Abstract

The angular spread is chosen as the characteristic parameter of beam directionality, and the directionality of Gaussian array beams propagating in atmospheric turbulence is studied. The consistency of the directionality of Gaussian array beams expressed in terms of the angular spread and the normalized far-field average intensity distribution is also examined. Closed-form expressions for the mean-squared beam width and angular spread of Gaussian array beams propagating in atmospheric turbulence are derived. It is found that for the case of coherent combination, under a certain condition array beams exist that may have the same directionality (i.e., the same angular spread) as a fully coherent Gaussian laser beam in free space and also in turbulence. However, the directionality of Gaussian array beams in terms of the angular spread is not consistent with that in terms of the normalized far-field average intensity distribution in free space, whereas they may be consistent as a result of turbulence.

© 2009 Optical Society of America

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References

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  1. 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]
  2. P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256-260 (1979).
    [CrossRef]
  3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
  4. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598 (2002).
    [CrossRef]
  5. H. T. Eyyuboğlu and Y. Baykal, “Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Express 12, 4659-4674 (2004).
    [CrossRef] [PubMed]
  6. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353-1367 (2006).
    [CrossRef] [PubMed]
  7. 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]
  8. 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]
  9. 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).
    [CrossRef] [PubMed]
  10. X. Chen and X. Ji, “Directionality of partially coherent annular flat-topped beams propagating through atmospheric turbulence,” Opt. Commun. 281, 4765-4770 (2008).
    [CrossRef]
  11. R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers,” Proc. SPIE 1224, 239-253 (1990).
    [CrossRef]
  12. G. L. Schuster and J. R. Andrews, “Coherent summation of saturated AlGaAs amplifiers,” Opt. Lett. 16, 913-915 (1991).
    [PubMed]
  13. X. Ji, E. Zhang, and B. Lü, “Propagation of multi-Gaussian beams in incoherent combination through turbulent atmosphere and their beam quality,” J. Mod. Opt. 53, 2111-2127 (2006).
    [CrossRef]
  14. Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
    [CrossRef]
  15. X. Ji, E. Zhang, and B. Lü, “Superimposed partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. B 25, 825-833 (2008).
    [CrossRef]
  16. J. T. Foley and M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297-300 (1978).
    [CrossRef]
  17. S. Lü and B. Lü, “The directionality of partially coherent beams expressed in terms of the far-field divergence angle and of the far-field radiant intensity distribution,” Opt. Commun. 281, 3514-3521 (2008).
    [CrossRef]
  18. 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]
  19. A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2-14 (1990).
    [CrossRef]
  20. J. C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175-185 (1978).
    [CrossRef]
  21. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794-1802 (2002).
    [CrossRef]
  22. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
    [CrossRef]
  23. H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264, 25-34 (2006).
    [CrossRef]
  24. X. Chu, Z. Liu, and Y. Wu, “Propagation of a general multi-Gaussian beam in turbulent atmosphere in a slant path,” J. Opt. Soc. Am. A 25, 74-79 (2008).
    [CrossRef]

2008 (6)

2007 (1)

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

2006 (3)

H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264, 25-34 (2006).
[CrossRef]

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

X. Ji, E. Zhang, and B. Lü, “Propagation of multi-Gaussian beams in incoherent combination through turbulent atmosphere and their beam quality,” J. Mod. Opt. 53, 2111-2127 (2006).
[CrossRef]

2004 (2)

H. T. Eyyuboğlu and Y. Baykal, “Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Express 12, 4659-4674 (2004).
[CrossRef] [PubMed]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

2003 (1)

2002 (2)

1991 (1)

1990 (2)

R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers,” Proc. SPIE 1224, 239-253 (1990).
[CrossRef]

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

1979 (2)

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

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]

1978 (3)

J. T. Foley and M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297-300 (1978).
[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]

J. C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175-185 (1978).
[CrossRef]

Altay, S.

H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264, 25-34 (2006).
[CrossRef]

Andrews, J. R.

Andrews, L. C.

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

Baykal, Y.

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264, 25-34 (2006).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Express 12, 4659-4674 (2004).
[CrossRef] [PubMed]

Bernard, J. M.

R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers,” Proc. SPIE 1224, 239-253 (1990).
[CrossRef]

Cai, Y.

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

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

Chen, X.

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]

X. Chen and X. Ji, “Directionality of partially coherent annular flat-topped beams propagating through atmospheric turbulence,” Opt. Commun. 281, 4765-4770 (2008).
[CrossRef]

Chodzko, R. A.

R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers,” Proc. SPIE 1224, 239-253 (1990).
[CrossRef]

Chu, X.

Collett, E.

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]

Davidson, F. M.

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.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

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]

Eyyuboglu, H. T.

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264, 25-34 (2006).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Express 12, 4659-4674 (2004).
[CrossRef] [PubMed]

Foley, J. T.

J. T. Foley and M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297-300 (1978).
[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.

Ji, X.

Korotkova, O.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

Leader, J. C.

Lin, Q.

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Liu, Z.

Lü, B.

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).
[CrossRef] [PubMed]

X. Ji, E. Zhang, and B. Lü, “Superimposed partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. B 25, 825-833 (2008).
[CrossRef]

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]

S. Lü and B. Lü, “The directionality of partially coherent beams expressed in terms of the far-field divergence angle and of the far-field radiant intensity distribution,” Opt. Commun. 281, 3514-3521 (2008).
[CrossRef]

X. Ji, E. Zhang, and B. Lü, “Propagation of multi-Gaussian beams in incoherent combination through turbulent atmosphere and their beam quality,” J. Mod. Opt. 53, 2111-2127 (2006).
[CrossRef]

Lü, S.

S. Lü and B. Lü, “The directionality of partially coherent beams expressed in terms of the far-field divergence angle and of the far-field radiant intensity distribution,” Opt. Commun. 281, 3514-3521 (2008).
[CrossRef]

Mirels, H.

R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers,” Proc. SPIE 1224, 239-253 (1990).
[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.

Ricklin, J. C.

Salem, M.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

Schuster, G. L.

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.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

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]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598 (2002).
[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]

Wu, Y.

Yang, A.

Zhang, E.

Zubairy, M. S.

J. T. Foley and M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297-300 (1978).
[CrossRef]

J. Mod. Opt. (1)

X. Ji, E. Zhang, and B. Lü, “Propagation of multi-Gaussian beams in incoherent combination through turbulent atmosphere and their beam quality,” J. Mod. Opt. 53, 2111-2127 (2006).
[CrossRef]

J. Opt. Soc. Am. (2)

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

J. Opt. Soc. Am. B (1)

Opt. Commun. (7)

J. T. Foley and M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297-300 (1978).
[CrossRef]

S. Lü and B. Lü, “The directionality of partially coherent beams expressed in terms of the far-field divergence angle and of the far-field radiant intensity distribution,” Opt. Commun. 281, 3514-3521 (2008).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

X. Chen and X. Ji, “Directionality of partially coherent annular flat-topped beams propagating through atmospheric turbulence,” Opt. Commun. 281, 4765-4770 (2008).
[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]

H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264, 25-34 (2006).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Proc. SPIE (2)

R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers,” Proc. SPIE 1224, 239-253 (1990).
[CrossRef]

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

Waves Random Media (1)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Mean-squared width w ( z ) versus propagation distance z. The calculation parameter is λ = 1.06 μ m and the other parameters are listed in Table 1. Solid curve: C n 2 = 10 14 m 2 3 ; dashed curve: C n 2 = 0 . a: Fully coherent Gaussian laser beam; b,c: Gaussian array beams for the coherent case; d,e: Gaussian array beams for the incoherent case.

Fig. 2
Fig. 2

Normalized average intensity distributions I ( x , z ) I ( 0 , z ) in accordance with Table 1, where λ = 1.06 μ m and z = 30 km . a: Fully coherent Gaussian laser beam; b,c: Gaussian array beams for the coherent case. (a) C n 2 = 0 , (b) C n 2 = 10 14 m 2 3 .

Fig. 3
Fig. 3

Normalized average intensity distributions I ( x , z ) I ( 0 , z ) in accordance with Table 1, where λ = 1.06 μ m and z = 30 km . a: Fully coherent Gaussian laser beam; d,e: Gaussian array beams for the incoherent case. (a) C n 2 = 0 , (b) C n 2 = 10 14 m 2 3 .

Fig. 4
Fig. 4

Mean-squared width w ( z ) versus propagation distance z. The calculation parameters are the same as in Fig. 1. (a) Using Eqs. (A5, A7); (b) using Eqs. (16, 26).

Tables (1)

Tables Icon

Table 1 Beam Parameters Relating to Figs. 1, 2, 3, 4

Equations (45)

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W ( 0 ) ( x 1 , x 2 , z = 0 ) = j 1 = ( N 1 ) 2 ( N 1 ) 2 j 2 = ( N 1 ) 2 ( N 1 ) 2 exp [ ( x 1 j 1 x d ) 2 + ( x 2 j 2 x d ) 2 w 0 2 ] ,
I ( x , z ) = k 2 π z d x 1 d x 2 W ( 0 ) ( x 1 , x 2 , z = 0 ) exp { i k 2 z [ ( x 1 2 x 2 2 ) 2 ( x 1 x 2 ) x ] } × exp [ ψ * ( x , x 1 , z ) + ψ ( x , x 2 , z ) ] m ,
exp [ ψ ( x , x 1 , z ) + ψ * ( x , x 2 , z ) ] m exp [ ( x 1 x 2 ) 2 ρ 0 2 ] ,
u = x 2 + x 1 2 , v = x 2 x 1 .
I ( x , z ) = k 2 π z j 1 = ( N 1 ) 2 ( N 1 ) 2 j 2 = ( N 1 ) 2 ( N 1 ) 2 d u d v exp [ 2 u 2 + v 2 2 2 ( j 1 + j 2 ) x d u + ( j 1 j 2 ) x d v + ( j 1 2 + j 2 2 ) x d 2 w 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 I ( x , z ) d x I ( x , z ) d x .
w 2 ( z ) = F 1 F 2 ,
F 1 = 4 x 2 I ( x , z ) d x ,
F 2 = I ( x , z ) d x .
F 1 = 4 ( z k ) 2 j 1 = ( N 1 ) 2 ( N 1 ) 2 j 2 = ( N 1 ) 2 ( N 1 ) 2 d u d v exp [ 2 u 2 + v 2 2 2 ( j 1 + j 2 ) x d u + ( j 1 j 2 ) x d v + ( j 1 2 + j 2 2 ) x d 2 w 0 2 ] × exp ( v 2 ρ 0 2 ) exp ( i k z u v ) δ ( v ) ,
x 2 exp ( i 2 π x s ) d x = 1 ( 2 π ) 2 δ ( s ) .
exp ( β 2 x 2 + γ x ) d x = π β exp ( γ 2 4 β 2 ) ,
f ( x ) δ ( x ) d x = f ( 0 ) ,
F 1 = π 2 w 0 j 1 = ( N 1 ) 2 ( N 1 ) 2 j 2 = ( N 1 ) 2 ( N 1 ) 2 exp [ ( j 1 j 2 ) 2 x d 2 2 w 0 2 ] × { [ w 0 2 + ( j 1 + j 2 ) 2 x d 2 ] + 4 k 2 w 0 2 [ 1 ( j 1 j 2 ) 2 x d 2 w 0 2 ] z 2 + [ 8 ( 0.545 C n 2 ) 6 5 k 2 5 ] z 16 5 } .
F 2 = W ( 0 ) ( x , x , z = 0 ) d x = π 2 w 0 j 1 = ( N 1 ) 2 ( N 1 ) 2 j 2 = ( N 1 ) 2 ( N 1 ) 2 exp [ ( j 1 j 2 ) 2 x d 2 2 w 0 2 ] .
w 2 ( z ) = A + B z 2 + F z 16 5 ,
A = j 1 = ( N 1 ) 2 ( N 1 ) 2 j 2 = ( N 1 ) 2 ( N 1 ) 2 C [ w 0 2 + ( j 1 + j 2 ) 2 x d 2 ] D ,
B = j 1 = ( N 1 ) 2 ( N 1 ) 2 j 2 = ( N 1 ) 2 ( N 1 ) 2 4 C w 0 2 k 2 [ 1 1 w 0 2 ( j 1 j 2 ) 2 x d 2 ] D ,
F = 8 ( 0.545 C n 2 ) 6 5 k 2 5 ,
C = exp [ 1 2 w 0 2 ( j 1 j 2 ) 2 x d 2 ] ,
D = j 1 = ( N 1 ) 2 ( N 1 ) 2 j 2 = ( N 1 ) 2 ( N 1 ) 2 C .
θ sp ( z ) lim z w ( z ) z = B + F z 6 5 .
w 2 ( z ) Gs = w 0 Gs 2 + 4 w 0 Gs 2 k 2 z 2 + F z 16 5 ,
θ sp ( z ) Gs = 4 w 0 Gs 2 k 2 + F z 6 5 .
B = 4 w 0 Gs 2 k 2 ,
w 2 ( z ) = w 0 2 + ( N 2 1 ) x d 2 3 + 4 w 0 2 k 2 z 2 + F z 16 5 .
I ( x , z ) = T j 1 = ( N 1 ) 2 ( N 1 ) 2 j 2 = ( N 1 ) 2 ( N 1 ) 2 exp ( P x 2 + Q x + R ) ,
T = H 1 w 0 4 + 2 w 0 2 ρ 0 2 + H 2 ,
P = H 2 [ ( T C G H ) 2 + ( 1 C ) 2 ] ,
Q = i H x d ( T 2 C 2 D G H 2 2 j 1 w 0 2 C 2 ) ,
R = x d 2 [ j 1 2 + j 2 2 w 0 2 + ( T C D 2 H ) 2 + ( j 1 w 0 2 C ) 2 ] ,
H = k 2 z ,
C = 1 w 0 2 + 1 ρ 0 2 i H ,
D = 2 j 2 w 0 2 + 2 j 1 w 0 2 ρ 0 2 C 2 ,
G = 1 1 ρ 0 2 C 2 .
I ( x , z ) = T j 1 = ( N 1 ) 2 ( N 1 ) 2 exp [ 2 T 2 ( x w 0 m x d w 0 ) 2 ] .
W ( 0 ) ( r 1 , r 2 , z = 0 ) = exp [ y 1 2 + y 2 2 w 0 2 ] j 1 = ( N 1 ) 2 ( N 1 ) 2 j 2 = ( N 1 ) 2 ( N 1 ) 2 exp [ ( x 1 j 1 x d ) 2 + ( x 2 j 2 x d ) 2 w 0 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 } ,
w 2 ( z ) = 4 x 2 I ( r , z ) d x d y I ( r , z ) d x d y .
w 2 ( z ) = A + B z 2 + F z 3 ,
F = 8 3 π 2 0 κ 3 Φ n ( κ ) d κ .
w 2 ( z ) = w 0 2 + ( N 2 1 ) x d 2 3 + 4 w 0 2 k 2 z 2 + F z 3 .
Φ n ( κ ) = 0.033 C n 2 κ 11 3 exp ( κ 2 κ m 2 ) ,
F = 20.293 C n 2 .

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