Abstract

The effects of turbulence on the beam quality of apertured partially coherent beams have been studied both analytically and numerically. Taking the Gaussian Schell-model (GSM) beam as a typical example of partially coherent beams, closed-form expressions for the average intensity, mean-squared beam width, power in the bucket, β parameter, and Strehl ratio of apertured partially coherent beams propagating through atmospheric turbulence are derived. It is shown that the smaller the beam truncation parameter is, the less affected by turbulence the apertured partially coherent beams are. Furthermore, the apertured partially coherent beams are less sensitive to the effects of turbulence than unapertured ones. The main results are interpreted physically.

© 2008 Optical Society of America

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References

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  1. R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics, Vol. XXII, E.Wolf, ed. (Elsevier, 1985), Chap. VI.
    [Crossref]
  2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
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    [Crossref]
  4. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598 (2002).
    [Crossref]
  5. A. Dogariu and S. Amarande, “Propagation of partially coherent beam: turbulence-induced degradation,” Opt. Lett. 28, 10-12 (2003).
    [Crossref] [PubMed]
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    [Crossref]
  7. 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]
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    [Crossref]
  9. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611-1618 (2005).
    [Crossref]
  10. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of partially coherent beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
    [Crossref]
  11. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
    [Crossref]
  12. H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,” Opt. Commun. 241, 11-15 (2004).
    [Crossref]
  13. X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
    [Crossref]
  14. J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355-1363 (1991).
    [Crossref]
  15. W. Lu, L. Liu, J. Sun, Q. Yang and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1-8 (2007).
    [Crossref]
  16. 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]
  17. X. Chu, Y. Ni, and G. Zhou, “Variations of on-axis average intensity along the propagation path in turbulent atmosphere for plane wave excitation limited by a circular aperture,” Opt. Commun. 267, 32-35 (2006).
    [Crossref]
  18. X. Chu, Y. Ni, and G. Zhou, “Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere,” Opt. Commun. 274, 274-280 (2007).
    [Crossref]
  19. X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 87, 547-552 (2007).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  28. A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713-720 (1988).
    [Crossref]

2008 (1)

2007 (4)

W. Lu, L. Liu, J. Sun, Q. Yang and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1-8 (2007).
[Crossref]

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]

X. Chu, Y. Ni, and G. Zhou, “Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere,” Opt. Commun. 274, 274-280 (2007).
[Crossref]

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 87, 547-552 (2007).
[Crossref]

2006 (2)

X. Chu, Y. Ni, and G. Zhou, “Variations of on-axis average intensity along the propagation path in turbulent atmosphere for plane wave excitation limited by a circular aperture,” Opt. Commun. 267, 32-35 (2006).
[Crossref]

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
[Crossref]

2005 (1)

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611-1618 (2005).
[Crossref]

2004 (3)

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of partially coherent beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[Crossref]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[Crossref]

H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,” Opt. Commun. 241, 11-15 (2004).
[Crossref]

2003 (1)

2002 (3)

2000 (1)

1998 (1)

A. Garay, “Continuous wave deuterium fluoride laser beam diagnostic system,” Proc. SPIE 888, 17-22 (1998).

1997 (1)

1991 (1)

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)

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

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

1988 (2)

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[Crossref]

A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713-720 (1988).
[Crossref]

1979 (1)

Amarande, S.

Andrews, L. C.

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

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]

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[Crossref]

Chen, S.

Chen, X.

Chu, X.

X. Chu, Y. Ni, and G. Zhou, “Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere,” Opt. Commun. 274, 274-280 (2007).
[Crossref]

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 87, 547-552 (2007).
[Crossref]

X. Chu, Y. Ni, and G. Zhou, “Variations of on-axis average intensity along the propagation path in turbulent atmosphere for plane wave excitation limited by a circular aperture,” Opt. Commun. 267, 32-35 (2006).
[Crossref]

Davidson, F. M.

Dogariu, A.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[Crossref]

A. Dogariu and S. Amarande, “Propagation of partially coherent beam: turbulence-induced degradation,” Opt. Lett. 28, 10-12 (2003).
[Crossref] [PubMed]

Fante, R. L.

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics, Vol. XXII, E.Wolf, ed. (Elsevier, 1985), Chap. VI.
[Crossref]

Friberg, A. T.

Garay, A.

A. Garay, “Continuous wave deuterium fluoride laser beam diagnostic system,” Proc. SPIE 888, 17-22 (1998).

Gbur, G.

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]

Ishimaru, A.

Ji, X.

Korotkova, O.

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of partially coherent beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[Crossref]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[Crossref]

Li, S.

Li, X.

Liu, L.

W. Lu, L. Liu, J. Sun, Q. Yang and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1-8 (2007).
[Crossref]

Lu, W.

W. Lu, L. Liu, J. Sun, Q. Yang and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1-8 (2007).
[Crossref]

Lü, B.

Mandel, L.

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

Ni, Y.

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 87, 547-552 (2007).
[Crossref]

X. Chu, Y. Ni, and G. Zhou, “Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere,” Opt. Commun. 274, 274-280 (2007).
[Crossref]

X. Chu, Y. Ni, and G. Zhou, “Variations of on-axis average intensity along the propagation path in turbulent atmosphere for plane wave excitation limited by a circular aperture,” Opt. Commun. 267, 32-35 (2006).
[Crossref]

Phillips, R. L.

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

Plonus, M. A.

Ponomarenko, S. A.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611-1618 (2005).
[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]

Ricklin, J. C.

Roychowdhury, H.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611-1618 (2005).
[Crossref]

H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,” Opt. Commun. 241, 11-15 (2004).
[Crossref]

Salem, M.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of partially coherent beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[Crossref]

Siegman, A. E.

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

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Applications and Issues, M.W.Dowley, ed., Vol. 17 of OSA Trends in Optics and Phonotic Series (OSA, 1998), pp. 184-199.

Sun, J.

W. Lu, L. Liu, J. Sun, Q. Yang and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1-8 (2007).
[Crossref]

Turunen, J.

Wang, S. C. H.

Wen, J. J.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[Crossref]

Wolf, E.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611-1618 (2005).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of partially coherent beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[Crossref]

H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,” Opt. Commun. 241, 11-15 (2004).
[Crossref]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[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]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598 (2002).
[Crossref]

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

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]

Yan, H.

Yang, Q.

W. Lu, L. Liu, J. Sun, Q. Yang and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1-8 (2007).
[Crossref]

Zhang, D.

Zhang, E.

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
[Crossref]

Zhou, G.

X. Chu, Y. Ni, and G. Zhou, “Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere,” Opt. Commun. 274, 274-280 (2007).
[Crossref]

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 87, 547-552 (2007).
[Crossref]

X. Chu, Y. Ni, and G. Zhou, “Variations of on-axis average intensity along the propagation path in turbulent atmosphere for plane wave excitation limited by a circular aperture,” Opt. Commun. 267, 32-35 (2006).
[Crossref]

Zhu, Y.

W. Lu, L. Liu, J. Sun, Q. Yang and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1-8 (2007).
[Crossref]

Appl. Opt. (2)

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

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 87, 547-552 (2007).
[Crossref]

J. Acoust. Soc. Am. (1)

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[Crossref]

J. Mod. Opt. (3)

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]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change of polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611-1618 (2005).
[Crossref]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (7)

X. Chu, Y. Ni, and G. Zhou, “Variations of on-axis average intensity along the propagation path in turbulent atmosphere for plane wave excitation limited by a circular aperture,” Opt. Commun. 267, 32-35 (2006).
[Crossref]

X. Chu, Y. Ni, and G. Zhou, “Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere,” Opt. Commun. 274, 274-280 (2007).
[Crossref]

W. Lu, L. Liu, J. Sun, Q. Yang and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1-8 (2007).
[Crossref]

H. Roychowdhury and E. Wolf, “Invariance of spectrum of light generated by a class of quasi-homogeneous sources on propagation through turbulence,” Opt. Commun. 241, 11-15 (2004).
[Crossref]

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259, 1-6 (2006).
[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]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of partially coherent beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[Crossref]

Opt. Lett. (1)

Proc. SPIE (2)

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

A. Garay, “Continuous wave deuterium fluoride laser beam diagnostic system,” Proc. SPIE 888, 17-22 (1998).

Waves Random Media (1)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[Crossref]

Other (4)

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics, Vol. XXII, E.Wolf, ed. (Elsevier, 1985), Chap. VI.
[Crossref]

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

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Applications and Issues, M.W.Dowley, ed., Vol. 17 of OSA Trends in Optics and Phonotic Series (OSA, 1998), pp. 184-199.

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

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Figures (6)

Fig. 1
Fig. 1

Relative average intensity distribution I ( x , z ) I 0 max of apertured and unapertured GSM beams with α = 0.6 at the plane z = 10 km : (a) δ = 0.3 and (b) δ . Solid curve, C n 2 = 10 14 m 2 3 ; dashed curve, C n 2 = 0 .

Fig. 2
Fig. 2

Relative mean-squared beam width w ( z ) w ( 0 ) of apertured and unapertured GSM beams with α = 1 versus propagation distance z. Solid curve, C n 2 = 10 14 m 2 3 ; dashed curve, C n 2 = 0 .

Fig. 3
Fig. 3

PIB curves of apertured and unapertured GSM beams with α = 0.5 at the plane z = 10 km . Solid curve, C n 2 = 10 14 m 2 3 ; dashed curve, C n 2 = 0 .

Fig. 4
Fig. 4

Plot of β parameter of apertured GSM beams in turbulence ( C n 2 = 10 14 m 2 3 ) at the plane z = 10 km versus beam truncation parameter δ.

Fig. 5
Fig. 5

Strehl ratio S R of apertured GSM beams in turbulence ( C n 2 = 10 14 m 2 3 ) at the plane z = 10 km versus beam truncation parameter δ.

Fig. 6
Fig. 6

Mean-squared beam width w of apertured GSM beams with α = 1 in free space at the plane z = 10 km versus the beam truncation parameter δ.

Equations (22)

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W ( 0 ) ( x 1 , x 2 , z = 0 ) = exp ( x 1 2 + x 2 2 w 0 2 ) exp [ ( x 1 x 2 ) 2 2 σ 0 2 ] ,
I ( x , z ) = k 2 π z h h h h d x 1 d x 2 W ( 0 ) ( x 1 , x 2 , z = 0 ) exp { i k 2 z [ ( x 1 2 x 2 2 ) 2 ( x 1 x 2 ) x ] } exp [ ψ ( x 1 , x ) + ψ * ( x 2 , x ) ] m ,
exp [ ψ ( x 1 , x ) + ψ * ( x 2 , x ) ] m = exp [ 0.5 D Ψ ( x 1 x 2 ) ] exp [ ( x 1 x 2 ) 2 ρ 0 2 ] ,
T ( x ) = { 1 x h 0 x > h .
T ( x ) = i = 1 N F i exp ( G i x 2 h 2 ) ,
I ( x , z ) = k 2 π z i = 1 N j = 1 N F i F j * d x 1 d x 2 W ( 0 ) ( x 1 , x 2 , z = 0 ) exp ( G i x 1 2 h 2 ) exp ( G j * x 2 2 h 2 ) exp { i k 2 z [ ( x 1 2 x 2 2 ) 2 ( x 1 x 2 ) x ] } exp [ ψ ( x 1 , x ) + ψ * ( x 2 , x ) ] m .
exp ( C 2 x 2 + D x ) d x = π C exp ( D 2 4 C 2 ) ,
I ( x , z ) = k 2 z i = 1 N j = 1 N F i F j * β exp [ k 2 4 β 2 z 2 ( G i + G j * w 0 2 δ 2 + 2 w 0 2 ) x 2 ] ,
β = [ G i G j * w 0 4 δ 4 + ( 1 w 0 2 + 1 2 w 0 2 α 2 + 1 ρ 0 2 ) ( G i + G j * w 0 2 δ 2 ) + i k 2 z ( G i G j * w 0 2 δ 2 ) + 1 w 0 4 + 2 w 0 2 ( 1 2 w 0 2 α 2 + 1 ρ 0 2 ) + k 2 4 z 2 ] 1 2 .
I ( x , z ) unapertured = k 2 z β exp [ k 2 2 w 0 2 z 2 β 2 x 2 ] ,
β = [ 1 w 0 4 + 2 w 0 2 ( 1 2 w 0 2 α 2 + 1 ρ 0 2 ) + k 2 4 z 2 ] 1 2 .
w 2 ( z ) = 4 x 2 I ( x , z ) d x I ( x , z ) d x .
w 2 ( z ) = 8 z 2 k 2 i = 1 N j = 1 N F i F j * β 2 ( G i + G j * w 0 2 δ 2 + 2 w 0 2 ) 3 2 i = 1 N j = 1 N F i F j * ( G i + G j * w 0 2 δ 2 + 2 w 0 2 ) 1 2 .
w 2 ( z ) unapertured = w 0 2 + 4 k 2 w 0 2 ( 1 + 1 α 2 ) z 2 + 8 ( 0.545 C n 2 ) 6 5 k 2 5 z 16 5 .
PIB = a a I ( x , z ) d x I ( x , z ) d x ,
PIB = i = 1 N j = 1 N F i F j * ( G i + G j * w 0 2 δ 2 + 2 w 0 2 ) 1 2 E r f [ k a 2 β z ( G i + G j * w 0 2 δ 2 + 2 w 0 2 ) 1 2 ] i = 1 N j = 1 N F i F j * ( G i + G j * w 0 2 δ 2 + 2 w 0 2 ) 1 2 ,
PIB unapertured = E r f [ k a 2 w 0 z β ] .
β = A A 0 ,
S R = I max I 0 max ,
S R = i = 1 N j = 1 N F i F j * β i = 1 N j = 1 N F i F j * β free ,
β free = [ G i G j * w 0 4 δ 4 + ( 1 + 1 2 α 2 ) ( G i + G j * w 0 4 δ 2 ) + i k 2 z ( G i G j * w 0 2 δ 2 ) + 1 w 0 4 + 1 w 0 4 α 2 + k 2 4 z 2 ] 1 2 .
S R unapertured = [ 1 w 0 4 + 1 w 0 4 α 2 + k 2 4 z 2 1 w 0 4 + 2 w 0 2 ( 1 ρ 0 2 + 1 2 w 0 2 α 2 ) + k 2 4 z 2 ] 1 2 .

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