Abstract

The closed-form expression for the mean-squared beam width of partially coherent Hermite–Gaussian (H-G) beams propagating through atmospheric turbulence is derived. The influence of turbulence on the spreading of partially coherent H-G beams is studied quantitatively by examining the mean-squared beam width. It is found that the smaller the coherence length σ0 of the source is, and the larger the beam order m and the wavelength λ are, the less partially coherent H-G beams are affected by the turbulence, although the beams with smaller σ0, larger m, and larger λ have greater spreading in free space. In addition, it is shown that two partially coherent H-G beams may generate the same angular spread and that there exist equivalent partially coherent H-G beams that may have the same directionality as a fully coherent Gaussian beam in free space and also in turbulence. The results are illustrated by examples, and a comparison with previous work is also made.

© 2008 Optical Society of America

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  1. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
  2. J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
    [Crossref]
  3. G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002).
    [Crossref]
  4. A. Dogariu and S. Amarande, "Propagation of partially coherent beam: turbulence-induced degradation," Opt. Lett. 28, 10-12 (2003).
    [Crossref] [PubMed]
  5. T. Shirai, A. Dogariu, and E. Wolf, "Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 20, 1094-1102 (2003).
    [Crossref]
  6. M. Salem, T. Shirai, A. Dogariu, and E. Wolf, "Long-distance propagation of partially coherent beams through atmospheric turbulence," Opt. Commun. 216, 261-265 (2003).
    [Crossref]
  7. X. Ji and B. Lü, "Turbulence-induced quality degradation of partially coherent beams," Opt. Commun. 251, 231-236 (2005).
    [Crossref]
  8. 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]
  9. 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]
  10. P. De Santis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
    [Crossref]
  11. 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]
  12. T. Shirai, A. Dogariu, and E. Wolf, "Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence," Opt. Lett. 28, 610-612 (2003).
    [Crossref] [PubMed]
  13. X. Ji, E. Zhang, and B. Lü, "Spreading of partially coherent flattened Gaussian beams propagating through turbulent media," J. Mod. Opt. 53, 1753-1763 (2006).
    [Crossref]
  14. M. Zahid and M. S. Zubairy, "Directionality of partially coherent Bessel-Gaussian," Opt. Commun. 70, 361-364 (1989).
    [Crossref]
  15. Y. Qiu, H. Guo, and Z. Chen, "Paraxial propagation of partially coherent Hermite-Gauss beams," Opt. Commun. 245, 21-26 (2005).
    [Crossref]
  16. S. C. H. Wang and M. A. Plonus, "Optical beam propagation for a partially coherent source in the turbulent atmosphere," J. Opt. Soc. Am. 69, 1297-1304 (1979).
    [Crossref]
  17. A. Ishimaru, "Phase fluctuations in a turbulent medium," Appl. Opt. 16, 3190-3192 (1977).
    [Crossref] [PubMed]
  18. A. E. Siegman, "New developments in laser resonators," Proc. SPIE 1224, 2-14 (1990).
    [Crossref]

2006 (1)

X. Ji, E. Zhang, and B. Lü, "Spreading of partially coherent flattened Gaussian beams propagating through turbulent media," J. Mod. Opt. 53, 1753-1763 (2006).
[Crossref]

2005 (2)

Y. Qiu, H. Guo, and Z. Chen, "Paraxial propagation of partially coherent Hermite-Gauss beams," Opt. Commun. 245, 21-26 (2005).
[Crossref]

X. Ji and B. Lü, "Turbulence-induced quality degradation of partially coherent beams," Opt. Commun. 251, 231-236 (2005).
[Crossref]

2003 (4)

2002 (1)

1998 (1)

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

1990 (2)

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
[Crossref]

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

1989 (1)

M. Zahid and M. S. Zubairy, "Directionality of partially coherent Bessel-Gaussian," Opt. Commun. 70, 361-364 (1989).
[Crossref]

1980 (1)

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 (2)

S. C. H. Wang and M. A. Plonus, "Optical beam propagation for a partially coherent source in the turbulent atmosphere," J. Opt. Soc. Am. 69, 1297-1304 (1979).
[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]

1978 (2)

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]

1977 (1)

Amarande, S.

Andrews, L. C.

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

Chen, Z.

Y. Qiu, H. Guo, and Z. Chen, "Paraxial propagation of partially coherent Hermite-Gauss beams," Opt. Commun. 245, 21-26 (2005).
[Crossref]

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.

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]

Guo, H.

Y. Qiu, H. Guo, and Z. Chen, "Paraxial propagation of partially coherent Hermite-Gauss beams," Opt. Commun. 245, 21-26 (2005).
[Crossref]

Ishimaru, A.

Ji, X.

X. Ji, E. Zhang, and B. Lü, "Spreading of partially coherent flattened Gaussian beams propagating through turbulent media," J. Mod. Opt. 53, 1753-1763 (2006).
[Crossref]

X. Ji and B. Lü, "Turbulence-induced quality degradation of partially coherent beams," Opt. Commun. 251, 231-236 (2005).
[Crossref]

Lü, B.

X. Ji, E. Zhang, and B. Lü, "Spreading of partially coherent flattened Gaussian beams propagating through turbulent media," J. Mod. Opt. 53, 1753-1763 (2006).
[Crossref]

X. Ji and B. Lü, "Turbulence-induced quality degradation of partially coherent beams," Opt. Commun. 251, 231-236 (2005).
[Crossref]

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]

Phillips, R. L.

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

Plonus, M. A.

Qiu, Y.

Y. Qiu, H. Guo, and Z. Chen, "Paraxial propagation of partially coherent Hermite-Gauss beams," Opt. Commun. 245, 21-26 (2005).
[Crossref]

Salem, M.

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, "Long-distance propagation of partially coherent beams through atmospheric turbulence," Opt. Commun. 216, 261-265 (2003).
[Crossref]

Shirai, T.

Siegman, A. E.

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

Wang, S. C. H.

Wolf, E.

Wu, J.

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
[Crossref]

Zahid, M.

M. Zahid and M. S. Zubairy, "Directionality of partially coherent Bessel-Gaussian," Opt. Commun. 70, 361-364 (1989).
[Crossref]

Zhang, E.

X. Ji, E. Zhang, and B. Lü, "Spreading of partially coherent flattened Gaussian beams propagating through turbulent media," J. Mod. Opt. 53, 1753-1763 (2006).
[Crossref]

Zubairy, M. S.

M. Zahid and M. S. Zubairy, "Directionality of partially coherent Bessel-Gaussian," Opt. Commun. 70, 361-364 (1989).
[Crossref]

Appl. Opt. (1)

J. Mod. Opt. (2)

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
[Crossref]

X. Ji, E. Zhang, and B. Lü, "Spreading of partially coherent flattened Gaussian beams propagating through turbulent media," J. Mod. Opt. 53, 1753-1763 (2006).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (7)

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, "Long-distance propagation of partially coherent beams through atmospheric turbulence," Opt. Commun. 216, 261-265 (2003).
[Crossref]

X. Ji and B. Lü, "Turbulence-induced quality degradation of partially coherent beams," Opt. Commun. 251, 231-236 (2005).
[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]

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]

M. Zahid and M. S. Zubairy, "Directionality of partially coherent Bessel-Gaussian," Opt. Commun. 70, 361-364 (1989).
[Crossref]

Y. Qiu, H. Guo, and Z. Chen, "Paraxial propagation of partially coherent Hermite-Gauss beams," Opt. Commun. 245, 21-26 (2005).
[Crossref]

Opt. Lett. (3)

Proc. SPIE (1)

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

Other (1)

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

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Figures (5)

Fig. 1
Fig. 1

Relative width w ( z ) w ( 0 ) of partially coherent H-G beams versus propagation distance z for (a) different values of σ 0 , and m = 1 , λ = 0.6328 μ m , w 0 = 0.02 m ; (b) different values of m, and σ 0 = 0.01 m , λ = 0.6328 μ m , w 0 = 0.02 m ; (c) different values of λ, and m = 1 , σ 0 = 0.01 m , w 0 = 0.02 m . Solid curve, C n 2 = 10 14 m 2 3 ; dashed, C n 2 = 0 .

Fig. 2
Fig. 2

Relative width w ( z ) w ( 0 ) of partially coherent H-G beams versus propagation distance z for different values of m. The turbulence distance z T is indicated by dashed–dotted line. The calculation parameters are w 0 = 0.01 m , σ 0 = 0.01 m , λ = 0.6328 μ m . z T z R .

Fig. 3
Fig. 3

Same as Fig. 2, except calculation parameters are w 0 = 0.1 m , σ 0 = 0.03 m , λ = 0.6328 μ m , and z T z R .

Fig. 4
Fig. 4

Relative width w ( z ) w ( 0 ) of partially coherent H-G beams versus propagation distance z for different values of λ. The turbulence distance z T is indicated by the dashed–dotted line. The calculation parameters are w 0 = 0.01 m , σ 0 = 0.01 m , m = 3 , and z T z R . Solid curve, C n 2 = 10 14 m 2 3 ; dashed, C n 2 = 0 .

Fig. 5
Fig. 5

Mean-squared beam width w ( z ) of the equivalent partially coherent H-G beams, the equivalent GSM beam, the equivalent fully coherent H-G beam, and the corresponding fully coherent Gaussian beam versus propagation distance z. The calculation parameter is λ = 0.6328 μ m , and the other calculation parameters are listed in Table 1. Solid curve, C n 2 = 10 14 m 2 3 ; dashed, C n 2 = 0 .

Tables (1)

Tables Icon

Table 1 Parameters of Equivalent Partially Coherent H-G, Equivalent GSM, Equivalent Fully Coherent H-G, and Corresponding Fully Coherent Gaussian Beams

Equations (51)

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W ( 0 ) ( x 1 , x 2 , z = 0 , ω ) = B m 2 H m ( 2 x 1 w 0 ) H m ( 2 x 2 w 0 ) exp ( x 1 2 + x 2 2 w 0 2 ) exp ( ( x 1 x 2 ) 2 2 σ 0 2 ) ,
B m 2 = 2 π 2 m w 0 m ! .
W ( x 1 , x 2 , z , ω ) = k 2 π z d x 1 d x 2 W 0 ( x 1 , x 2 , z = 0 , ω ) × exp { ( i k 2 z ) [ ( x 1 2 x 2 2 ) 2 ( x 1 x 1 x 2 x 2 ) + ( x 1 2 x 2 2 ) ] } exp [ ψ ( x 1 , x 1 ) + ψ * ( x 2 , x 2 ) ] m ,
exp [ ψ ( x 1 , x 1 ) + ψ * ( x 2 , x 2 ) ] m exp [ ( x 1 x 2 ) 2 + ( x 1 x 2 ) ( x 1 x 2 ) + ( x 1 x 2 ) 2 ρ 0 2 ] ,
ρ 0 = ( 0.545 C n 2 k 2 z ) 3 5 .
u = x 2 + x 1 2 , v = x 2 x 1 ,
S ( x , z , ω ) = W ( x , x , z , ω ) = k 2 π z B m 2 d u d v H m [ 2 w 0 ( u + 1 2 v ) ] H m [ 2 w 0 ( u 1 2 v ) ] exp ( 2 u 2 w 0 2 ) exp ( v 2 2 w 0 2 ) × exp ( v 2 2 σ 0 2 ) exp ( v 2 ρ 0 2 ) exp ( i k z u v ) exp ( i k z x v ) .
w 2 ( z ) = 4 x 2 S ( x , z , ω ) d x S ( x , z , ω ) d x .
w ( z ) = 2 1 + 2 m 4 w 0 2 + 1 k 2 ( 1 + 2 m w 0 2 + 1 σ 0 2 ) z 2 + 2 ( 0.545 C n 2 ) 6 5 k 2 5 z 16 5 .
w ( z ) w ( 0 ) = 1 + 4 k 2 w 0 2 [ 1 w 0 2 + 1 ( 1 + 2 m ) σ 0 2 ] z 2 + 8 ( 0.545 C n 2 ) 6 5 k 2 5 ( 1 + 2 m ) w 0 2 z 16 5 .
w ( z ) w ( 0 ) free = 1 + 4 k 2 w 0 2 [ 1 w 0 2 + 1 ( 1 + 2 m ) σ 0 2 ] z 2 .
w ( z ) GSM = 2 1 4 w 0 2 + 1 k 2 ( 1 w 0 2 + 1 σ 0 2 ) z 2 + 2 ( 0.545 C n 2 ) 6 5 k 2 5 z 16 5 .
w ( z ) HG = 2 1 + 2 m 4 w 0 2 + 1 + 2 m k 2 w 0 2 z 2 + 2 ( 0.545 C n 2 ) 6 5 k 2 5 z 16 5 ,
w ( z ) w ( 0 ) HG = 1 + 4 k 2 w 0 4 z 2 + 8 ( 0.545 C n 2 ) 6 5 k 2 5 ( 1 + 2 m ) w 0 2 z 16 5 .
w ( z ) Gs = 2 1 4 w 0 2 + 1 k 2 w 0 2 z 2 + 2 ( 0.545 C n 2 ) 6 5 k 2 5 z 16 5 .
w 2 ( z ) = w p 2 + w d 2 z 2 + F z 16 5 ,
w p 2 = ( 1 + 2 m ) w 0 2 ,
w d 2 = 4 k 2 ( 1 + 2 m w 0 2 + 1 σ 0 2 ) ,
F = 8 ( 0.545 C n 2 ) 6 5 k 2 5 .
w 2 ( z ) free = w p 2 + w d 2 z 2 .
z R = w p w d .
w 2 ( z T ) w 2 ( z T ) free w 2 ( z T ) = 0.1 .
w 2 ( z ) w d 2 z 2 + F z 16 5 ,
w 2 ( z ) free w d 2 z 2 .
z T ( w d 2 9 F ) 5 6 .
w 2 ( z ) w p 2 + F z 16 5 ,
w 2 ( z ) free w p 2 .
z T ( w p 2 9 F ) 5 16 .
θ sp ( z ) lim z w ( z ) z = 2 1 k 2 ( 1 + 2 m w 0 2 + 1 σ 0 2 ) + 2 ( 0.545 C n 2 ) 6 5 k 2 5 z 6 5 .
1 + 2 m 1 w 01 2 + 1 σ 01 2 = 1 + 2 m 2 w 02 2 + 1 σ 02 2
θ sp ( z ) GSM = 2 1 k 2 ( 1 w 0 2 + 1 σ 0 2 ) + 2 ( 0.545 C n 2 ) 6 5 k 2 5 z 6 5 .
1 + 2 m w 0 2 + 1 σ 0 2 = 1 w 0 GSM 2 + 1 σ 0 GSM 2
θ sp ( z ) HG = 2 1 + 2 m k 2 w 0 2 + 2 ( 0.545 C n 2 ) 6 5 k 2 5 z 6 5 .
1 + 2 m w 0 2 + 1 σ 0 2 = 1 + 2 m HG w 0 HG 2
θ sp ( z ) Gs = 2 1 k 2 w 0 2 + 2 ( 0.545 C n 2 ) 6 5 k 2 5 z 6 5 .
1 + 2 m w 0 2 + 1 σ 0 2 = 1 w 0 GSM 2 + 1 σ 0 GSM 2 = 1 + 2 m HG w 0 HG 2 = 1 w 0 Gs 2
w 2 ( z ) = 4 F 1 F 2 ,
F 1 = x 2 S ( x , z , ω ) d x ,
F 2 = S ( x , z , ω ) d x .
x 2 exp ( i 2 π x s ) d x = 1 ( 2 π ) 2 δ ( s ) ,
F 1 = ( z k ) 2 B m 2 d u d v H m [ 2 w 0 ( u + 1 2 v ) ] H m [ 2 w 0 ( u 1 2 v ) ] exp ( 2 u 2 w 0 2 ) exp ( v 2 2 w 0 2 ) × exp ( v 2 2 σ 0 2 ) exp ( v 2 ρ 0 2 ) exp ( i k z u v ) δ ( v ) .
exp ( x 2 ) H m ( x + y ) H m ( x + z ) d x = 2 m π L m ( 2 y z ) ,
F 1 = ( z k ) 2 1 m ! f ( v ) δ ( v ) d v ,
f ( v ) = exp ( A v 2 ) L m ( B v 2 ) ,
A = ( 1 2 w 0 2 + 1 2 σ 0 2 + 1 ρ 0 2 + k 2 w 0 2 8 z 2 ) ,
B = 1 w 0 2 + k 2 w 0 2 4 z 2 .
F 1 = ( z k ) 2 1 m ! f ( 0 ) .
f ( x ) δ ( x ) d x = f ( 0 ) ,
F 1 = 1 + 2 m 4 w 0 2 + 1 k 2 ( 1 + 2 m w 0 2 + 1 σ 0 2 ) z 2 + 2 ( 0.545 C n 2 ) 6 5 k 2 5 z 16 5 .
F 2 = S ( x , z , ω ) d x = W ( 0 ) ( x , x , z = 0 , ω ) d x = 1 .
w ( z ) = 2 1 + 2 m 4 w 0 2 + 1 k 2 ( 1 + 2 m w 0 2 + 1 σ 0 2 ) z 2 + 2 ( 0.545 C n 2 ) 6 5 k 2 5 z 16 5 .

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