Abstract

A nonuniformly polarized beam array (NUBPA) is modeled by coherent superposition of a pair of orthogonally polarized spatial modes. The propagation of a NUPBA in turbulent atmosphere is investigated based on the extended Huygens–Fresnel method. An analytical expression for the intensity profile of a NUBPA in turbulent atmosphere is presented. The influence of polarization degree, intensity of turbulence, array number, and distance between adjacent elements on the intensity profile in the receiving plane is evaluated numerically and analyzed in detail.

© 2011 Optical Society of America

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2010

2009

2008

2007

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

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

2006

H. T. Eyyuboğlu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405 (2006).
[CrossRef]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117(2006).
[CrossRef]

Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568–570 (2006).
[CrossRef] [PubMed]

2005

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

2004

2003

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]

B. Li and B. Lü, “Characterization of off-axis superposition of partially coherent beams,” J. Opt. A Pure Appl. Opt. 5, 303–307 (2003).
[CrossRef]

2000

1999

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

1998

J. M. Movilla, G. Piquero, R. Martinez-Herrero, and P. M. Mejias, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

1985

1979

1978

Baykal, Y.

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
[CrossRef] [PubMed]

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98, 557–565 (2010).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91, 265–271 (2008).
[CrossRef]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16, 7665–7673 (2008).
[CrossRef] [PubMed]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “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, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405 (2006).
[CrossRef]

Bloom, G.

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert–Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

Cai, Y.

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91, 265–271 (2008).
[CrossRef]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16, 7665–7673 (2008).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “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, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568–570 (2006).
[CrossRef] [PubMed]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117(2006).
[CrossRef]

Carras, M.

Chen, X.

X. Li, X. Chen, and X. Ji, “Influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite–Gaussian beams,” Opt. Commun. 282, 7–13(2009).
[CrossRef]

Chen, Y.

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

Chen, Z.

Cheng, W.

Chu, X.

Clark, R. G.

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[CrossRef]

Dogariu, A.

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]

Du, X.

Eyyuboglu, H. T.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98, 557–565 (2010).
[CrossRef]

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
[CrossRef] [PubMed]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91, 265–271 (2008).
[CrossRef]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16, 7665–7673 (2008).
[CrossRef] [PubMed]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “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, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405 (2006).
[CrossRef]

Fan, T. Y.

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

Gbur, G.

Goodno, G. D.

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert–Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

Gu, Y.

Guattari, G.

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

Haus, J. W.

He, S.

Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568–570 (2006).
[CrossRef] [PubMed]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117(2006).
[CrossRef]

Injeyan, H.

J. Marmo, H. Injeyan, H. Komine, S. McNaught, J. Machan, and J. Sollee, “Joint high power solid state laser program advancements at Northrop Grumman,” Proc. SPIE 7195, 719507 (2009).
[CrossRef]

Jeong, Y.

Ji, X.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98, 557–565 (2010).
[CrossRef]

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
[CrossRef] [PubMed]

X. Li, X. Chen, and X. Ji, “Influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite–Gaussian beams,” Opt. Commun. 282, 7–13(2009).
[CrossRef]

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]

Jones, D. C.

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[CrossRef]

Komine, H.

J. Marmo, H. Injeyan, H. Komine, S. McNaught, J. Machan, and J. Sollee, “Joint high power solid state laser program advancements at Northrop Grumman,” Proc. SPIE 7195, 719507 (2009).
[CrossRef]

Korotsova, O.

Lallier, E.

Larat, C.

Leader, J. C.

Li, B.

B. Li and B. Lü, “Characterization of off-axis superposition of partially coherent beams,” J. Opt. A Pure Appl. Opt. 5, 303–307 (2003).
[CrossRef]

Li, X.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98, 557–565 (2010).
[CrossRef]

X. Li, X. Chen, and X. Ji, “Influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite–Gaussian beams,” Opt. Commun. 282, 7–13(2009).
[CrossRef]

Lin, Q.

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16, 7665–7673 (2008).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “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.

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]

B. Li and B. Lü, “Characterization of off-axis superposition of partially coherent beams,” J. Opt. A Pure Appl. Opt. 5, 303–307 (2003).
[CrossRef]

Ma, H.

Ma, Y.

Machan, J.

J. Marmo, H. Injeyan, H. Komine, S. McNaught, J. Machan, and J. Sollee, “Joint high power solid state laser program advancements at Northrop Grumman,” Proc. SPIE 7195, 719507 (2009).
[CrossRef]

Marcadet, X.

Marmo, J.

J. Marmo, H. Injeyan, H. Komine, S. McNaught, J. Machan, and J. Sollee, “Joint high power solid state laser program advancements at Northrop Grumman,” Proc. SPIE 7195, 719507 (2009).
[CrossRef]

Martinez-Herrero, R.

J. M. Movilla, G. Piquero, R. Martinez-Herrero, and P. M. Mejias, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

McComb, T. S.

McNaught, S.

J. Marmo, H. Injeyan, H. Komine, S. McNaught, J. Machan, and J. Sollee, “Joint high power solid state laser program advancements at Northrop Grumman,” Proc. SPIE 7195, 719507 (2009).
[CrossRef]

McNaught, S. J.

Mejias, P. M.

J. M. Movilla, G. Piquero, R. Martinez-Herrero, and P. M. Mejias, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

Movilla, J. M.

J. M. Movilla, G. Piquero, R. Martinez-Herrero, and P. M. Mejias, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

Nilsson, J.

Payne, D. N.

Piquero, G.

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert–Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
[CrossRef]

J. M. Movilla, G. Piquero, R. Martinez-Herrero, and P. M. Mejias, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

Plonus, M. A.

Pu, J.

T. Wang and J. Pu, “Propagation of non-uniformly polarized beams in a turbulent atmosphere,” Opt. Commun. 281, 3617–3622 (2008).
[CrossRef]

Rothenberg, J. E.

Sahu, J. K.

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]

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert–Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

Scott, A. M.

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[CrossRef]

Sermutlu, E.

H. T. Eyyuboğlu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405 (2006).
[CrossRef]

Shellan, J. B.

Shirai, T.

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]

Sollee, J.

J. Marmo, H. Injeyan, H. Komine, S. McNaught, J. Machan, and J. Sollee, “Joint high power solid state laser program advancements at Northrop Grumman,” Proc. SPIE 7195, 719507 (2009).
[CrossRef]

Stace, C.

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[CrossRef]

Stacey, C. D.

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[CrossRef]

Stone, S. M.

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[CrossRef]

Thielen, P. A.

Turner, A. J.

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[CrossRef]

Wang, S. C. H.

Wang, T.

T. Wang and J. Pu, “Propagation of non-uniformly polarized beams in a turbulent atmosphere,” Opt. Commun. 281, 3617–3622 (2008).
[CrossRef]

Wang, X.

Weber, M. E.

Wickham, M. G.

Wolf, E.

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]

Wu, Y.

Xu, X.

Zhan, Q.

Zhang, E.

Zhao, D.

Zhou, P.

Zhu, Y.

Appl. Opt.

Appl. Phys. B

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98, 557–565 (2010).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91, 265–271 (2008).
[CrossRef]

Appl. Phys. Lett.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117(2006).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

J. Opt. A Pure Appl. Opt.

B. Li and B. Lü, “Characterization of off-axis superposition of partially coherent beams,” J. Opt. A Pure Appl. Opt. 5, 303–307 (2003).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

X. Li, X. Chen, and X. Ji, “Influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite–Gaussian beams,” Opt. Commun. 282, 7–13(2009).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405 (2006).
[CrossRef]

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]

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the one-dimensional NUPBA.

Fig. 2
Fig. 2

Effect of γ on polarization distribution of the beam source: (a)  κ = 1 , (b)  κ = 0.5 , and (c)  κ = 0.85 .

Fig. 3
Fig. 3

Intensity distributions along different propagation distances when κ = 0.85 and C n 2 = 1 × 10 14 m 2 / 3 : (a)  z = 5 km , (b)  z = 50 km , and (c)  z = 150 km .

Fig. 4
Fig. 4

Intensity distributions along different propagation distances when κ = 1 and C n 2 = 1 × 10 14 m 2 / 3 : (a)  z = 5 km , (b)  z = 50 km , and (c)  z = 150 km .

Fig. 5
Fig. 5

Intensity distributions along different propagation distance in a specified turbulence with C n 2 = 5 × 10 14 m 2 / 3 : (a)  z = 5 km , (b)  z = 50 km , and (c)  z = 150 km .

Fig. 6
Fig. 6

Intensity distributions along different propagation with array number of M = 7 : (a)  z = 5 km , (b)  z = 50 km , and (c)  z = 150 km .

Fig. 7
Fig. 7

Intensity distributions in a specified turbulence with different distances between adjacent elements in the beam array when M = 7 , κ = 1 , C n 2 = 3 × 10 14 m 2 / 3 , and z = 50 km : (a)  x d = 2 w 0 , (b)  x d = 3 w 0 , and (c)  x d = 5 w 0 .

Equations (13)

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J ( r 1 , r 2 , z = 0 ) = [ J x x ( r 1 , r 2 ) J x y ( r 1 , r 2 ) J y x ( r 1 , r 2 ) J y y ( r 1 , r 2 ) ] ,
J x x ( r 1 , r 2 ) = E x * ( r 1 ) E x ( r 2 ) ,
J y y ( r 1 , r 2 ) = E y * ( r 1 ) E y ( r 2 ) ,
J x y ( r 1 , r 2 ) = E x * ( r 1 ) E y ( r 2 ) = J y x * ( r 1 , r 2 ) .
E m , x ( r ) = exp [ ( x x m ) 2 w 0 2 ] , E m , y ( r ) = κ H 1 [ 2 ( x x m ) w 0 ] exp [ ( x x m ) 2 w 0 2 ] .
E x ( r ) = m E m , x ( r ) = m = M 1 2 M 1 2 exp [ ( x m x d ) 2 w 0 2 ] , E y ( r ) = m E m , y ( r ) = m = M 1 2 M 1 2 κ H 1 [ 2 ( x m x d ) w 0 ] exp [ ( x m x d ) 2 w 0 2 ] .
I ( r ) = J x x ( r , r ) + J y y ( r , r ) .
J ( r 1 , r 2 , z ) = ( k 2 π z ) 2 J ( r 1 , r 2 , z = 0 ) exp [ i k 2 z ( r 1 r 1 ) 2 + i k 2 z ( r 2 r 2 ) 2 ] × exp [ ψ ( r 1 , r 1 , z ) + ψ * ( r 2 , r 2 , z ) ] d r 1 d r 2 ,
I total ( r , z ) = I x ( z ) + I y ( z ) = J x x ( r , z ) + J y y ( r , z ) .
J x x ( x , z ) = k 2 z A m = M 1 2 M 1 2 n = M 1 2 M 1 2 exp ( T x 2 + Q x + R ) ,
A = 1 w 0 4 + 2 w 0 2 ρ 0 2 + ( k 2 z ) 2 , C = 1 w 0 2 + 1 ρ 0 2 i k 2 z , D = 2 n w 0 2 + 2 m w 0 2 ρ 0 2 C 2 , G = 1 1 ρ 0 2 C 2 , T = ( k 2 z ) 2 [ ( C G A ) 2 + ( 1 C ) 2 ] , Q = i k x d 2 z ( C 2 D G A 2 2 m w 0 2 C 2 ) , R = x d [ m 2 + n 2 w 0 2 + ( C D 2 A ) 2 + ( m w 0 2 C ) 2 ] ;
J y y ( x , z ) = κ 2 k 2 z A D m = M 1 2 M 1 2 n = M 1 2 M 1 2 1 2 w 0 2 A 2 × exp { R k 2 4 z 2 x 2 + [ ( i k E z D 2 2 ) ( m n ) x d R C ] i k x 2 z + S } T ,
A = 1 w 0 2 + 1 ρ 0 2 i k 2 z , D = 1 w 0 2 + 1 ρ 0 2 + i k 2 z 1 ρ 0 4 A 2 , E = 1 1 ρ 0 2 A 2 , R = 1 A 2 + E 2 D 2 , C = i k z m x d 2 ρ 0 2 ( m n ) x d , F = 1 / A w 0 2 A 2 2 , S = R C 2 4 + i k 2 z ( m 2 n 2 ) x d 2 ( k 2 4 z 2 D 2 + 1 ρ 0 2 ) ( m n ) 2 x d 2 i k E C 2 z D 2 ( m n ) x d , T = r = 0 1 l = 0 min ( 1 , r ) ( 1 r ) ( 1 l ) r ! ( r l ) ! 2 1 + 2 l 2 ( 2 w 0 ρ 0 2 F ) r ( 1 2 w 0 2 D 2 ) 1 + r 2 l 2 H 1 r ( i k z F x + F C ) H 1 + r 2 l ( i k E x + i k ( m n ) x d E C z 2 ( D 2 w 0 2 2 ) z ) .

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