Abstract

The spreading of partially coherent flat-topped vortex beams in non-Kolmogorov atmospheric turbulence and the evolution of the average intensity and coherence vortices are studied based on the extended Huygens–Fresnel principle and the non-Kolmogorov model introduced by Toselli et al. [Proc. SPIE 6551, 65510E (2007)], where the effect of non-Kolmogorov power spectrum on the beam propagation properties and the evolution behavior is stressed. The results are illustrated numerically and interpreted physically. A comparison with previous work is also made.

© 2011 Optical Society of America

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References

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  1. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
  2. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
    [CrossRef]
  3. J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
    [CrossRef]
  4. 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]
  5. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10–12(2003).
    [CrossRef] [PubMed]
  6. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–230 (2008).
    [CrossRef]
  7. T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002(2008).
    [CrossRef]
  8. J. Li and B. Lü, “Propagation of GSM vortex beams through atmosphere turbulence and evolution of coherent vortices,” J. Opt. A: Pure Appl. Opt. 11, 045710 (2009).
    [CrossRef]
  9. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
    [CrossRef]
  10. R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16(1995).
    [CrossRef]
  11. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferreroa, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
    [CrossRef]
  12. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferreroa, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003(2008).
    [CrossRef]
  13. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian–Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35, 715–717 (2010).
    [CrossRef] [PubMed]
  14. X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283, 3398–3403 (2010).
    [CrossRef]
  15. F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
    [CrossRef]
  16. Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007–1009 (2002).
    [CrossRef]
  17. G. Wu, H. Guo, and D. Deng, “Paraxial propagation of partially coherent flat-topped beam,” Opt. Commun. 260, 687–690 (2006).
    [CrossRef]
  18. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).
  19. 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]
  20. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  21. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
    [CrossRef]
  22. I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
    [CrossRef] [PubMed]

2010 (2)

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283, 3398–3403 (2010).
[CrossRef]

G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian–Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35, 715–717 (2010).
[CrossRef] [PubMed]

2009 (1)

J. Li and B. Lü, “Propagation of GSM vortex beams through atmosphere turbulence and evolution of coherent vortices,” J. Opt. A: Pure Appl. Opt. 11, 045710 (2009).
[CrossRef]

2008 (3)

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002(2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferreroa, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003(2008).
[CrossRef]

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–230 (2008).
[CrossRef]

2007 (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferreroa, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

2006 (1)

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

2005 (1)

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
[CrossRef]

2003 (4)

2002 (2)

1995 (2)

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16(1995).
[CrossRef]

1994 (1)

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef] [PubMed]

1990 (1)

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

Amarande, S.

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferreroa, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003(2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferreroa, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

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

Beland, R. R.

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16(1995).
[CrossRef]

Chen, Z.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002(2008).
[CrossRef]

Chu, X.

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283, 3398–3403 (2010).
[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.

Feng, X.

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283, 3398–3403 (2010).
[CrossRef]

Ferreroa, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferreroa, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003(2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferreroa, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

Flossmann, F.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
[CrossRef]

Freund, I.

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef] [PubMed]

Gbur, G.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

Guo, H.

G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian–Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35, 715–717 (2010).
[CrossRef] [PubMed]

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

Li, J.

J. Li and B. Lü, “Propagation of GSM vortex beams through atmosphere turbulence and evolution of coherent vortices,” J. Opt. A: Pure Appl. Opt. 11, 045710 (2009).
[CrossRef]

Li, Y.

Lü, B.

J. Li and B. Lü, “Propagation of GSM vortex beams through atmosphere turbulence and evolution of coherent vortices,” J. Opt. A: Pure Appl. Opt. 11, 045710 (2009).
[CrossRef]

Luo, B.

Maier, M.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
[CrossRef]

Mandel, L.

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

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferreroa, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003(2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferreroa, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

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

Pu, J.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002(2008).
[CrossRef]

Qiao, C.

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283, 3398–3403 (2010).
[CrossRef]

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

Schwarz, U. T.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
[CrossRef]

Shirai, T.

Shvartsman, N.

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef] [PubMed]

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferreroa, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003(2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferreroa, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

Tyson, R. K.

Visser, T. D.

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[CrossRef]

Wang, T.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002(2008).
[CrossRef]

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Wolf, E.

Wu, G.

G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian–Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35, 715–717 (2010).
[CrossRef] [PubMed]

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

Yu, S.

J. Mod. Opt. (1)

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

J. Opt. A: Pure Appl. Opt. (1)

J. Li and B. Lü, “Propagation of GSM vortex beams through atmosphere turbulence and evolution of coherent vortices,” J. Opt. A: Pure Appl. Opt. 11, 045710 (2009).
[CrossRef]

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

Opt. Commun. (4)

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[CrossRef]

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283, 3398–3403 (2010).
[CrossRef]

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
[CrossRef]

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

Opt. Eng. (2)

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002(2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferreroa, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003(2008).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (1)

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef] [PubMed]

Proc. SPIE (3)

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16(1995).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferreroa, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

Other (3)

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

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

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

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Figures (4)

Fig. 1
Fig. 1

Evolution of normalized intensity distribution of a partially coherent flat-topped vortex beam with | m | = 1 and a partially coherent flat-topped vortex-free beam propagating through non-Kolmogorov atmospheric turbulence. The calculation parameters are seen in the text.

Fig. 2
Fig. 2

T ( α , z ) versus α for different values of the propagation distance z.

Fig. 3
Fig. 3

Beam width of a partially coherent flat-topped vortex beam with | m | = 1 propagating through non-Kolmogorov atmospheric turbulence versus α for different values of the propagation distance z.

Fig. 4
Fig. 4

Position and number of coherence vortices of a partially coherent flat-topped vortex beam with m = 1 propagating through non- Kolmogorov atmospheric turbulence for different values of L 0 and l 0 . (a)  L 0 = 1 m , l 0 = 0.01 m , (b)  L 0 = 10 m , l 0 = 0.01 m , (c)  L 0 = 1 m , l 0 = 0.002 m . The black and white dots denote m = + 1 and m = 1 , respectively.

Tables (2)

Tables Icon

Table 1 z dip , z flat , and z Gau for Different Values of α

Tables Icon

Table 2 Conservation Distance z c for Different Values of α

Equations (39)

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U ( ρ , z = 0 ) = u ( ρ ) [ ρ x + i sgn ( m ) ρ y ] | m | ,
U ( ρ , z = 0 ) = n = 1 N ( 1 ) n 1 N ( N n ) exp [ n ( ρ x 2 + ρ y 2 ) 4 w 0 2 ] [ ρ x + i sgn ( m ) ρ y ] | m | ,
W ( 0 ) ( ρ 1 , ρ 2 , 0 ) = [ ρ 1 x ρ 2 x + ρ 1 y ρ 2 y + i sgn ( m ) ( ρ 1 x ρ 2 y ρ 2 x ρ 1 y ) ] | m | × n 1 = 1 N n 2 = 1 N ( 1 ) n 1 + n 2 2 N 4 ( N n 1 ) ( N n 2 ) exp ( n 1 ρ 1 2 + n 2 ρ 2 2 4 w 0 2 ) × exp [ ( n 1 + n 2 ) ( ρ 1 ρ 2 ) 2 4 σ 0 2 ] ,
W ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 d 2 ρ 1 d 2 ρ 2 W ( 0 ) ( ρ 1 , ρ 2 , 0 ) exp { i k 2 z [ ( ρ 1 ρ 1 ) 2 ( ρ 2 ρ 2 ) 2 ] } exp [ ψ * ( ρ 1 , ρ 1 ) + ψ ( ρ 2 , ρ 2 ) ] ,
exp [ ψ * ( ρ 1 , ρ 1 ) + ψ ( ρ 2 , ρ 2 ) ] = exp { 4 π 2 k 2 z 0 1 0 d κ d ξ κ Φ n ( κ , α ) × [ 1 J 0 ( κ | ( 1 ξ ) ( ρ 1 ρ 2 ) + ξ ( ρ 1 ρ 2 ) | ) ] } = exp { π 2 k 2 z 3 0 κ 3 Φ n ( κ , α ) d κ [ ( ρ 1 ρ 2 ) 2 + ( ρ 1 ρ 2 ) ( ρ 1 ρ 2 ) + ( ρ 1 ρ 2 ) 2 ] } = exp { T ( α , z ) [ ( ρ 1 ρ 2 ) 2 + ( ρ 1 ρ 2 ) ( ρ 1 ρ 2 ) + ( ρ 1 ρ 2 ) 2 ] } ,
T ( α , z ) = π 2 k 2 z 3 0 κ 3 Φ n ( κ , α ) d κ ,
Φ n ( κ , α ) = A ( α ) C ˜ n 2 exp ( κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) α / 2 , ( 0 κ < , 3 < α < 4 )
A ( α ) = 1 4 π 2 Γ ( α 1 ) cos ( α π 2 ) , ( Γ ( x ) -Gamma function ) ,
κ 0 = 2 π L 0 , ( L 0 - outer scale of the atmospheric turbulence ) ,
κ m = c ( α ) l 0 , ( l 0 - inner scale of the atmospheric turbulence ) ,
c ( α ) = [ Γ ( 5 α 2 ) A ( α ) 2 π 3 ] 1 α 5 ,
T ( α , z ) = π 2 k 2 z 6 ( α 2 ) A ( α ) C ˜ n 2 × { 2 κ 0 ( 4 α ) + exp ( κ 0 2 κ m 2 ) κ m ( 2 α ) [ ( α 2 ) κ m 2 + 2 κ 0 2 ] Γ ( 2 α 2 , κ 0 2 κ m 2 ) } .
W ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 exp [ i k 2 z ( ρ 1 2 ρ 2 2 ) ] exp [ T ( α , z ) ( ρ 1 ρ 2 ) 2 ] × n 1 = 1 N n 2 = 1 N ( 1 ) n 1 + n 2 2 N 4 ( N n 1 ) ( N n 2 ) d 2 u d 2 v × [ ( u 2 v 2 4 ) ± i ( u x v y u y v x ) ] exp ( A v 2 B u v ) exp ( n 1 + n 2 4 w 0 2 u 2 ) × [ i k z ( ρ 1 ρ 2 ) u ] exp { v [ i k 2 z ( ρ 1 + ρ 2 ) T ( α , z ) ( ρ 1 ρ 2 ) ] } ,
A = n 1 + n 2 16 w 0 2 + T ( α , z ) + n 1 + n 2 4 σ 0 2 ,
B = n 1 n 2 4 w 0 2 + i k z ,
x n exp ( p x 2 + 2 q x ) d x = n ! exp ( q 2 p ) π p ( q p ) n k = 0 E [ n 2 ] 1 ( n 2 k ) ! k ! ( p 4 q 2 ) k , ( n = 0 , 1 , 2 ) ,
W ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 exp [ i k 2 z ( ρ 1 2 ρ 2 2 ) ] exp { T ( α , z ) [ ( ρ 1 x ρ 2 x ) 2 + ( ρ 1 y ρ 2 y ) 2 ] } n 1 N n 2 N ( 1 ) n 1 + n 2 2 N 4 ( N n 1 ) ( N n 2 ) [ ( M 1 M 2 ) ± i ( M 3 M 4 ) ] ,
M 1 = π 2 A C E x E y exp ( F x 2 + F y 2 C ) ( F x 2 + F y 2 C 2 + 1 C ) ,
M 2 = π 2 w 0 2 ( n 1 + n 2 ) D exp { k 2 w 0 2 ( n 1 + n 2 ) z 2 [ ( ρ 1 x ρ 2 x ) 2 + ( ρ 1 y ρ 2 y ) 2 ] } × exp ( G x 2 + G y 2 D ) ( G x 2 + G y 2 D 2 + 1 D ) ,
M 3 = 2 π 2 w 0 ( n 1 + n 2 ) A C D E x F x G y C D exp [ k 2 w 0 2 ( n 1 + n 2 ) z 2 ( ρ 1 y ρ 2 y ) 2 ] exp ( F x 2 C + G y 2 D ) ,
M 4 = 2 π 2 w 0 ( n 1 + n 2 ) A C D E y F y G x C D exp [ k 2 w 0 2 ( n 1 + n 2 ) z 2 ( ρ 1 x ρ 2 x ) 2 ] exp ( F y 2 C + G x 2 D ) ,
C = n 1 + n 2 4 w 0 2 B 2 4 A ,
D = A B 2 w 0 2 ( n 1 + n 2 ) ,
E x = exp { 1 4 A [ i k 2 z ( ρ 1 x + ρ 2 x ) T ( α , z ) ( ρ 1 x ρ 2 x ) ] 2 } ,
F x = 1 2 [ i k z ( ρ 1 x ρ 2 x ) i k B 4 A z ( ρ 1 x + ρ 2 x ) + B 2 A T ( α , z ) ( ρ 1 x ρ 2 x ) ] ,
G x = 1 2 [ i k 2 z ( ρ 1 x + ρ 2 x ) T ( α , z ) ( ρ 1 x ρ 2 x ) i 2 k B w 0 2 ( n 1 + n 2 ) z ( ρ 1 x ρ 2 x ) ] .
I ( ρ , z ) = W ( ρ , ρ , z ) = ( k w 0 2 z ) 2 n 1 = 1 N n 2 = 1 N ( 1 ) n 1 + n 2 2 N 4 ( N n 1 ) ( N n 2 ) 1 ( n 1 + n 2 ) D × [ P + Q ( ρ x 2 + ρ y 2 ) ] exp [ β ( ρ x 2 + ρ y 2 ) ] ,
β = k 2 4 z 2 D ,
P = 4 C 1 D ,
Q = k 2 4 z 2 ( 1 D 2 B 2 A 2 C 2 ) .
w ( z ) = ρ 2 I ( ρ , z ) d 2 ρ I ( ρ , z ) d 2 ρ ,
x 2 m exp ( β x 2 n ) d x = Γ ( 2 m + 1 2 n ) / n β 2 m + 1 2 n ,
w ( z ) = ( w 1 w 2 ) 1 / 2 ,
w 1 = n 1 = 1 N n 2 = 1 N ( 1 ) n 1 + n 2 2 N 4 ( N n 1 ) ( N n 2 ) 1 ( n 1 + n 2 ) D β 3 ( 2 Q + P β ) ,
w 2 = n 1 = 1 N n 2 = 1 N ( 1 ) n 1 + n 2 2 N 4 ( N n 1 ) ( N n 2 ) 1 ( n 1 + n 2 ) D β 2 ( Q + P β ) .
w ( z ) = [ 4.70 w 0 2 + ( 0.86 w 0 2 + 2.72 σ 0 2 ) z 2 k 2 + 4 k 2 T ( α , z ) z 2 ] 1 / 2 = ( I + II + III ) 1 / 2 .
μ ( ρ 1 , ρ 2 , z ) = W ( ρ 1 , ρ 2 , z ) [ I ( ρ 1 , z ) I ( ρ 2 , z ) ] 1 / 2 ,
Re [ μ ( ρ 1 , ρ 2 , z ) ] = 0 ,
Im [ μ ( ρ 1 , ρ 2 , z ) ] = 0 ,

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