Abstract

The concept of pseudo-Bessel correlated beams is introduced, and their scintillation properties on propagation through turbulence are investigated. By using the Rytov approximation, the scintillation index of pseudo-Bessel correlated beams is formulated in weak turbulence. The study of scintillation is extended into strong turbulence by numeric simulations. It is shown that by choosing an appropriate coherence parameter, pseudo-Bessel correlated beams have lower scintillation than comparable fully coherent beams in both weak and strong turbulence. In addition, the configuration of pseudo-Bessel correlated beams is modified by adding a horizontal beamlet; the scintillation properties of these modified beams are also discussed.

© 2010 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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2009 (4)

2008 (1)

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuation in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 93, 605–611 (2008).
[CrossRef]

2007 (3)

2006 (2)

2005 (2)

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

T. J. Schulz, “Optimal beams for propagation through random media,” Opt. Lett. 30, 1093–1095 (2005).
[CrossRef] [PubMed]

2004 (1)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom,” Opt. Eng. 43, 330–341 (2004).
[CrossRef]

2003 (1)

2002 (2)

2001 (1)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

2000 (1)

1998 (1)

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
[CrossRef]

1996 (1)

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

1995 (2)

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

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995).

1991 (2)

J. Turunen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
[CrossRef]

A. T. Friberg, A. Vasara, and J. Turunen, “Partially coherent propagation-invariant beams: Passage through paraxial optical systems,” Phys. Rev. A 43, 7079–7082 (1991).
[CrossRef] [PubMed]

1988 (1)

1987 (3)

F. Gori, G. Guattaria, and C. Padovani, “Modal expansion of J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

1983 (2)

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 1054–1059 (1983).

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Sci. 18, 551–556 (1983).
[CrossRef]

1979 (1)

Andrews, L. C.

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

O. Korotkova, L. C. Andrews, and R. L. Phillips, “A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom,” Opt. Eng. 43, 330–341 (2004).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995).

Banakh, V. A.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 1054–1059 (1983).

Baykal, Y.

Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Appl. Opt. 48, 1943–1954 (2009).
[CrossRef] [PubMed]

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuation in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 93, 605–611 (2008).
[CrossRef]

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Sci. 18, 551–556 (1983).
[CrossRef]

Belmonte, A.

Borghi, R.

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Bouchal, Z.

Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002).
[CrossRef]

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
[CrossRef]

Buldakov, V. M.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 1054–1059 (1983).

Cai, Y.

Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Appl. Opt. 48, 1943–1954 (2009).
[CrossRef] [PubMed]

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuation in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 93, 605–611 (2008).
[CrossRef]

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
[CrossRef]

Cincotti, G.

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Davidson, F. M.

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Eyyuboglu, H. T.

Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Appl. Opt. 48, 1943–1954 (2009).
[CrossRef] [PubMed]

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuation in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 93, 605–611 (2008).
[CrossRef]

Flatté, S. M.

Friberg, A. T.

J. Turunen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
[CrossRef]

A. T. Friberg, A. Vasara, and J. Turunen, “Partially coherent propagation-invariant beams: Passage through paraxial optical systems,” Phys. Rev. A 43, 7079–7082 (1991).
[CrossRef] [PubMed]

Gbur, G.

Gori, F.

F. Gori, G. Guattaria, and C. Padovani, “Modal expansion of J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Gu, Y.

Guattaria, G.

F. Gori, G. Guattaria, and C. Padovani, “Modal expansion of J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

Klein, L.

Korotkova, O.

Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34, 2261–2263 (2009).
[CrossRef] [PubMed]

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuation in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 93, 605–611 (2008).
[CrossRef]

G. Gbur and O. Korotkova, “Angular spectrum representation for the propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence,” J. Opt. Soc. Am. A 24, 745–752 (2007).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom,” Opt. Eng. 43, 330–341 (2004).
[CrossRef]

Leader, J. C.

Mandel, L.

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

Martin, J. M.

Miceli, J. J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Mironov, V. L.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 1054–1059 (1983).

Moloney, J. V.

Padovani, C.

F. Gori, G. Guattaria, and C. Padovani, “Modal expansion of J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Palma, C.

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Peleg, A.

Phillips, R. L.

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

O. Korotkova, L. C. Andrews, and R. L. Phillips, “A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom,” Opt. Eng. 43, 330–341 (2004).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

Plonus, M. A.

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Sci. 18, 551–556 (1983).
[CrossRef]

Polynkin, P.

Qian, X.

Rao, R.

Rhoadarmer, T.

Ricklin, J. C.

Schulz, T. J.

Sermutlu, E.

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuation in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 93, 605–611 (2008).
[CrossRef]

Turunen, J.

A. T. Friberg, A. Vasara, and J. Turunen, “Partially coherent propagation-invariant beams: Passage through paraxial optical systems,” Phys. Rev. A 43, 7079–7082 (1991).
[CrossRef] [PubMed]

J. Turunen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
[CrossRef]

Vasara, A.

J. Turunen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
[CrossRef]

A. T. Friberg, A. Vasara, and J. Turunen, “Partially coherent propagation-invariant beams: Passage through paraxial optical systems,” Phys. Rev. A 43, 7079–7082 (1991).
[CrossRef] [PubMed]

Visser, T.

Voelz, D.

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
[CrossRef]

Wang, S. J.

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Sci. 18, 551–556 (1983).
[CrossRef]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995).

Wolf, E.

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

Xiao, X.

Zhu, W.

Appl. Opt. (4)

Appl. Phys. B: Lasers Opt. (1)

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuation in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 93, 605–611 (2008).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

A. Peleg and J. V. Moloney, “Scintillation reduction by use of multiple Gaussian laser beams with different wavelengths,” IEEE Photonics Technol. Lett. 19, 883–885 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Commun. (4)

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
[CrossRef]

Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002).
[CrossRef]

F. Gori, G. Guattaria, and C. Padovani, “Modal expansion of J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Opt. Eng. (1)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom,” Opt. Eng. 43, 330–341 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Opt. Spectrosc. (1)

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 1054–1059 (1983).

Phys. Rev. A (1)

A. T. Friberg, A. Vasara, and J. Turunen, “Partially coherent propagation-invariant beams: Passage through paraxial optical systems,” Phys. Rev. A 43, 7079–7082 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Radio Sci. (1)

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Sci. 18, 551–556 (1983).
[CrossRef]

Other (4)

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

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[CrossRef]

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

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995).

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Figures (8)

Fig. 1
Fig. 1

Configuration of a pseudo-Bessel correlated beam. k n is the wave vector of the nth beamlet. k n = k u n , and its direction is specified by the unit vector u n whose projection in the source plane is u n . θ = 2 arcsin ( | u n | ) is the vertex of cone.

Fig. 2
Fig. 2

On-axis scintillation of a 2-beamlet pseudo-Bessel correlated beam as a function of the relative correlation length r 0 w 0 . Here the wavelength is taken to be λ = 1.55 μ m , and the width of the beam is taken to be w 0 = 0.05 m . The turbulence strength parameter is C n 2 = 10 15 m 2 3 and the propagation distance is L = 2 km .

Fig. 3
Fig. 3

On-axis scintillation of a pseudo-Bessel correlated beam as a function of the number of its constituent beamlets N. The relative correlation length is taken to be r 0 w 0 = 0.33 . The rest of the parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

On-axis scintillation of a 16-beamlet pseudo-Bessel correlated beam as a function of the relative correlation length r 0 w 0 . The parameters are the same as in Fig. 2.

Fig. 5
Fig. 5

On-axis scintillation of a pseudo-Bessel correlated beam as a function of the relative correlation length r 0 w 0 . The wavelength is also λ = 1.55 μ m and the width of the beam is w 0 = 0.05 m . Here the turbulence strength parameter is C n 2 = 10 14 m 2 3 and the propagation distance is L = 3 km .

Fig. 6
Fig. 6

On-axis scintillation of an 8-beamlet pseudo-Bessel correlated beam as a function of the Rytov variance σ 1 2 = 1.23 C n 2 k 7 6 L 11 6 . The relative correlation length is r 0 w 0 = 0.28 and the other parameters are the same as in Fig. 5.

Fig. 7
Fig. 7

(a) Dashed curve shows the on-axis scintillation of a 16-beamlet pseudo-Bessel correlated beam as a function of the relative correlation length r 0 w 0 , while the solid curve shows the minimum on-axis scintillation of the corresponding modified pseudo-Bessel correlated beam. The optimal amplitude E 0 is shown in (b). The parameters are the same as in Fig. 4.

Fig. 8
Fig. 8

(a) Dashed curve shows the on-axis scintillation of an 8-beamlet pseudo-Bessel correlated beam as a function of the relative correlation length r 0 w 0 , while the solid curve shows the minimum on-axis scintillation of the corresponding modified pseudo-Bessel correlated beam. The optimal amplitude E 0 is shown in (b). The parameters are the same as in Fig. 5.

Equations (38)

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

W ( ρ 1 , ρ 2 , ω ) = U ( ρ 1 , ω ) U * ( ρ 2 , ω ) ,
W ( ρ 1 , ρ 2 , ω ) = S ( ρ 1 , ω ) S ( ρ 2 , ω ) μ ( ρ 1 , ρ 2 , ω ) ,
μ ( ρ 1 , ρ 2 ) = J 0 ( | ρ 1 ρ 2 | r 0 ) ,
J 0 ( | ρ 1 ρ 2 | r 0 ) = 1 2 π 0 2 π exp [ i k u ( ρ 1 ρ 2 ) ] d φ u ,
W ( ρ 1 , ρ 2 ) n = 1 N A n ( ρ 1 ) A n * ( ρ 2 ) ,
A n ( ρ ) = 1 N S ( ρ ) exp ( i k u n ρ ) ,
S ( ρ ) = exp ( 2 ρ 2 w 0 2 ) ,
A n ( ρ , z ) = A 0 n ( ρ , z ) exp [ ψ n 1 ( ρ , z ) + ψ n 2 ( ρ , z ) + ] ,
I ( ρ , L ) = n = 1 N I n ( ρ , L ) ,
σ 2 ( ρ , L ) = I 2 ( ρ , L ) I ( ρ , L ) 2 1 ,
σ 2 ( ρ , L ) = m = 1 N n = 1 N I m ( ρ , L ) I n ( ρ , L ) ( n = 1 N I n ( ρ , L ) ) 2 1 .
I n ( ρ , L ) = | A 0 n ( ρ , L ) | 2 exp { 2 Re [ E 1 n ( ρ , L ) ] + E 2 n n ( ρ , L ) } ,
I m ( ρ , L ) I n ( ρ , L ) = I m ( ρ , L ) I n ( ρ , L ) × exp { 2 Re [ E 2 m n ( ρ , L ) ] + 2 Re [ E 3 m n ( ρ , L ) ] } ,
E 1 n ( ρ , L ) = π k 2 exp [ i k L u n 2 2 p ( L ) ] Φ n ( κ ) d 2 κ × 0 L exp [ i k η u n 2 2 p ( η ) ] exp [ i k γ ( η ) ( L η ) u n 2 2 p 2 ( η ) ] d η ,
E 2 m n ( ρ , L ) = 2 π k 2 exp [ i k L u m 2 2 p ( L ) ] exp [ i k L u n 2 2 p * ( L ) ] × 0 L exp { i k 2 p ( η ) [ η + γ ( η ) ( L η ) p ( η ) ] u m 2 } × exp { i k 2 p * ( η ) [ η + γ * ( η ) ( L η ) p * ( η ) ] u n 2 } d η × exp { i [ γ ( η ) γ * ( η ) ] ( L η ) κ 2 2 k } exp { i [ γ ( η ) γ * ( η ) ] κ ρ } × exp { i ( L η ) κ [ u m p ( L ) u n p * ( L ) ] } Φ n ( κ ) d 2 κ ,
E 3 m n ( ρ , L ) = 2 π k 2 exp [ i k L u m 2 2 p ( L ) ] exp [ i k L u n 2 2 p ( L ) ] × 0 L exp { i k 2 p ( η ) [ η + γ ( η ) ( L η ) p ( η ) ] ( u m 2 + u n 2 ) } d η exp [ i ( L η ) p ( L ) κ ( u m u n ) ] exp [ i γ ( η ) ( L η ) κ 2 k ] Φ n ( κ ) d 2 κ ,
Φ n ( κ ) = 0.033 C n 2 exp ( κ 2 κ m 2 ) ( κ 2 + κ 0 2 ) 11 6 ,
σ 1 2 = 1.23 C n 2 k 7 6 L 11 6 .
E ( ρ , z = 0 ) = E 0 exp ( ρ 2 w 0 2 ) ,
σ min 2 = { σ pb 2 σ h 2 ( σ pb , h 2 ) 2 σ pb 2 + σ h 2 2 σ pb , h 2 if σ pb , h 2 < min [ σ pb 2 , σ h 2 ] min [ σ pb 2 , σ h 2 ] otherwise } ,
E 0 = I pb ( σ pb 2 σ pb , h 2 ) I h ( σ h 2 σ pb , h 2 ) if σ pb , h 2 < min [ σ pb 2 , σ h 2 ] .
σ pb , h 2 = I pb I h I pb I h 1 .
σ n , h 2 = I n I h I n I h 1
E 1 n ( ρ , L ) = ψ n 2 ( ρ , L ) + 1 2 ψ n 1 ( ρ , L ) 2 ,
E 2 m n ( ρ , L ) = ψ m 1 ( ρ , L ) ψ n 1 * ( ρ , L ) ,
E 3 m n ( ρ , L ) = ψ m 1 ( ρ , L ) ψ n 1 ( ρ , L ) .
A 0 n ( ρ , L ) = 1 N p ( L ) exp [ i k L ρ 2 p ( L ) w 0 2 ] exp [ i ( 2 k u n ρ k L u n 2 ) 2 p ( L ) ] .
ψ n 1 ( ρ , L ) = k 2 2 π 0 L d z exp [ i k ( L z ) + i k | s ρ | 2 2 ( L z ) ] A 0 n ( s , z ) A 0 n ( ρ , L ) n 1 ( s , z ) L z d 2 s .
n 1 ( ρ , z ) = exp ( i κ ρ ) d v ( κ , z ) ,
ψ n 1 ( ρ , L ) = i k exp [ i k L u n 2 2 p ( L ) ] 0 L exp { i k 2 p ( z ) [ z + γ ( z ) ( L z ) p ( z ) ] u n 2 } d z × exp [ i γ ( z ) ( L z ) κ 2 2 k ] exp [ i γ ( z ) κ ρ ] × exp [ i γ ( z ) ( L z ) p ( z ) κ u n ] d v ( κ , z ) ,
d v ( κ , z ) d v * ( κ , z ) = F n ( κ , | z z | ) δ ( κ κ ) d 2 κ d 2 κ ,
E 2 m n ( ρ , L ) = k 2 exp [ i k L u m 2 2 p ( L ) ] exp [ i k L u n 2 2 p * ( L ) ] × 0 L 0 L exp { i k 2 p ( z ) [ z + γ ( z ) ( L z ) p ( z ) ] u m 2 } × exp { i k 2 p * ( z ) [ z + γ * ( z ) ( L z ) p * ( z ) ] u n 2 } d z d z × exp { i 2 k [ γ ( z ) ( L z ) γ * ( z ) ( L z ) ] κ 2 } exp { i κ [ L z p ( L ) u m L z p * ( L ) u n ] } exp { i [ γ ( z ) γ * ( z ) ] κ ρ } F n ( κ , | z z | ) d 2 κ .
Φ n ( κ ) = 1 2 π F n ( κ , μ ) d μ ,
E 2 m n ( ρ , L ) = 2 π k 2 exp [ i k L u m 2 2 p ( L ) ] exp [ i k L u n 2 2 p * ( L ) ] × 0 L exp { i k 2 p ( η ) [ η + γ ( η ) ( L η ) p ( η ) ] u m 2 } × exp { i k 2 p * ( η ) [ η + γ * ( η ) ( L η ) p * ( η ) ] u n 2 } d η × exp { i [ γ ( η ) γ * ( η ) ] ( L η ) κ 2 2 k } exp { i [ γ ( η ) γ * ( η ) ] κ ρ } exp { i ( L η ) κ [ u m p ( L ) u n p * ( L ) ] } Φ n ( κ ) d 2 κ ,
d v ( κ , z ) d v ( κ , z ) = F n ( κ , | z z | ) δ ( κ + κ ) d 2 κ d 2 κ .
E 1 n ( ρ , L ) = k 2 2 π 0 L d z exp [ i k ( L z ) + i k | s ρ | 2 2 ( L z ) ] × A 0 n ( s , z ) A 0 n ( ρ , L ) ψ n 1 ( s , z ) n 1 ( s , z ) L z d 2 s .
E 1 n ( ρ , L ) = k 2 exp [ i k L u n 2 2 p ( L ) ] 0 L d z 0 z d z exp { i k 2 [ z p ( z ) + γ ( z z ) p 2 ( z ) + γ ( L z ) p 2 ( z ) ] u n 2 } × d v ( κ , z ) d v ( κ , z ) exp [ i γ ( z z ) κ 2 2 k ] exp [ i γ ( z z ) p ( z ) κ u n ] exp [ i γ ( L z ) | κ + γ κ | 2 2 k ] exp [ i γ ( L z ) p ( z ) ( κ + γ κ ) u n ] exp [ i γ ( κ + γ κ ) ρ ] ,
E 1 n ( ρ , L ) = π k 2 exp [ i k L u n 2 2 p ( L ) ] Φ n ( κ ) d 2 κ × 0 L exp [ i k η u n 2 2 p ( η ) ] exp [ i k γ ( η ) ( L η ) u n 2 2 p 2 ( η ) ] d η ,

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