Abstract

By using the generalized beam formulation, the scintillation index is derived and evaluated for cosh-Gaussian beams in a turbulent atmosphere. Comparisons are made to cos-Gaussian and Gaussian beam scintillations. The variations of scintillations against propagation length at different values of displacement and focusing parameters are examined. The dependence of scintillations on source size at different propagation lengths is also investigated. Two-dimensional scintillation index distributions covering the entire transverse receiver planes are given. From the graphic illustrations, it is found that in comparison to pure Gaussian beams cosh-Gaussian beams have lower on-axis scintillations at smaller source sizes and longer propagation distances. The focusing effect appears to impose more reduction on the cosh-Gaussian beam scintillations than those of the Gaussian beam. The distribution of the off-axis scintillation index values of the Gaussian beams appears to be uniform over the transverse receiver plane, whereas that of the cosh-Gaussian beam is arranged according to the position of the slanted axis.

© 2007 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.
  3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, Bellingham, 2005).
    [CrossRef]
  4. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
    [CrossRef]
  5. V. A. Banakh and V. L. Mironov, "Influence of the diffraction size of a transmitting aperture and the turbulence spectrum on the intensity fluctuations of laser radiation," Sov. J. Quantum Electron. 8, 875-878 (1978).
    [CrossRef]
  6. V. P. Lukin and B. V. Fortes, "Phase correction of turbulent distortions of an optical wave propagating under conditions of strong intensity fluctuations," Appl. Opt. 41, 5616-5624 (2002).
    [CrossRef] [PubMed]
  7. W. B. Miller, J. C. Ricklin, and L. C. Andrews, "Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam," J. Opt. Soc. Am. A 11, 2719-2726 (1994).
    [CrossRef]
  8. L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave," Waves Random Media 11, 271-291 (2001).
    [CrossRef]
  9. R. L. Fante, "Comparison of theories for intensity fluctuations in strong turbulence," Radio Sci. 11, 215-220 (1976).
    [CrossRef]
  10. V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, and R. Sh. Tsvik, "Focused-laser-beam scintillations in the turbulent atmosphere," J. Opt. Soc. Am. 64, 516-518 (1974).
  11. F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, "Scintillation: theory vs. experiment," in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
    [CrossRef]
  12. O. Korotkova, "Control of the intensity fluctuations of random electromagnetic beams on propagation in weak atmospheric turbulence," in Free-Space Laser Communication Technologies XVIII, G. S. Mecherle, ed., Proc. SPIE 6105, 61050V (2006).
    [CrossRef]
  13. Y. Baykal and M. A. Plonus, "Intensity fluctuations due to a partially coherent source in atmospheric turbulence as predicted by Rytov's method," J. Opt. Soc. Am. A 2, 2124-2132 (1985).
    [CrossRef]
  14. F. S. Vetelino and L. C. Andrews, "Annular Gaussian beams in turbulent media," in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 86-97 (2004).
    [CrossRef]
  15. Y. Baykal and H. T. Eyyuboǧlu, "Scintillation index of flat-topped-Gaussian beams," Appl. Opt. 45, 3793-3797 (2006).
    [CrossRef] [PubMed]
  16. D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, "Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model," in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B (2006).
    [CrossRef]
  17. Y. Baykal, "Beams with arbitrary field profiles in turbulence," in Thirteenth Joint International Symposium on Atmospheric and Ocean Optics. Atmospheric Physics, G. G. Matvienko and V. A. Banakh, eds., Proc. SPIE 6522, 652209 (2006).
    [CrossRef]
  18. Y. Baykal, "Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere," J. Opt. Soc. Am. A 21, 1290-1299 (2004).
    [CrossRef]
  19. H. T. Eyyuboǧlu and Y. Baykal, "Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere," Appl. Opt. 44, 976-983 (2005).
    [CrossRef] [PubMed]
  20. Y. Baykal, "Formulation of correlations for general-type beams in atmospheric turbulence," J. Opt. Soc. Am. A 23, 889-893 (2006).
    [CrossRef]
  21. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).
  22. H. T. Eyyuboǧlu and Y. Baykal, "Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere," Opt. Express 12, 4659-4674 (2004).
    [CrossRef] [PubMed]

2006 (5)

O. Korotkova, "Control of the intensity fluctuations of random electromagnetic beams on propagation in weak atmospheric turbulence," in Free-Space Laser Communication Technologies XVIII, G. S. Mecherle, ed., Proc. SPIE 6105, 61050V (2006).
[CrossRef]

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, "Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model," in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B (2006).
[CrossRef]

Y. Baykal, "Beams with arbitrary field profiles in turbulence," in Thirteenth Joint International Symposium on Atmospheric and Ocean Optics. Atmospheric Physics, G. G. Matvienko and V. A. Banakh, eds., Proc. SPIE 6522, 652209 (2006).
[CrossRef]

Y. Baykal, "Formulation of correlations for general-type beams in atmospheric turbulence," J. Opt. Soc. Am. A 23, 889-893 (2006).
[CrossRef]

Y. Baykal and H. T. Eyyuboǧlu, "Scintillation index of flat-topped-Gaussian beams," Appl. Opt. 45, 3793-3797 (2006).
[CrossRef] [PubMed]

2005 (2)

H. T. Eyyuboǧlu and Y. Baykal, "Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere," Appl. Opt. 44, 976-983 (2005).
[CrossRef] [PubMed]

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, "Scintillation: theory vs. experiment," in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

2004 (3)

2002 (1)

2001 (1)

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave," Waves Random Media 11, 271-291 (2001).
[CrossRef]

1994 (1)

1985 (1)

1978 (1)

V. A. Banakh and V. L. Mironov, "Influence of the diffraction size of a transmitting aperture and the turbulence spectrum on the intensity fluctuations of laser radiation," Sov. J. Quantum Electron. 8, 875-878 (1978).
[CrossRef]

1976 (1)

R. L. Fante, "Comparison of theories for intensity fluctuations in strong turbulence," Radio Sci. 11, 215-220 (1976).
[CrossRef]

Al-Habash, M. A.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave," Waves Random Media 11, 271-291 (2001).
[CrossRef]

Andrews, L. C.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, "Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model," in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B (2006).
[CrossRef]

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, "Scintillation: theory vs. experiment," in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

F. S. Vetelino and L. C. Andrews, "Annular Gaussian beams in turbulent media," in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 86-97 (2004).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave," Waves Random Media 11, 271-291 (2001).
[CrossRef]

W. B. Miller, J. C. Ricklin, and L. C. Andrews, "Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam," J. Opt. Soc. Am. A 11, 2719-2726 (1994).
[CrossRef]

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

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

Banakh, V. A.

V. A. Banakh and V. L. Mironov, "Influence of the diffraction size of a transmitting aperture and the turbulence spectrum on the intensity fluctuations of laser radiation," Sov. J. Quantum Electron. 8, 875-878 (1978).
[CrossRef]

V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, and R. Sh. Tsvik, "Focused-laser-beam scintillations in the turbulent atmosphere," J. Opt. Soc. Am. 64, 516-518 (1974).

Baykal, Y.

Clare, B.

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, "Scintillation: theory vs. experiment," in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

Corbett, K.

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, "Scintillation: theory vs. experiment," in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

Cowan, D. C.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, "Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model," in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B (2006).
[CrossRef]

Eyyuboglu, H. T.

Fante, R. L.

R. L. Fante, "Comparison of theories for intensity fluctuations in strong turbulence," Radio Sci. 11, 215-220 (1976).
[CrossRef]

Fortes, B. V.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

Grant, K.

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, "Scintillation: theory vs. experiment," in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

Hopen, C. Y.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave," Waves Random Media 11, 271-291 (2001).
[CrossRef]

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

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.

Khmelevtsov, S. S.

V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, and R. Sh. Tsvik, "Focused-laser-beam scintillations in the turbulent atmosphere," J. Opt. Soc. Am. 64, 516-518 (1974).

Korotkova, O.

O. Korotkova, "Control of the intensity fluctuations of random electromagnetic beams on propagation in weak atmospheric turbulence," in Free-Space Laser Communication Technologies XVIII, G. S. Mecherle, ed., Proc. SPIE 6105, 61050V (2006).
[CrossRef]

Krekov, G. M.

V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, and R. Sh. Tsvik, "Focused-laser-beam scintillations in the turbulent atmosphere," J. Opt. Soc. Am. 64, 516-518 (1974).

Lukin, V. P.

Miller, W. B.

Mironov, V. L.

V. A. Banakh and V. L. Mironov, "Influence of the diffraction size of a transmitting aperture and the turbulence spectrum on the intensity fluctuations of laser radiation," Sov. J. Quantum Electron. 8, 875-878 (1978).
[CrossRef]

V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, and R. Sh. Tsvik, "Focused-laser-beam scintillations in the turbulent atmosphere," J. Opt. Soc. Am. 64, 516-518 (1974).

Phillips, R. L.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave," Waves Random Media 11, 271-291 (2001).
[CrossRef]

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

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

Plonus, M. A.

Recolons, J.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, "Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model," in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B (2006).
[CrossRef]

Ricklin, J. C.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Tsvik, R. Sh.

V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, and R. Sh. Tsvik, "Focused-laser-beam scintillations in the turbulent atmosphere," J. Opt. Soc. Am. 64, 516-518 (1974).

Vetelino, F. S.

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, "Scintillation: theory vs. experiment," in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

F. S. Vetelino and L. C. Andrews, "Annular Gaussian beams in turbulent media," in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 86-97 (2004).
[CrossRef]

Young, C.

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, "Scintillation: theory vs. experiment," in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

Young, C. Y.

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, "Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model," in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B (2006).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. A (4)

Opt. Express (1)

Proc. SPIE (5)

F. S. Vetelino, C. Young, L. C. Andrews, K. Grant, K. Corbett, and B. Clare, "Scintillation: theory vs. experiment," in Atmospheric Propagation II, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 5793, 166-177 (2005).
[CrossRef]

O. Korotkova, "Control of the intensity fluctuations of random electromagnetic beams on propagation in weak atmospheric turbulence," in Free-Space Laser Communication Technologies XVIII, G. S. Mecherle, ed., Proc. SPIE 6105, 61050V (2006).
[CrossRef]

D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, "Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model," in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B (2006).
[CrossRef]

Y. Baykal, "Beams with arbitrary field profiles in turbulence," in Thirteenth Joint International Symposium on Atmospheric and Ocean Optics. Atmospheric Physics, G. G. Matvienko and V. A. Banakh, eds., Proc. SPIE 6522, 652209 (2006).
[CrossRef]

F. S. Vetelino and L. C. Andrews, "Annular Gaussian beams in turbulent media," in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 86-97 (2004).
[CrossRef]

Radio Sci. (1)

R. L. Fante, "Comparison of theories for intensity fluctuations in strong turbulence," Radio Sci. 11, 215-220 (1976).
[CrossRef]

Sov. J. Quantum Electron. (1)

V. A. Banakh and V. L. Mironov, "Influence of the diffraction size of a transmitting aperture and the turbulence spectrum on the intensity fluctuations of laser radiation," Sov. J. Quantum Electron. 8, 875-878 (1978).
[CrossRef]

Waves Random Media (1)

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave," Waves Random Media 11, 271-291 (2001).
[CrossRef]

Other (6)

V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, and R. Sh. Tsvik, "Focused-laser-beam scintillations in the turbulent atmosphere," J. Opt. Soc. Am. 64, 516-518 (1974).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.

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

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

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

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Figures (7)

Fig. 1
Fig. 1

Scintillation index of cosh-Gaussian beam at selected values of displacement parameters and scintillation index of a single cos-Gaussian beam versus propagation distance.

Fig. 2
Fig. 2

Scintillation index of beams from Fig. 1 with focusing introduced.

Fig. 3
Fig. 3

Scintillation index of beams from Fig. 1, off-axis case with p x = p y = 0.5   cm .

Fig. 4
Fig. 4

Scintillation index of cosh-Gaussian, Gaussian, and cos-Gaussian beams versus source size at selected values of propagation lengths.

Fig. 5
Fig. 5

Scintillation index distribution of collimated and focused cosh-Gaussian and Gaussian beams over the receiver plane.

Fig. 6
Fig. 6

Intensity distribution of collimated cosh-Gaussian beam from to Fig. 5 at source and receiver planes.

Fig. 7
Fig. 7

Scintillation index distribution of cosh-Gaussian beam over the receiver plane at selected inner and outer scales of turbulence.

Equations (10)

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u s ( s ) = u s ( s x , s y ) = l = 1 N A l   exp ( i θ l ) × exp [ ( 0 .5 k α x l s x 2 + i V x l s x ) ] × exp [ ( 0 .5 k α y l s y 2 + i V y l s y ) ] ,
α x l = 1 / ( k α s x l 2 ) + i / F x l ,   α y l = 1 / ( k α s y l 2 ) + i / F y l ,
m 2 ( p , L ) = 4 B χ ( p , L ) = 4 π Re { 0 L d η 0 κ d κ 0 2 π d θ [ S 1 ( p , L , η , κ , θ ) + S 2 ( p , L , η , κ , θ ) Φ n ( κ ) ] } ,
S 1 ( p , L , η , κ , θ ) = S N ( p , L , η , κ , θ ) S N ( p , L , η , κ , θ ) D 2 ( p , L ) ,
S 2 ( p , L , η , κ , θ ) = S N ( p , L , η , κ , θ ) S N * ( p , L , η , κ , θ ) | D ( p , L ) | 2 ,
S N ( p , L , η , κ , θ ) = l = 1 N A l e i θ l i k 1 + i α l L   exp [ k ( p x 2 + p y 2 ) α l 2 ( 1 + i α l L ) ] exp [ i ( V x l p x + V y l p y ) 1 + i α l L ] × exp [ i ( V x l 2 + V y l 2 ) L 2 k ( 1 + i α l L ) ] exp [ i ( L η ) ( V x l   cos   θ + V y l   sin   θ ) κ k ( 1 + i α l L ) ] × exp [ i ( 1 + i α l η ) ( p x   cos   θ + p y   sin   θ ) κ 1 + i α l L ] exp [ 0.5 i ( L η ) ( 1 + i α l η ) κ 2 k ( 1 + i α l L ) ] ,
D ( p , L ) = l = 1 N A l e i θ l 1 1 + i α l L   exp [ k ( p x 2 + p y 2 ) α l 2 ( 1 + i α l L ) ] exp [ i ( V x l p x + V y l p y ) 1 + i α l L ] × exp [ i ( V x l 2 + V y l 2 ) L 2 k ( 1 + i α l L ) ] .
Φ n ( κ ) = 0.033 C n 2   exp ( κ 2 / κ m 2 ) / ( κ 2 + κ 0 2 ) 11 / 6 ,
m 2 ( p , L ) = 1.3028 C n 2 k 2   Re [ | D ( p , L ) | 2 l 1 = 1 N l 2 = 1 N r = 0 A l 1 A l 2 *   exp [ i ( θ l 1 θ l 2 ) ] 1 ( 1 + i α l 1 L ) 1 ( 1 i α l 2 * L ) ( 0.25 ) r r ! × κ 0 2 r ( 5 / 3 )   exp [ k ( p x 2 + p y 2 ) α l 1 2 ( 1 + i α l 1 L ) k ( p x 2 + p y 2 ) α l 2 * 2 ( 1 i α l 2 * L ) ] exp [ i ( V x l 1 p x + V y l 1 p y ) 1 + i α l 1 L + i ( V x l 2 * p x + V y l 2 * p y ) 1 i α l 2 * L ] × exp { i ( V x l 1 2 + V y l 1 2 ) L 2 k ( 1 + i α l 1 L ) + i [ ( V x l 2 2 ) * + ( V y l 2 2 ) * ] L 2 k ( 1 i α l 2 * L ) } ( 0 L d η { [ i ( L η ) V x l 1 k ( 1 + i α l 1 L ) + i ( 1 + i α l 1 η ) p x 1 + i α l 1 L i ( L η ) V x l 2 * k ( 1 i α l 2 * L ) i ( 1 i α l 2 * η ) p x 1 i α l 2 * L ] 2 + [ i ( L η ) V y l 1 k ( 1 + i α l 1 L ) + i ( 1 + i α l 1 η ) p y 1 + i α l 1 L i ( L η ) V y l 2 * k ( 1 i α l 2 * L ) i ( 1 i α l 2 * η ) p y 1 i α l 2 * L ] 2 } r U { r + 1 , r + 1 / 6 , 0.5 ( L η ) [ i ( 1 + i α l 1 η ) k ( 1 + i α l 1 L ) i ( 1 i α l 2 * η ) k ( 1 i α l 2 * L ) + 1 κ m 2 ] κ 0 2 } )
D 2 ( p , L ) l 1 = 1 N l 2 = 1 N r = 0 A l 1 A l 2   exp [ i ( θ l 1 + θ l 2 ) ] 1 ( 1 + i α l 1 L ) 1 ( 1 + i α l 2 L ) ( 0.25 ) r r ! κ 0 2 r ( 5 / 3 ) × exp [ k ( p x 2 + p y 2 ) α l 1 2 ( 1 + i α l 1 L ) k ( p x 2 + p y 2 ) α l 2 2 ( 1 + i α l 2 L ) ] exp [ i ( V x l 1 p x + V y l 1 p y ) ( 1 + i α l 1 L ) i ( V x l 2 p x + V y l 2 p y ) ( 1 + i α l 2 L ) ] × exp [ i ( V x l 1 2 + V y l 1 2 ) L 2 k ( 1 + i α l 1 L ) i ( V x l 2 2 + V y l 2 2 ) L 2 k ( 1 + i α l 2 L ) ] ( 0 L d η { [ i ( L η ) V x l 1 k ( 1 + i α l 1 L ) + i ( 1 + i α l 1 η ) p x 1 + i α l 1 L i ( L η ) V x l 2 k ( 1 + i α l 2 L ) i ( 1 + i α l 2 η ) p x 1 + i α l 2 L ] 2 + [ i ( L η ) V y l 1 k ( 1 + i α l 1 L ) + i ( 1 + i α l 1 η ) p y 1 + i α l 1 L i ( L η ) V y l 2 k ( 1 + i α l 2 L ) i ( 1 + i α l 2 η ) p y 1 + i α l 2 L ] 2 } r U { r + 1 , r + 1 / 6 , 0.5 ( L η ) [ i ( 1 + i α l 1 η ) k ( 1 + i α l 1 L ) + i ( 1 + i α l 2 η ) k ( 1 + i α l 2 L ) + 1 κ m 2 ] κ 0 2 } ) ] , (9)

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