Abstract

The Wigner distribution function (WDF) has been used to study the beam propagation factor (M 2-factor) for partially coherent flat-topped (PCFT) beams with circular symmetry in a turbulent atmosphere. Based on the extended Huygens–Fresnel principle and the definition of the WDF, an expression for the WDF of PCFT beams in turbulence has been given. By use of the second-order moments of the WDF, the analytical formulas for the root-mean-square (rms) spatial width, the rms angular width, and the M 2-factor of PCFT beams in turbulence have been derived, which can be applied to cases of different spatial power spectra of the refractive index fluctuations. The rms angular width and the M 2-factor of PCFT beams in turbulence have been discussed with numerical examples. It can be shown that the M 2-factor of PCFT beams in turbulence depends on the beam order, degree of global coherence of the source, waist width, wavelength, spatial power spectrum of the refractive index fluctuations, and propagation distance.

© 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]
  13. R. Martínez-Herrero, G. Piquero, and P. M. Mejías, "On the propagation of the kurtosis parameter of general beams," Opt. Commun. 115, 225-232 (1995).
    [CrossRef]
  14. A. E. Siegman, "New developments in laser resonators," Proc. SPIE. 1224, 2-14 (1990).
    [CrossRef]
  15. F. Gori, M. Santarsiero, and A. Sona, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
    [CrossRef]
  16. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, "Spreading properties of beams radiated by partially coherent Schell-model sources," J. Opt. Soc. Am. A 16, 106-112 (1999).
    [CrossRef]
  17. C. Palma, P. De Santis, G. Cincotti, and G. Guattari, "Propagation and coherence evolution of optical beams in gain media," J. Mod. Opt. 43, 139-153 (1996).
    [CrossRef]
  18. B. Zhang, Q. Wen, and X. Guo, "Beam propagation factor of partially coherent beams in gain or absorbing media," Optik 117, 123-127 (2006).
    [CrossRef]
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    [CrossRef]
  20. Y. Li, "Light beams with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002).
    [CrossRef]
  21. Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
    [CrossRef]
  22. Y. Dan, B. Zhang, and P. Pan, "Propagation of partially coherent flat-topped beams through a turbulent atmosphere," J. Opt. Soc. Am. A 25, 2223-2231 (2008).
    [CrossRef]
  23. X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, "Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams," J. Opt. Soc. Am. A 24, 3554-3563 (2007).
    [CrossRef]
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    [CrossRef]
  25. A. Starikov and E. Wolf, "Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields," J. Opt. Soc. Am. 72, 923-928 (1982).
    [CrossRef]
  26. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
  27. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).
  28. X. Ji and B. Lü, "Turbulence-induced quality degradation of partially coherent beams," Opt. Commun. 251, 231-236 (2005).
    [CrossRef]
  29. G. Wu, H. Guo, and D. Deng, "Paraxial propagation of partially coherent flat-topped beam," Opt. Commun. 260, 687-690 (2006).
    [CrossRef]
  30. H. T. Eyyubo?lu, C. Arpali, and Y. Baykal, "Flat topped beams and their characteristics in turbulent media," Opt. Express 14, 4196-4207 (2006).
    [CrossRef] [PubMed]
  31. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, New York, 1980).

2008 (2)

M. H. Mahdieh, "Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor," Opt. Commun. 281, 3395-3402 (2008).
[CrossRef]

Y. Dan, B. Zhang, and P. Pan, "Propagation of partially coherent flat-topped beams through a turbulent atmosphere," J. Opt. Soc. Am. A 25, 2223-2231 (2008).
[CrossRef]

2007 (1)

2006 (3)

B. Zhang, Q. Wen, and X. Guo, "Beam propagation factor of partially coherent beams in gain or absorbing media," Optik 117, 123-127 (2006).
[CrossRef]

G. Wu, H. Guo, and D. Deng, "Paraxial propagation of partially coherent flat-topped beam," Opt. Commun. 260, 687-690 (2006).
[CrossRef]

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

2005 (1)

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

2003 (2)

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]

1999 (1)

1996 (1)

C. Palma, P. De Santis, G. Cincotti, and G. Guattari, "Propagation and coherence evolution of optical beams in gain media," J. Mod. Opt. 43, 139-153 (1996).
[CrossRef]

1995 (1)

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, "On the propagation of the kurtosis parameter of general beams," Opt. Commun. 115, 225-232 (1995).
[CrossRef]

1991 (3)

J. Serna, R. Martínez-Herrero, and P. M. Mejías, "Parametric characterization of general partially coherent beams propagating through ABCD optical systems," J. Opt. Soc. Am. A 8, 1094-1098 (1991).
[CrossRef]

F. Gori, M. Santarsiero, and A. Sona, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[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]

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

1986 (1)

1982 (1)

1977 (1)

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

1968 (1)

Amarande, S.

Arpali, Eyyuboglu

Bastiaans, M. J.

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]

Borghi, R.

Chen, S.

Chen, X.

Cincotti, G.

M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, "Spreading properties of beams radiated by partially coherent Schell-model sources," J. Opt. Soc. Am. A 16, 106-112 (1999).
[CrossRef]

C. Palma, P. De Santis, G. Cincotti, and G. Guattari, "Propagation and coherence evolution of optical beams in gain media," J. Mod. Opt. 43, 139-153 (1996).
[CrossRef]

Dan, Y.

De Santis, P.

C. Palma, P. De Santis, G. Cincotti, and G. Guattari, "Propagation and coherence evolution of optical beams in gain media," J. Mod. Opt. 43, 139-153 (1996).
[CrossRef]

Deng, D.

G. Wu, H. Guo, and D. Deng, "Paraxial propagation of partially coherent flat-topped beam," Opt. Commun. 260, 687-690 (2006).
[CrossRef]

Dogariu, A.

Eyyubo?lu, H. T.

Gbur, G.

Gori, F.

M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, "Spreading properties of beams radiated by partially coherent Schell-model sources," J. Opt. Soc. Am. A 16, 106-112 (1999).
[CrossRef]

F. Gori, M. Santarsiero, and A. Sona, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[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]

Guattari, G.

C. Palma, P. De Santis, G. Cincotti, and G. Guattari, "Propagation and coherence evolution of optical beams in gain media," J. Mod. Opt. 43, 139-153 (1996).
[CrossRef]

Guo, H.

G. Wu, H. Guo, and D. Deng, "Paraxial propagation of partially coherent flat-topped beam," Opt. Commun. 260, 687-690 (2006).
[CrossRef]

Guo, X.

B. Zhang, Q. Wen, and X. Guo, "Beam propagation factor of partially coherent beams in gain or absorbing media," Optik 117, 123-127 (2006).
[CrossRef]

Ishimaru, A.

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

Ji, X.

Li, X.

Li, Y.

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]

Lü, B.

Mahdieh, M. H.

M. H. Mahdieh, "Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor," Opt. Commun. 281, 3395-3402 (2008).
[CrossRef]

Martínez-Herrero, R.

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, "On the propagation of the kurtosis parameter of general beams," Opt. Commun. 115, 225-232 (1995).
[CrossRef]

J. Serna, R. Martínez-Herrero, and P. M. Mejías, "Parametric characterization of general partially coherent beams propagating through ABCD optical systems," J. Opt. Soc. Am. A 8, 1094-1098 (1991).
[CrossRef]

Mejías, P. M.

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, "On the propagation of the kurtosis parameter of general beams," Opt. Commun. 115, 225-232 (1995).
[CrossRef]

J. Serna, R. Martínez-Herrero, and P. M. Mejías, "Parametric characterization of general partially coherent beams propagating through ABCD optical systems," J. Opt. Soc. Am. A 8, 1094-1098 (1991).
[CrossRef]

Palma, C.

C. Palma, P. De Santis, G. Cincotti, and G. Guattari, "Propagation and coherence evolution of optical beams in gain media," J. Mod. Opt. 43, 139-153 (1996).
[CrossRef]

Pan, P.

Piquero, G.

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, "On the propagation of the kurtosis parameter of general beams," Opt. Commun. 115, 225-232 (1995).
[CrossRef]

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]

Santarsiero, M.

M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, "Spreading properties of beams radiated by partially coherent Schell-model sources," J. Opt. Soc. Am. A 16, 106-112 (1999).
[CrossRef]

F. Gori, M. Santarsiero, and A. Sona, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

Serna, J.

Shirai, T.

Siegman, A. E.

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

Sona, A.

F. Gori, M. Santarsiero, and A. Sona, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

Starikov, A.

Vahimaa, P.

Walther, A.

Wen, Q.

B. Zhang, Q. Wen, and X. Guo, "Beam propagation factor of partially coherent beams in gain or absorbing media," Optik 117, 123-127 (2006).
[CrossRef]

Wolf, E.

Wu, G.

G. Wu, H. Guo, and D. Deng, "Paraxial propagation of partially coherent flat-topped beam," Opt. Commun. 260, 687-690 (2006).
[CrossRef]

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]

Zhang, B.

Y. Dan, B. Zhang, and P. Pan, "Propagation of partially coherent flat-topped beams through a turbulent atmosphere," J. Opt. Soc. Am. A 25, 2223-2231 (2008).
[CrossRef]

B. Zhang, Q. Wen, and X. Guo, "Beam propagation factor of partially coherent beams in gain or absorbing media," Optik 117, 123-127 (2006).
[CrossRef]

Zubairy, M. S.

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

J. Mod. Opt. (3)

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]

C. Palma, P. De Santis, G. Cincotti, and G. Guattari, "Propagation and coherence evolution of optical beams in gain media," J. Mod. Opt. 43, 139-153 (1996).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (7)

Opt. Commun. (8)

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]

F. Gori, M. Santarsiero, and A. Sona, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, "On the propagation of the kurtosis parameter of general beams," Opt. Commun. 115, 225-232 (1995).
[CrossRef]

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

M. H. Mahdieh, "Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor," Opt. Commun. 281, 3395-3402 (2008).
[CrossRef]

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

G. Wu, H. Guo, and D. Deng, "Paraxial propagation of partially coherent flat-topped beam," Opt. Commun. 260, 687-690 (2006).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (2)

Optik (1)

B. Zhang, Q. Wen, and X. Guo, "Beam propagation factor of partially coherent beams in gain or absorbing media," Optik 117, 123-127 (2006).
[CrossRef]

Proc. IEEE (1)

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

Proc. SPIE. (1)

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

Other (5)

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, Bellingham, 2005).
[CrossRef]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, New York, 1980).

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

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

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Figures (6)

Fig. 1.
Fig. 1.

(a). rms angular width and (b) normalized rms angular width versus propagation distance for different values of beam order N, respectively. The calculation parameters are α=2, λ=850 nm, w 0=0.05 m, C 2 n=10-15 m-2/3.

Fig. 2.
Fig. 2.

Normalized rms angular width versus propagation distance for different values of α. The calculation parameters are N=10, λ=850 nm, w 0=0.05 m, C 2 n=10-15 m-2/3.

Fig. 3.
Fig. 3.

(a). M 2-factor and (b) normalized M 2-factor versus propagation distance for different values of beam order N, respectively. The calculation parameters are α=2, λ=850 nm, w 0=0.05 m, C 2 n=10-15 m-2/3.

Fig. 4.
Fig. 4.

Normalized M 2-factor versus propagation distance for different values of α. The calculation parameters are N=10, λ=850 nm, w 0=0.05 m, C 2 n=10-15 m-2/3.

Fig. 5.
Fig. 5.

Normalized M 2-factor versus w 0 for different values of beam order N. The calculation parameters are α=2, z=1 km, C 2 n=10-15 m-2/3, λ=850 nm.

Fig. 6.
Fig. 6.

Normalized M 2-factor versus wave length λ for different values of beam order N. The calculation parameters are α=2, z=1 km, C 2 n=10-15 m-2/3, w 0=0.05 m.

Equations (64)

Equations on this page are rendered with MathJax. Learn more.

E N ( ρ ) = m = 1 N α m exp [ ( m p N ρ 2 w 0 2 ) ]
α m = ( 1 ) m + 1 N ! m ! ( N m ) !
p N = 2 m = 1 N m = 1 N α m α m m + m .
Γ ( ρ 1 , ρ 2 , 0 ) = < E ( ρ 1 , 0 ) E * ( ρ 2 , 0 ) > m
= m = 1 N m = 1 N α m α m exp [ ( m p N ρ 1 2 w 0 2 + m p N 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 { ik z [ ( ρ ρ ) · ( ρ d ρ d ) ] H ( ρ d , ρ d , z ) } d 2 ρ d 2 ρ d ,
W ( ρ , ρ d , 0 ) = Γ ( ρ 1 , ρ 2 , 0 ) = Γ ( ρ + ρ d 2 , ρ ρ d 2 , 0 )
ρ = ( ρ 1 + ρ 2 ) 2 ,   ρ d = ρ 1 ρ 2 ,
ρ= ( ρ 1 + ρ 2 ) 2 ,   ρ d = ρ 1 ρ 2 .
H ( ρ d , ρ d , z ) = 4 π 2 k 2 z 0 1 0 [ 1 J 0 ( κ ρ d ξ + ( 1 ξ ) ρ d ) ] Φ n ( κ ) κ d κ ,
h ( ρ , θ , z ) = ( k 2 π ) 2 W ( ρ , ρ d , z ) exp ( ik θ · ρ d ) d 2 ρ d
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 .
f ( x ) δ ( x ) d x = f ( 0 ) ,
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 k κ d , 0 ) = m = 1 N m = 1 N α m α m exp ( { p N w 0 2 ( m + m ) ρ 2
+ p N w 0 2 ( m m ) ρ · ( ρ d + z k κ d )
+ [ p N 4 w 0 2 ( m + m ) + 1 2 σ 0 2 ] ρ d + z k κ d 2 } ) .
exp ( s 2 x 2 ± q x ) d x = π s exp ( q 2 4 s 2 ) ,   ( s > 0 )
h ( ρ , θ , z ) = w 0 2 k 2 16 π 3 p N m = 1 N m = 1 N [ α m α m m + m
× exp ( a ρ d 2 b κ d 2 + c ρ d · κ d i ρ · κ d ik θ · ρ d H ) d 2 κ d d 2 ρ d ] ,
H = 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 ( mm p N m + m + 1 2 α 2 ) ,
b = w 0 2 4 ( m + m ) p N + z 2 k 2 w 0 2 ( mm p N m + m + 1 2 α 2 ) iz 2 k ( m m m + m ) ,
c = 2 z k w 0 2 ( mm p N m + m + 1 2 α 2 ) + i 2 ( m m m + m ) ,
α = σ 0 w 0
< x n 1 y n 2 θ x m 1 θ y m 2 > = 1 P x n 1 y n 2 θ x m 1 θ y m 2 h ( ρ , θ , z ) d 2 ρ d 2 θ
δ ( s ) = 1 2 π exp ( isx ) d x ,
δ ( n ) ( s ) = 1 2 π ( ix ) n exp ( isx ) d x , ( n = 1 , 2 )
f ( x ) δ ( n ) ( x ) d x = ( 1 ) n f ( n ) ( 0 ) , ( n = 1 , 2 )
< ρ 2 > = < x 2 > + < y 2 >
= 2 w 0 2 p N 2 [ m = 1 N m = 1 N α m α m ( m + m ) 2 ] + 2 k 2 w 0 2 [ 1 α 2
+ 4 m = 1 N m = 1 N mm α m α m ( m + m ) 2 ] z 2 + 4 3 π 2 T z 3 ,
< θ 2 > = < θ x 2 > + < θ y 2 >
= 2 k 2 w 0 2 [ 1 α 2 + 4 m = 1 N m = 1 N mm α m α m ( m + m ) 2 ] + 4 π 2 Tz ,
< ρ · θ > = < x θ x > + < y θ y >
= 2 k 2 w 0 2 [ 1 α 2 + 4 m = 1 N m = 1 N mm α m α m ( m + m ) 2 ] z + 2 π 2 Tz 2 ,
T = 0 Φ n ( κ ) κ 3 d κ
w N ( z ) ( < ρ < ρ > 2 > ) 1 2 = ( < ρ 2 > ) 1 2
= { 2 w 0 2 p N 2 [ m = 1 N m = 1 N α m α m ( m + m ) 2 ] + 2 k 2 w 0 2 [ 1 α 2
+ 4 m = 1 N m = 1 N mm α m α m ( m + m ) 2 ] z 2 + 4 3 π 2 T z 3 } 1 2 ,
θ N ( z ) ( < θ < θ > 2 > ) 1 2 = ( < θ 2 > ) 1 2
= { 2 k 2 w 0 2 [ 1 α 2 + 4 m = 1 N m = 1 N mm α m α m ( m + m ) 2 ] + 4 π 2 Tz } 1 2 .
θ r N ( z ) θ N ( z ) θ N ( 0 )
= { 1 + 2 π 2 k 2 w 0 2 [ 1 α 2 + 4 m = 1 N m = 1 N mm α m α m ( m + m ) 2 ] 1 Tz } 1 2 .
w 1 ( z ) = [ w 0 2 2 + 2 k 2 w 0 2 ( 1 + 1 α 2 ) z 2 + 4 3 π 2 T z 3 ] 1 2 ,
θ 1 ( z ) = [ 2 k 2 w 0 2 ( 1 + 1 α 2 ) + 4 π 2 T z ] 1 2 ,
θ r 1 ( z ) = [ 1 + 2 π 2 k 2 w 0 2 ( 1 α 2 + 1 ) 1 Tz ] 1 2 .
M 2 ( z ) = k ( < ρ 2 > < θ 2 > < ρ · θ > 2 ) 1 2
= k [ ( < x 2 > + < y 2 > ) ( < θ x 2 > + < θ y 2 > ) ( < x θ x > + < y θ y > ) 2 ] 1 2 .
M N 2 ( z ) = [ M N 4 ( 0 ) + k 2 w 0 2 ATz + B w 0 1 Tz 3 + 4 3 π 4 k 2 T 2 z 4 ] 1 2
M N 4 ( 0 ) = 3 AB 16 π 4 ,
A = 8 π 2 p N 2 m = 1 N m = 1 N α m α m ( m + m ) 2 ,
B = 8 π 2 3 [ 1 α 2 + 4 m = 1 N m = 1 N mm α m α m ( m + m ) 2 ] .
M r N 2 ( z ) M N 2 ( z ) M N 2 ( 0 )
= ( 1 + 16 π 4 k 2 w 0 2 3 B Tz + 16 π 4 3 w 0 2 A Tz 3 + 64 π 8 k 2 9 AB T 2 z 4 ) 1 2 .
M 1 2 ( z ) = [ ( 1 + 1 α 2 ) + 2 π 2 k 2 w 0 2 T z
+ 8 π 2 3 w 0 2 ( 1 + 1 α 2 ) T z 3 + 4 π 4 k 2 3 T 2 z 4 ] 1 2 ,
M r 1 2 ( z ) = [ 1 + 2 π 2 k 2 w 0 2 ( 1 + 1 α 2 ) 1 T z
+ 8 π 2 3 w 0 2 T z 3 + 4 π 4 k 2 3 ( 1 + 1 α 2 ) 1 T 2 z 4 ] 1 2 .
Φ n ( κ ) = 0 . 033 C n 2 κ 11 3 exp ( κ 2 κ m 2 )
T = 0 . 1661 C n 2 l 0 1 3

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