Abstract

Based on the modified beam model for flat-topped beams and the Schell model for partially coherent light, an expression for partially coherent flat-topped (PCFT) beams has been proposed. The propagation characteristics of PCFT beams with circular symmetry through a turbulent atmosphere have been studied. By using the generalized Huygens–Fresnel integral and Fourier transform method, the expressions for the cross-spectral density function and the average intensity have been given and the analytical expression for the root-mean-square width has been derived. The effects of the beam order, the spatial coherence, and the turbulent parameter on the intensity distributions and beam spreading have been discussed in detail. Our results show that the on-axis intensity of the beams decreases with increasing turbulence and decreasing coherence of the source, whereas the on-axis intensity of the beams in the far field decreases slightly with increasing beam order. The relative spreading of PCFT beams is smaller for beams with a higher order, a lower degree of global coherence of the source, a larger inner scale, and a smaller outer scale of the turbulence.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  11. A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071-1079 (1992).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  28. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
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  30. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1986).

2008 (3)

M. Alavinejad, B. Ghafary, and F. D. Kashani “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46, 1-5 (2008).
[CrossRef]

M. Alavinejad, B. Ghafary, and D. Razzaghi, “Spectral changes of partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 281, 2173-2178 (2008).
[CrossRef]

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46, 357-362 (2008).
[CrossRef]

2007 (1)

2006 (2)

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

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

2003 (3)

2002 (4)

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

S. A. Ponomarenko, J.-J Greffet, and E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1-8 (2002).
[CrossRef]

Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007-1009 (2002).
[CrossRef]

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225-234 (2002).
[CrossRef]

1994 (1)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335-341 (1994).
[CrossRef]

1992 (1)

A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071-1079 (1992).
[CrossRef]

1991 (1)

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

1990 (1)

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

1989 (1)

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361-364 (1989).
[CrossRef]

1988 (2)

Y. Li, “Focusing properties of the Gaussian beam generated by optical resonators with Gaussian reflectivity mirrors,” J. Mod. Opt. 35, 1833-1846 (1988).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, and O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172-1177 (1988).
[CrossRef]

1982 (1)

1979 (1)

1978 (1)

1977 (1)

A. Ishimaru, “Theory and application of wave propagation and scattering in random media,” Proc. IEEE 65, 1030-1061 (1977).
[CrossRef]

1976 (1)

Alavinejad, M.

M. Alavinejad, B. Ghafary, and F. D. Kashani “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46, 1-5 (2008).
[CrossRef]

M. Alavinejad, B. Ghafary, and D. Razzaghi, “Spectral changes of partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 281, 2173-2178 (2008).
[CrossRef]

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46, 357-362 (2008).
[CrossRef]

Amarande, S.

Andrews, L. C.

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

Arpali, Ç.

Baykal, Y.

Boardman, A. D.

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1986).

Cai, Y.

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

Chen, S.

Chen, X.

De Silvestri, S.

S. De Silvestri, P. Laporta, V. Magni, and O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172-1177 (1988).
[CrossRef]

Dogariu, A.

Eyyuboglu, H. T.

Fante, R. L.

Gbur, G.

Ghafary, B.

M. Alavinejad, B. Ghafary, and F. D. Kashani “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46, 1-5 (2008).
[CrossRef]

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46, 357-362 (2008).
[CrossRef]

M. Alavinejad, B. Ghafary, and D. Razzaghi, “Spectral changes of partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 281, 2173-2178 (2008).
[CrossRef]

Gori, F.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335-341 (1994).
[CrossRef]

Greffet, J.-J

S. A. Ponomarenko, J.-J Greffet, and E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1-8 (2002).
[CrossRef]

Ishimaru, A.

A. Ishimaru, “Theory and application of wave propagation and scattering in random media,” Proc. IEEE 65, 1030-1061 (1977).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Ji, X.

Kashani, F. D.

M. Alavinejad, B. Ghafary, and F. D. Kashani “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46, 1-5 (2008).
[CrossRef]

Laporta, P.

S. De Silvestri, P. Laporta, V. Magni, and O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172-1177 (1988).
[CrossRef]

Lavigne, P.

A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071-1079 (1992).
[CrossRef]

Leader, J. C.

Li, X.

Li, Y.

Y. Li, “Flat-topped beam with non-circular cross-sections,” J. Mod. Opt. 50, 1957-1966 (2003).
[CrossRef]

Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007-1009 (2002).
[CrossRef]

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225-234 (2002).
[CrossRef]

Y. Li, “Focusing properties of the Gaussian beam generated by optical resonators with Gaussian reflectivity mirrors,” J. Mod. Opt. 35, 1833-1846 (1988).
[CrossRef]

Lü, B.

Magni, V.

S. De Silvestri, P. Laporta, V. Magni, and O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172-1177 (1988).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Morin, M.

A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071-1079 (1992).
[CrossRef]

Parent, A.

A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071-1079 (1992).
[CrossRef]

Phillips, R. L.

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

Plonus, M. A.

Ponomarenko, S. A.

S. A. Ponomarenko, J.-J Greffet, and E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1-8 (2002).
[CrossRef]

Razzaghi, D.

M. Alavinejad, B. Ghafary, and D. Razzaghi, “Spectral changes of partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 281, 2173-2178 (2008).
[CrossRef]

Shirai, T.

Starikov, A.

Svelto, O.

S. De Silvestri, P. Laporta, V. Magni, and O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172-1177 (1988).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Springfield, 1971).

Wang, S. C. H.

Wolf, E.

Wu, J.

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

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-Gauss beams,” Opt. Commun. 70, 361-364 (1989).
[CrossRef]

Zubairy, M. S.

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361-364 (1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. De Silvestri, P. Laporta, V. Magni, and O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172-1177 (1988).
[CrossRef]

J. Mod. Opt. (4)

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

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

Y. Li, “Flat-topped beam with non-circular cross-sections,” J. Mod. Opt. 50, 1957-1966 (2003).
[CrossRef]

Y. Li, “Focusing properties of the Gaussian beam generated by optical resonators with Gaussian reflectivity mirrors,” J. Mod. Opt. 35, 1833-1846 (1988).
[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. (4)

J. Opt. Soc. Am. A (3)

Opt. Commun. (5)

S. A. Ponomarenko, J.-J Greffet, and E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1-8 (2002).
[CrossRef]

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225-234 (2002).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335-341 (1994).
[CrossRef]

M. Alavinejad, B. Ghafary, and D. Razzaghi, “Spectral changes of partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 281, 2173-2178 (2008).
[CrossRef]

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361-364 (1989).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (2)

M. Alavinejad, B. Ghafary, and F. D. Kashani “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46, 1-5 (2008).
[CrossRef]

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46, 357-362 (2008).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071-1079 (1992).
[CrossRef]

Proc. IEEE (1)

A. Ishimaru, “Theory and application of wave propagation and scattering in random media,” Proc. IEEE 65, 1030-1061 (1977).
[CrossRef]

Other (5)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Springfield, 1971).

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1986).

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Figures (9)

Fig. 1
Fig. 1

Transverse profiles of (a) FT beams and (b) modified FT beams for different values of beam order M.

Fig. 2
Fig. 2

On-axis intensity distributions. In each figure, solid curves, C n 2 = 10 15 m 2 3 , dashed curves, C n 2 = 0 .

Fig. 3
Fig. 3

Transverse intensity distributions of PCFT beams for different beam orders at several propagation distances. In each figure, solid curves, α = 1.5 ; dashed curves, α and M = 1 (a coherent Gaussian beam). All curves are based on C n 2 = 10 15 m 2 3 .

Fig. 4
Fig. 4

Transverse intensity distributions of PCFT beams for different values of turbulent parameter C n 2 at several propagation distances. In each figure, solid curves, M = 10 ; dashed curves, M = 1 (GSM beams). All curves are based on α = 1.5 .

Fig. 5
Fig. 5

Transverse intensity distributions of PCFT beams for different values of coherent parameter α at several propagation distances. In each figure, solid curves, M = 10 ; dashed curves, M = 1 (GSM beams). All curves are based on C n 2 = 10 15 m 2 3 .

Fig. 6
Fig. 6

Variations of coefficient B with the inner scale and outer scale of turbulence.

Fig. 7
Fig. 7

Normalized rms width of PCFT beams propagating through atmospheric turbulence for different beam orders. Solid curves, α = 1.5 ; dashed curves, α and M = 1 (a coherent Gaussian beam).

Fig. 8
Fig. 8

Normalized rms width of PCFT beams propagating through atmospheric turbulence for different values of turbulent parameter C n 2 . Solid curves, M = 10 ; dashed curves, M = 1 (GSM beams).

Fig. 9
Fig. 9

Normalized rms width of PCFT beams propagating through atmospheric turbulence for different values of coherent parameter α. Solid curves, M = 10 ; dashed curves, M = 1 (GSM beams).

Equations (48)

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F M ( ρ ) = F 0 M { 1 [ 1 exp ( ρ 2 w 0 2 ) ] M } ,
E M ( ρ ) = 1 [ 1 exp ( p M ρ 2 w 0 2 ) ] M ,
E M ( ρ ) = m = 1 M α m exp [ ( m p M ρ 2 w 0 2 ) ] ,
α m = ( 1 ) m + 1 M ! m ! ( M m ) ! .
0 E M ( ρ ) ρ d ρ = 0 E 1 ( ρ ) ρ d ρ = w 0 2 2 .
p M = p M L = m = 1 M α m m .
0 [ E M ( ρ ) ] 2 ρ d ρ = 0 [ E 1 ( ρ ) ] 2 ρ d ρ = w 0 2 4 .
p M = p M P = 2 m = 1 M m = 1 M α m α m m + m .
W ( ρ 1 , ρ 2 , 0 ) = m = 1 M m = 1 M α m α m exp { [ ( m p M P ρ 1 2 w 0 2 ) + ( m p M P ρ 2 2 w 0 2 ) + ρ 1 ρ 2 2 2 σ 0 2 ] } ,
W ( ρ , ρ d , z ) = ( k 2 π z ) 2 W ( ρ , ρ d , 0 ) exp { i k z [ ( ρ ρ ) ( ρ d ρ d ) ] H ( ρ d , ρ d , z ) } d 2 ρ d 2 ρ d ,
H ( ρ d , ρ d , z ) = 4 π 2 k 2 z 0 1 d ξ 0 [ 1 J 0 ( κ ρ d ξ + ( 1 ξ ) ρ d ) ] Φ n ( κ ) κ d κ ,
δ ( ρ ρ ) = 1 ( 2 π ) 2 exp [ i κ d ( ρ ρ ) ] d 2 κ d ,
W ( ρ , ρ d , 0 ) 1 ( 2 π ) 2 W ( ρ , ρ d , 0 ) exp [ i κ d ( ρ ρ ) ] d 2 κ d d 2 ρ .
W ( ρ , ρ d , z ) = 1 ( 2 π ) 2 W ( ρ , ρ d + z k κ d , 0 ) exp [ i ρ κ d i κ d ρ H ( ρ d , ρ d + z k κ d , z ) ] d 2 κ d d 2 ρ .
W ( ρ , ρ d , z ) = w 0 2 4 π p M P m = 1 M m = 1 M α m α m ( m + m ) exp ( a ρ d 2 b κ d 2 + c ρ d κ d + i ρ κ d H ) d 2 κ d ,
H H ( ρ d , ρ d + z k κ d , z ) = 4 π 2 k 2 z 0 1 d ξ 0 [ 1 J 0 ( κ z k κ d ξ + ρ d ) ] Φ n ( κ ) κ d κ ,
a = 1 w 0 2 ( m m p M P m + m + 1 2 α 2 ) ,
b = w 0 2 4 ( m + m ) p M P + z 2 k 2 w 0 2 ( m m p M P m + m + 1 2 α 2 ) i z 2 k ( m m m + m ) ,
c = 2 z k w 0 2 ( m m p M P m + m + 1 2 α 2 ) + i 2 ( m m m + m ) ,
α = σ 0 w 0
W ( ρ , ρ d , z ) = w 0 2 8 π exp ( a ρ d 2 b κ d 2 + c ρ d κ d + i ρ κ d H ) d 2 κ d ,
a = 1 2 w 0 2 ( 1 + 1 α 2 ) ,
b = w 0 2 8 + z 2 2 k 2 w 0 2 ( 1 + 1 α 2 ) ,
c = z k w 0 2 ( 1 + 1 α 2 ) ,
a = m m p M P w 0 2 ( m + m ) ,
b = w 0 2 4 ( m + m ) p M P + z 2 k 2 w 0 2 ( m m p M P m + m ) i z 2 k ( m m m + m ) ,
c = 2 z k w 0 2 ( m m p M P m + m ) ,
exp [ H ( ρ d , ρ d , z ) ] exp [ ( 1 ρ 0 2 ) ( ρ d 2 + ρ d ρ d + ρ d 2 ) ] ,
H = 1 ρ 0 2 ( 3 ρ d 2 + 3 z k ρ d κ d + z 2 k 2 κ d 2 ) .
W ( ρ , ρ d , z ) = w 0 2 p M P m = 1 M m = 1 M b 1 α m α m ( m + m ) exp ( a 1 ρ d 2 b 1 ρ 2 + i c 1 ρ d ρ ) ,
a 1 = 1 w 0 2 ( m m p M P m + m + 1 2 α 2 ) + 3 ρ 0 2 c 1 2 4 b 1 ,
b 1 = [ w 0 2 ( m + m ) p M P + 4 z 2 k 2 w 0 2 ( m m p M P m + m + 1 2 α 2 ) i 2 z k ( m m m + m ) + 4 z 2 ρ 0 2 k 2 ] 1 ,
c 1 = 2 b 1 [ 2 z k w 0 2 ( m m p M P m + m + 1 2 α 2 ) + i 2 ( m m m + m ) 3 z ρ 0 2 k ] .
I ( ρ , z ) = W ( ρ , 0 , z ) = w 0 2 4 π p M P m = 1 M m = 1 M α m α m ( m + m ) exp ( b κ d 2 + i ρ κ d H ) d 2 κ d ,
H = 4 π 2 k 2 z 0 1 d ξ 0 [ 1 J 0 ( z k κ d κ ξ ) ] Φ n ( κ ) κ d κ .
Φ n ( κ ) = 0.033 C n 2 ( κ 2 + 1 L 0 2 ) 11 6 exp ( κ 2 κ m 2 ) ,
H = 0.547 k 1 3 C n 2 z 8 3 κ d 5 3 .
I ( ρ , z ) = w 0 2 2 p M P m = 1 M m = 1 M α m α m ( m + m ) 0 κ d J 0 ( κ d ρ ) cos [ z ( m m ) 2 k ( m + m ) κ d 2 ] exp ( b κ d 2 H ) d κ d ,
b = w 0 2 4 ( m + m ) p M P + z 2 k 2 w 0 2 ( m m p M P m + m + 1 2 α 2 ) .
I ( ρ , z ) = w 0 2 p M P m = 1 M m = 1 M b 1 α m α m ( m + m ) exp ( b 1 ρ 2 ) .
w M ( z ) = [ ρ 2 I M ( ρ , z ) d 2 ρ I M ( ρ , z ) d 2 ρ ] 1 2 .
exp ( i ρ κ d ) d 2 ρ = ( 2 π ) 2 δ ( κ d )
( x 2 + y 2 ) exp [ i ( x κ d x + y κ d y ) ] d x d y = ( 2 π ) 2 [ δ ( κ d x ) δ ( κ d y ) + δ ( κ d x ) δ ( κ d y ) ] ,
w M ( z ) = { 2 w 0 2 p M P 2 [ m = 1 M m = 1 M α m α m ( m + m ) 2 ] + [ w 0 2 2 α 2 + 2 w 0 2 m = 1 M m = 1 M m m α m α m ( m + m ) 2 ] ( 2 k w 0 2 ) 2 z 2 + 4 3 π 2 z 3 0 Φ n ( κ ) κ 3 d κ } 1 2 .
w M f ( z ) = { 2 w 0 2 p M P 2 [ m = 1 M m = 1 M α m α m ( m + m ) 2 ] + [ w 0 2 2 α 2 + 2 w 0 2 m = 1 M m = 1 M m m α m α m ( m + m ) 2 ] ( 2 k w 0 2 ) 2 z 2 } 1 2 .
w M ( N ) ( z ) w M ( z ) w M f ( z ) = [ 1 + B w M f 2 ( z ) C n 2 z 3 ] 1 2 ,
B = 4 π 2 3 C n 2 0 Φ n ( κ ) κ 3 d κ .
w M ( N ) ( z ) = [ 1 + 2.186 l 0 1 3 w M f 2 ( z ) C n 2 z 3 ] 1 2 .

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