 , v = r 2 r 1 ,
I ( r , z ) = ( k 2 π z ) 2 d 2 u d 2 v W ( 0 ) ( u , v ) exp ( i k u · v z ) exp ( i k r · v z )
× exp { 4 π 2 k 2 z 0 1 0 κ Φ n [ 1 J 0 ( κ ξ v ) ] d κ d ξ } .
w x 2 ( z ) = 4 F 1 F 2 ,
F 1 = x 2 I ( r , z ) d x d y ,
F 2 = I ( r , z ) d x d y .
x 2 exp ( i 2 π x s ) d x = δ ( s ) ( 2 π ) 2 , exp ( i 2 π x s ) d x = δ ( s ) .
F 1 = ( z k ) 2 Σ i 1 = 0 + Σ j 1 = 0 + Σ i 2 = 0 + Σ j 2 = 0 + λ i 1 , j 1 2 1 2 ( π 2 i 1 + j 1 w 0 h 2 i l ! j l ! ) 1 2
× λ i 2 , j 2 2 1 2 ( π 2 i 2 + j 2 w 0 h 2 i 2 ! j 2 ! ) 1 2 d 2 u d 2 v exp ( 2 u x 2 w 0 h 2 ) exp [ v x 2 2 w 0 h 2 ]
× H i 1 [ 2 1 2 ( u x + v x 2 ) w 0 h ] H j 1 [ 2 1 2 ( u x v x 2 ) w 0 h ] exp ( 2 u y 2 w 0 h 2 )
× exp [ v y 2 ( 2 w 0 h 2 ) ] H i 2 [ 2 1 2 ( u y + v y 2 ) w 0 h ] H j 2 [ 2 1 2 ( u y v y 2 ) w 0 h ] exp ( i k u · v z )
× exp { 4 π 2 k 2 z 0 1 0 κ Φ n ( κ ) [ 1 J 0 ( κ ξ v ) ] d κ d ξ } δ ( v x ) δ ( v y ) .
f ( x ) δ ( x ) d x = f ( 0 ) , exp ( x 2 ) H i ( x + y ) H j ( x + z ) d x = { 0 ( i j ) 2 i i ! π ( i = j ) ,
exp ( x 2 ) H i ( x + y ) H j ( x + z ) d x = 2 j π 1 2 i ! z j i L i j i ( 2 y z ) , ( i j ) ,
+ f ( x ) δ ( x ) d x = f ( 0 ) ,
F 1 = ( z k ) 2 i 2 = 0 + λ i 2 , i 2 i 1 = 0 + j 1 = 0 + λ i 1 , j 1 ( 2 j 1 i 1 i 1 ! j 1 ! ) 1 2 ( 1 2 1 2 w 0 h i k w 0 h 2 3 2 z ) j 1 i 1 f ( 0 ) ,
A = 1 w 0 h 2 + k 2 w 0 h 2 ( 4 z 2 ) ,
f ( v ) = v j 1 i 1 exp ( A v 2 2 ) L i 1 j 1 i 1 ( A v 2 ) exp { 4 π 2 k 2 z 0 1 0 + κ Φ n ( κ ) [ 1 J 0 ( κ ξ v ) ] d κ d ξ } .
F 1 = ( w 0 h 2 4 ) i 2 = 0 + λ i 2 , i 2 i 1 = 0 + { ( 1 + 2 i 1 ) λ i 1 , i 1 + 2 [ ( i 1 + 1 ) ( i 1 + 2 ) ] 1 2 λ i 1 , i 1 + 2 }
+ [ z 2 ( k 2 w 0 h 2 ) ] i 2 = 0 + λ i 2 , i 2 i 1 = 0 + { ( 1 + 2 i 1 ) λ i 1 , i 1 2 [ ( i 1 + 1 ) ( i 1 + 2 ) ] 1 2 λ i 1 , i 1 + 2 }
+ [ ( 2 π 2 3 ) z 3 0 + κ 3 Φ n ( κ ) d κ ] i 2 = 0 + λ i 2 , i 2 i = 0 + λ i 1 , i 1 .
F 2 = i 1 = 0 + λ i 1 , i 1 i 2 = 0 + λ i 2 , i 2 .
w x 2 ( z ) = w 0 h 2 i = 0 + { ( 1 + 2 i ) β i , i + 2 [ ( i + 1 ) ( i + 2 ) ] 1 2 β i , i + 2 }
+ 4 z 2 ( k 2 w 0 h 2 ) i = 0 + { ( 1 + 2 i ) β i , i 2 [ ( i + 1 ) ( i + 2 ) ] 1 2 β i , i + 2 } + ( 8 π 2 3 ) z 3 0 κ 3 Φ n ( κ ) d κ ·
θ x ( z ) lim z w x ( z ) / z = { 4 ( k 2 w 0 h 2 ) i = 0 + { ( 1 + 2 i ) β i , i 2 [ ( i + 1 ) ( i + 2 ) ] 1 2 β i , i + 2 } + ( 8 π 2 3 ) z 0 + κ 3 Φ n ( κ ) d κ } 1 2 .
w x 2 ( z ) Gs = w 0 Gs 2 + 4 z 2 ( k Gs 2 w 0 Gs 2 ) + ( 8 π 2 3 ) z 3 0 + κ 3 Φ n ( κ ) d κ ,
θ x ( z ) Gs = [ 4 ( k Gs 2 w 0 Gs 2 ) + ( 8 π 2 3 ) z 0 + κ 3 Φ n ( κ ) d κ ] 1 2 .
( k 2 w 0 h 2 ) 1 i = 0 i = + { ( 1 + 2 i ) β i , i 2 [ ( i + 1 ) ( i + 2 ) ] 1 2 β i , i + 2 } = ( k 0 Gs 2 w 0 Gs 2 ) 1
Φ n ( κ ) = 0.033 C n 2 ( κ 2 + 1 L 0 2 ) 11 6 exp ( κ 2 κ m 2 ) ,
F = ( 8 π 2 3 ) 0 + κ 3 Φ n ( κ ) d κ = 18.846 C n 2 .
W ( 0 ) ( x 1 , x 2 ) = exp ( x 1 2 w 0 2 ) sin h ( Ω 0 x 1 ) exp ( x 2 2 w 0 2 ) sin h ( Ω 0 x 2 ) ,
λ i , j = ( π 2 ) 1 2 γ w 0 h ( 2 + γ ) ( i ! j ! ) 1 2 ( 2 γ 4 + 2 γ ) ( i + j ) 2 exp ( δ 1 + γ ) H i [ γ δ ( 4 γ 2 ) 1 2 ] H j [ γ δ ( 4 γ 2 ) 1 2 ] ,
W ( 0 ) ( x 1 , x 2 ) = exp [ ( x 1 2 + x 2 2 ) w 0 2 ] exp [ ( x 1 x 2 ) 2 ( 2 σ 0 2 ) ] ,
λ i , j = λ 0 { ( [ 1 + 1 ( 2 α ) ] 2 γ 2 4 ) 1 2 1 + 1 ( 2 α ) + γ 2 } i ( i ! j ! ) 1 2 2 i + j r i ( w 0 h 2 ) j p ( i + j ) 2
× k = 0 min ( i , j ) [ ( 1 ) k + ( i + j ) 2 2 k k ! [ ( i k ) 2 ] ! [ ( j k ) 2 ] ! ( 2 r 2 p 1 ) ( i k ) 2 ( w 0 h 2 p 2 1 ) ( j k ) 2 ] ,
λ 0 = [ π γ p 1 + 1 ( 2 α ) + γ 2 ] 1 2 , p = 4 α 2 [ 1 + 1 ( 2 α ) + γ 2 ] 2 1 4 α 2 w 0 2 [ 1 + 1 ( 2 α ) + γ 2 ] ,
r = α w 0 h { [ 1 + 1 ( 2 α ) ] 2 γ 2 4 } 1 2 , α = ( σ 0 w 0 ) 2 .

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